=o)cosh(A„Z„) + / ( 0 = 2 » - 1 ) s i n h ( A n / n ) A i (CPn + ACw>=o)cosh(An/n) + AC(0=2"- 1 )s inh(A n / n )) f(o) = T — ( - r - cosh(Aiyi) + -f- sinh(Ajyj) + C, 1 ^p. \ Ai Ai Pi (5.74) (5.75) 1 2A 2B cosh(Aj|/j) - 1 - sinh(Ajyi) + C s A,; A,-(5.76) 5.2 One-dimensional Analysis of Tapered Double-doubler Joints and Reinforcements 134 The stresses for the negative sections of the double-doubler joint can then be determined utilizing the additional boundary conditions: M - i O = - r a \ ) (5-77) V - * ) = ( 5 - 7 8 ) PA-Vi) = X(y,z) t a(y) Figure 5.5: Distr ibut ion of the shear stress through the thickness of a double-sided joint or reinforcement Note that the nomenclature of the previously determined shear stress in the adhesive ra(y) is kept. In general the shear stress in a bonded joint or reinforcement r(y,z) is given by: ,dv{y,z) T(y,z) = Gj(y,z) « G-Evaluating the shear stress for the patch gives: 1 ft dz (5.81) dvv{y,z) rP(y,z) = — y-^ + ia + *p - zj ra(y) = Gp—^-(5.82) The in-plane displacement vp(y,z) is then determined through integration dvp(y,z) Vp{y,z) -I -dz Gptp \ 2+ t & + tp~2> Ta(2/) + k ( y ) dz z it* (5.83) with the integration constant h{y) being determined as a function of the midplane displace-ment u p ( j / , z = ^ + t a + ^ r ) of the patch, i.e. 1 fu — 4- t 4- — T' G p t p \ 2 + t & + 2 / 4 + 2 + 4 t p J ™ (5.84) 5.2 One-dimensional Analysis of Tapered Double-doubler Joints and Reinforcements 136 Substituting h(y) into equation (5.83) and evaluating the in-plane displacement vp(y,z) at the patch/adhesive interface yields: 3 t Vpiy^f+ti) = U p f o ^ + t a + k ) - -77-ra(») . (5.85) O U p The in-plane displacement vs(y,z) at the substrate/adhesive interface can be determined in a similar manner. Based on Figure Figure 5.5, the shear stress at any point in the substrate is evaluated as: 2z ^ dv%(y,z) rs(y,z) = — ra(„) = G s - ^ - . (5.86) The in-plane displacement of the substrate vs{y,z) can also be determined through integration as: { 2 * ' (5.87) = 7 r r r a ( y ) + 9(y) U S I S where the integration constant g(y) can be assessed as a function of the midplane displace-ment vs(y,z=o) of the substrate, i.e. 9(y) = vs(y,z=o) . (5.88) Substituting g(y) in equation (5.87) and evaluating the in-plane displacement vs{y,z) at the substrate/adhesive interface gives: w s(v,*=£) = vs(y,z=o) - 7 ^ - r a ( » ) . (5.89) Applying the assumption of constant shear stress through the adhesive yields: Q ra(y) = - p (yp(.y,z=$+tj - va(y,z=$)) . (5.90) The relationship between the midplane displacements of the adherends and the adhesive shear stress can be obtained by substituting equations (5.85) and (5.89) into equation (5.90) 5.2 One-dimensional Analysis of Tapered Double-doubler Joints and Reinforcements 137 to give: Ta(tf) = (5.91) This result can easily be incorporated into the basic one-dimensional analysis by modifying the elastic shear stress distribution parameter A. If equation (5.5) is replaced by equation (5.91), the elastic shear stress distribution parameter A for a constant thickness double-sided reinforcement or a double-doubler joint yields: The remaining one-dimensional analysis is unaffected by accounting for the shear-lag effect in the adherends. Thus the elastic shear stress distribution parameter Aj for tapered or stepped joints and reinforcements is obtained simply by indexing the appropriate variables. This section describes the application of the derived equations associated with the various joint and reinforcement types for a specific set of composite patch and metal substrate material properties representing cross sections of the sandwich type A M R L specimen. The honeycomb which is used in this particular specimen to eliminate out-of-plane bending is neglected in this case and the two face sheets are combined as the substrate in the analysis. The stress distributions for uniform thickness adherends versus tapered edges of the patch are then directly compared. Boron/epoxy 5521/4 was chosen as patch material, aluminum 2024-T3 as substrate and FM73M as film adhesive with the following properties: (5.92) 5 . 2 . 8 R e s u l t s 5.2 One-dimensional Analysis of Tapered Double-doubler Joints and Reinforcements 138 Boron/epoxy 5521/4 (0°-Orientation): Ep = 210 GPa Gp = 7.24 GPa ap = 4.61 pe/°C Aluminum 2024-T3: Es = 72.4 GPa Gs = 27.2 GPa ap = 23.45 pe/"C FM73M Film Adhesive G& = 844 MPa . The applied loads for the reinforcements and the double-doubler joint were chosen identi-cally as thermal loads representing thermal residual stress loading of the A M R L specimen after the curing cycle. The temperature differential is given as the difference between the effective stress free temperature and ambient temperature. Note that the shear modulus at ambient temperature was used in the subsequent calculations. Applied Load: AT = - 6 4 . 8 ° C P = 0 m m Double-sided reinforcement - Uniform versus tapered To compare the effect of tapering for a double-sided reinforcement, the following geometric parameters were employed. Double-sided reinforcement with uniform thickness: ts = 6.35 mm t a = 0.25 mm t p = 0.924 mm I = 75 mm Double-sided reinforcement with tapered edges L = 6.35 mm U = 0.25 mm tD = 1.056 mm—i-0.132 mm hl*<7 1<'<7 ^ = 57 mm h** EPx and uPyx being the major Poisson's ratio: 1 1 - K c - ^ p J + a p x A T (5.97) e p y = jr-(-vPyxvPx + a P y) + aPyAT . (5.98) To simplify the estimation of the thermal residual stress, a rigid bond is assumed based on ° the small load transfer length in comparison to the overlap length (1/A V < (_L_ + M (_L_ + JL) ( 5 , „ 6 ) Epytp EstsJ \EPxtp EstsJ \EPytp Ests (a. - a p J A T ( -?f + j « (a. - c* P y)AT ( + ^ F r ) (5-107) V -C 'Pytp A ^ S / V ^ P X ^ P &sts The thermal residual stresses can be expressed as: „ 2(as-aPx)AT 2(o. - O p J A T + j f c ) f-^- + — 1 t (^— + —) (—— + \ ^ P X * P £ > s * s / s y ^ p x ' p EstsJ yEpytp 2 Ests (5.108) „ 2 ( a s - a P y ) A T a S y = - . • (5-109) Note that equation (5.109) is identical to the result from the one-dimensional solution at the center of a double-sided reinforcement with 1 / A < I . The loss in accuracy by reducing the two-dimensional analysis to a one-dimensional problem for the given material combination 5.3 Application of the Rose Model for Thermal Residual Stress Loading 147 is well below 1% for the thermal residual stress in the substrate along the fiber direction (i.e. o~s ) which is the main concern with respect to the performance of the bonded repair. In addition, the one-dimensional analysis of a double-sided reinforcement allows one to easily include the influence of the adhesive and the associated load transfer zone using equation (5.25). Thus the stress in the fiber direction at the prospective crack location in the center of the reinforced region due to the elevated temperature cure is given by: Having determined the stress at the prospective crack location a0, the crack is introduced in the second stage of the analysis. The load which was transmitted across the crack is now partly transferred into the patch and partly redistributed in the plate. The relative magnitudes of this redistribution primarily depend on the crack length, i.e. the longer the crack, the more load is transferred into the reinforcement [78]. The stress intensity factor Kr for a crack with a crack length a in a bonded repair is estimated by interpolation between two upper bounds, Ku and for short and long cracks respectively. An upper bound for Kr for short cracks can be derived on the assumption that no load is transferred into the reinforcement [23]: Rose gave the equations for a center crack, where Y = 1, while the A M R L specimen contains an edge crack. Based on the assumption that the edge crack is small in comparison to the width of the specimen, the correction factor for this geometry is Y = 1.12. (5.110) 5.3.2 Stage II: Upper Bounds for Kv (5.111) 5.3 Application of the Rose Model for Thermal Residual Stress Loading 148 Rose showed that the relationship between energy release rate G and stress intensity factor K also holds for a reinforced cracked structure, i.e. that the body force field corresponding to the shear tractions does not affect the relation between G and K [78]. The crack in the isotropic substrate will not always be under plane stress conditions as it might be expected in a repair of an aircraft skin. The crack in a bonded repair under only thermal residual stress loading will generally be under plane strain conditions while combined with high external loading it will be under plane stress conditions. The relationships between the energy release rate and the stress intensity factor are given for the crack in the isotropic substrate under plane stress and plane strain conditions as: Thus the corresponding energy release rate Gu can be determined using equation (5.112) or (5.113). This upper bound will be close to the correct value for short cracks and represents an asymptote with first order contact in the limit a —» 0 for Kx of an infinite patched plate with a crack [78]. Note that the energy release rate Gu is linearly increasing with the crack length, although Gu cannot exceed the energy release rate Goo f ° r a semi-infinite crack. Rose showed that the limiting value for the energy release rate G^ can be adequately estimated using one-dimensional modeling which ignores the stress variation across the thickness [79]. This upper value for the energy release rate can be estimated by the change in potential energy when introducing a cut in the substrate of a double-sided reinforced plate converting the specimen to a double-doubler joint. Thus Goo is the work extracted (per unit thickness and per unit crack growth) on the stress a0 at the prospective crack location relaxed to zero through the displacement v0 on both crack faces [79]: G = — (Plane stress) (5.112) G = — (1 - vl) (Plane strain) E (5.113) (5.114) 5.3 Application of the Rose Model for Thermal Residual Stress Loading 149 The stress at the prospective crack location a0 was determined using equation (5.3.1) in stage I of the Rose model using a one-dimensional model of a double-sided reinforcement. The displacement v0 as a result of introducing a cut in the double-sided reinforcement can be easily determined using superposition. According to Figure 5.12, the displacement v0 is P P Gs(y=0)=ao Figure 5.12: Illustration of the superposition principle given as the displacement of one joint or crack face for a pressurized cut in a double-doubler joint. The general analysis as given in Section 5.2.1 has to be slightly altered to account for the different boundary conditions. The compatibility equations for the adherends and the adhesive are now expressed as: dvp(y) _ Tp(y) €p(») C 8 (») = dy dy Ests ra(y) = G V M y ) « ^(vpiy) - vB(y)) (5.115) (5.116) (5.117) Differentiating equation (5.117) with respect to y and substituting equation (5.115) and 5.3 Application of the Rose Model for Thermal Residual Stress Loading 150 (5.116) yields: d T a ( 2 / ) C a / dvp(y) dvs(y) dy ta V dy dy , C a / 7 » Taiy)- ( 5 - 1 1 8 ) \Eptp Ests The characteristic differential equation remains identical. The two integration constants Cs and Cp can be determined using the overall force equilibrium 2TP(») + Ts(y) = 2CP + C s = 0 (5.119) and the first derivative of the shear stress ra(j/) with respect to y, which can be equated to equation (5.118) as shown below: - AX cosh(Ay) + BAsinh(Ay) _ C a (Tp{y) T s ( ^ ( 5 ' 1 2 0 ) t a y Eptp Ests Substituting the relationships for the stress resultants (5.11, 5.12) and equation (5.119) yields AXcosh(Xy) + BXsinh(Xy) = % (-J— + J (Acosh(Ay) + Ssinh(Ay)) i a A \Eptp EstsJ 1 9 - n ta, \ Eptp Esta J Solving for C p and subsequently for C s gives Cp = Ca = 0 . (5.122) The boundary conditions for the stress resultants of an internally pressurized double-doubler joint are evaluated at the center of the sheet and at the edge of the patch as follows. Boundary condition at the center 2A Ts („=o) = -a0ts = - — (5.123) 5.3 Application of the Rose Model for Thermal Residual Stress Loading 151 Boundary conditions at the positive edge AP BP Tp(y=i) = - T - cosh(AZ) + — sinh(A — \ T—-—~T\ = Ya0V^A (Plane strain) . (5.135) V C1 _ vi) Note that the differentiation between plane stress and plane strain conditions for G^ was introduced due to the definition of A in K^. Furthermore, Rose showed that this result is also the first term in the asymptotic expansion of Kx for A 2 /=0) = 0 (5.149) rxy(-2aoo) = ka0 (5.151) oy (^x2 +y 2 ->oo) = a0 (5.152) Txy( \A 2 +2/ 2 ">-oo) = 0 . (5.153) Sih gave the corrected Westergaard type stress function, which fulfills the above stated boundary conditions as [85]: where v ^ > A = ( l - k ) ^ . (5.155) The derivative of Z{z) with respect to z can be easily determined as: dZ(z) a0a2 (5.156) dz (z2 + 2az)-2 In the second step of the analysis, the real and imaginary part of Z(z) and the derivative dZ(z)/dz have to be determined. It is convenient to switch to polar coordinates using: z = reie (5.157) 5.4 Estimation of the Stress Field around the Crack Tip 159 where r = y/x2 4 y2 (5.158) eie = cos0 4 i s i n 0 . (5.159) Thus Z(z) can be rewritten as: zM = -7^z + a) ^{z + a)2 - a2 V ' 2 a0(reie + a) aQ - (!-«)• vV 2e i 2« + 2 a r e i e 2 a0(reie + a) y/(r2 cos 20 4 2ar cos 0) 4 i(r2 sin 20 4- 2ar sin 0) ^ -J^Q^ _ a 0 (re i e 4- a) -^(r 2 cos 20 4- 2ar cos 9) - i(r2 sin 29 4 2ar sin 0) y/(r2 cos 20 4 2ar cos 9)2 4 (r 2 sin 20 4- 2ar sin 0)2 In order to allow the separation of Z(z) in real and imaginary parts as well as to reduce the size of the expression, r*(r,e) and 9*(r,e) are defined as: r*(r,9) = {(r2 cos 20 4- 2ar cos 9)'2 4- (r 2 sin 20 4 2ar sin 0)2] < = [r4 4 4 a V 4 4ar 3 cos 0]* (5.161) r 2 cos 20 4- 2ar cos 0 0 (r,9) = arccos( ) . (5.162) v r*(r,e)2 ' v ; Note that the arccos() is used to maintain a unique solution for 0 < 0 < ir. Dealing with quarter symmetry, it is important to keep in mind that r is limited to: — cos 0 2 for | < 0 < 7r (5.163) Based on this expression it is resolved that 9*{r,e) reaches its maximum value of 7r along the line x — —a, y > 0 and at the crack face. The parameter 9*(r,e) has its minimum value of 5.4 Estimation of the Stress Field around the Crack Tip 160 0 along the line x > 0, y = 0. Using the relationships for r*(r,9) and 0*(r,e), Z(z) yields: o-0(reie + a)r*(r,8)e-l^{r'e) u , a 0 Zi(z) = 1 — A; — r*(r,e)2 v ' 2 = " « f e ' < , " i " M ) , + ^ - ' 9 " M ) ) - ( 1 - * ' f = (^cos("- + 7^r{\9"{-e))) (5'164) - < i - * ) £ ( r 1 a 1 \ + i a0[ — — sin(0 - -0*(r.*)) + sin(-0*(r,«)) . \r*(r,e) 2 r*(r,e) 2 / Finally Re[.Zf»] and Im[Z(z)] can be expressed as: Re[Z<«)] = oQ cos(9 - \»-(,,)) + _ | - c o s ( i t l ' <,,.))) (5.165) Im[Z(2)] = a 0 sin(0 - ±0>,*) ) + sin( Vr^ ))) . (5.166) \T (r,6j Z T (r,9) Z J The evaluation of the real and imaginary part of dZ(z)/dz is carried out in a similar fashion: dZ(z) a0a2 dz ~ . (z2 + 2az)i ,2 onar (r2ei2B + 2arei6)* a0a2 [(r2 cos 29 + 2ar cos 9) + i(r2 sin 29 + 2arsin9)] 2 ^ a0a2[(r2 cos 20 + 2ar cos0) — i(r2 sin 20 + 2arsm0)]t (r 2 cos 20 + 2ar cos 0)2 + (r 2 sin 20 + 2arsm0)2] 2 «2 " -i\0*{r,0) -ne 521 fie 527 Ate 558 ± 1 Ate 562 ± 1 Ate B / E p 26 -632 Ate B / E p 32 -647 fie -634 Ate -661 ± 1 Ate N / A B / E p 35 -622 Table 5.1: Comparison of the thermal residual strains at 21°C between experimental re-sults, the one-dimensional analysis and the fracture mechanics analysis cially location C where the thermal residual strains would increase to approximately 660 Ate, which is certainly not the case. The thermal residual strains are predicted at location C quite well assuming no disbond. Therefore a disbond can be ruled out as a likely cause. Neglecting that the patch covers only 47% of the specimen width leads to some inaccuracy in the estimation of the stress at the prospective crack location. The stress estimate at the 5.5 Comparison between Closed Form Theoretical and Experimental Results 172 Location Material Gauge No. Experimental Measurement Experimental Measurement Average 1-D Analysis Fracture Mechanics Analysis A Al 2 38 fie 38 fie 1 8 4 ± 2 3 / x e N / A B / E p B / E p 23 43 -1469 lie -1808 fie -1639 /ie -1528 ± 195 lie N / A B Al 3 1504 //e 1504 fie 1065 ±6 lie 1122 ± 2 6 /ie B / E p 24 -1262 /ie B / E p 30 -1258 fie -1224 /ie -1261 ± 7 /ie N / A B / E p 33 -1152 /ie C Al 12 1215 lie 1215 lie 1212 ± 1/ie 1268 ± 3 fie B / E p 25 -1456 /ie B / E p 31 -1343 /ie -1364 /ie -1435 ± 1/ze N / A B / E p 42 -1293 lie D Al A l 10 22 1145 /ie 1154 fie 1150 /ie 1177 ± 2 /xe 1194 ± 1 /ie B / E p 28 -1399 /ie B / E p 29 -1386 /ie -1356 fie -1395 ± 2/ie N / A B / E p 40 -1282 fie G Al 7 758 /ie 758 /ie 291 ± 1 /ie N / A B / E p B / E p 27 37 -1602 lie -1719 /ie -1661 fie -2261 ± 1 /ie N / A H Al Al 5 19 1161 /ie 1115 lie 1138 /ie 1210 ± 1/ie 1219 ± 1/ie B / E p 26 -1371 /ie B / E p 32 -1417 lie -1396 /ie -1434 ± l^e N / A B / E p 35 -1399 /ie Table 5.2: Comparison of the thermal residual strains at — 56.5°C between experimental results, the one-dimensional analysis and the fracture mechanics analysis prospective crack location can be improved over the one-dimensional analysis by carrying out a finite element analysis of the A M R L specimen without a crack. The adhesive does not need to be modeled in this particular analysis since virtually no load transfer occurs at the prospective crack location. The finite element analysis of the A M R L specimen as discussed in the following chapter 5.5 Comparison between Closed Form Theoretical and Experimental Results 173 gives additional insight into the complex stress distribution in the vicinity of the crack thus giving a better understanding of the cause for the high tensile residual strains in the aluminum at location B . The stress and strain distributions as caused by the crack tip stress intensity factor are affected by the load transfer into the patch due to the crack as shown in the transverse shear stress plot for the A M R L specimen (see Figure 6.32). This influence is not accounted for in the fracture mechanics approach. The predictions at location G are not accurate either (Ae = 193 /Lie for the aluminum). The explanation for the discrepancy is given by the gradient of the thermal residual strains in the fiber direction through the tapered section. Additionally, in-plane bending has to be expected. The finite element model gives an excellent prediction for the thermal residual strains in the aluminum in this particular region. The thermal residual strains at the tapered location A are predicted reasonably well. After accounting for position uncertainty of the strain gauge, the thermal residual stresses are well predicted for the boron/epoxy. The thermal residual stresses in the aluminum are somewhat overpredicted (84fie at ambient temperature and 184//e at an operating temperature of —56.5°C). The prediction of the thermal residual strains in the boron/epoxy patch at location B , C , D and H are quite good. The average discrepancy is only 20 fie at ambient temperature and 46 pe at an operating temperature of —56.5°C. The thermal residual stress field in the bonded repair will change with crack growth when the test specimen is subjected to fatigue loading. In addition to crack growth, disbonds around the crack face are quite common. Initial flaws such a inadequate surface prepara-tion can also lead to large disbonds. The influence of disbonds on the stress and strain distribution can be quite large. A more generalized Rose model, which can account for disbonds, is therefore developed in the following section and represents one of the major achievements of this research work. 5.6 Generalization of the Rose Model for Partially Disbonded Patches 174 5.6 G e n e r a l i z a t i o n o f t h e R o s e M o d e l f o r P a r t i a l l y D i s b o n d e d P a t c h e s The accurate prediction of the effect of a disbond on crack propagation in the metallic sub-strate requires elaborate calculations [5, 73, 76, 81]. Baker [13] presented the development of a useful estimate of the influence of disbond size by adapting the Rose model [79] giving a ratio between the stress intensity factor for a repair without a disbond and the stress intensity accounting for a symmetric disbond. The approach in this section expands on Baker's initial formulation by adding an additional term to improve the determination of the crack opening displacement as well as accounting for an imbalance in the stiffness of the patches and substrate and their thermal mismatch. The approach is first carried out for uniform thickness patches and then extended to tapered patches. The analysis presents a significantly improved estimate for the change of the stress intensity factor in the presence of a disbond. Figure 5.20: Illustration of the superposition principle for a composite repair with disbonds 5.6 Generalization of the Rose Model for Partially Disbonded Patches 175 As discussed in Section 5.3, the stress at the prospective crack location in the presence of a disbond has to be determined first using the stage I analysis of the Rose model. The previously outlined method to determine the stress at the prospective crack location for a repair without a disbond utilizing a double-sided reinforcement can be used for a disbonded patch only if the two patches and substrate have identical stiffnesses and equal coefficients of thermal expansion. The reinforcement would be modeled from the free edge to the edge of the disbond. The requirement of identical stiffness and C T E of the adherends fulfills the requirement for zero shear stress at the edge of the disbond thus being equivalent to the center of a reinforcement without a disbond. The requirement for zero shear stress at the edge of the disbond can also be fulfilled for specific combinations of adherend stiffnesses and C T E mismatches. In any other case, it is necessary to derive a one-dimensional model, which takes the disbond as a boundary condition into account. The half crack opening displacement v0 as a result of introducing a cut in the substrate of the double-sided reinforcement can then be assessed using superposition. Superposition can be used to determine the joint opening displacement v* in the presence of a disbond which is required to determine the upper bound similar to the approach presented in Section 5.3. Figure 5.20 shows the superposition scheme with symmetric disbonds. 5 . 6 . 1 D o u b l e - s i d e d U n i f o r m T h i c k n e s s R e p a i r w i t h D i s b o n d s Stage I: Stress at the prospective crack location The first step of the Rose model requires the determination of the stress at the prospective crack location. Figure 5.21 shows the free body diagram for half of a reinforcement with a disbond. In order to be able to use the system of equations derived for a reinforcement without a disbond, two coordinate systems were introduced. 5.6 Generalization of the Rose Model for Partially Disbonded Patches 176 The y - coordinate starts at the end of the disbond, while the y* - coordinate originates in the center of the reinforcement. The following approach is directed toward deriving the influence of a disbond as a boundary condition. The disbond generally results in a shear stress in the adhesive at y* = b in comparison to zero shear stress in the center of a reinforcement without a disbond. The length changes in the patch and substrate due to both a direct stress resultant and a temperature change in the disbonded area and are given by: Eptp AL = eJ) -E's 4 ^b + asATb . (5.188) (5.189) Using the definition of the elastic shear stress distribution parameter A and the force equi-librium given in equation (5.13), the shear stress at y = 0 can be expressed as: R d ra (»=o) A 2 y Eptp Ests J A 2 \ Eptp Ests J {vP(y=o) - Vs(y=0)) (Alp - Als) =x*b(T;<-^;{as-aiAT y Eptp Ests J (5.190) (5.191) (5.192) AT lpo 'so lpo lpo Lso lpo AT - • y * \ M 1 ^ 1 ^ 1 • y Figure 5.21: Free body diagram for a reinforcement with a disbond 5.6 Generalization of the Rose Model for Partially Disbonded Patches 177 The shear stress expression simplifies even further using the integration constant Cp as given in equation (5.16). raRd(,=o) = A 2 6 ( T p R o d - C p ) (5.193) A proper boundary condition can now be established by substituting T p R d and rRd(y=o). The direct stress resultant for the patch T P o can be evaluated using equation (5.11) as • Rd ,Rd ^.Rd A T;o = T ; V O ) = — + Cp . (5.194) The shear stress rRd(y=o) is then determined using equation (5.10). r a R V o ) = 73Rd (5.195) The boundary condition for the disbonded reinforcement is given by substituting equation (5.194) and (5.195) into equation (5.193) thus yielding: BRd = ARdXb . (5.196) Note that this boundary condition for b = 0 is identical to the boundary at the center of a double-sided reinforcement without a disbond. The second boundary condition for the double-sided partly disbonded reinforcement is evaluated at the edge of the reinforcement (y - l) as: . Rd „ Rd Tp\y=i) = — cosh(AZ) + - — sinh(Af) + C p = 0 (5.197) A A where y4.Rd can be determined by substituting BRd in equation (5.197) yielding: ARd = — (5.198) icosh(AZ) + 6sinh(A0 Wfs + ( « s - « P ) A T + (icosh(A0+6sinh(A/)) ' (5.199) 5.6 Generalization of the Rose Model for Partially Disbonded Patches 178 The second coefficient B and then be calculated using equation (5.196): D R d Cp , = ~ ^ c o s h ( A 0 + is inh(A0 ( 5 - 2 0 0 ) + (^cosh(A/) + Isinh(A/)) (5.201) Having determined A and B , the stress distributions for the adhesive and adherends can be readily determined. The stress at the prospective crack location in the presence of a disbond is given as: 2ARd o0 = - I ^ 1- Cs (5.202) = 2 , Est, + ("s - « P ) A T \ 1 t (_k |_ _2__ \ / cosh(AZ) + 6Asinh(A(!) S \ ^P*p &sts ) / p -2(aa-ap)AT (5.203) S t a g e I I : U p p e r b o u n d s KT After determining the stress at the prospective crack location o~d, the crack is introduced into the second stage of the analysis. The analysis is, for the most part, identical with the approach given in Section 5.3.2. The main difference is the contribution of three components to the half crack opening rather than one. The strains in the disbonded sections of the patches and substrate lead to increased displacements, in addition to the crack opening due to shear in the adhesive. An upper bound for the stress intensity factor for short cracks assuming no load transfer into the reinforcement is given as [23]: K* = Yo~dy/ira . (5.204) 5.6 Generalization of the Rose Model for Partially Disbonded Patches 179 The corresponding energy release rate Gu can then be determined based on plane stress or plane strain conditions: This upper bound will be close to the correct value for short cracks and represents an asymptote with first order contact in the limit a - » 0 for K* of an infinite patched plate with a crack [78]. Note that the energy release rate Gdu is linearly increasing with crack length, although cannot exceed the energy release rate G^ for a semi-infinite crack. The limiting value for the energy release rate G^ is again modeled using the one-dimensional approach which ignores the stress variation across the thickness [79]. This upper value for the energy release rate is estimated by the change in potential energy when introducing a cut in the substrate of a double-sided reinforced plate with disbonds, therefore converting the specimen to a double-doubler joint with disbonds centered around the joint. Thus as given in equation (5.114) is the work extracted (per unit thickness and per unit crack growth) on the stress at the prospective crack location crd to relax to zero through the displacement v* on both crack faces [79]. The stress at the prospective crack location o* was determined in equation (5.203) as stage I of the Rose model using a one-dimensional model of a double-sided partially disbonded reinforcement. The displacement v% as a result of introducing a cut in the double-sided partially disbonded reinforcement can then be easily determined using superposition as shown in Figure 5.20. According to Figure 5.20, the displacement v* is given as the displacement of one joint or crack face for a pressurized cut in a partially disbonded double-doubler joint. The general analysis of the shear stress in the adhesive as given in Section 5.3.2 is still applicable in contrast to the partially disbonded double-sided reinforcement. (Plane stress) (5.205) (1 — u2) (Plane strain) . (5.206) 5.6 Generalization of the Rose Model for Partially Disbonded Patches 180 Thus the shear stress at the edge of the disbond for an internally pressurized partially disbonded double-doubler joint can be written as: ^ = - x A J m - ( 5 ' 2 0 7 ) The displacement vdol due to the shear displacement in the adhesive can be evaluated using equation (5.117). Therefore the displacement vdol is given by equation (5.129) as: d V * ~ 2 S UP'P + EstJ Atanh(AJ) ' ( 5 , 2 ° 8 ) As stated above, it is necessary to include the displacements due to the strain in the disbonded section of the patch and substrate. These additional displacements are given as: d vdom = Ufb (5.210) which sum to provide the total half crack opening displacement as: vl = vl\ + %n + %ni (5.211) o\U f 1 2 1 2 \Evtp E3tJ Atanh(AZ) 2 Eptp Es ^ 4 (5.213) 2 \Eptp EstJ \Atanh(A0 Thus the upper limit for the energy release rate can be expressed for plane stress and plane strain conditions as follows [18]. Plane stress ,d d d Yza„ where Plane strain '2 ~E. Gl = v0v0 = ^ T T A (5.214) s A = ^ ( 1 + 2 ^ ) ( n ^ 5 ( A o + 6 l ( 5 - 2 1 5 ) Gl = o'X = ^ T A ( 1 - vj) (5.216) 5.6 Generalization of the Rose Model for Partially Disbonded Patches 181 where Based on the validity of equation (5.112) and (5.113), the stress intensity factor for this upper bound is: = y/EsG^ = y < 7 „ V T T A (Plane stress) (5.218) = ] J = Y a o ^ (Plane s t r a i n ) • (5-219) As before, this result can be used as the first term in the asymptotic expansion of Kr for A < a. The Rose model suggests an analytical approximation by interpolating between the asymptotes Kd and which is generalized for different crack shapes below [18] K t - Y ^ ^ (5.220, with the corresponding crack extension force Kd2 Gd = -±- (Plane stress) (5.221) E* ^ 2 Gd = ^ - ( 1 - v2s) (Plane strain) . (5.222) The accuracy of the analysis can also be improved by accounting for the shear stiffness of the adherends. This can be readily implemented by using Ash e ar-iag as given in equation (5.92) instead of A. The results of this analysis scheme are presented for three different cases: • Thermal residual stress loading (AT = — 64.8°C) • Remote stress loading (0^ = 40 MPa) • Combined thermal residual stress and remote stress loading ( A T = —64.8°C, aoo = 40 MPa) 5.6 Generalization of the Rose Model for Partially Disbonded Patches 182 Note that the stress intensity factors are determined for all cases using the plane stress solution to allow for a better comparison between the three cases. Plane stress conditions are typically present under a combination of high external loading and thermal residual stresses in a real application. The material and geometric data used is this analysis are given below: Substrate Material Young's modulus Shear modulus Coef. of Therm. Expan. Thickness Edge crack length Correction factor Patch Material Young's modulus (||) Shear modulus Coef. of Therm. Expan. Thickness Length Adhesive Shear modulus Thickness Ea = 72.4 GPa 23.45 ^ Gs = 27.2 GPa a s = ts = 6.35 mm a = 20 mm Y = 1.12 Ep = 210 GPa Gr 7.24 GPa ap = 4.61 ^ 7J P = 6.35 mm lp = 75.0 mm C a = 0.844 GPa 7 j a = 0.25 mm The stress intensity factors K^, Kd and K* were determined for each of the loading conditions for the entire range of b, i.e. from no disbond (6=0 mm) to completely disbonded (6=75 mm). Note that K* is dependent on the chosen crack length and will therefore shift up or down for different crack lengths. K* indicates for which disbond length or K* is dominant. 5.6 Generalization of the Rose Model for Partially Disbonded Patches 183 20 -r 17.5 - -15 - • I 12.5 - -(0 10 - -Q . s 7.5 - -5 - ; 2.5 - -0 - -35 40 b [mm] Figure 5.22: Stress intensity factor dependence on disbond length for a bonded repair under thermal residual stress loading (AT = —64.8°C) Figure 5.22 through Figure 5.24 show some interesting features when looking at large disbonds. The stress intensity factors drop to zero for the case of thermal residual stress loading when approaching a complete disbond, which should be expected. This effect seems to be dominant only when the remaining bonded length is approximately equal to the load transfer length, which in this case is 6.5 mm. The opposite behaviour can be seen for the I ra Q . s 201 17.5 + 15 + 12.5 10 7.5 5 2.5 0 K 10 K -f- + 15 20 25 30 35 40 b [mm] 45 50 55 60 65 70 75 Figure 5.23: Stress intensity factor dependence on disbond length for a bonded repair under remote stress loading (O-QO = 40 MPa) 5.6 Generalization of the Rose Model for Partially Disbonded Patches 184 35 40 b [mm] Figure 5.24: Stress intensity factor dependence on disbond length for a bonded repair under combined thermal residual stress and remote stress loading (AT = — 64.8°C, (Too = 40 MPa) case of a remotely applied stress (Figure 5.23). The stress intensity factor converges to infinity, which is to be expected due to using a pressurized double-doubler joint model, i.e. the substrate would separate. K* reaches its limiting value of an unpatched specimen, which is also properly reflected by K*. The behavior of the stress intensity factors for the combined loading case (Figure 5.24) also satisfies expectations. is again converging to infinity caused by the remaining remote stress after the complete disbond, while K* is again limited by the unpatched scenario. K* reflects the change between the limiting stress intensity factors well. Rose's patching efficiency definition [78] leading to zero patching efficiency for the un-patched crack and 1.0 for elimination of the crack i.e. a reduction in stress intensity to 0 MPai /m shows the problem of disbonds clearly: RK = 1 K: (5.223) K(b=h). This particular definition cannot be used for the case where only thermal residual stresses are present, i.e. K*(b=ip) equal to zero. But the definition reflects the patch efficiency 5.6 Generalization of the Rose Model for Partially Disbonded Patches 185 quite well for partially disbonded patches with a loading combination of thermal residual stresses and remote stresses, which is most common for test specimens (Figure 5.25). Under the given boundary and loading conditions, the patching leads only to a reduction in the absolute stress intensity factor to a disbonding length of approximately 19.1 mm. Larger disbonds represent a worse case than the unrepaired specimen due to the thermal residual stresses. Note that this particular disbond length shifts to a higher value if the crack length increases. For a crack length of 50 mm the disbond can be up to approximately 58.7 mm before patching will lead to an increase in the stress intensity factor. The influence of thermal residual stresses on the stress intensity factor for real bonded repair applications with disbonds is generally not as severe as shown here for an unrestrained specimen due to the lower effective coefficient of thermal expansion of the substrate. The patching efficiency is much closer to the case of remote stress loading (Figure 5.25). 1 x Remote Stress 0.8 +. Thermal,Residual Stress & Remote Stress 0.6 + 0.4-k 0.2 + 0 5 10 15 20 55 60 65 70 '5 -0.2 + -0.4 -L b [mm] Figure 5.25: Patch efficiency for partially disbonded patches The results show how readily an idealized disbond can be included in the analysis. In comparison to the basic analysis carried out in Section 5.3, the revised results for the determination of an upper bound for the stress intensity factor including disbonds have 5.6 Generalization of the Rose Model for Partially Disbonded Patches 186 only the additional disbond length term b. If b = 0, the original solution without disbonds is obtained. In this respect, the presented solution is a more generalized estimation for the performance of bonded repairs and represents one of the main achievements of this research work. The solution can be improved further by accounting for plastic strain in the adhesive although the influence of the adhesive shear deformation towards the crack opening dis-placement is reduced from 100% of the total crack opening displacement for a fully bonded repair to only 10% for a repair containing large disbonds using the fully elastic analysis. Even for a significant increase in the shear displacement by accounting for plastic strain, the change in the stress intensity factor will be relatively small making the elastic solution an excellent first order approximation. Additional experimental work as well as higher order analytical work is required to determine the overall performance of the method. 5 . 6 . 2 D o u b l e - s i d e d T a p e r e d R e p a i r w i t h D i s b o n d s The stress intensity factor change with disbond growth can also determined for a repair involving a patch with a tapered edge. The direct closed form method for determining the stress distribution for tapered reinforcements and double-doubler joints was presented in Section 5.2. This method can be easily adapted for analyzing the stress at the prospective crack location using a partially disbonded reinforcement model and the shear stress at the edge of the disbond in a pressured double-doubler joint. Note that this approach is only valid until the disbond reaches the last step adjacent to the edge. Then the model reduces to a simpler analysis similar to the uniform thickness patch but must include the thickness change of the patch within the disbonded region. Using this two step model, the influence of a taper on disbond tolerance can be investigated for thermal, mechanical and 5.6 Generalization of the Rose Model for Partially Disbonded Patches 187 Figure 5.26: Free body diagram for a tapered reinforcement with a disbond a combination of both load types. The model is fully elastic and can be extended using a numerical approach for elastic-plastic behaviour in the adhesive. Stage I: Stress at the prospective crack location Determination of the stress at the prospective crack location is carried out in a manner similar to that of the uniform thickness repair with a disbond. In the case of a tapered edge, the boundary condition for determining the stress at the prospective crack location must consider the change in thickness over the partial disbond. Figure 5.26 shows the free body diagram for this particular case. Note that there is no need for an interface condition between steps when only the last step ln remains bonded, thus simplifying the analysis. In order to use the previously developed system of equations for tapered reinforcements, the notation for the different steps in the disbonded section is indexed 2-eL..-1,0,1 where 1 indicates the identical geometry as the first section of the remaining bonded area and d is the number of disbonded steps. The stress at the prospective crack location for a disbond with a fully bonded last step is determined in the first part of the stage I analysis of the Rose model. 5.6 Generalization of the Rose Model for Partially Disbonded Patches 188 As before, the change in length in the patch and substrate can be evaluated using: i i k=2-d Als = esb = TS1 R d k=2-d P k Pfc + asATb where b= J2 (5.224) (5.225) (5.226) k=2-d Using the definition of the elastic shear stress distribution parameter A and the force equi librium given in equation (5.13), the shear stress at yi = 0 can be expressed as: A? R d T a i ( W i = ° ) = A? ( V I / I = O ) - ws(yi=o)) (A/ p - A/ s ) Af6 _,Rd fc=2-d " P i yEP1tP1 EstsJ yEPltPl Ests J (5.227) (5.228) (5.229) The shear stress expression can be simplified by defining the constant H as H = (5.230) and using the integration constant Cp from Section 5.2 as given by equation (5.16): R d r a i {yi=o) = \2b(T™H-CPl) (5.231) A proper boundary condition can now be established by substituting and r™(yi=o). The direct stress resultant for the patch can be evaluated using equation (5.11). . R d R d R d A1 TPI = T P I (wi=°) = ~\7' + C ^ (5.232) 5.6 Generalization of the Rose Model for Partially Disbonded Patches 189 The shear stress is determined using equation (5.10) R d „ R d „ r a i (Vl=o) = B1 (5.233) thus the boundary condition yields: B™ = A^XibH + CPlXlb(H - 1) . (5.234) The additional equation to solve for ARd and BRd is given by the boundary condition at the edge, thus: TPn(yn=in) = Y± cosh(A„Z„) + sinh(A„/ n ) + CPn = 0 . (5.235) An An Using equation (5.52) and (5.53) developed in Section 5.2, the boundary condition at the edge yields: . R d R d /(0=o) cosh(A„Z„) 4- / O ^ " - 1 ) sinh(An/n) A \ = -Bi -Xi /(=2^-\) s i n h ( A n / n ) (5 253) a° l2 /r>=o)cosh(A nJ n) + / ( ^ 2 " - 1 ) s i n h ( A n Z n ) = - ^ A i | j r - (5-254) The corresponding half crack opening displacement vdQl is then calculated using equation (5.8) and (5.117): 0 1 2 ( i ? P l £ P l £ s £ s ) A]Ff ^ ^ 5.6 Generalization of the Rose Model for Partially Disbonded Patches 193 The additional displacements due to strains in the substrate and patch are given as: uoIl 7oIII 2 ^—' E t k=2-d p"lpk d (5.256) (5.257) which are then used to calculate the total half crack opening displacement as follows: d d d d = «oi + v o i i + vom 1 2 + 2 \EP1 "'Pi -^s^s d , 1 2 + \ 1 d 1 d i _ J _ _ L a°ta 1 i . , a o . 2 \ £ , p 1 7 j p i E'gt, (5.258) (5.259) (5.260) As previously discussed, it is necessary to determine the half crack opening for a disbond extending underneath the last step of the patch separately. The half crack opening due to the shear displacement in the adhesive can be determined using equation (5.208) from the uniform thickness patch analysis while the displacements due to the strain in the substrate and patch are given in equation (5.256) and (5.257). Thus the total half crack opening displacement for a disbond extending underneath the last step of the patch is given by: crj: o"s EPltPl + E^ts A i tanh(Ai_Zi) ( 1 _|_ 2 \ \ (5.261) The upper limit for the stress intensity factor under plane stress conditions can be expressed for the partially disbonded tapered repair as (5.262) 5.6 Generalization of the Rose Model for Part ia l ly Disbonded Patches 194 where A is given for disbonds not extending underneath the last step of the patch as A = 1 1 + Eat a vrF 2 V 2£P1tP1 V •iFt + (_1_ + _2_) \ ^ P l ^Pl Ests J (5.263) and for disbonds extending underneath the last step of the patch as A 1 / Ests TTY2 2EPltPl i kSJk U*1**+ *0 Ai tanh(Aiii) ( I _| 2_\ \ (5.264) A factor of 1/(1 — f 2 ) needs to be included in A in the case of plane strain conditions. Using and as asymptotes, as suggested by the original Rose model, the stress intensity factor and the corresponding energy release rate for different crack lengths are then derived by interpolation using equation (5.220) and (5.221). It is noted that the accuracy of this analysis was improved by accounting for the shear stiffness of the adherends by using Ashear-iag (equation (5.92)) instead of A. The results of this analysis scheme are presented for the same three cases as presented in Section 5.6.1, i.e.: • Thermal residual stress loading ( A T = — 64.8°C) • Remote stress loading (O-QO = 40 M P a ) • Combined thermal residual stress and remote stress loading ( A T = —64.8°C, (Too = 40 M P a ) The only difference between the uniform thickness patch analysis and the tapered patch analysis is the thickness change of the patch near the edge. A l l other material parameters remain unchanged. 5.6 Generalization of the Rose Model for Partially Disbonded Patches 195 The patch thickness is defined as: P l < i < 7 = 1.056 mm — i 0.132 mm (5.265) with step lengths of: lx = 57mm and . I, 2** « = ° - 3 3 = aSz = 23.29 / ie /°C a 5 y = 23.45 nt/°C F M 73M: Eax = Eay = Eaz = 2.19 GPa Gaxy = Gayz = Gaxz = 844 MPa "axy = Vayz = "axz - 0.3 « a x = Oiay — a a z = 50 / ie /°C Note: The y-direction is aligned with the boron fiber and aluminum grain orientation. 6.2 Verification Models 204 6 .2 V e r i f i c a t i o n M o d e l s In total, five verification models were created. The initial model representing a uniform thickness reinforcement was used to determine the mesh size in regions of high shear stress. This model served as a basic reference case due to the availability of a good theoretical model for comparison. A tapered reinforcement was chosen as a second model to optimize the required number of elements in the tapered region. Both normal and reversed stacking sequence were investigated to assess the importance of detailed representation of the actual structure. Next, a model of the substrate in the cracked section was investigated. The focus of this model was placed on the stress intensity factor prediction. The mesh for the crack region had to be sufficiently fine along the crack face to predict the adhesive shear stress properly once the adhesive and patch were added to the model, i.e. the next step. The results of this model were then compared to the developed closed form approximation for the stress field around a patched crack. The A M R L specimen exhibits additional in-plane bending due to the partial patching across the specimen width which are not considered in the theoretical approach making this patched crack model an assessment tool for the theoretical crack tip stress field model. The half circular geometry of the patch also complicated the finite element analysis. It was required to check if the aspect ratios found for the tapered reinforcement were acceptable for a circular patch due to the change in the material orientation with respect to the element orientation. Finally the entire A M R L specimen was modeled using the mesh specifications derived by the different verification models. The number of elements required to get an accurate representation of the A M R L specimen was so large that an attempt to verify the mesh size in the full model was too time consuming and resources were quickly exhausted. 6.2 Verification Models 205 6.2.1 Constant Thickness Double-sided Reinforcement This first model was intended to confirm the input data into MSC/Nastran as well as compare the results with the proven one-dimensional model of L.J . Hart-Smith. In order to avoid any stress concentration effects, the model had identical areas for the substrate, adhesive and patch. In-plane quarter symmetry as well as symmetry through the constant thickness model were used. The material thickness were selected to represent the inner section of the A M R L specimen: 3.175 mm thick aluminum face sheet, 0.25 mm thick adhesive layer and the 0.924 mm thick seven layers of boron/epoxy. This investigation was also useful for determining the required element size for the reinforced crack region in the A M R L specimen since the shear stress distribution away from the crack tip can be approximated by the shear stress distribution in a double-doubler joint. It should be noted that the shear stress distribution near the center of a double-doubler joint under thermal loading is identical to the shear stress distribution near the free edge of reinforcement considering a sufficiently long overlap length. The aluminum, adhesive and patch were modeled initially using two elements through the thickness of each material. Five elements were used across the half-width of 12.5 mm. The specimen half-length was 75 mm. The element size was reduced from 5 mm in the center of the specimen with zero shear stress to 0.25 mm, 0.0625 mm and 0.03125 mm respectively along the length to the specimen edge with the highest shear stresses. The last four elements adjacent to the free edge had the same element length when an element length of 0.25 mm and 0.0625 mm was used. Eight elements with equal length were chosen for the model with an element length of 0.03125 mm. The chosen element lengths were based on fractions of the adhesive thickness between ta and ta/8 covering Callinan's recommendation of a maximum element length of rja/5 near the edge [24]. A temperature change of — 64.8°C was applied as a loading condition representing the cooling stage from the stress free temperature to room temperature. 6.2 Verification Models 206 Figure 6.1 shows the shear stress in the mid-plane of the adhesive along the centerline and close to the end of the specimen for the three F E M analyses and Hart-Smith's theoretical solution. The theoretical analysis shows excellent agreement up to 0.5 mm away from the free edge. This one-dimensional approach cannot account for the drop to zero shear stress at the free edge. The coarse F E M model shows the highest shear stress (21.4 MPa) 0.25 mm away from the free edge while the two finer meshes show a peak shear stress (20.0 MPa) at approximately 74.85 mm from the center of the specimen. It should be noted that the one-dimensional analysis predicts nearly the same value (19.7 MPa). Changing from an element length equivalent of 25% to 12.5% of the adhesive thickness results in only minor differences in the peak shear stress and its location. 22 20 18 16 14 12 | 8 " 6 H 4 2 0 -2 -4 -6 -8 -1-D Theory -0.25 mm - 0.0625 mm -0.03125 mmm 74.125 74.25 74.375 74.5 y [mm] 74.625 74.75 74.875 Figure 6.1: Adhesive shear stress for a uniform thickness reinforcement using element lengths between ta and ta/8 The mesh with an element length of 12.5% of the adhesive thickness shows a shear stress of -5.0 MPa at the free edge while the finer mesh drops to -6.1 MPa. Both meshing schemes show a significant error which is not surprising considering the large stress gradient near the free edge. The shear stress prediction for an element length equivalent to 1/4 of the adhesive thickness shows the better result for this particular finite element verification model. 6.2 Verification Models 207 In addition, a model with a larger number of elements through the thickness was tested. The mesh with the smallest element length of (12.5%) ta was used with 6 elements through the thickness for each material. Figure 6.2 shows the variation in shear stress results for 2 and 6 elements through the thickness. 22 20 18 16 14 12 ^ 10 Q- 8 a - 2 Elements - 6 Elements 74.125 74.25 74.375 74.5 y [mm] 74.625 74.75 74.875 75 Figure 6.2: Adhesive shear stress for a uniform thickness reinforcement with a different number of elements through the thickness The position of the peak shear stress shifted by only 0.04 mm and the peak stress was reduced by 0.5 MPa. The shear stress at the free edge was less than 0.1 MPa, i.e. nearly the theoretical value of 0 MPa. The larger number of elements through the thickness would be preferred near the free edge, but it would increase the total number of elements since a transition should not be made in sections with high stress gradients. Based on the importance of predicting the peak shear stress and its location, it was decided to use an element length equivalent to 25% of the adhesive thickness. Generally, two elements through the thickness were used for each material except in the aluminum near the crack where four elements through the thickness were employed to improve the prediction of the crack opening. 6.2 Verification Models 208 6 . 2 . 2 T a p e r e d R e i n f o r c e m e n t w i t h a R e v e r s e d S t a c k i n g S e q u e n c e A tapered reinforcement was chosen as the second verification model. It was important to verify that a sufficient number of elements has been used for each step to determine the thermal residual strains in the adherends and the adhesive shear stress properly. The smallest number of elements was desirable to reduce the overall degrees of freedom for the A M R L sandwich type model. In the past, tapered edges have been modeled using tapered elements or a normal stacking sequence, not necessarily representing the real lay-up sequence [19, 24]. Specially tapered elements reduce the required number of elements but they do not represent the true structural details of the actual taper. As part of this verification model, a normal stacking sequence was compared to the actual reversed stacking sequence of the A M R L specimen (Figure 6.3) showing the importance of proper structural representation and design choices under given loading conditions. Reversed Stacking Normal Stacking Figure 6.3: Reverse and normal stacking sequence The reversed stacking sequence is significantly more complex to model than the normal stacking sequence if the change in material orientation in the thickness direction is included. The size of the adhesive fillet was measured using a precured patch where the peel ply was not removed. The fillet shows up as a smooth region versus the typical fabric texture of the peel ply. The size of the fillet varies with respect to the location along the outer edge in the case of a half circular patch as used for the A M R L sandwich type specimen. The fillet length is approximately 1 to 1.25 mm for boron/epoxy if the taper is aligned with the fiber orientation, while it is only 0.25 mm where the fiber orientation is parallel 6.2 Verification Models 209 to the taper. The carrier fabric of the boron/epoxy prepreg causes a small fillet where the fibers are perpendicular to the taper direction while the larger fillet is a result of the stiffness of the boron fibers where the fibers are parallel to the taper orientation. In order to reduce the complexity and the required number of elements to account for the fillet properly, a triangular fillet with a length of 1 mm was chosen regardless of the location in the patch. This approach can be justified by the relatively long distance from the locations of the locally thicker adhesive to the regions of high shear stress in case where the fiber orientation is not parallel to the taper orientation, thus not affecting the results in a critical area. Figure 6.4 through Figure 6.9 show results of the convergence study which was carried out for the reversed stacking sequence. The shear stress in the mid-plane of the adhesive as well as the thermal residual strains at the top and bottom surface of the patch and substrate were evaluated. The effect of smaller elements as well as more elements through the thickness are shown in these figures. Figure 6.4: Transverse shear stress at the adhesive mid-plane in the tapered region for different F E M meshes 6.2 Verification Models 210 Typical ly seven elements through the thickness of the boron/epoxy were chosen to simplify the modeling while the adhesive and aluminum were represented by 2 elements through the thickness as shown in Figure 6.5. L i i J L . _ l l f g = a a , Figure 6.5: Detailed view of the finite element mesh in the tapered section-reversed stacking The number of elements was doubled to study the element number dependence through the thickness. These figures show the extreme cases of this convergence study for element length along the taper with the smallest element size of both 0.125 m m and 0.03125 mm at the ply drop-off. Furthermore, the number of elements was reduced along the taper for the Figure 6.6: Thermal residual strains at the top surface of the patch in the tapered region for different F E M meshes case of seven elements through the thickness of the boron/epoxy, two elements through the adhesive and two elements through the aluminum thickness. The smallest element length 6.2 Verification Models 211 Figure 6.7: Thermal residual strains at the bottom surface of the patch in the tapered region for different F E M meshes was 0.125 mm in comparison to the curve for the model with twice as many elements through the thickness. The labels in the plots identify the number of elements for each component and the smallest element size at the ply drop-offs. 600 500 400 300 J i 200 100 0 -100 -200 - - - - 7 B/en-2 FM73M - 2 Al-0.03125 mm 1 7 4 B/ep - 4 FM73M - 2 Al -B/ep - 2 FM73M - 2 Al - 0 0.125 mm 125 mm 48 51 54 57 60 63 y [mm] 66 69 72 75 78 Figure 6.8: Thermal residual strains at the top surface of the substrate in the tapered region for different F E M meshes 6.2 Verification Models 212 7 1 B/ep - 2 FM73M - 2 Al - 0 4 B/ep - 4 FM73M - 4 Al -B/ep - 2 FM73M - 2 Al - 0 03125 mm D.125 mm 7 125 mm I 48 51 54 57 60 63 66 69 72 75 78 y [ m m ] Figure 6.9: Thermal residual strains at the bottom surface of the substrate in the tapered region for different F E M meshes All curves in Figure 6.4 through Figure 6.9 show good agreement, especially the ther-mal residual strains at the aluminum substrate bottom (see Figure 6.9). Location A in Figure 6.6 shows the effect of reducing the number of elements along the taper in some non-critical regions. The smaller element length at the ply drop-offs emphasizes the singu-larities in the thermal residual strains in the boron/epoxy (see location B in Figure 6.7). The mesh with seven elements through the patch and two elements through the thickness of the adhesive as well as the substrate and an element length of 0.125 mm at the ply drop-off was chosen to model the tapered region of the A M R L specimen based on the good agreement with the finer meshes. A comparison between the normal and reverse stacking sequence was carried out next. In addition, the results of the one-dimensional analysis are plotted. Figure 6.10 shows how well the one-dimensional analysis can predict the shear stresses in the mid-plane of the adhesive especially for the normal stacking sequence. The general trend is predicted properly, even for the reversed stacking sequence but the finite element analysis shows clearly the influence of local increases in the adhesive thickness. The peak adhesive shear stress is overpredicted 6.2 Verification Models 213 14.0 13.0 12.0 11.0 10.0 9.0 8.0 a" 7.0 2 6.0 £ 5.0 4.0 3.0 2.0 1.0 0.0 -1.0 -2.0 I Reverse Stacking i Normal Stacking /A 1 -D Theory /// A f A A J J / / / . TI \ V \ \i \J \ V \ > 48 51 54 57 60 63 y [mm] 66 69 72 75 78 Figure 6.10: Transverse shear stress at the adhesive mid-plane for reversed and normal stacking sequence by the one-dimensional analysis, which is caused by extending the adhesive layer past the last boron/epoxy layer. Figure 6.1 shows how well the one-dimensional theory predicts the peak adhesive shear stress if the patch and adhesive end at the same location. It should be noted that the mid-plane does not show the highest shear stress using the reversed stacking sequence. The singularities at the ply drop-offs lead to an increase in the shear stress close to the patch/adhesive interface. Figure 6.11 and Figure 6.12 show the sensitivity of the thermal residual strains in the patch towards the choice of stacking sequence. A large discrepancy between the finite element results and the one-dimensional analysis is visible. The magnitude of the thermal residual strains predicted by the one-dimensional analysis compares only reasonably well to the strains determined in the F E M analysis at the patch/adhesive interface with a patch utilizing the normal stacking sequence. The large spikes in the finite element results for the thermal residual strains using either stacking sequence are caused by the stress singularities at the ply drop-offs. This effect is not included in the theoretical one-dimensional analysis. A significant reduction of over 6.2 Verification Models 214 200 Figure 6.11: Thermal residual strains at the top surface of the patch for reversed and normal stacking sequence 500 fie in the peak compressive strain through the thickness of the patch is evident for the reversed stacking sequence. In addition the location of the peak compressive strain also shifts from near the ply drop-off at the bottom of the patch to the end of the adhesive fillet at the top surface of the patch. For the normal stacking sequence for the patch lay-up, Figure 6.12: Thermal residual strains at the bottom surface of the patch for reversed and normal stacking sequence 6.2 Verification Models 215 the reduction in compressive thermal residual strains through the thickness of the patch is even more significant (up to 700/ie) for this particular loading case. Figure 6.13 and Figure 6.14 show the results for the thermal residual strains in the substrate. The effect of the stacking sequence on the aluminum substrate is not significant especially at the bottom side of the 3.175 mm thick aluminum sheet where the deviation in thermal residual strains between the normal and reversed stacking sequence is less than 15 fie. The result of the one-dimensional analysis is also quite close to the F E M analysis with generally less than 30 fie variation regardless of the stacking sequence at this location. It should also be noted that the thermal residual stresses in the substrate don't drop to zero at the patch edge due to the extended adhesive layer. The effect of the reversed stacking sequence on the thermal residual strains in the substrate are more pronounced at the substrate/adhesive interface. The thermal residual strains are reduced in the regions of increased adhesive thickness while they are in excellent agreement in the mid-section of a tapered step. 600 500 400 300 Jr 200 100 0 -100 -200 \ Reverse S tacking acking 1 Normal St. 1 -D Theon48 51 54 57 60 63 y [mm] 66 69 72 75 78 Figure 6.13: Thermal residual strain at the top surface of the substrate for reversed and normal stacking sequence 6.2 Verification Models 216 600 Figure 6.14: Thermal residual strain at the bottom surface of the substrate for reversed and normal stacking sequence One of the most important findings are the high peel stresses in the adhesive due to the reversed stacking sequence (see Figure 6.15). The main cause is the orientation change of the upper plies passing over a ply drop-off resulting in peel forces. The peel stresses develop at different locations depending on tension or compression in the patch. The peak peel stress in the adhesive mid-plane occurs at the ply drop off in case of compressive stresses in the 10.0 8.0 6.0 4.0 2.0 0.0 -2.0 -4.0 -6.0 -8.0 -10.0 ll il Jt / / / / / / / J / / / / / / / / / \c / / / / / / 7 / / \ r if i I I 48 51 54 57 60 63 y [mm] 66 69 72 75 78 Figure 6.15: Peel stresses in the adhesive mid-plane for the reversed stacking sequence 6.2 Verification Models 217 patch (see location A in Figure 6.16) while for tensile stresses the peak occurs at location B. Tensile loading of the bonded repair will therefore generally change the location and reduce the magnitude of the peel stresses introduced by the thermal residual stresses, while compressive loading will reverse the peel stresses and thus may contribute to the failure of the bonded repair. A fractographic investigation by the National Research Council showed a change in failure mode at the ply drop using a reversed stacking sequence. This failure mode change was not present when using a normal stacking sequence [37]. Based on the findings in the finite element analysis, peel stresses can be associated with this failure mode change. t t A B Figure 6.16: Critical location for peel stresses in the adhesive mid-plane for the reversed stacking sequence Two conclusions can be drawn from the prediction of high peel stresses: The stresses and strains in a tapered patch with a reversed stacking sequence should not be analysed by approximating the taper using a normal stacking sequence nor a tapered element. Addi-tionally special attention should be paid to the peel stress in the tapered region when using a reversed stacking sequence under compressive loading of the bonded repair. The stresses in the substrate and in the adhesive are generally the main concern for bonded repairs. Most crack patching applications arise from cracking in sections loaded in tension. The one-dimensional analysis predicts the thermal residual strains for the aluminum sub-strate and the adhesive shear stress very well. Small discrepancies arise due to specific de-6.2 Verification Models 218 sign choices, such as extending the adhesive layer beyond the patch. Patch design choices such as the stacking sequence become far more important for reinforcements or cracked sections of components loaded in compression based on the high peel stresses found in the taper using a reversed stacking sequence. 6 . 2 . 3 C r a c k M e s h V e r i f i c a t i o n The crack mesh verification model was the first step in generating an appropriate finite element model for the cracked region of the A M R L specimen. This model deals only with the metal substrate due to the availability of a closed form solution allowing the verification of the finite element results rather than a convergence study. A large number of crack meshes were investigated. One of the main difficulties in this investigation was the small element length requirement perpendicular to the crack in order to allow for the proper determination of the adhesive shear stresses and thermal residual stresses in the adherends. This requirement was complicated by the lack of a crack tip element in MSC/Nastran for Windows. For this reason three approaches were investigated. First, a finite element mesh was generated using only parabolic elements. In a second approach, additional bulk data cards were submitted as part of the analysis sequence creating MSC/Nastran CRAC3D crack tip elements which are not supported in the user interface. The third approach utilized the typical quarter point method where the mid-side nodes of the elements surrounding the crack tip are moved to the quarter point location. This third approach was ruled out due to the number of warnings issued by MSC/Nastran with respect to the determination of the stresses, i.e. crack tip stress reached 10000 GPa. The MSC/Nastran user guide noted that this approach might also lead to an early abortion of the calculation. Another significant disadvantage of moving the mid-side is the connec-6.2 Verification Models 219 tivity to the bonded patch. If the mid-side node in the patch is moved to the quarter point location, the stress distribution will be highly distorted. The other option is to make a transition in the aluminum or adhesive which will lead to uncertain answers due to the unknown element behaviour. Significant verification work would be required to study such element behaviour. The advantage of using the regular parabolic elements is a better spatial stress distribution. The shape of the constant stress contour lines are closer to the theoretical estimation espe-cially for elements with high through thickness aspect ratios (25:1). In order to get a good spatial distribution with the MSC/Nastran CRAC3D elements, a through thickness aspect ratio of 6:1 was required, thus using significantly more elements through the thickness. The use of the CRAC3D elements is significantly more complex and the implementation is time consuming. In addition, the elements cannot be visually checked and results cannot be displayed although they were available in the T06 output file. It should also be noted that the computed stress intensity factor by M S C / N A S T R A N varied by approximately 13% de-pending on the use of the full modeling option or the symmetry option. The listed element stresses for these two modeling options varied as well based on different natural coordi-nates. The stresses, strains and displacements for the surrounding elements were compared and found identical despite the differences in the output of the crack tip element. The stress intensity predictions given by M S C / N A S T R A N seemed rather low in comparison to extrapolation of stress intensity factor plots based on the crack opening displacement. The theoretical stress intensity factor for this particular verification model can be calculated using [23]: (6.1) where Y = 1.12-0.23 + - « • " ( £ ) ' (6.2) 6.2 Verification Models 220 and was determined to be Kr = 2\.\2MPa^/m for a specimen containing a 20 mm long crack and a width of 50 mm under 40 MPa of remote tensile loading. This theoretical value was used to assess the quality of the finite element crack tip mesh by comparing it to the stress intensity factor calculated from the finite element results. The finite element model length was chosen to be 100 mm. It was decided to use a sheet thickness of 1.5875 mm to achieve plane stress conditions under the chosen remote loading. The choice of plate thickness also improved the aspect ratio of the elements in the crack tip region to approximately 1:6 by utilizing symmetry through the thickness and with respect to the crack. Two elements were placed to represent the half sheet thickness. The crack opening displacements determined by the finite element analysis were used to calculate the stress intensity factor as given by [25]: K IFEM 2TTE r 4 (6.3) Note that the equation for KIpEM is shown for plane stress conditions. Figure 6.17 shows the stress intensity factors calculated for the finite element crack opening displacements utilizing the different crack tip modeling approaches. The results for the stress intensity 23.5 23 22.5 22 21.5 21 QL 20.5 \ / — •or f * Crack tip element (CRAC3D) Crack tip element (Quarter point node) No crack tip element (Parabolic element) • —A— ,i o ; ^ 5 f 3 1 0 1 2 1 4 1 6 1 8 2 20 19.5 19 18.5 18 17.5 r [mm] Figure 6.17: Stress intensity factors for different crack tip elements 6.2 Verification Models 221 factor based on the crack opening displacements are within 1% of each other for the different crack tip modeling approaches for locations which are more than 10% of the crack length away from the crack tip. Larger deviations occur closer to the crack tip. The MSC/Nastran CRAC3D element shows a spike after the second element from the crack tip. The reason for this spike is possibly the different displacement formulation used by MSC/Nastran for the crack tip element in comparison to the parabolic elements. The quarter point approach results also in a spike in the stress intensity factor at the interface between the crack tip element and the adjacent parabolic elements. The model with only parabolic elements shows a gentle decrease in the stress intensity factor close to the crack tip. Chan et.al.'s [25] approach of using a linear extrapolation is applied rather than calculating K I F B M just from the crack tip elements. Good linear sections of the stress intensity factor versus r are found from approximately 15% to 50% of the crack length. The extrapolated stress intensity factors are within 0.5% of the theoretical estimation for the three modeling procedures representing excellent results. This can be largely contributed to the small element size of approximately 0.3% of the crack length in the crack tip vicinity. The stress in the loading direction was investigated as a second parameter to check the performance of the crack tip modeling. Figure 6.18 shows the stress in the loading direction (ay) ahead of the crack along the through thickness center line. Negative coordinates refer in this case to the cracked section. The nodal stress for the CRAC3D element are not provided in the post-processor thereby not allowing nodal stress averaging with the adjacent elements. Generally this is not a concern with a fine finite element mesh, but the large stress gradient present at a crack tip it will distort the results. Therefore the nodal stresses in the elements next to the crack tip element were not displayed. It is apparent that the stresses for the model with the CRAC3D element are higher ahead of the crack tip than the results given by the two other modeling approaches. It should 6.2 Verification Models 222 3000 2800 2600 2400 2200 2000 1800 „ 1600 £ 1400 ~ 1200 0 1000 800 600 400 200 0 -200 -400 X Crack tip element (CRAC3D) — • — Crack tip element (Quarter point node) —A— No crack tip element (Parabolic element) v -j - & / s-x^ k 1 t 4_ v w w w w w ^ J\— ML M A M. A 2 -1 75 -1 .5 -1 25 -1 -0 75 -0 .5 -0 25 0 0. 25 0 5 0. 75 1. 25 1 5 1. 75 2 r [mm] Figure 6.18: Stress perpendicular to the crack (ay) for different crack tip elements be noted that this difference only becomes evident for stresses exceeding the yield stress of the material (310 MPa) very close to the crack tip ( « 0.35 mm). The modeling approach using the quarter point method and no crack tip element have excellent agreement ahead of the crack tip except that the stresses for the quarter node element approach generate extremely large values (> lOOOOGPa) in the crack tip element while the parabolic element adjacent to the crack tip reaches nine times the yield stress of the material. A point of verification is that the stress along the crack face should be zero. All modeling approaches reach this value within 1-1.5% of the total crack length from the crack tip. The parabolic element shows the highest over shoot in the element adjacent to the crack tip, which is not surprising considering the steep gradient and the element formulation. It should be noted that the crack tip displacements the in x-direction (ahead of the crack tip) using the CRAC3D element were not satisfactory. The crack front through the thickness appeared jagged and possibly unreliable. 6.2 Verification Models 223 Having investigated these different modeling approaches, it was decided to use parabolic elements at the crack tip. All three methods can be used to determine the stress intensity factor quite accurately utilizing Chan's [25] extrapolation technique. The quarter node method was rejected due to the difficulties in connectivity to the patch. The CRAC3D element approach showed generally good results but required a tedious input as well as a larger number of elements through the thickness to avoid distortion of the in-plane stress field. Additionally the jagged displacements found along the crack front were unrealistic. The behaviour of the parabolic elements was quite good due to the small element size in the crack tip vicinity. The parabolic element also handled higher aspect ratios through the thickness better. The only short fall was the approximately 10% overshoot in comparison to the peak stress along the crack front. Having established a good mesh scheme using parabolic elements for the crack region of the substrate, the adhesive layer and the patch were added to the model as shown in Figure 6.19. 6 . 2 . 4 P a t c h e d C r a c k M o d e l The patched crack verification model was chosen to evaluate the theoretical crack tip stress field model for bonded repairs. The previous crack mesh verification model provided con-fidence in a proper mesh for the substrate. A constant thickness patch of 0.924 mm as well as the 0.25 mm thick adhesive layer were added to the crack tip verification model after increasing the thickness of the substrate to 3.175 mm thereby representing the center section of A M R L sandwich type specimen. The honeycomb used in the A M R L specimen is neglected since the theoretical model doesn't account for it. A uniform thickness patch over the entire substrate was chosen to avoid any additional bending due to partial coverage of the patch across the width of the model. The model length of 100 mm was chosen long enough to avoid an interaction between the load transfer zone at the free edge of the model and the high stress region around the crack tip. No mesh refinement was carried out near 6.2 Verification Models 224 the free edge due to the focus on the stress distribution in the vicinity of the crack tip. Two elements were used through the thickness of the adhesive, while four elements through the thickness of the seven ply patch were applied. The first layer of elements was used to represent the boron/epoxy layer next to the adhesive. The three remaining element layers for the patch always represented two boron/epoxy plies each. The mesh for the aluminum substrate was left identical to the previous verification model. Figure 6.19 shows the finite element mesh used in this investigation: One of the most difficult decisions to be made in this finite element approach was the modeling of the adhesive along and across the crack. Modeling the adhesive continuously across the crack implies that no crack opening occurs at the adhesive/substrate interface based on no crack opening in the initial unstressed state. In order to model the adhesive continuously across the crack using a discrete modeling approach such as finite elements, a small crack opening displacement must be present and the adhesive must be extended past the crack face towards the symmetry line of the repair. A plasticity analysis can then be carried out to get a proper result for the crack opening displacement in the substrate. The approach taken in this research work is based on a discontinuous adhesive layer across the crack. Its advantage is that the analysis can be carried out using a linear-elastic Figure 6.19: Finite element mesh for the patched crack verification model 6.2 Verification Models 225 approach thus reducing the computational requirements. Additionally it presents an upper bound approach for the crack opening displacement. Failure of the adhesive in tension on top of the crack is a realistic possibility. Planned fractographic investigations of the region around the crack after fatigue failure will hopefully give a better indication about the true failure sequence. Unfortunately, papers dealing with the finite element modeling of this critical region have not discussed this problem in detail. The approach taken in this research work is similar to finite element models of bonded repairs developed by other researchers using spring elements to represent the adhesive [19]. A spring model implies that the adhesive is discontinuous across the crack. These models have been successfully used to predict the crack growth rate in bonded repairs. The finite element stress distributions for ax, ay and rxy along 9 = 0° and 90° are presented and compared to the closed form theoretical analysis in Figure 6.20 through Figure 6.25. All graphs show the theoretical results for plane stress and plane strain conditions. The stress intensity factor caused by the thermal residual stresses alone will generally lead to plane strain conditions for the crack analysis while combined with other external loading plane stress conditions at the crack tip are common. The finite element results for the substrate are displayed at the bottom and top surface of the aluminum face-sheet. The substrate bottom surface is the critical case due to the larger crack opening displacement. The adhesive restricts the crack opening displacement at the top surface of the substrate to a significant degree. The difference in the crack opening displacement between the top and bottom surface of one face-sheet at the edge of the A M R L specimen is approximately 22% which compares well with the 19% given by Baker for the F E M analysis of a simple double-doubler joint [18]. Figure 6.20 shows excellent agreement with the theoretical stresses in the loading direction along 9 = 0° for plane strain conditions and the finite element results for the substrate bottom. Note that the substrate bottom coincides with the through thickness symmetry 6.2 Verification Models 226 plane due to neglecting the honeycomb. The difference between the fracture mechanics analysis for plane stress and plane strain conditions is approx. 5% for stresses below 150 MPa. Significant difference is visible between the finite element results for the top and bottom surface of the substrate. Good agreement between the closed form theoretical analysis and the F E M analysis for the symmetry plane can be attributed to accounting for the shear-lag in the adherends. The discrepancy in the stresses at locations more than 17 mm from the crack tip is due to modeling an edge crack in the F E M analysis and using the center crack assumption in the theoretical approach. The estimation of the stress intensity factor for the substrate using the finite element results imposes some difficulties. Typically the crack opening displacement is used to estimate the stress intensity factor for a cracked sheet. The questions that arises is, can this approach be applied to bonded repairs? Inspecting the stress field behind the crack tip (6 > 90°) for a bonded repair reveals differences in comparison to the typical stress field of a crack in an unreinforced sheet due to the load transfer into the patch rather than through the remaining 150 100 CO I 50 o II — 0 -50 -100 Fracture Mechanics Theory (Plane Strain) Fracture Mechanics Theory (Plane Stress) FEM Substrate Symmetry Plane FEM Substrate Top -20 -15 -10 10 15 20 25 30 Figure 6.20: Stress in loading direction (ay) for the patched substrate along 9 = 0° 6.2 Verification Models 227 uncracked section. This finding makes the application of the crack opening displacement method based on an unreinforced sheet questionable. On the other hand, the stress field observed ahead of the crack tip for a bonded repair shows the typical stress distribution. It seems therefore more appropriate to use a stress extrapolation method to determine the stress intensity factor for different through thickness locations in the substrate. This approach is also supported due to the stresses ahead of the crack tip being the driving force for the crack rather than the crack opening displacement. The stress extrapolation method can be based on the usual one term approximation if data points are selected close to the crack tip. The corresponding stress intensity factor for a given stress in loading direction at a distance r from the crack tip along 9 = 0° can be determined using: Kr = V27rray (6.4) Alternatively the correct elastic solution for a center crack in an infinite plate can be used to determine the stress intensity factor. In addition to the local stress 2.5 mm. This can be explained by the load transfer into the patch close to the crack rather than redistribution through the remaining section, and shows clearly the limitations of the developed theoretical approach. The theoretical approach provides good results ahead of the crack tip but the model does not account for the local load transfer into the patch. 6.2 Verification Models 229 150 125 Q. s O 75 at '° II O 50 25 I I I I I • Fracture Mechanics Theory (Plane Strain) Fracture Mechanics Theory (Plane Stress) • F E M Substrate Symmetry Plane F E M Substrate Top 2.5 7.5 10 12.5 15 r [mm] 17.5 20 22.5 25 27.5 30 Figure 6.21: Stress perpendicular to crack (ay) in the patched substrate for 9 = 90° Thus care should be exercised when using the model close to the load transfer zone. Note that the finite element results are only shown for r < 12.5 mm due to the lack of nodes along that line. Unfortunately MSC/Nastran for Windows does not offer postprocessing along arbitrary lines. The stresses in the x-direction are also quite well predicted by the closed form analysis especially close to the crack tip as shown in Figure 6.22 and Figure 6.23. It should be noted that the finite element analysis accounts for the small difference in the coefficient of thermal expansion between the patch and substrate in the x-direction. This influence was neglected in the theoretical analysis. The predicted stress parallel to the crack using the theoretical model is in-between the F E M results for the top and bottom surface of the substrate along 0 = 0° and 9 = 90° and close to the crack tip. The drop into compressive stresses for ax along 9 — 90° near the crack tip is also properly indicated. The remote stresses are also in good agreement. The finite element analysis generally predicts higher stresses for 2 mm < r < 10 mm. The stress ax along the crack face cannot really be 6.2 Verification Models 230 Fracture Mechanics Theory (Plane Strain) Fracture Mechanics Theory (Plane Stress) F E M Substrate Symmetry Plane F E M Substrate T o p Figure 6.22: Stress parallel to the crack (ax) in the patched substrate for 9 — 0° predicted using the theoretical model. The theoretical model is based on a center crack and an effective crack length for the bonded repair which is only a fraction of the real crack. 150 125 100 cs Q. 75 o II 5 50 - Fracture Mechanics Theory (Plane Strain) - Fracture Mechanics Theory (Plane Stress) - F E M Substrate Symmetry Plane F E M Substrate T o p 25 -25 2.5 7.5 10 12.5 15 r [mm] 17.5 20 22.5 25 27.5 30 Figure 6.23: Stress parallel to the crack (ax) in the patched substrate for 9 = 90° 6.2 Verification Models 231 The theoretical prediction for the in-plane shear stresses, rxy, show excellent agreement with the F E M analysis. The shear stress along 9 = 0° must be zero. Figure 6.24 shows clearly the error in the F E M analysis which must be expected at the singularity considering the use of parabolic elements and the high stress gradient near the crack tip. Figure 6.25 shows again just slightly better agreement between F E M results for the substrate bottom and the theoretical approach. 150 125 100 co Q. O II 75 S 50 25 — Fracture Mechanics Theory (Plane Stress/Plane Strain) — F E M Substrate Symmetry Plane — F E M Substrate T o p -25 -20 -15 -10 5 r [mm] 10 15 20 25 30 Figure 6.24: Shear stress (rxy) in the patched substrate for 9 = 0° In general the developed closed form theoretical model for the stress field in a bonded repair gives a good estimate for the stresses in the substrate considering the limitations of the model. The most important limitation is imposed by the load transfer across the crack from the substrate into the patch. The load transfer itself is accounted for in the reduction of the stress intensity factor but it is not accounted for in the stress field prediction. Thus the stress field estimate especially for the stress in loading direction, is certainly not well applicable for 9 > 80° to 90° due to the local load transfer as well as modeling an effective crack length instead of the real crack length with the restraint of the patch. 6.2 Verification Models 232 150 125 100 75 'a [MP 50 o O 25 o> II CO 1r 0 -25 -50 -75 100 _ re Mechanics Theory (Plane Strain) -re Mechanics Theory (Plane Stress) ubstrate Symmetry Plane Fractu Fractu F E M S - - - - reivi ouDSiraie i op if 2.5 7.5 10 12.5 15 r [mm] 17.5 20 22.5 25 27.5 30 Figure 6.25: Shear stress (rxy) in the patched substrate for 6 = 90° The stress field ahead of the crack tip is predicted quite well although it doesn't account for the localized load transfer into the patch around the stress singularity. In this case the load transfer is linked to the stress increase due to the singularity thus limiting the distortion of the stress field as predicted for a cracked sheet without a reinforcement. The reduction in the stress intensity factor due to the load transfer caused by the stress rise ahead of the crack tip is accounted for in Rose's model. The finite element results also showed that the transverse shear stress ahead of the crack tip is quite low i.e. less than 1 MPa. The through thickness variation of the in-plane stresses is not addressed in the stress field approach. The Rose model accounts for the shear-lag in the adherends using a linear relationship thus showing generally better agreement with the stresses at the substrate bottom, which is the through thickness symmetry plane of the model. Some improvement should be expected if a higher order approach is taken for the transverse shear distribution. A significant improvement in the stress field prediction should be achieved if proper stress functions are used which address the interaction with the patch. The developed theoretical 6.2 Verification Models 233 model can be used for preliminary design work and experimental work such as the placement of strain gauges. 6.2.5 Half Circular Patch Model The half circular patch model is the last requirement prior to modeling the sandwich type A M R L specimen. The objective is the establishment of the number of elements needed along the perimeter of the patch. The model consists of a two ply patch with one step using the reversed stacking sequence. Only two plies were chosen to reduce the modeling time. The substrate is modeled with the same in-plane dimensions as the longest ply, i.e. 75 mm. No crack was introduced in this model and symmetry conditions were applied to reduce the model to a 90° patched section. To check for convergence, runs of 8, 16 and 32 elements were used along the 90° patched section. Figure 6.26 shows the mesh used for the half circular patch. verification model with 32 elements along the circular perimeter. Figure 6.27 shows the through thickness meshing FH44444IIIIIIIII I I I I I I l ± H Figure 6.26: F E M mesh for circular patch verification model 6.2 Verification Models 234 scheme at the tapered edge. The two bottom element layers represent the aluminum. The adjacent two element layers are used to model the adhesive. In addition, the fillet in the tapered step has adhesive material characteristics. The top element layer to the right of the ply drop-off and the two top layers left of the ply drop-off represent the boron/epoxy composite layers. The adhesive fillet can be better seen in Figure 6.34. Figure 6.27: Detailed view of the tapered edge in the circular patch verification model The transverse shear stress ryz at a distance of 0.125 mm away from the free edge was chosen as the convergence parameter. Figure 6.28 shows the results of this convergence study. The highest shear stress occurs for 9 = 90° as expected, while it is nearly zero for 6 = 0°. Less than a 0.2% difference in the shear stress prediction is visible between 20 and 90 degrees using different number of elements along the perimeter. The error increases to approximately 1% for smaller angles which is an excellent result. Based on these results, 20 elements were chosen along the perimeter of the A M R L specimen patch. In order to account better for free edge effects, the element size was reduced for four elements near the free edge. The same refinement was used along the symmetry line at y = 0 mm to ensure accurate results for the outer taper step with the shortest fiber length. Element sizes 13% larger than the presented case of 16 elements for the 90° section were chosen for the section between 6° < 9 < 78°. 6.3 A M R L Sandwich Type Specimen Model 235 6 .3 A M R L S a n d w i c h T y p e S p e c i m e n M o d e l By utilizing the results of the previously discussed refinement models, a finite element model for the A M R L specimen can now be created without the need of further mesh refinements. Figure 6.29 shows the final mesh for the A M R L sandwich type specimen using double-symmetry. A representation of the honeycomb was added in order to get a better comparison with the experimental data. Although the honeycomb limits the out-of-plane bending, it does not eliminate it. The honeycomb was modeled as an isotropic material with an elastic modulus equivalent to the compressive modulus of the honeycomb in thickness direction (1.517 GPa [48]) and a coefficient of thermal expansion identical to the C T E of the aluminum face-sheet. Although the assumed in-plane elastic modulus of the honeycomb is only 2% of the aluminum face-sheet elastic modulus, it is still somewhat higher than the true moduli. An attempt to use significantly lower values for the in-plane modulus 6.3 A M R L Sandwich Type Specimen Model 236 Honeycomb = 6.35 mm -H h-Aluminum = 3.175 mm Figure 6.29: F E M mesh for A M R L sandwich type specimen 6.3 A M R L Sandwich Type Specimen Model 237 of the honeycomb caused numerical problems, i.e. pivot ratio of the stiffness matrix was not within the specifications of the finite element solver. Accounting for the honeycomb in the finite element model leads to changes of up to 9% in the thermal residual strains at the strain gauged locations. It should be noted that the maximum out-of-plane deflection does not occur in the center of a patch but near the tapered region for this particular honeycomb. The A M R L sandwich type specimen is modeled using 18806 elements with 254670 degrees of freedom. The purpose of the finite element analysis was to provide a better understanding of the detailed stress and strain distributions while the experimental work gave a good foundation for the analysis assumptions. This goal has been fulfilled as illustrated by the following results. Figure 6.30 and Figure 6.31 show the thermal residual strain fields for the mid-plane of the substrate and the top layer of the boron/epoxy patch in longitudinal direction giving examples for the complexity of the stress and strain fields in this particular application. Additionally, the transverse shear stress in the mid-plane of the adhesive caused by the mismatch in the coefficients of thermal expansion in the adherends is shown in Figure 6.32 since it is one of the important design variables. The peel stresses caused by the thermal loading are presented due to their importance with respect to disbonding. The top surface of the adhesive layer was chosen for this representation based on the magnitude of the peel stress at this interface. Finite element results are only presented for the cooling cycle from the effective stress free temperature to ambient temperature, i.e. A T = 64.8°, since the stress and strain distributions don't change assuming the coefficient of thermal expansion remains constant. The measured changes in these material coefficients with temperature are not large enough to cause any significant changes in the stress and strain distributions. Figure 6.30 shows the thermal residual strains parallel to the fiber orientation in the mid-plane of the aluminum face-sheet. The stress field ahead of the crack tip shows the typical stress field solution of a center crack in an infinite plate. The stress field gets more dis-6.3 A M R L Sandwich Type Specimen Model 238 0. 10. 20. 30. 40. 50. 60. 70. 80. 90. 100. 110. 120. 130. 140. 150. 160. Figure 6.30: Thermal residual strains parallel to the fiber orientation (y-direction) for the A M R L sandwich type specimen in [fie] at the mid-plane of the substrate torted wi th increasing distance from the crack tip as shown by the 500 fie contour line in Figure 6.30. The load transfer from the substrate into the patch near the crack face occurs approximately parallel to the crack and does not show the typical distribution of an unpatched crack where the load in carried by the remaining section ahead of the crack. The bending influence of the specimen is clearly visible for x > 117 mm. A n increase in the strains outside the patched region is found along the center line of the specimen although its magnitude is not cri t ical . 6.3 A M R L Sandwich Type Specimen Model 239 0. 10. 20. 30. 40. 50. 60. 70. 80. x Figure 6.31: Thermal residual strains parallel to the fiber orientation (y-direction) for the A M R L sandwich type specimen in [fie] in the top-surface of the patch The thermal residual strain distribution for the top layer of the boron/epoxy patch as presented in Figure 6.31 shows high strain gradients in the tapered region and near the crack. The thermal residual strain distribution in the tapered section along the y-axis in Figure 6.31 is very similar to the results of the tapered reinforcement verification model as presented in Figure 6.11. Moving from the y-axis to the x-axis, the thermal residual strain distribution changes ini t ia l ly just slightly but past 45° more significant changes occur in the 6.3 A M R L Sandwich Type Specimen Model 240 tapered region. The high gradients are significantly reduced as the symmetry line of the patch is approached. Due to the large element size in tangential direction, some inaccuracy in the tangential extent of contour lines should be expected. The convergence study for the circular patch showed that the chosen element size leads only to relatively small changes in the transverse shear stress distribution near the free edge. Large thermal residual strain changes can be identified near the crack face, where the load is transferred from the substrate into the patch. The compressive thermal residual strains decrease from approximately -700 fit (y = 22.5 mm) to -200 fie (y = 8 mm). Within 2 mm of the crack face, an additional increase in compressive stresses can be found at the top surface of the patch. Note that at the bottom surface of the patch, tensile strains are present leading to a high through thickness strain gradient in the patch over the crack. Figure 6.32 shows the transverse shear stress ryz in the mid-plane of the adhesive. The load transfer zone from the substrate into the patch extends approximately 30 mm perpendicular from the crack face. The transverse shear stress drops to near zero at the crack face fulfilling the boundary condition. It is important to note that the transverse shear stress ahead of the crack tip and up to 6 angles of 80° measured from the crack tip against the x-axis is nearly zero, i.e. below 1 MPa for this particular load case. This substantiates the assumption of negligible interaction between the patch and substrate ahead of the crack made for the estimation of the stress field. The increase in transverse shear stresses past approximately 80° shows the limitations in the stress field approach in the cracked section since the local load transfer is not accounted for in the theoretical approach. The transverse shear stress increases up to 15.5 MPa in the tapered section near the free edge of the patch, as also shown in Figure 6.10 for the tapered reinforcement verification model. Note that very low transverse shear stresses are present in the 2.5 mm section extending beyond the patch. The overall highest transverse shear stress is found close to the crack faces at the free edge 6.3 A M R L Sandwich Type Specimen Model 241 Figure 6.32: Transverse shear stress ryz for the A M R L sandwich type specimen in [MPa] in the mid-plane of the adhesive of the specimen. The transverse shear stress reaches nearly -20 MPa in the mid-plane of the adhesive at this location leading to the typical disbond pattern as presented by Baker [13]. One of the most important findings of the finite element analysis were the high tensile peel stresses in the adhesive at each tapered step caused by the reversed stacking sequence for the patch. The most critical locations are at the interface between the adhesive and the 6.3 A M R L Sandwich Type Specimen Model 242 0. 10. 20. 30. 40. 50. 60. 70. 80. Figure 6.33: Adhesive peel stress az for the A M R L sandwich type specimen in [MPa] at the adhesive/patch interface patch. Figure 6.33 shows the peel stress distribution and the critical locations in red on the top surface of the adhesive layer. Note that this view includes the adhesive fillets at the taper steps. A detailed isometric view of two tapered steps along the free edge of the specimen (R = 66 mm and R = 69 mm) is shown in Figure 6.34. This view also shows the modeling of the adhesive fillets well. Note that only one element is used through the 6.3 A M R L Sandwich Type Specimen Model 243 Figure 6.34: Detailed view of the adhesive peel stress az for the A M R L sandwich type specimen in [MPa] for the ply drop-offs at R = 66 m m and R = 69 m m thickness. Increasing the number of elements does not necessarily improve the accuracy since additional elements in the adjacent composite may lead to a misrepresentation. A more accurate determination of the stress and strain field would require modeling the actual fibers and matr ix of the composite material. The highest peel stresses in the taper are found at the t ip of the adhesive fillet in the current representation (see Figure 6.34). The overall 6.3 A M R L Sandwich Type Specimen Model 244 highest peel stresses are found near the crack face at the interface between the adhesive and patch. Figure 6.35: Fractographic investigation patch/adhesive interface at the ply drop-off The high peel stresses for the reversed stacking sequence at the steps in the tapered section could be the cause for the change in failure mode at a ply drop-off documented by the National Research Counci l of Canada [37]. Figure 6.35 [37] shows the finding in the frac-tographic investigation. This particular repair was subjected to high compressive loading which increases the peel stresses at the ply drop-off in addition to the peel stresses caused by the elevated temperature cure. Most bonded repairs are subjected to tensile loading which acts to reduce the peel stresses caused by the thermal residual loading, thus reducing the risk of failure init iat ion at this site. Based on this finding, it should be clear that the design of the tapered region of composite patches must account for the stacking sequence induced stresses at the tapered edge under combined thermal residual and applied loading. Table 6.1 shows the finite element results for the thermal residual strains at the strain gauged locations. The finite element results were obtained by averaging over the strain 6.3 A M R L Sandwich Type Specimen Model 245 gauge area using the contour plots for the thermal residual strains. By zooming in on the strain gauge area or sections of it, contour lines were plotted for every 2 fie (or less) thereby allowing the determination the average thermal residual strains quite accurately. An error of less than 20 fie is estimated for locations where the strain distribution varies in a highly non-linear fashion, e.g. locations A and G (see Figure 4.15). An error of less than 5 fie is estimated for locations with nearly linear thermal residual strain gradients, e.g. locations B - F and H . Generally good agreement between the finite element results and the experimental mea-surements has been achieved considering the assumptions made for the numerical modeling as well as the limitations of the experimental measurements. The two important assump-tions made in the finite element analysis are 1.) a uniform thickness adhesive layer exists under the patch except for the fillets in the tapered section and 2.) zero thickness of the strain gauges. It is known that some adhesive bleeding occurred at the free edge of the bonded repair, through the crack and starter notch during the curing process, however this is not believed to have had any significant effect on the stress distribution. Additionally, a local reduction in the adhesive thickness should be expected near the tapered edge. It is also known that the thickness of the patch was also been altered by embedding up to three strain gauges at one location. The placement of the gauges themselves were carefully planned to minimize any effect, but some increase in thickness cannot be avoided. It is assumed that the two gauges placed at the adhesive interface to the adherends were em-bedded into the adhesive thus reducing the increase in overall thickness. This has certainly had some limited effect on the local adhesive thickness which may also explain some of the discrepancy between the experimental and finite element results at locations with high transverse shear stresses, such as location A . The thermal residual strains for the aluminum substrate are quite well predicted for loca-tion A . The maximum discrepancy is only 31 fie, a good result considering the high strain 6.3 A M R L Sandwich Type Specimen Model 246 Locat ion Mater ia l Orient . Gauge No . F E M Exper imenta l Measurement Exper imenta l Measurement Average 1-D Analys i s Fracture Mechanics Analysis A A l I I 2 -24 tie 5 tie 5 /ze 84 ± 11/ze N / A A l X 1 -24 tie 7 fie 7 /ze N / A N / A B / E p B / E p I I I I 23 43 -746 tie -889 /ze -607 ize -665 /ze -636 /ze - 6 9 6 X 8 9 /ze N / A B / E p 1 44 129 tie 176 tie 176 tie N / A N / A B A l I I 3 606 tie 696 /ze 696 /ze 488 ± 3 fie 518 ± 12 fie A l J . 4 -174 txe -130 /ie -130 tie N / A N / A B / E p I I 24 -623 tie -598 /ze B / E p I I 30 -618 ue -593 /ze -574 /ze - 5 7 9 ± 4 / z e N / A B / E p I I 33 -594 tie -532 /ze B / E p X 34 34 fie -94 /ze -94 /ze N / A N / A C A l I I 12 512 / ie 566 /ze 566 fie 558 ± 1 fie 585 ± 2 /te A l X 11 -43 tie -78 /ze -78 /ze N / A N / A B / E p I I 25 -681 /ze -688 /ze B / E p I I 31 -682 tie -627 fie -633 fie - 6 6 1 ± 1/ze N / A B / E p I I 42 -693 /ze -585 /ze B / E p J_ 41 116 /ze -105 /ze -105 /ze N / A N / A D A l A l I I 10 22 496 tie 519 508 /ze 528 /ze 518 / i e 541 ± 1/ze 550 X 1 / ie A l A l X X 09 21 -164 fie -164 / ie -139 fie -112 /ze -126 /ie N / A N / A B / E p I I 28 -682 / ie -638 lie B / E p I I 29 -679 lie -641 /ze -616 / ie - 6 4 1 ± 1 tie N / A B / E p I I 40 -677 /ze -568 /ue B / E p X 39 5 tie -95 /ie -95 lie N / A N / A E A l I I 13 36 /ze 96 /ze 96 /ze N / A N / A A l X 14 -32 / i e -32 fie -32 /ze N / A N / A F A l I I 16 -87 lie -96 /ze -96 fie N / A N / A A l X 15 35 /ue 49 /ze 49 /ze N / A N / A G A l I I 7 317 tie 327 tie 327 tie 134 ± 1 / ie N / A A l X 8 -77 /ze -62 /ze -62 /ze N / A N / A B / E p B / E p I I I I 27 37 -866 l i e -860 /it -720 tie -748 / z e -734 /ze -1038 ± 1/ze N / A B / E p X 38 130 /ue 235 /ze 235 tie N / A N / A H A l A l I I I I 5 19 484 fie 477 /ze 532 tie 521 /ze 527 /ze 558 ± 1 fie 562 ± 1 fie A l A l X X 6 18 -130 /ze -132 fie -117 /ze -91 /ze -104 lie N / A N / A B / E p I I 26 -739 / ie -632 /ze B / E p I I 32 -739 fie -647 tie -634 / ie - 6 6 1 ± 1/ze N / A B / E p I I 35 -740 ize -622 / ie B / E p X 36 31 /Lie 30 lie 30 / i e N / A N / A Table 6.1: Comparison of the thermal residual strains at 21°C between finite element re-sults, experimental results, the one-dimensional analysis and the fracture me-chanics analysis gradient and the uncertainty in the shear modulus behaviour with temperature. A discrep-ancy of up to 25% between the finite element thermal residual strain prediction and the experimental measurements is present for the boron/epoxy patch at this location. Possible 6.3 A M R L Sandwich Type Specimen Model 247 causes are the position inaccuracy of the gauges, unaccounted changes in the local adhesive thickness especially in a high strain gradient area as well as local thickness increase due to the embedded gauges. The error for position inaccuracy is certainly large ( « 90 fie) as presented in Table 5.1. The uncertainty in the shear modulus behaviour with temperature might also be a contributing factor. The finite element analysis provided improved insight as to why high tensile thermal resid-ual strains for gauge 3 mounted on the aluminum face sheet in fiber orientation at location B were measured. The transverse shear distribution showed that the fracture mechanics model should only be applied to an angle 9 of approximately 80° or less to limit the influ-ence of the local load transfer for this particular load case. Gauge 3 is located in an area with increasing transverse shear stress. The increasing transverse shear stress indicates a distortion of the stress and strain distribution of an unreinforced crack which was assumed in the theoretical fracture mechanics analysis. In addition to the local increase in transverse shear stress in the adhesive, the placement of the strain gauges at the adhesive interfaces leads to a thinner adhesive layer, which will also affect the local stress distribution in com-parison the uniform thickness adhesive layer assumed in the finite element model. The thermal residual strain in the aluminum face-sheet at gauge 4 is reasonable well predicted with an discrepancy of only 44 fie (25%). The thermal residual strains measured in the fiber direction in the boron/epoxy patch near the adhesive interface shows excellent agreement with the finite element results with a discrepancy of only 4%. The thermal residual strains for the gauge mounted on top of the patch show larger differences between the numerical and experimental result which is likely caused by the local increase in thickness due to embedding strain gauges. This local section experiences some bending which decreases the strains in fiber orientation and increases the strains perpendicular to the fiber orientation under the given thermal residual loading conditions. 6.3 A M R L Sandwich Type Specimen Model 248 The quality of the finite element prediction at location C is quite similar. The gauges placed in or on top of the patch show again the effect of the local increase in thickness. The finite element analysis and the experimental measurement agree very well at the bottom surface of the patch (within « 1%). The thermal residual strains for gauge 31, which is mounted on the first boron/epoxy ply, show a discrepancy of 8% which is quite high for this particular location. A probable cause is the thickness of the underlying gauges including the solder connections. The highest difference was found for the thermal residual strains in the transverse direction on top of the patch. Two factors may have contributed to this result. The local increase in thickness leads to bending stresses under the applied tension due to the transverse mismatch in the coefficients of thermal expansion. Furthermore, the experimental strain measurements in the transverse direction show significantly more creep than the gauges in the fiber orientation at this location i.e. 60 fie vs. 10 fie in the fiber direction. It should be noted that the measured thermal residual strains are significantly smaller perpendicular in comparison to parallel to the fiber orientation thus making the inaccuracy due to creep more significant in the transverse direction. The thermal residual strains for the aluminum in fiber direction show a difference of just over 9% and perpendicular to the fiber orientation a discrepancy of 35 fie is found. The thermal residual strains are very well predicted for the aluminum in fiber direction at location D . The discrepancy between the prediction and measurement is less than 12 fie. The change through the thickness of the aluminum was predicted by including the honeycomb in the finite element analysis. The thermal residual strains in the transverse direction show larger discrepancies of 25fie and 52 fie. This larger difference might be due to a combination of the long lead wires whose effect was numerically corrected and the lower resistance of this gauge. The difference between the predicted and measured thermal residual strains for the boron/epoxy in fiber orientation shows again the effect of the local increase in thickness. The discrepancy for the two gauges close to the bottom of the patch 6.3 A M R L Sandwich Type Specimen Model 249 is approximately 6% increasing to 16% at the top. The thermal residual strains for the boron/epoxy in transverse direction show a similar trend as discussed for location B and C . The thermal residual strains measured at location E outside the patched area show a significant discrepancy of 60 fie in longitudinal direction between the F E M prediction and the experimental measurement while the transverse direction indicates perfect agreement. A possible cause might be the influence of the grip arrangement which was not modeled in the theoretical analysis. Furthermore, a small difference in the coefficients of thermal expansion might be present between the Al 6061-T6, which was used for the reinforcements and spacers in the grip section, and the Al 2024-T3 used for the face-sheets. The experimental measurements and finite element prediction for the thermal residual strains at location F show only a discrepancy of less than 14 fie, which is excellent. The magnitude certainly shows the in-plane bending generated by the thermal residual stresses. The thermal residual strains in the aluminum for the longitudinal and transverse direc-tion at location G show excellent agreement between the finite element results and the experimental measurements (< 15 fie). This is a significant improvement over the one-dimensional analysis previously used to get an estimate for the thermal residual strains. The finite element analysis also provides a better prediction of the thermal residual strain distribution for the boron/epoxy patch but still overpredicts their magnitude. Possible causes include the uncertainty in the shear modulus assumptions with temperature, the presence of creep especially for the transverse direction (150 fie), or possibly local increases in thickness as well as inaccuracy in the gauge position. The finite element predictions don't correspond to the experimental measurements as well as expected at location H . The through thickness change in thermal residual strains for 6.3 A M R L Sandwich Type Specimen Model 250 the aluminum in the longitudinal direction is well predicted due the inclusion of the hon-eycomb in the finite element model, but they disagree in the absolute magnitude by about 10%. The thermal residual strain in the transverse direction on top of the face-sheet is well predicted (discrepancy of only 13 fie) but for the bottom surface they disagree by 41 fie. Possible explanations are the long lead wires and the low resistance of the strain gauge installed at the bottom surface of the face-sheet combined with the lower measured strain. The discrepancy in the thermal residual strains between the experimental measurements and finite element results for the boron/epoxy patch is between 12 and 16%. Again the local increase in thickness due to embedded gauges is probably one reason for these dis-agreements. Other factors include misalignment of gauges, local reduction the adhesive thickness and the simplification of not modeling the grip section thus affecting the in-plane bending behaviour of the specimen. Overall, the thermal residual strains are well predicted for the aluminum in both longitudi-nal and transverse directions. The average discrepancy between the finite element results and the experimental results are only 36 fie and 27 fie, respectively. Although these mag-nitudes are small, it should be noted that the finite element analysis underpredicted the thermal residual strains for the aluminum in longitudinal direction which is not particu-larly desirable from a design viewpoint. One recommendation would be to increase the effective stress free temperature slightly such that a conservative finite element prediction is achieved. The thermal residual strains determined by the finite element analysis for the boron/epoxy are, on average, 88 fie higher than the experimental results. The most important cause is the local increase in thickness by embedding strain gauges, which is also applicable to the transverse thermal residual strains for the boron/epoxy patch. The aver-age discrepancy of even 100 fie in transverse direction can also be attributed to creep. The through thickness variation of the thermal residual strains in the patch will be influenced by the particular placement of each gauge as well as the solder connection to the lead wires. 6.3 A M R L Sandwich Type Specimen Model 251 Considering the influencing factors which were discussed in this section, good agreement between the experimental results and the finite element results has been achieved. These measurements suggest that for future experimental work, only a single gauge should be embedded at any given location. The finite element analysis revealed that significant through thickness effects are only present very close to the crack and in the tapered section. The use of strain gauges is limited in these regions due to their size. Very small gauges may be applied to the metallic substrate but are inadequate for composites since they would be affected by the local interaction between fibers and matrix. The other important achievement is the actual embedding of gauges in a bonded repair with reliable connections. Embedded gauges offer a good option for 'smart patches' since the gauge is protected from environmental influences. A M R L started to investigate the feasibility of 'smart patches' [36] by monitoring surface strains of the patch using an A M R L sandwich type specimen. The challenge is to find a location for the strain gauge where the gauge installation doesn't lead to disbonds around the gauge under fatigue loading while at the same time can indicate disbonds around the crack tip and/or crack propagation. A possible solution would be the placement of a gauge underneath the top layer of the patch. The optimum in-plane location needs to be determined by a number of finite element representations which model the crack growth and disbond growth as observed by A M R L [13]. Their surface strain measurements should give an excellent verification tool as well as an excellent initial guess for an appropriate location. 7 Discussion and Conclusion 252 Chapter 7 Discussion and Conclusion 7 .1 I n t r o d u c t i o n As outlined in Chapter 1, the main goals of this research were to identify the magnitude of thermal residual stresses in bonded composite repairs and to establish an appropriate thermal residual stress/strain model. These research goals were achieved through four stages: The first stage involved manufacturing six A M R L sandwich type composite bonded repair specimens to provide some of the specimens required for fatigue damage initiation testing in addition to one test specimen which was instrumented by placing 44 strain gauge at eight locations and at up to five different interfaces. Residual strains at ambient temperature combining thermal residual strains and other process induced strains were measured during the manufacturing process. In stage two, the stress free temperature of the bonded repair was experimentally determined and thermal residual strains were measured as a function of operating temperature. For stage three, a theoretical analysis was carried out to determine the thermal residual stress and strain distributions in bonded repairs which also addressed the effect of symmetrical disbonds around the crack as well as the stress field ahead of the crack. As final stage four, a detailed finite element analysis was carried out to assess the limitations of the theoretical analysis as well as providing a more detailed insight in the complex thermal residual stress and strain state of the A M R L sandwich type specimen. 7.2 A M R L Specimen Manufacturing Procedure 253 The conclusions and contributions of this research work are highlighted in the following sections. 7 .2 A M R L S p e c i m e n M a n u f a c t u r i n g P r o c e d u r e The objective of this first stage included the selection of a suitable test specimen for thermal residual stress or strain measurements as well as for future fatigue damage initiation test-ing. Appropriate process specifications had to be determined as part of the manufacturing procedure in order to create durable and representative specimens. The instrumentation lo-cations had to be determined based on critical failure locations as reported in the literature. The conclusions and contributions of this portion of the research are outlined below: • A representative specimen was selected, constructed and instrumented for determin-ing the residual strain distribution in a bonded repair. The sandwich type A M R L specimen (see Figure 3.1) was chosen for the experimental determination of the stress state in bonded repairs due to its ability to represent real aircraft structures and frequent use in the evaluation of bonded repairs. Improving the understanding of the stress distribution in the A M R L specimen is valuable for all researchers interpreting experimental results obtained from this specimen. • An effective manufacturing process for the A M R L sandwich type specimen has been developed and presented in detail, including jigs and fixtures to prepare this specimen. Proper process specifications represent the single most critical element of bonded composite repairs since they control the fatigue characteristics of the repair including the occurring failure modes. Due to the lack of detailed process specifications and guidelines in the literature, the manufacturing process for the A M R L sandwich type 7.3 Residual and Thermal Residual Strain Measurements 254 specimen was developed and documented in great detail to provide other researchers with the opportunity to build this specimen. The National Research Council of Canada provided the surface preparation procedure as part of a research collabora-tion. • An instrumentation technique has been developed for obtaining the thermal residual strain distribution as a function of temperature in a patched repair specimen. Strain gauges were chosen as non-destructive measuring devices based on availability, expense, reliability as well as applicability to composites. Techniques for embed-ding strain gauges and lead wire routing in the patch are presented. The innovative solutions provided for instrumented A M R L sandwich type specimen manufacturing procedures can be applied in smart patches. 7.3 Residual and Thermal Residual Strain Measurements Although thermal residual strains were the primary objective of this research phase, the residual strains due to the elevated temperature cure of a bonded repair consist of both ther-mal and process induced residual strains, the latter resulting from effects such as anisotropic shrinkage of the patch or adhesive, temperature gradients in the specimen during process-ing, and/or temperature gradients due to tooling and internal resin flow [50]. While thermal residual strains are affected by subsequent changes in temperature, process induced residual strains may not be affected. The contributions and conclusion of the research performed include the following: • The total residual strains were obtained during the assembly stage of the A M R L sandwich type specimen. 7.3 Residual and Thermal Residual Strain Measurements 255 The key finding from these measurements were the 60% lower residual strains in the fiber direction in comparison to standard thermal residual strain estimate which is based on the temperature differential between curing and ambient temperature. • The stress free temperature for FM73M was experimentally measured to be 101.2°C. This type of measurement was not been reported in the literature for bonded repairs. The difference between the stress free temperature and the curing temperature of 19.8°C shows the importance of this measurement. • Thermal strains were measured in the A M R L sandwich type specimen and the ther-mal residual strains were evaluated based on the previously determined stress free temperature. This experimental work represents the key measurement of this research work since neither thermal strains versus temperature nor thermal residual strains versus tem-perature have been documented in the literature. The experimental data represent an important element to improve current estimates of the stress state after patching as well as the the stress intensity factor and crack growth rates. • An effective stress free temperature of 85.8°C was defined based on the experimental thermal residual strain data in order to establish a linear-elastic thermal residual stress/strain model. An important finding was the significant change in the slope of the thermal residual strain versus temperature plots at approximately 80° C to 90° C instead of the expected cure temperature. Near linear behaviour was found below 80°C. Therefore an effective stress free temperature of 85.8°C was defined using a linear regression analysis of the thermal residual strains versus temperature plots. The error in the temperature 7.3 Residual and Thermal Residual Strain Measurements 256 differential employed in standard calculations for the thermal residual stresses and strains is 35.2°C based on the experimental measurements making this definition an excellent improvement over current practices. • A simple test scheme is proposed to evaluate the effective stress free temperature for different adhesives. It cannot be assumed that the same thermal behaviour is true for other adhesives. The presented experimental data and conclusions are only valid for the use of F M 73M as adhesive material. Therefore the proposed specimen presents an excellent tool for determining the the effective stress free temperature for other adhesives. • Process induced strains were assessed based on the difference between the measured residual strains and thermal residual strains and found to be negligible for FM73M. Based on the good agreement between the average residual strains (541 fie with a standard deviation of 14%) and the thermal residual strains (546 fie with a standard deviation of 16%) for the aluminum at ambient temperature, only negligible tem-perature independent process induced strains are present in the fiber direction. In contrast, the residual strain average due to the elevated temperature cure measured in the aluminum perpendicular to the fiber direction of the patch at the same locations (-164 fie with a standard deviation of 44%) is significantly higher in compression than the thermal residual strain average of -111 fie (with a standard deviation of 20%) indicating that process induced strains are substantially higher in this direction (see Table 6.1). The magnitude of the estimated temperature independent process induced strains shows that they can generally be neglected in future analysis, which has not been previously shown for bonded repairs. 7.4 Theoretical Analysis of Thermal Residual Stresses in Bonded Repair Specimens 257 7.4 Theoretical Analysis of Thermal Residual Stresses in Bonded Repair Specimens The research objectives for the third phase were to establish an appropriate thermal residual stress/strain model for bonded repairs, an estimate for the stress distribution near the crack tip and the influence of disbonds on the stress intensity factor under thermal residual stress loading. A l l of these objectives were achieved wi th a focus on practical use thus avoiding the need of specialized computer programs. The conclusions and contribution of the presented theoretical approach are provided below: • A solution algorithm allowing the direct determination of the elastic stress distribu-tion for a tapered patch was developed without the usual requirement for a specialized numerical code. This solution algorithm makes the linear-elastic approach from L . J . Hart -Smith for tapered and stepped joints more accessible for more engineers to get a basic under-standing of joint design. Furthermore this approach avoids the difficulties encountered in the numerical solution scheme as outlined by Hart-Smith. A correction for the shear-lag in the adherends as presented by A M R L researchers [18, 52] was included to improve Hart-Smith 's one-dimensional analysis. • A stress field model was introduced for the determination of the stresses in the crack tip region in order to improve the theoretical predictions near the crack tip. One of the key achievements was the development of a concise solution for the stress field of a center crack in an infinite plate representing a superior solution for this classic fracture mechanics problem. A complex stress field approach was used to solve this problem in a unique way replacing the need to use a large number of boundary conditions to determine a sufficient number of coefficients for an accurate result, or the more widely employed approximations wi th their l imitations. The 7.4 Theoretical Analysis of Thermal Residual Stresses in Bonded Repair Specimens 258 developed solution was then employed as a predictive tool for bonded repairs for the stress and strain field near the crack tip in bonded repairs. Applications for this type of approach include the determination of sensor locations for smart patches as well as other experimental work. • A generalized version of the Rose model was developed, which accounts for disbonds. The model can account for all disbond lengths including a complete disbond. The generalized Rose model maintains the simplicity of the original Rose model as one of its features. It was found that thermal residual stresses in combination with a partial disbond can lead to an increase in the stress intensity factor beyond the stress intensity factor of the unpatched specimen depending on the degree of restraint. This effect was found to be most severe for unrestrained specimens where a disbond length of less than 20 mm eliminated the benefit of the patching. Further experimental work is required to verify the predictive capabilities of the extended Rose model for disbonds. • Generally good agreement has been found between the experimental measurements and theoretical predictions using the one-dimensional as well as the stress field model within the limitations of these models. The presented models have been evaluated for the thermal residual stresses and strains present in the A M R L sandwich type specimen. Effects such as in-plane bending due to the smaller patch in comparison to the substrate and out-of-plane bending due to the limited stiffness of the honeycomb in case of the A M R L specimen have to be considered when assessing the predictive capabilities of the models. The key to the good agreement between the closed form analytical solution and the experimental data was the definition of the stress free temperature as determined through the experimental work. 7.5 Finite Element Analysis of the A M R L Specimen 259 7.5 Finite Element Analysis of the A M R L Specimen The objectives for the last phase of research were to establish the limitations of the theoret-ical model as well as to obtain a better insight into the complex thermal residual stress and strain distribution of the A M R L specimen. The stress distribution around the crack tip and the taper were the main focus. Due to the complexity of bonded repairs, a number of verification models were created to investigate each critical section in detail. This approach allowed one to determine a sufficiently fine mesh ensuring confidence in the results while reducing the overall degrees of freedom (and required solution time) for an entire model of the A M R L sandwich type specimen. A linear-elastic three-dimensional model for A M R L specimen was created utilizing 18806 elements with 254670 degrees of freedom. The conclusions and contribution of this final portion of the research are outlined below: • The stress distribution in both normal and reversed stacking sequence patch design was investigated and compared with respect to potential failure modes. High peel stresses were found at the ply drop-off due to the incline of the boron/epoxy plies using a reverse stacking sequence. The most critical case for the reversed stacking sequence with respect to peel stresses was identified as a remote compressive loading combined with compressive thermal residual stresses in the patch. This finding was substantiated by experimental observations from the National Research Council of Canada where a change in failure mode at the ply drop off was demonstrated un-der these loading conditions. Therefore the magnitude of the peel stresses should be carefully examined whenever compressive thermal residual stresses are combined with compressive remote loading for a patch with a reversed stacking sequence. As a general design rule, a normal stacking sequence should be used with remote com-pressive loading and a reversed stacking sequence with tensile remote loading for low C T E patches. 7.5 Finite Element Analysis of the A M R L Specimen 260 • Overall the finite element analysis predictions compare well with the experimentally measured thermal residual strains in the A M R L sandwich type specimen especially for the aluminum. The average discrepancy between the finite element results and the experimental results at room temperature for the aluminum in fiber orientation as well as perpen-dicular to the fiber orientation are only 36 fie (with a standard deviation of 27 fie) and 27 fie (with a standard deviation of 16 fie), respectively. The thermal residual strains determined by the finite element analysis for the boron/epoxy are on average 88 fie higher than the experimental results (with a standard deviation of 56 fie). The main cause for this discrepancy is presumed to be the local increase in thickness by embedding strain gauges. This is also applicable to the transverse thermal residual strains for the boron/epoxy patch in combination with creep leading to an average discrepancy of even 100 fie (with a standard deviation of 74 fie). The through thick-ness variation of the thermal residual strains in the patch will be influenced slightly by the particular placement of each gauge as well as the solder connection to the lead wires. Considering the influencing factors good agreement between the experimental results and the finite element results was achieved. • A finite element verification model utilizing a uniform thickness patch over the entire substrate region showed good agreement with the presented theoretical stress field approach ahead of the crack tip. The theoretical model has certain limitations for the stress field behind the crack tip since no load transfer mechanism into the patch is included into the stress field model. Furthermore, the effective crack length predicted by the Rose model is significantly smaller than the physical crack length. The theoretical stress field predictions for the A M R L sandwich type specimen shows additional discrepancies to the finite element 7.6 Implementation in a Design Procedure 261 results and experimental measurements since it does not account for in-plane bending and deflection of the honeycomb. • The finite element analysis substantiated that embedded strain gauges can be used to obtain excellent strain measurements in bonded repairs. Due to the overall good agreement between theoretical predictions and experimental measurements, it is concluded that embedded gauges offer a good option for 'smart patches' since the gauge is protected from environmental influences. Further experi-mental work is required to test the influence of damage initiation around embedded strain gauges. The finite element analysis also showed the influence of embedding of strain gauges at multiple interfaces. The local increase in thickness is considered to be the cause for the decrease in compressive residual strains towards the top surface. It is recommended that for future experimental strain measurements, the number of embedded strain gauges be limited to one gauge per location. 7.6 Implementation in a Design Procedure The experimental and theoretical achievements can readily be included in patch design procedures such as Baker's [18] design procedure for minimum patch thickness as presented in Chapter 2. Figure 7.1 shows the extension to Baker's original design scheme based on the results of the current work, and are discussed below. The design procedure remains unchanged through Step 5. Then the peak strain in the patch (Step 6) occurs typically in the tapered section or over the crack. The derived direct linear-elastic stress/strain solution for tapered joints and reinforcements provides a simple tool to substantiate the strain at both locations instead of just checking over the crack as presented by Baker [18]. Strain concentration factors, which can be evaluated using finite element models should be included in the peak strain estimate. 7.6 Implementation in a Design Procedure 262 Choosing the effective stress free temperature as determined in this research work instead of the curing temperature will give an accurate estimate of the thermal residual stress for Step 7. This improvement will also lead to a better prediction of the peak stress intensity factor (Step 9). Step l Step 2 Step 3 Step 4 Step 5 Step 6 Step 7 Step 8 Step 9 Step 10 Step 11 Step 12 Figure 7.1: Minimum patch thickness design scheme 7.7 Recommendations for Future Work 263 The design approach can be extended to design for a particular stress intensity factor and/or stress intensity factor range (Step 10 & 11). An improvement in the prediction of the stress intensity factor range can also addressed by including the stress ratio. The true stress ratio for a bonded repair can now be determined based on the experimental and theoretical determination of thermal residual stresses in this research work. The design to a particular stress intensity factor or stress intensity factor range should include allowable disbond sizes. The presented generalized Rose model allows the determination of the stress intensity factor with symmetrical disbonds in the bonded repair which also applies to Baker's proposed generic certification approach [11, 12]. The availability of an increasing database of experimental fatigue crack propagation data for different materials as well as environmental conditions allows the development of a predictive model for the crack propagation rate. Crack retardation effects due to overloads will generally be impossible to include in any certification approach since the stress history is unknown in most real repairs. It should be noted that Figure 7.1 shows an idealized schematic which is based on a re-duction of the crack opening displacement for an increase in patch thickness. This is not always guaranteed since the stress intensity factor can increase under thermal loading with increasing patch thickness. 7.7 Recommendations for Future Work The following recommendations can be made for future work in the area of thermal residual stresses: • Measuring the residual strains during the curing cycle would be quite valuable to improve our understanding of the adhesive behaviour during this stage. The proposed generic thermal residual strain specimen can be used for this research. 7.7 Recommendations for Future Work 264 • One of the remaining important questions is the magnitude of thermal residual stresses when a 80°C temperature cure is used. Based on Boykett's and Walker's estimated thermal residual strains and the true adhesive behaviour for F M 73M investigated in this research, it appears that the reduction in thermal residual strains due to lowering the curing temperature from 121°C and 80°C is quite small. The suggested generic thermal residual strain specimen is also well suited for this test. • The thermal residual stresses as a function of temperature should evaluated for the different adhesives in use. Again the generic thermal residual strain specimen can be used for this application. • Future thermal residual strain measurements and other strain measurements utilizing embedded strain gauges should limit the number of gauges preferably to one gauge at a single location to minimize changes in the local strain distribution due to the increase in thickness. • In order to assess the true thermal residual stresses for repair, the coefficient of thermal expansion should be determined at the repair location experimentally. The measurement can be carried out during the thermal survey for placing heat blan-kets. The strain gauge should be installed using an adhesive suitable for elevated temperature measurements. The following recommendations can be made for future work in the general area of com-posite repairs: • Experimental evaluation of the presented disbond model is suggested since its sim-plicity offers the potential for an excellent tool to evaluate the influence of disbonds in situ. • A detailed finite element analysis is suggested to evaluate an improvement of the normal stacking sequence by using a final composite layer which extends over the previously stacked plies. This final layer acts as a protective layer for the exposed 7.7 Recommendations for Future Work 265 edges of the tapered section. This study should consider different materials for this layer to minimize peel stresses. • The presented research work should be continued by carefully investigating damage initiation since the true stress state of the repair is now available. This investigation should focus on reinitiation of the crack as well as disbond initiation over the crack and possibly near the free edge. This research would allow one to identify the damage initiation life of a bonded repair, which can represent a major portion of the fatigue life of a bonded repair. • One of the key challenge for the widespread application of adhesive bonding is the need for a non-destructive durability test for adhesive joints. In addition, a non-destructive inspection technique is needed to verify proper surface preparation. While these issues can be somewhat addressed under laboratory conditions, any NDI techniques must be suitable for field conditions. • Further research is required to investigate the adhesive behaviour under elevated temperatures (inch temperatures above the curing temperature). A strong frequency dependency is suspected since tests at A M R L showed the F M 73 performs well un-der cyclic loading (0.5 and 5 Hz) at temperatures above 100°C, while quasi static tests during this research showed that the patch and substrate can expand nearly unrestrained at temperatures above 90° C. • Further research is required to establish smart patches as a optional design feature to simplify certification in critical repairs. Embedding sensors can be easily carried out as shown in this research. Early fatigue damage around the sensor is a concern which needs to be addressed. Furthermore, the development of suitable instrumen-tation is necessary. A M R L is currently working on the development of smart patch technology [11]. Bibliography 266 B i b l i o g r a p h y [1] A . M . Albat. Interim Report - Investigation of Failure Initiation Models for Bonded Composite Patches on Metallic Structures. Technical Report Contract No. 31184-6-0161/001/ST, University of British Columbia, 1996. [2] A . M . Albat. Final Report - Investigation of Failure Initiation Models for Bonded Composite Patches on Metallic Structures. Technical Report Contract No. 31184-6-0161/001/ST, University of British Columbia, 1997. [3] A . M . Albat, D.P. Romilly and M.D. Raizenne. Thermal Residual Strain Measurement in a Composite Repair on a Cracked Aluminum Structure. In Eleventh International Conference on Composite Materials - Proceedings, pp. VI-279-VI-288, 1997. [4] D.J. Allman. A Theory for Elastic Stresses in Adhesive bonded Lap Joints. Quarterly Journal of Mechanics and Applied Mathematics, Vol. X X X , No. 4, pp. 415-436, 1977. [5] J .M. Anderson, C S . Chu, and W . M . McGee. Growth characteristics of a fatigue crack approaching and growing beneath an adhesively bonded doubler. ASME paper 77- WA/Mat U, 1978. [6] D.R. Arnott. Field repair bonding surface treatments. 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A p p e n d i x A A M R L - S p e c i m e n D r a w i n g s 275 Appendix A AMRL-Specimen Drawings Appendix A A M R L - S p e c i m e n Drawings 276 Figure A . l : A M R L specimen (Face-sheets) Appendix A AMRL-Specimen Drawings 277 mo wo c oo J! 0 0 00 otn 00 Figure A.2: A M R L specimen (Spacer block) Figure A.3: A M R L specimen (Honeycomb) Appendix A A M R L - S p e c i m e n Drawings 279 OB -a CD a o cd CO 3 c i - cp * 6 CO co o CD CU CO a .2 cu £ 5 o 55 Figure A.4: A M R L specimen (Shims) Appendix A AMRL-Specimen Drawings 280 Figure A.5: A M R L specimen (Jig) Appendix A AMRL-Specimen Drawings 2 8 1 Figure A.6: A M R L specimen (Jig-bolts) Figure A.7: Instrumented A M R L specimen (Aluminum) Figure A . 8 : Instrumented A M R L specimen (Patch-adhesive interface) Figure A.9: Instrumented A M R L specimen (1st boron/epoxy ply) Figure A. 10: Instrumented A M R L specimen (Top boron/epoxy ply) Figure A . l l : Instrumented A M R L specimen (Superimposed drawing) Appendix A AMRL-Specimen Drawings 287 Figure A.12: Instrumented A M R L specimen (Backside of face-sheet 7) Figure A.13: Instrumented A M R L specimen (Gauge locations) Appendix A AMRL-Specimen Drawings 289 Figure A. 14: Instrumented A M R L specimen (Backside gauge locations) Appendix A AMRL-Specimen Drawings 290 Figure A.15: Instrumented A M R L specimen (Terminal locations) Appendix B Pre-cracking Documentation 291 Appendix B P r e - c r a c k i n g D o c u m e n t a t i o n Appendix B . l Pre-crack Data 292 B.l Pre-crack Data Sequence a fmm] [N] [N] a fmm] ACT [MPa] A K [MPax/m] da dN r j x m i i cycle -I 0 50000 55351 55352 95603 95604 100000 137762 150315 150316 200000 213098 229329 500000 512500 512501 600000 630490 630491 650000 700000 734099 744615 744616 750000 765000 780000 795000 810000 875000 931000 938000 950000 1000000 1015000 1030000 1045000 1060000 1075000 1084100 1090000 2.94 3750 4.74 3750 5.27 3750 5.27 3000 7.62 3000 7.62 2500 7.84 2500 9.41 2500 9.98 2500 9.98 2000 11.58 2000 12.09 Overload 12.42 2000 12.42 2000 13.10 2828 13.10 1750 14.32 1750 15.09 1750 15.09 1500 15.42 1500 16.45 1500 17.30 1500 17.53 1500 17.53 1250 17.59 Overload 17.97 1250 18.07 1250 18.17 1250 18.18 1250 18.23 1250 18.37 1750 18.57 1500 18.65 1250 19.19 1250 19.24 1250 19.47 1250 19.57 1250 19.75 1250 19.90 1250 19.94 1250 20.05 1250 37500 37500 37500 30000 30000 25000 25000 25000 25000 20000 20000 20000 20000 28280 17500 17500 17500 15000 15000 15000 15000 15000 12500 12500 12500 12500 12500 12500 17500 15000 12500 12500 12500 12500 12500 12500 12500 12500 12500 Crack Initiation 5.01 66.44 1.130 9.41 Sequence Change Retardation Sequence Change Retardation 8.63 9.70 44.29 44.29 1.139 1.141 8.30 8.82 Sequence Change Retardation 11.84 35.43 1.147 7.84 Retardation Retardation Retardation Increased Load to overcome Retardation Retardation 14.71 31.00 1.155 7.70 Sequence Change Retardation Sequence Change Retardation Retardation Retardation Retardation Retardation Retardation Retardation Increased Load to overcome Retardation Retardation Retardation 0.0990 0.0416 0.0454 0.0389 0.0253 15.94 26.57 1.158 6.89 0.0206 16.88 26.57 1.161 7.10 0.0249 17.42 26.57 1.162 7.23 0.0219 19.22 22.15 1.168 6.35 0.0033 19.36 22.15 1.168 6.38 0.0153 19.52 22.15 1.168 6.41 0.0067 19.66 22.15 1.169 6.43 0.0120 19.83 22.15 1.169 6.46 0.0100 19.92 22.15 1.170 6.48 0.0044 20.00 22.15 1.170 6.49 0.0186 Table B . l : Pre-crack data for face-sheet 1 Appendix B . l Pre-crack Data 293 Sequence 0 50000 55351 95603 95604 114580 141462 146366 146367 191528 197503 197504 225600 250000 267611 272787 272788 300000 321218 350000 359318 359319 375000 400000 450000 465000 469088 469089 480000 521223 546198 580000 617581 648339 681273 701941 727472 750000 765000 772955 780000 a [mm] 2.86 3.69 3.99 4.89 4.89 5.57 7.31 7.60 7.60 10.15 10.15 10.83 11.80 12.34 12.56 12.56 13.18 13.82 14.71 15.01 15.01 15.38 15.76 16.96 17.39 17.62 17.62 17.63 17.85 18.10 18.34 18.55 18.89 19.04 19.20 19.43 19.63 19.73 19.88 20.01 [N] 3750 3750 3750 3750 3000 3000 3000 3000 2500 2500 2500 2000 2000 2000 2000 2000 1750 1750 1750 1750 1750 1500 1500 1500 1500 1500 1500 1250 1250 1250 1250 1250 1250 1250 1250 1250 1250 1250 1250 1250 1250 [N] 37500 37500 37500 37500 30000 30000 30000 30000 25000 25000 25000 20000 20000 20000 20000 20000 17500 17500 17500 17500 17500 15000 15000 15000 15000 15000 15000 12500 12500 12500 12500 12500 12500 12500 12500 12500 12500 12500 12500 12500 12500 a [mml ACT [MPa] Crack Initiation 3.84 4.44 66.44 66.44 Sequence Change Retardation 6.44 7.46 53.15 53.15 Sequence Change Retardation 10.02 44.29 Sequence Change Retardation 11.32 12.07 12.45 35.43 35.43 35.43 Sequence Change Retardation 13.50 14.26 14.86 31.00 31.00 31.00 Sequence Change Retardation 15.57 16.36 17.18 17.51 26.57 26.57 26.57 26.57 Sequence Change Retardation 17.74 17.98 18.22 18.45 18.72 18.97 19.12 19.32 19.53 19.68 19.81 19.95 22.15 22.15 22.15 22.15 22.15 22.15 22.15 22.15 22.15 22.15 22.15 22.15 AK [MPav/m] 1.127 1.128 1.133 1.136 1.142 1.146 1.148 1.149 1.152 1.154 1.155 1.157 1.159 1.162 1.163 1.163 1.164 1.165 1.165 1.166 1.167 1.167 1.168 1.169 1.169 1.169 1.170 8.22 8.85 8.57 9.24 8.97 7.65 7.92 8.05 7.35 7.57 7.74 6.80 6.99 7.17 7.25 6.08 6.13 6.17 6.21 6.26 6.31 6.34 6.37 6.41 6.44 6.46 6.48 Table B.2: Pre-crack data for face-sheet 2 Appendix B . l Pre-crack Data 294 Sequence [ 1 N [ ] a [mm] 1 mm [N] Pmax [N] a [mm] ACT [MPa] c [ 1 A K [MPav/rH] da dN [JSLl L cycle J 1 0 2.97 3750 37500 Crack Initiation 30000 3.32 3750 37500 3.49 66.44 1.126 7.83 0.0330 40000 3.65 3750 37500 3.92 66.44 1.127 8.31 0.0540 50000 4.19 3750 37500 4.61 66.44 1.129 9.03 0.0840 60000 5.03 3750 37500 Sequence Change 2 60001 5.03 3000 30000 Retardation 85750 5.93 3000 30000 6.55 53.15 1.134 8.64 0.0589 106619 7.16 3000 30000 7.41 53.15 1.136 9.21 0.0799 112873 7.66 3000 30000 Sequence Change 3 112874 7.66 2500 25000 Retardation 120000 7.98 2500 25000 8.55 44.29 1.139 8.26 0.0575 139637 9.11 2500 25000 9.26 44.29 1.140 8.61 0.0280 150000 9.40 2500 25000 9.72 44.29 1.142 8.84 0.1028 156223 10.04 2500 25000 Sequence Change 4 156224 10.04 2000 20000 Retardation 162500 10.06 2000 20000 11.06 35.43 1.145 7.56 0.0379 215000 12.05 2000 20000 12.23 35.43 1.148 7.97 0.0360 225000 12.41 2000 20000 12.47 35.43 1.149 8.05 0.0788 226396 12.52 2000 20000 Sequence Change 5 226397 12.52 1750 17500 Retardation 250000 12.62 1750 17500 13.18 31.00 1.151 7.26 0.0297 287688 13.74 1750 17500 14.29 31.00 1.154 7.58 0.0367 317400 14.83 1750 17500 14.98 31.00 1.156 7.77 0.1027 320322 15.13 1750 17500 Sequence Change 6 320323 15.13 1500 15000 Retardation 357438 15.70 1500 15000 16.11 26.57 1.159 6.93 0.0293 385389 16.52 1500 15000 16.99 26.57 1.161 7.13 0.0293 417519 17.46 1500 15000 17.49 26.57 1.163 7.24 0.0148 420904 17.51 1500 15000 Sequence Change 7 420905 17.51 1250 12500 Retardation 450000 17.91 1250 12500 18.05 22.15 1.164 6.14 0.0134 470118 18.18 1250 12500 18.42 22.15 1.165 6.21 0.0157 500000 18.65 1250 12500 18.75 22.15 1.166 6.27 0.0118 516885 18.85 1250 12500 19.14 22.15 1.167 6.34 0.0125 562500 19.42 1250 12500 19.47 22.15 1.168 6.40 0.0080 575000 19.52 1250 12500 19.61 22.15 1.169 6.42 0.0136 587500 19.69 1250 12500 19.75 22.15 1.169 6.45 0.0096 600000 19.81 1250 12500 19.92 22.15 1.170 6.48 0.0191 610971 20.02 1250 12500 Table B.3: Pre-crack data for face-sheet 3 Appendix B . l Pre-crack Data 295 Sequence N a L mm Pmax [ ] [ ] [mm] [N] [N] 1 0 2.78 3750 37500 30000 3.09 3750 37500 50000 3.91 3750 37500 57757 4.87 3750 37500 62303 5.07 3750 37500 2 62304 5.07 3000 30000 85000 5.25 3000 30000 120000 7.23 3000 30000 122810 7.41 3000 30000 124236 7.57 3000 30000 3 124237 7.57 2500 25000 146688 8.38 2500 25000 150101 8.59 2500 25000 172366 10.06 2500 25000 4 172367 10.06 2000 20000 175000 10.13 2000 20000 225000 12.11 2000 20000 236357 12.50 2000 20000 5 236358 12.50 Overload 249185 12.92 1750 17500 300000 14.73 1750 17500 305979 15.00 1750 17500 6 305980 15.00 1500 15000 325000 15.43 1500 15000 350000 16.20 1500 15000 375000 16.97 1500 15000 391819 17.52 1500 15000 7 391820 - 17.52 1250 12500 422459 17.95 1250 12500 450000 18.40 1250 12500 475000 18.73 1250 12500 500000 19.16 1250 12500 512500 19.36 1250 12500 525000 19.61 1250 12500 537500 19.89 1250 12500 543882 20.00 1250 12500 a [mm] ACT [MPa] Crack Initiation 3.50 4.39 4.97 66.44 66.44 66.44 Sequence Change Retardation 6.24 7.32 7.49 53.15 53.15 53.15 Sequence Change Retardation 8.49 9.33 44.29 44.29 Sequence Change Retardation 11.12 12.31 35.43 35.43 Sequence Change Retardation Retardation 14.87 31.00 Sequence Change Retardation 15.82 16.59 17.25 26.57 26.57 26.57 Sequence Change Retardation 18.18 18.57 18.95 19.26 19.49 19.75 19.95 22.15 22.15 22.15 22.15 22.15 22.15 22.15 A K [MPav/rH] 1.126 1.128 1.130 1.133 1.135 1.136 1.138 1.141 1.145 1.148 1.155 1.158 1.160 1.162 1.165 1.166 1.167 1.168 1.168 1.169 1.170 7.85 8.80 9.38 8.43 9.15 9.26 8.23 8.65 7.58 ;.oo 7.74 7.04 7.19 6.16 6.23 6.30 6.36 6.40 6.45 6.48 Table B.4: Pre-crack data for face-sheet 4 Appendix B . l Pre-crack Data 296 Sequence N a 1 min Pmax [ ] [ ] [mm] [N] [N] 1 0 2.82 3750 37500 40000 3.15 3750 37500 50000 3.27 3750 37500 60000 3.66 3750 37500 80000 4.56 3750 37500 87763 5.15 3750 37500 2 87764 5.15 3000 30000 110524 6.19 3000 30000 125000 6.90 3000 30000 129575 7.42 3000 30000 130684 7.54 3000 30000 3 130685 7.54 2500 25000 150000 8.32 2500 25000 166528 9.36 2500 25000 175000 9.95 2500 25000 176683 10.11 2500 25000 4 176684 10.11 2000 20000 200000 10.66 2000 20000 225000 11.66 2000 20000 241759 12.53 2000 20000 5 241760 12.53 1750 17500 250000 12.67 1750 17500 300000 13.91 1750 17500 320000 14.40 1750 17500 336069 15.08 1750 17500 6 336070 15.08 1500 15000 350000 15.43 1500 15000 375000 16.02 1500 15000 400000 16.85 1500 15000 410825 17.21 1500 15000 420564 17.58 1500 15000 7 420565 17.58 1250 12500 450000 17.98 1250 12500 490000 18.60 1250 12500 516383 19.00 1250 12500 530000 19.32 1250 12500 543893 19.70 1250 12500 550000 19.83 1250 12500 555808 20.01 1250 12500 a [mml ACT [MPa] Crack Initiation 3.21 3.47 4.11 66.44 66.44 66.44 66.44 Sequence Change Retardation 6.55 7.16 7.48 53.15 53.15 53.15 Sequence Change Retardation 8.84 9.66 10.03 44.29 44.29 44.29 Sequence Change Retardation 11.16 12.10 35.43 35.43 Sequence Change Retardation 13.29 14.16 14.74 31.00 31.00 31.00 Sequence Change Retardation 15.73 16.44 17.03 17.40 26.57 26.57 26.57 26.57 Sequence Change Retardation 18.29 18.80 19.16 19.51 19.77 19.92 22.15 22.15 22.15 22.15 22.15 22.15 AK [MPax/E] 1.125 1.126 1.128 1.129 1.134 1.135 1.136 1.139 1.141 1.142 1.145 1.148 1.151 1.153 1.155 1.158 1.160 1.161 1.162 1.165 1.166 1.167 1.168 1.169 1.170 7.51 7.81 8.51 9.27 8.64 9.05 9.25 8.41 8.80 8.98 7.60 7.93 7.29 7.54 7.71 6.84 7.00 7.14 7.22 6.18 6.28 6.34 6.41 6.45 6.48 Table B.5: Pre-crack data for face-sheet 5 Appendix B . l Pre-crack Data 297 Sequence N a r min Pmax [ ] [ ] [mm] [N] [N] 1 0 2.74 3750 37500 40000 3.43 3750 37500 50000 3.69 3750 37500 60000 4.01 3750 37500 70000 4.59 3750 37500 75092 5.00 3750 37500 2 75093 5.00 3000 30000 95000 5.72 3000 30000 115562 6.83 3000 30000 124105 7.30 3000 30000 125545 7.52 3000 30000 3 125546 7.52 2500 25000 150000 8.42 2500 25000 170015 9.45 2500 25000 175004 9.78 2500 25000 179155 10.03 2500 25000 4 179156 10.03 2000 20000 200000 10.56 2000 20000 219850 11.21 2000 20000 225000 11.40 2000 20000 245014 12.12 2000 20000 250000 12.30 2000 20000 255875 12.53 2000 20000 5 255876 12.53 1750 17500 275000 13.04 1750 17500 300000 13.62 1750 17500 316237 14.05 1750 17500 332544 14.57 1750 17500 347588 15.09 1750 17500 6 347589 15.09 1500 15000 387131 15.79 1500 15000 415000 16.38 1500 15000 430000 16.72 1500 15000 447712 17.00 1500 15000 462500 17.42 1500 15000 466926 17.51 1500 15000 7 466927 17.51 1500 15000 500000 17.84 1250 12500 522783 18.01 1250 12500 550000 18.24 1250 12500 596529 18.60 1250 12500 629665 18.84 1250 12500 659800 19.00 1250 12500 700000 19.62 1250 12500 725000 19.75 1250 12500 748217 20.00 1250 12500 a [mm] Aa [MPa] c [ ] AK [MPax/m] da dN L cycle J Crack Initiation 3.56 66.44 1.126 7.91 0.0260 3.85 66.44 1.127 8.23 0.0320 4.30 66.44 1.128 8.71 0.0580 4.80 66.44 1.129 9.21 0.0805 Sequence Change Retardation 6.28 53.15 1.133 8.45 0.0540 7.07 53.15 1.135 8.99 0.0550 7.41 53.15 1.136 9.21 0.1528 Sequence Change Retardation 8.94 44.29 1.140 8.46 0.0515 9.62 44.29 1.141 8.79 0.0661 9.91 44.29 1.142 8.92 0.0602 Sequence Change Retardation 10.89 35.43 1.145 7.50 0.0327 11.31 35.43 1.146 7.65 0.0369 11.76 35.43 1.147 7.81 0.0360 12.21 35.43 1.148 7.97 0.0361 12.42 35.43 1.149 8.04 0.0391 Sequence Change Retardation 13.33 31.00 1.151 7.30 0.0232 13.84 31.00 1.152 7.45 0.0265 14.31 31.00 1.154 7.58 0.0319 14.83 31.00 1.155 7.73 0.0346 Sequence Change Retardation 16.09 26.57 1.159 6.92 0.0212 16.55 26.57 1.160 7.03 0.0227 16.86 26.57 1.161 7.10 0.0158 17.21 26.57 1.162 7.18 0.0284 17.47 26.57 1.163 7.24 0.0203 Sequence Change Retardation 17.93 22.15 1.164 6.12 0.0075 18.13 22.15 1.164 6.15 0.0085 18.42 22.15 1.165 6.21 0.0077 18.72 22.15 1.166 6.26 0.0072 18.92 22.15 1.167 6.30 0.0053 19.31 22.15 1.168 6.37 0.0154 19.69 22.15 1.169 6.44 0.0052 19.88 22.15 1.170 6.47 0.0108 Table B.6: Pre-crack data for face-sheet 6 Appendix B . l Pre-crack Data 298 Sequence N a [mm] [N] Pmax [N] a [mm] ACT [MPa] AK [MPax/m] 0 40000 50000 60000 70000 80000 80001 100000 129273 129274 150000 175000 182233 183852 183853 225000 245006 264530 264531 275000 300000 325000 349060 349061 370126 400000 425000 450000 459931 459932 525000 625682 650000 670000 695310 710000 735000 776829 2.72 3.18 3.53 3.83 4.35 5.00 5.00 5.70 7.49 7.49 8.18 9.44 9.75 10.03 10.03 10.92 11.70 12.53 12.53 12.72 13.45 14.21 15.14 15.14 15.53 16.17 16.67 17.24 17.58 17.58 17.92 18.80 18.99 19.13 19.37 19.48 19.68 20.00 3750 3750 3750 3750 3750 3750 3000 3000 3000 2500 2500 2500 2500 2500 2000 2000 2000 2000 1750 1750 1750 1750 1750 1500 1500 1500 1500 1500 1500 1250 1250 1250 1250 1250 1250 1250 1250 1250 37500 37500 37500 37500 37500 37500 30000 30000 30000 25000 25000 25000 25000 25000 20000 20000 20000 20000 17500 17500 17500 17500 17500 15000 15000 15000 15000 15000 15000 12500 12500 12500 12500 12500 12500 12500 12500 12500 Crack Initiation 3.36 66.44 1.126 7.68 0.0350 3.68 66.44 1.127 8.05 0.0300 4.09 66.44 1.128 8.49 0.0520 4.68 66.44 1.129 9.09 0.0650 Sequence Change Retardation 6.60 53.15 1.134 8.67 0.0611 Sequence Change Retardation 8.81 44.29 1.139 8.39 0.0504 9.60 44.29 1.141 8.78 0.0429 9.89 44.29 1.142 8.92 0.1729 Sequence Change Retardation 11.31 35.43 1.146 7.65 0.0390 12.12 35.43 1.148 7.93 0.0425 Sequence Change Retardation 13.09 31.00 1.150 7.23 0.0292 13.83 31.00 1.152 7.45 0.0304 14.68 31.00 1.155 7.69 0.0387 Sequence Change Retardation 15.85 26.57 1.158 6.87 0.0214 16.42 26.57 1.160 7.00 0.0200 16.96 26.57 1.161 7.12 0.0228 17.41 26.57 1.162 7.22 0.0342 Sequence Change Retardation 18.36 22.15 1.165 6.20 0.0087 18.90 22.15 1.167 6.29 0.0078 19.06 22.15 1.167 6.32 0.0070 19.25 22.15 1.168 6.36 0.0095 19.43 22.15 1.168 6.39 0.0075 19.58 22.15 1.169 6.42 0.0080 19.84 22.15 1.169 6.47 0.0077 Table B.7: Pre-crack data for face-sheet 7 Appendix B . l Pre-crack Data 299 Sequence N 0 30000 40000 50000 60000 70000 80000 80001 100000 129785 130573 130574 175000 187125 187126 200000 236358 252235 265223 265224 275000 300000 322439 340654 340655 350000 400000 427468 450000 454396 454397 475000 500000 575146 628607 700000 707545 715226 a [mml 2.85 3.16 3.35 3.54 4.00 4.27 4.99 4.99 5.60 7.42 7.50 7.50 9.38 10.05 10.05 10.25 11.40 12.02 12.64 12.64 12.78 13.60 14.40 15.05 15.05 15.26 16.23 16.78 17.38 17.50 17.50 17.67 17.84 18.43 18.94 19.75 19.89 20.03 p . A min [N] 3750 3750 3750 3750 3750 3750 3750 3000 3000 3000 3000 2500 2500 2500 2000 2000 2000 2000 2000 1750 1750 1750 1750 1750 1500 1500 1500 1500 1500 1500 1250 1250 1250 1250 1250 1250 1250 1250 [N] 37500 •37500 37500 37500 37500 37500 37500 30000 30000 30000 30000 25000 25000 25000 20000 20000 20000 20000 20000 17500 17500 17500 17500 17500 15000 15000 15000 15000 15000 15000 12500 12500 12500 12500 12500 12500 12500 12500 a [mml ACT [MPa] Crack Initiation 3.26 3.45 3.77 4.14 4.63 66.44 66.44 66.44 66.44 66.44 Sequence Change Retardation 6.51 7.46 53.15 53.15 Sequence Change Retardation 9.72 44.29 Sequence Change Retardation 10.83 11.71 12.33 35.43 35.43 35.43 Sequence Change Retardation 13.19 14.00 14.73 31.00 31.00 31.00 Sequence Change Retardation 15.75 16.51 17.08 17.44 26.57 26.57 26.57 26.57 Sequence Change Retardation 17.76 18.14 18.69 19.35 19.82 19.96 22.15 22.15 22.15 22.15 22.15 22.15 AK [MPav/rH] 1.126 1.126 1.127 1.128 1.129 1.133 1.136 1.142 1.144 1.147 1.148 1.151 1.153 1.155 1.158 1.160 1.161 1.162 1.163 1.164 1.166 1.168 1.169 1.170 7.56 7.78 8.15 8.54 9.05 8.62 9.24 8.83 7.48 7.79 8.01 7.26 7.50 7.70 6.84 7.02 7.15 7.23 6.08 6.16 6.26 6.38 6.46 6.49 Table B.8: Pre-crack data for face-sheet 8 Appendix B . l Pre-crack Data 3 0 0 Sequence N a p . 1 mm Pmax [ ] [ ] [mm] [N] [N] 1 0 2.86 3750 37500 30000 3.26 3750 37500 40000 3.49 3750 37500 50000 3.86 3750 37500 60000 4.28 3750 37500 70000 4.72 3750 37500 75822 5.18 3750 37500 2 75823 5.18 3000 30000 80000 5.31 3000 30000 110012 6.50 3000 30000 126379 7.54 3000 30000 3 126380 7.54 2500 25000 150000 8.56 2500 25000 173211 10.01 2500 25000 4 173212 10.01 2000 20000 200000 10.58 2000 20000 225081 11.48 2000 20000 251137 12.53 2000 20000 5 251138 12.53 1750 17500 285892 13.65 1750 17500 309877 14.46 1750 17500 325770 15.12 1750 17500 6 325771 15.12 1500 15000 350000 15.65 1500 15000 380431 16.42 1500 15000 400000 17.11 1500 15000 410523 17.53 1500 15000 7 410524 17.53 1250 12500 447205 17.96 1250 12500 475002 18.39 1250 12500 500002 18.76 1250 12500 525000 19.21 1250 12500 540308 19.52 1250 12500 550000 19.71 1250 12500 566001 20.03 1250 12500 a [mml ACT [MPa] Crack Initiation 3.38 3.68 :.07 4.50 4.95 66.44 66.44 66.44 66.44 66.44 Sequence Change Retardation 5.91 7.02 53.15 53.15 Sequence Change Retardation 9.29 44.29 Sequence Change Retardation 11.03 12.01 35.43 35.43 Sequence Change Retardation 14.06 14.79 31.00 31.00 Sequence Change Retardation 16.04 16.77 17.32 26.57 26.57 26.57 Sequence Change Retardation 18.18 18.58 18.99 19.37 19.62 19.87 22.15 22.15 22.15 22.15 22.15 22.15 AK [MPav/m] 1.126 1.127 1.128 1.129 1.130 1.132 1.135 1.140 1.145 1.148 1.153 1.155 1.159 1.161 1.162 1.165 1.166 1.167 1.168 1.169 1.170 7.70 8.04 8.47 8.91 9.36 8.19 8.96 8.63 7.55 7.90 7.51 7.72 6.91 7.08 7.20 6.16 6.24 6.31 6.38 6.43 6.47 Table B.9: Pre-crack data for face-sheet 9 Appendix B . l Pre-crack Data 301 Sequence N a r min Pmax [ ] [ ] [mm] [N] [N] 1 0 2.67 3750 37500 30000 3.04 3750 37500 40000 3.24 3750 37500 50000 3.83 3750 37500 60000 4.51 3750 37500 66224 5.06 3750 37500 2 66225 5.06 3000 30000 108775 7.63 3000 30000 3 108776 7.63 2500 25000 125000 8.15 2500 25000 150000 9.53 2500 25000 153352 9.73 2500 25000 157413 10.05 2500 25000 4 157414 10.05 2000 20000 175000 10.65 2000 20000 200000 11.47 2000 20000 219470 12.38 2000 20000 222700 12.50 2000 20000 5 222701 12.50 1750 17500 250000 13.22 1750 17500 275000 14.17 1750 17500 287685 14.80 1750 17500 295449 15.05 1750 17500 6 295450 15.05 1500 15000 321872 15.80 1500 15000 357004 16.67 1500 15000 375000 17.25 1500 15000 383855 17.52 1500 15000 7 383856 17.52 1250 12500 417915 17.91 1250 12500 450000 18.46 1250 12500 467615 18.79 1250 12500 475000 18.86 1250 12500 500000 19.28 1250 12500 512500 19.54 1250 12500 525000 19.77 1250 12500 534156 19.91 1250 12500 538561 20.02 1250 12500 a [mm] [MPa] Crack Initiation 3.14 3.54 4.17 4.79 66.44 66.44 66.44 66.44 Sequence Change Retardation Sequence Change Retardation 8.84 9.63 9.89 44.29 44.29 44.29 Sequence Change Retardation 11.06 11.93 12.44 35.43 35.43 35.43 Sequence Change Retardation 13.70 14.49 14.93 31.00 31.00 31.00 Sequence Change Retardation 16.24 16.96 17.39 26.57 26.57 26.57 Sequence Change Retardation 18.19 18.63 18.83 19.07 19.41 19.66 19.84 19.97 22.15 22.15 22.15 22.15 22.15 22.15 22.15 22.15 AK [MPav/iri] 1.125 1.126 1.128 1.129 1.139 1.141 1.142 1.145 1.147 1.149 1.152 1.154 1.155 1.159 1.161 1.162 1.165 1.166 1.166 1.167 1.168 1.169 1.169 1.170 7.43 7.89 8.58 9.20 8.41 8.79 8.92 7.56 7.87 8.05 7.41 7.63 7.76 6.96 7.12 7.22 6.16 6.25 6.28 6.33 6.39 6.43 6.47 6.49 Table B.10: Pre-crack data for face-sheet 10 Appendix B . l Pre-crack Data 302 Sequence [ ] N [ ] a [mm] p . L min [N] Pmax [N] a [mm] Aa [MPa] c [ ] AK [MPayS] da dN rj£m_l L cycleJ 1 0 2.77 3750 37500 Crack Initiation 30000 3.10 3750 37500 3.26 66.44 1.126 7.56 0.0310 40000 3.41 3750 37500 3.77 66.44 1.127 8.14 0.0710 50000 4.12 3750 37500 4.46 66.44 1.128 8.87 0.0670 60000 4.79 3750 37500 4.90 66.44 1.130 9.31 0.1570 61338 5.00 3750 37500 Sequence Change 2 61339 5.00 3000 30000 Retardation 70000 5.31 3000 30000 6.08 53.15 1.132 8.31 0.0574 96633 6.84 3000 30000 7.06 53.15 1.135 8.98 0.0746 102529 7.28 3000 30000 7.41 53.15 1.136 9.21 0.0670 106262 7.53 3000 30000 Sequence Change 3 106262 7.53 2500 25000 Retardation 120000 8.08 2500 25000 8.95 44.29 1.140 8.46 0.0577 150000 9.81 2500 25000 9.91 44.29 1.142 8.92 0.0818 152324 10.00 2500 25000 Sequence Change 4 152325 10.00 2000 20000 Retardation 175000 10.57 2000 20000 11.41 35.43 1.146 7.69 0.0422 214848 12.25 2000 20000 12.38 35.43 1.149 8.02 0.0360 221784 12.50 2000 20000 Sequence Change 5 221785 12.50 1750 17500 Retardation 250000 13.35 1750 17500 13.78 31.00 1.152 7.43 0.0344 275000 14.21 1750 17500 14.66 31.00 1.155 7.68 0.0444 295063 15.10 1750 17500 Sequence Change 6 295064 15.10 1500 15000 Retardation 325000 15.72 1500 15000 16.23 26.57 1.159 6.96 0.0270 362796 16.74 1500 15000 17.13 26.57 1.162 7.16 0.0296 389144 17.52 1500 15000 Sequence Change 7 389145 17.52 1250 12500 Retardation 415703 17.94 1250 12500 18.18 22.15 1.165 6.16 0.0153 446372 18.41 1250 12500 18.64 22.15 1.166 6.25 0.0198 469082 18.86 1250 12500 19.11 22.15 1.167 6.33 0.0191 495230 19.36 1250 12500 19.66 22.15 1.169 6.43 0.0257 518558 19.96 1250 12500 19.98 22.15 1.170 6.49 0.0270 520041 20.00 1250 12500 Table B . l l : Pre-crack data for face-sheet 11 Appendix B . l Pre-crack Data 303 Sequence N a 1 min Pmax [ ] [ ] [mm] [N] [N] 1 0 2.79 3750 37500 30000 3.03 3750 37500 40000 3.36 3750 37500 50000 3.64 3750 37500 60000 3.99 3750 37500 70000 4.59 3750 37500 75430 5.26 3750 37500 2 75431 5.26 3000 30000 118697 6.98 3000 30000 126417 7.54 3000 30000 3 126418 7.54 2500 25000 157619 8.59 2500 25000 176726 9.62 2500 25000 182041 10.19 2500 25000 4 182042 10.19 2000 20000 200000 10.29 2000 20000 250000 11.94 2000 20000 264716 12.51 2000 20000 5 264717 12.51 1750 17500 300000 13.64 1750 17500 344220 15.00 1750 17500 6 344221 15.00 1500 15000 373940 15.46 1500 15000 392269 15.87 1500 15000 418803 16.46 1500 15000 445077 17.19 1500 15000 455123 17.54 1500 15000 7 455124 17.54 1250 12500 475000 17.58 1250 12500 521582 18.17 1250 12500 545192 18.49 1250 12500 581820 18.93 1250 12500 624711 19.70 1250 12500 642532 20.01 1250 12500 a [mml ACT [MPa] A K [MPa%/nT] da dN 1 cycle-I Crack Initiation 3.20 3.50 3.82 4.29 4.93 66.44 66.44 66.44 66.44 66.44 Sequence Change Retardation 7.26 53.15 Sequence Change Retardation 9.11 9.91 44.29 44.29 Sequence Change Retardation 11.12 12.23 35.43 35.43 Sequence Change Retardation 14.32 31.00 Sequence Change Retardation 15.67 16.17 16.83 17.37 26.57 26.57 26.57 26.57 Sequence Change Retardation 17.! 18.33 18.71 19.32 19.86 22.15 22.15 22.15 22.15 22.15 1.125 1.126 1.127 1.128 1.130 1.135 1.140 1.142 1.145 1.148 1.154 1.157 1.159 1.161 1.162 1.164 1.165 1.166 1.168 1.169 7.49 7.85 8.20 8.70 9.33 9.11 8.54 8.92 7.58 7.97 7.59 6.82 6.94 7.09 7.21 6.11 6.19 6.26 6.37 6.47 0.0330 0.0280 0.0350 0.0600 0.1234 0.0725 0.0539 0.1072 0.0330 0.0387 0.0308 0.0224 0.0222 0.0278 0.0348 0.0127 0.0136 0.0120 0.0180 0.0174 Table B.12: Pre-crack data for face-sheet 12 Appendix B . l Pre-crack Data 304 Sequence N a p . 1 min Pmax [ 1 [ 1 [mm] [N] [N] 1 0 2.87 3750 37500 30000 3.29 3750 37500 40000 3.82 3750 37500 50000 4.58 3750 37500 54334 5.13 3750 37500 2 54335 5.13 3000 30000 75000 5.90 3000 30000 89553 6.71 3000 30000 100000 7.65 3000 30000 3 100001 7.65 2500 25000 120863 8.42 2500' 25000 133617 9.10 2500 25000 146021 9.85 2500 25000 149749 10.03 2500 25000 4 149750 10.03 2000 20000 177096 10.72 2000 20000 195245 11.40 2000 20000 208364 12.04 2000 20000 218916 12.51 2000 20000 5 218917 12.51 1750 17500 238809 13.25 1750 17500 267029 14.28 1750 17500 275000 14.54 1750 17500 287980 15.10 1750 17500 6 287981 15.10 1500 15000 309564 15.57 1500 15000 343578 16.53 1500 15000 375749 17.55 1500 15000 7 375750 17.55 1250 12500 436513 18.19 1250 12500 450000 18.35 1250 12500 472526 18.68 1250 12500 488216 19.05 1250 12500 501939 19.18 1250 12500 523300 19.62 1250 12500 544815 20.03 1250 12500 a [mm] A<7 [MPa] Crack Initiation 3.56 4.20 4.86 66.44 66.44 66.44 Sequence Change Retardation 6.31 7.18 53.15 53.15 Sequence Change Retardation 8.76 9.48 9.94 44.29 44.29 44.29 Sequence Change Retardation 11.06 11.72 12.28 35.43 35.43 35.43 Sequence Change Retardation 13.77 14.41 14.82 31.00 31.00 31.00 Sequence Change Retardation 16.05 17.04 26.57 26.57 Sequence Change Retardation 18.27 18.52 18.87 19.12 19.40 19.83 22.15 22.15 22.15 22.15 22.15 22.15 AK [MPa^ rrT] 1.126 1.128 1.129 1.133 1.135 1.139 1.141 1.142 1.145 1.147 1.148 1.152 1.154 1.155 1.159 1.161 1.165 1.166 1.167 1.167 1.168 1.169 7.91 8.61 9.27 8.47 9.06 8.37 8.72 8.94 7.56 7.80 7.99 7.43 7.61 7.73 6.91 7.14 6.18 6.23 6.29 6.33 6.39 6.46 Table B.13: Pre-crack data for face-sheet 13 Appendix B . l Pre-crack Data 305 Sequence [ ] N [ ] a [mm] 1 min [N] Pmax [N] a [mm] ACT [MPa] c [ ] A K [MPax/rH] da dN r J t E L i L cycle-1 1 0 2.59 3750 37500 Crack Initiation 30000 3.28 3750 37500 3.38 66.44 1.126 7.70 0.0190 40000 3.47 3750 37500 3.56 66.44 1.126 7.91 0.0180 50000 3.65 3750 37500 3.89 66.44 1.127 8.27 0.0470 60000 4.12 3750 37500 4.43 66.44 1.128 8.84 0.0610 70000 4.73 3750 37500 4.92 66.44 1.130 9.33 0.1436 72576 5.10 3750 37500 Sequence Change 2 72577 5.10 3000 30000 Retardation 100000 6.07 3000 30000 6.79 53.15 1.134 8.80 0.0576 125000 7.51 3000 30000 Sequence Change 3 125001 7.51 2500 25000 Retardation 150000 8.36 2500 25000 8.75 44.29 1.139 8.36 0.0524 164899 9.14 2500 25000 9.29 44.29 1.140 8.63 0.0287 175000 9.43 2500 25000 9.76 44.29 1.142 8.85 0.0725 184106 10.09 2500 25000 Sequence Change 4 184107 10.09 2000 20000 Retardation 219067 10.50 2000 20000 10.95 35.43 1.145 7.52 0.0329 246451 11.40 2000 20000 11.70 35.43 1.147 7.79 0.0425 260580 12.00 2000 20000 12.27 35.43 1.148 7.99 0.0414 273639 12.54 2000 20000 Sequence Change 5 273640 12.54 2000 20000 Retardation 300000 13.22 2000 20000 13.55 35.43 1.152 8.42 0.0243 327178 13.88 2000 20000 14.22 35.43 1.153 8.64 0.0410 343529 14.55 2000 20000 14.79 35.43 1.155 8.82 0.0329 358117 15.03 2000 20000 Sequence Change 6 358118 15.03 1500 15000 Retardation 375000 15.24 1500 15000 15.47 26.57 1.157 6.78 0.0159 403933 15.70 1500 15000 16.08 26.57 1.159 6.92 0.0163 450000 16.45 1500 15000 16.69 26.57 1.160 7.06 0.0188 475000 16.92 1500 15000 17.21 26.57 1.162 7.18 0.0208 502941 17.50 1500 15000 Sequence Change 7 502942 17.50 1250 12500 Retardation 546800 17.82 1250 12500 18.02 22.15 1.164 6.13 0.0081 596065 18.22 1250 12500 18.40 22.15 1.165 6.20 0.0067 650000 18.58 1250 12500 18.73 22.15 1.166 6.26 0.0079 688092 18.88 1250 12500 19.09 22.15 1.167 6.33 0.0089 735506 19.30 1250 12500 19.53 22.15 1.169 6.41 0.0130 770135 19.75 1250 12500 19.91 22.15 1.170 6.48 0.0086 807262 20.07 1250 12500 Table B.14: Pre-crack data for face-sheet 14 Appendix B.2 Pre-crack Diagrams 306 B.2 Pre-crack Diagrams o T3 0.200 0.100 0.080 0.060 0.040 0.020 0.010 0.008 0.006 0.004 0.002 • • • • • • • • • 0 • • • a < 17.5 mm • a> 17.5 mm 6.0 7.0 8.0 9.0 AK [MPafin] Figure B . l : Crack propagation rates for face-sheet 1 10.0 0.200 o.ioo 0.080 0.060 J£ 0.040 o % z 0.020 0.010 0.008 0.006 0.004 0.002 • • • • • • • • 1 • m • • • • • a< 17.5 mm • a> 17.5 mm 6.0 7.0 8.0 AK [MPafir?] 9.0 10.0 Figure B.2: Crack propagation rates for face-sheet 2 Appendix B.2 Pre-crack Diagrams 307 0.200 0.100 0.080 0.060 0.040 o 15 o.oio 0.002 • • _ I a • • • • • • • • • • a< 17.5 mm • a > 17.5 mm 6.0 7.0 8.0 9.0 10.0 AK [MPafm] Figure B.3: Crack propagation rates for face-sheet 3 • • • • • • _ • • • • • • • • • • a< 17.5 mm • a> 17.5 mm 6.0 7.0 8.0 AK [MPa fin] 9.0 10.0 Figure B.4: Crack propagation rates for face-sheet 4 Appendix B.2 Pre-crack Diagrams 308 0.200 0.100 0.080 0.060 3? 0.040 o >> % 0.020 a o.oio "° 0.008 0.006 0.004 0.002 • • • • • • • • • • • • • • • • • • • • • • a< 17.5 mm • a > 17.5 mm 6.0 7.0 8.0 9.0 AK [MPafin] Figure B.5: Crack propagation rates for face-sheet 5 10.0 0.200 o.ioo 0.080 0.060 0.040 0.020 0.010 0.008 0.006 0.004 0.002 • m • • • • • • • • • • • • • • • * • a < 17.5 mm • a> 17.5 mm 6.0 9.0 7.0 8.0 AK [MPafin] Figure B.6: Crack propagation rates for face-sheet 6 10.0 Appendix B.2 Pre-crack Diagrams 309 • n • H • • • • • • • • a< 17.5 mm • a> 17.5 mm 6.0 9.0 7.0 8.0 AK [MPafnT] Figure B.7: Crack propagation rates for face-sheet 7 10.0 • • • • • • • • • • • • 1 • • • • a < 17.5 mm • a > 17.5 mm 6.0 7.0 8.0 AK [MPaVnf] 9.0 10.0 Figure B.8: Crack propagation rates for face-sheet 8 Appendix B.2 Pre-crack Diagrams 310 • • • " • • • • • • • ** • • • a < 17.5 mm • a> 17.5 mm 6.0 7.0 9.0 8.0 AK [MPaVrrT] Figure B.9: Crack propagation rates for face-sheet 9 10.0 • B~~ • • • • • . • • • • • * • • • a < 17.5 mm • a > 17.5 mm 6.0 7.0 8.0 AK [MPafnT] 9.0 10.0 Figure B.10: Crack propagation rates for face-sheet 10 Appendix B.2 Pre-crack Diagrams 311 • • • • • • • • • • 1 • • • • • a < 17.5 mm • a > 17.5 mm 6.0 9.0 7.0 8.0 AK [MPafhT] Figure B . l l : Crack propagation rates for face-sheet 11 10.0 0.200 o.ioo 0.080 0.060 0.040 o >. u 0.020 3^ 0.010 7 3 0.008 0.006 0.004 0.002 • • • • • • • B • • • • • • • • a < 17.5 mm • a> 17.5 mm 6.0 7.0 8.0 9.0 10.0 AK [MPafin] Figure B.12: Crack propagation rates for face-sheet 12 Appendix B.2 Pre-crack Diagrams 312 • • u • • u u • m • • • • • • • • • a< 17.5 mm • a > 17.5 mm 6.0 9.0 7.0 8.0 AK [MPafm] Figure B.13: Crack propagation rates for face-sheet 13 10.0 • • • • • • • • • • • • • • • • • • • a< 17.5 mm • a> 17.5 mm 6.0 7.0 8.0 AK [MPafhT] 9.0 10.0 Figure B.14: Crack propagation rates for face-sheet 14 Appendix B.3 Pre-crack Pictures B.3 Pre-crack Pictures 313 Figure B.17: Face-sheet 3 pre-crack Figure B.18: Face-sheet 4 pre-crack Appendix B.3 Pre-crack Pictures 314 Figure B.19: Face-sheet 5 pre-crack Figure B.20: Face-sheet 6 pre-crack Figure B.21: Face-sheet 7 pre-crack Figure B.22: Face-sheet 8 pre-crack Appendix B.3 Pre-crack Pictures 315 Figure B.26: Face-sheet 12 pre-crack Appendix B.3 Pre-crack Pictures Figure B.28: Face-sheet 14 pre-crack Appendix C Ultrasonic C-Scans 317 Appendix C Ultrasonic C-Scans Appendix C . l Ultrasonic C-Scans of the Precured Patches 318 C . l Ultrasonic C-Scans of the Precured Patches Figure C.2: C-scan of patch 2 Figure C.3: C-scan of patch 3 A p p e n d i x C l U l t r a s o n i c C - S c a n s o f t h e P r e c u r e d P a t c h e s 319 F i g u r e C .6 : C - s c a n o f p a t c h 6 Appendix C . l Ultrasonic C-Scans of the Precured Patches 3 2 0 Figure C .9: C-scan of patch 9 Appendix C l Ultrasonic C-Scans of the Precured Patches Figure C.12: C-scan of patch 12 Appendix C . l Ultrasonic C-Scans of the Precured Patches 322 Figure C.14: C-scan of patch 14 Appendix D Mater ia l Properties 323 Appendix D Material Properties Appendix D . l Mechanical Properties of Boron/epoxy 5521/4 Laminates 324 D . l M e c h a n i c a l P r o p e r t i e s o f B o r o n / e p o x y 5 5 2 1 / 4 L a m i n a t e s The following material properties for Textron Specialty Materials Boron 5521/4 for dry environmental condition were used in this research work [45, 52, 87]. Thickness: Longitudinal Elastic Modulus: Transverse Elastic Modulus: Shear Moduli: Longitudinal Poisson's Ratio: Transverse Poisson's Ratios: Longitudinal CTE @21°C: Transverse CTE @21°C: Longitudinal CTE @-56.5°C: Transverse CTE @-56.5°C: EPy = BL. = G G Pxy Pxz a Py aPx = aPv = 0.132 mm (0.0052 in) 210 GPa (30.5 msi) EPz = 25 GPa (3.6 msi) = GPyz =7.24 GPa (1050 ksi) w 1 GPa (145 ksi) = 0.025 0.21 0.17 4.61 ne/°C (2.56 ftefF) aPz = 25.87 /ie/°C (14.37 ne/°F) 4.16 /xe/°C (2.31 /ie/°F) aPz = 25.40 /xe/°C (14.11 fie/°F) The following coefficients of thermal expansion were measured as part of this research work: 5.25 20 40 Temperature [°C] 120 Figure D . l : Coefficient of thermal expansion for boron/epoxy 5521/4 parallel to the fiber direction based on different reference temperatures Appendix D . l Mechanical Properties of Boron/epoxy 5521/4 Laminates 325 26.50 24.50 -I 1 1 1 1 1 1 1 1 1 -60 -40 -20 0 20 40 60 80 100 120 Temperature [°C] Figure D.2: Coefficient of thermal expansion for boron/epoxy 5521/4 perpendicular to the fiber direction based on different reference temperatures Appendix D.2 Mechanical Properties of Aluminum 2024-T3 326 D.2 Mechanical Properties of Aluminum 2024-T3 The following material properties for 2024-T3 aluminum were used in this research work [91]: Thickness: Elastic Moduli: Shear Moduli: Poisson's Ratio: Longitudinal CTE @21°C: Transverse CTE @21°C: Longitudinal CTE @-56.5°C: Transverse CTE @-56.5°C: Tensile Yield Stress: Compressive Yield Stress: ts = 3.175 mm (0.125 in) E** = Esy = ESz = 72.4 GPa (10.5 msi) °sxy = Gsyz = GSxz = 27.2 GPa (3.95 msi) * V = "s«* = "s„ = 0.33 aSy = 23.45 ne/°C (12.94 /ue/°F) «s, = aSz = 23.29 ne/°C (12.93 /ie/°F) aSy = 22.77 fie/°C (12.65 M e/°F) «s, = 20 40 Temperature [°C] 120 Figure E.10: Gauge 10 - Thermal and thermal residual strains [Aluminum 2500 2000 1500 1000 500 0 -500 -1000 -1500 -2000 -2500 Thermal Strain Thermal Residual Strain -60 -40 -20 20 40 Temperature [°C] 60 80 100 120 Figure E . l l : Gauge 11 - Thermal and thermal residual strains [Aluminum _L] 20 40 Temperature [°C] 120 Figure E.12: Gauge 12 - Thermal and thermal residual strains [Aluminum || Appendix E Measured Thermal and Thermal Residual Strains 334 Thermal Strain Thermal Residual Strain -60 -40 -20 20 40 Temperature [°C] 60 80 100 120 Figure E.13: Gauge 13 - Thermal and thermal residual strains [Aluminum || 2500 2000 1500 1000 500 c 0 E CO -500 -1000 -1500 -2000 -2500 Thermal Strain Thermal Residual Strain -60 -40 -20 20 40 Temperature [°C] 60 80 100 120 Figure E.14: Gauge 14 - Thermal and thermal residual strains [Aluminum _L] 2500 2000 1500 1000 500 c 0 'co CO -500 -1000 -1500 -2000 -2500 • — Thermal Strain Thermal Residual Strain -60 -40 -20 20 40 Temperature [°C] 60 80 100 120 Figure E.15: Gauge 15 - Thermal and thermal residual strains [Aluminum _L] Appendix E Measured Thermal and Thermal Residual Strains 335 2500 2000 1500 1000 To ^_ 500 c 0 "5 CO -500 -1000 -1500 -2000 -2500 Thermal Strain Thermal Residual Strain -60 -40 -20 20 40 Temperature [°C] 60 80 100 120 Figure E.16: Gauge 16 - Thermal and thermal residual strains [Aluminum || Thermal Strain Thermal Residual Strain -60 -40 -20 20 40 Temperature [°C] 60 80 100 120 Figure E.17: Gauge 17 - Thermal and thermal residual strains [Aluminum 45° 2500 2000 1500 1000 717 500 c 0 '5 -500 -1000 -1500 -2000 -2500 Thermal Strain Thermal Residual Strain -60 -40 -20 20 40 Temperature [°C] 60 80 100 120 Figure E.18: Gauge 18 - Thermal and thermal residual strains [Aluminum _L Appendix E Measured Thermal and Thermal Residual Strains 336 20 40 Temperature [°C] 100 120 Figure E.19: Gauge 19 - Thermal and thermal residual strains [Aluminum 2500 2000 1500 1000 500 c 0 2 4— CO -500 -1000 -1500 -2000 -2500 -60 Thermal Strain Thermal Residual Strain -40 -20 20 40 Temperature [°C] 60 80 100 120 Figure E.20: Gauge 20 - Thermal and thermal residual strains [Aluminum 45° 2500 2000 1500 1000 500 c 0 "55 55 -500 -1000 -1500 -2000 -2500 Thermal Strain Thermal Residual Strain -60 -40 -20 20 40 Temperature [°C] 60 80 100 120 Figure E.21: Gauge 21 - Thermal and thermal residual strains [Aluminum JL] Appendix E Measured Thermal and Thermal Residual Strains 337 20 40 Temperature [°C] 100 120 Figure E.22: Gauge 22 - Thermal and thermal residual strains [Aluminum 20 40 Temperature [°C] 120 Figure E.23: Gauge 23 - Thermal and thermal residual strains [Boron/epoxy 20 40 Temperature [°C] 120 Figure E.24: Gauge 24 - Thermal and thermal residual strains [Boron/epoxy ||] Appendix E Measured Thermal and Thermal Residual Strains 338 20 40 Temperature [°C] 120 Figure E.25: Gauge 25 - Thermal and thermal residual strains [Boron/epoxy 20 40 Temperature [°C] 120 Figure E.26: Gauge 26 - Thermal and thermal residual strains [Boron/epoxy 20 40 Temperature [°C] 120 Figure E.27: Gauge 27 - Thermal and thermal residual strains [Boron/epoxy ||] Appendix E Measured Thermal and Thermal Residual Strains 339 20 40 Temperature [°C] 120 Figure E.28: Gauge 28- Thermal and thermal residual strains [Boron/epoxy 20 40 Temperature [°C] 120 Figure E.29: Gauge 29 - Thermal and thermal residual strains [Boron/epoxy 20 40 Temperature [°C] Figure E.30: Gauge 30 - Thermal and thermal residual strains [Boron/epoxy Appendix E Measured Thermal and Thermal Residual Strains 340 20 40 Temperature [°C] 120 Figure E.31: Gauge 31 - Thermal and thermal residual strains [Boron/epoxy 20 40 Temperature [°C] 120 Figure E.32: Gauge 32 - Thermal and thermal residual strains [Boron/epoxy 20 40 Temperature [°C] 120 Figure E.33: Gauge 33 - Thermal and thermal residual strains [Boron/epoxy || Appendix E Measured Thermal and Thermal Residual Strains 341 c 2500 2000 1500 1000 500 0 -500 -1000 -1500 -2000 -2500 Thermal Strain Thermal Residual Strain -60 -40 -20 20 40 Temperature [°C] 60 80 100 120 Figure E.34: Gauge 34 - Thermal and thermal residual strains [Boron/epoxy _L] 20 40 Temperature [°C] 120 Figure E.35: Gauge 35 - Thermal and thermal residual strains [Boron/epoxy 2500 2000 1500 1000 500 c 0 2 CO -500 -1000 -1500 -2000 -2500 Thermal Strain Thermal Residual Strain -60 -40 -20 20 40 60 Temperature [°C] 80 100 120 Figure E.36: Gauge 36 - Thermal and thermal residual strains [Boron/epoxy _L] Appendix E Measured Thermal and Thermal Residual Strains 342 20 40 Temperature [°C] 120 Figure E.37: Gauge 37 - Thermal and thermal residual strains [Boron/epoxy 2500 2000 1500 1000 500 0 -500 -1000 -1500 -2000 -2500 -60 Thermal Strain Thermal Residual Strain -40 -20 20 40 Temperature [°C] 60 80 100 120 Figure E.38: Gauge 38 - Thermal and thermal residual strains [Boron/epoxy J_] 2500 2000 1500 1000 500 c 0 ' r a 5) -500 -1000 -1500 -2000 -2500 Thermal Strain Thermal Residual Strain -60 -40 -20 20 40 Temperature [°C] 60 80 100 120 Figure E.39: Gauge 39 - Thermal and thermal residual strains [Boron/epoxy _L] f Appendix E Measured Thermal and Thermal Residual Strains 343 20 40 Temperature [°C] Figure E.40: Gauge 40 - Thermal and thermal residual strains [Boron/epoxy 2500 2000 1500 1000 To 500 c 0 •5 -500 -1000 -1500 -2000 -2500 Thermal Strain Thermal Residual Strain -60 -40 -20 20 40 Temperature [°C] 60 80 100 120 Figure E.41: Gauge 41 - Thermal and thermal residual strains [Boron/epoxy _L] 20 40 Temperature [°C] Figure E.42: Gauge 42 - Thermal and thermal residual strains [Boron/epoxy || Appendix E Measured Thermal and Thermal Residual Strains 344 Thermal Strain Thermal Residual Strain -60 -40 -20 20 40 Temperature [°C] 60 80 100 120 Figure E.43: Gauge 43 - Thermal and thermal residual strains [Boron/epoxy ||] 2500 2000 1500 1000 r — i CO 500 c 0 2 CO -500 -1000 -1500 -2000 -2500 Thermal Strain Thermal Residual Strain -60 -40 -20 20 40 Temperature [°C] 60 80 100 120 Figure E.44: Gauge 44 - Thermal and thermal residual strains [Boron/epoxy ± ] *