(zj) and \p (z^ in the above expressions are given by, t(Zj) = -ma * W 2(srs2) ZJ+x/\Z-]-a2(l + sy} H Z 2 ) = JB \u00C2\u00B00r*f 2(srs2) Z2+v/{Zj-a2(l + sy} (3.41) where Sj and s2 are the complex roots of the equation cl/~ 2c16^ + (2c12+c66>s2- 2c26+ c22 = 0 <142) whose coefficients C^j s are the coefficients of the plate compliance matrix with respect to the coordinate axes. The complex variables Zj and Z2 in (3.40) and (3.41) are given by, Mathematical Analysis 48 Zj = x + sjy Z2 = x + s2y (3.43) The solutions of equations (3.40) through (3.43) yields the in-plane stresses around a hole in a composite laminate, when the coefficients of the laminate elastic compliance matrix are determined from LPT as; Cjj = lA.. U ij J (3.44) The stresses a Y , a v and T y v resulting from the exact solution are now transformed to polar coordinates using the following stress transformation; a, r \u00C2\u00B0e \u00E2\u0080\u0094 sin26 cos2e sin26 cos26 \u00E2\u0080\u00A2 sin 8 cos 6 sin 8 cos 8 2sin 8 cos 8 - 2sin 8 cos 8 cos28-sin28 xy (3.45) For orthotopic laminates employing the point stress method, the stresses o r, OQ and T ^ derived at a radial distance / from the hole boundary are taken as the uniform in-plane laminate stresses of the approximating straight edge laminate. For the average stress methods however, a closed form solution of the laminate stresses is not available. The stresses are averaged numerically over the radial distance /. In the present work Simpson's Rule is applied, which fits a second order polynomial into the radial distribution. The accuracy of the method improves greatly if a large number of intervals is used. Mathematical Analysis 49 3.6 RESIDUAL T H E R M A L STRESSES:-If a laminate is subjected to a constant temperature change AT from its cure temperature, thermal stresses are induced which alter the stress state everywhere, including near. the hole boundary. The changes that occur in the stress state are uniform throughout the laminate, except near the hole boundary, where interlaminar stresses are present These interlaminar stresses can still be derived using the methods suggested in Sec. 3.1., if the thermal effects are included in the approximate straight edge laminate stress analysis. Once the laminate stresses due to applied mechanical loads are defined for the approximating straight edge laminate, as outlined in Sec. 3.1 (using the exact plane stress solutions), it can be analysed for individual layer stresses incorporating both mechanical and thermal effects. The following treatment is for a typical straight edge laminate approximating the circular, edge at the hole boundary. from both mechanical and thermal loads are given by e , the ply stresses in each layer are found by LPT as; where a.j are the coefficients of the various layers with respect to the laminate principal axes, and A T is the temperature rise from the curing temperature. The other terms in (3.46) have the same meanings as described earlier in Sec. 3.2. Summing (3.46) through all the plies we obtain, If the net mid-plane strains in the straight edge laminate resulting (3.46) n (3.47) Mathematical Analysis SO where iVz- are the force resultants defined as the force per unit length. Equation (3.47) is usually written as Ni + Nj = Ai}e\u00C2\u00B0. (3.48) T where N. are the thermal forces given by k=l l J 1 llj = h I CTna^T (3.49) Solving for the mid-plane strains and substituting in (3.46), the ply stresses in the k th ply are found to be given by, ok = (\u00C2\u00A3\u00E2\u0080\u00A2\u00E2\u0080\u00A2[*' \Nm + NT)- a.AT] (3.50) / ij j m m m J v / k The additional ply radial stress & which contributes to the r interlaminar normal stress oz and the interlaminar shear stress r r z is that which is generated by the tangential and shear laminate stresses ah. and \u00E2\u0080\u009E. This is found by u ru setting the laminate radial stress ( or, correspondingly the force resultant ) in T equation (3.50) to zero. But N ^, which is not allowed for by the plane stress solution k remains. Similarly, the additional ply shear stress f ^ which contributes to the interlaminar shear stress T Q Z on the other hand results from the laminate tangential and radial stresses OQ and and is thus found by setting N$ in the above expression to zero. Again, A 7 ^ contributes to the additional ply shear stress. Thus in both cases, given that the thermal stresses are always additional to the plane elasticity solution, all three thermal stress components contribute to each of the interlaminar stresses. In the modified stress methods the deviatoric ply stresses are found by first calculating the ply stresses due to all three components of laminate stresses from (3.50), and then substracting out the corresponding laminate stress. Thus, when calculating Mathematical Analysis 51 the deviatoric ply radial stress 5 , (allowing for both mechanical and thermal loads) the L k laminate radial stress a is substracted out from the ply radial stress a . The deviatoric r r r ply shear stress f ^ on the otherhand is calculated by substracting the laminate shear L k stress T \u00E2\u0080\u009E from the ply shear stress T n . Thus, in both cases, all of the terms in Nm rd rd m T or N contribute to the interlaminar stresses. m Mathematical Analysis Fig. 3.1. laminate configuration. 3.2. Straight edge approximation of a laminate hole. Mathematical Analysis 54 Mathematical Analysis 55 2-D Approximation of the hole problem How additional ply stresses due to lamination generate interlaminar stresses Calculation of lar the straight edg Point and Avera, ninate stresses for e laminate using ge stress methods Models of approxi Stress die in the trans\ .mate interlaminar >tributions rerse direction Exact plane stress solution of in-plane laminate stresses around the hole Fig. 3:5. Flow diagram of the method of analysis. Mathematical Analysis 56 ^1 Mathematical Analysis 58 Mathematical Analysis 59 Fig. 3.9. Approximate distribution of r near the free-edge. 60 CHAPTER TV COMPARISONS WITH LITERATURE In this chapter, the results of the analytical technique presented in the previous Chapter are compared with theoretical and experimental results from the literature. Comparisons are made with the calculations of interlaminar stresses and observations of delamination damage obtained by several authors. A fair agreement was observed for a wide range of laminates, as discussed in the following sections. 4.1 THEORETICAL COMPARISONS:-There are very few numerical solutions for the hole problem in the literature, especially when compared to the large number of solutions for the straight edge problem. This is mainly due to the complexity of the hole problem. The stress fields around holes are fully three dimensional and functions of all three space variables. Solution of this problem, therefore, requires the use of numerical methods, such as finite element Although the costs associated with the formulation and the use of three dimensional finite element programs are high, such calculations offer the most popular means of laminate stress analysis The solutions given by Raju and Crews (1982), Rybicki and Schmueser (1978), and Whitcomb (1981) employ three dimensional finite element stress Comparisons with Literature 61 analysis to calculate the stresses around the hole. The closed-form analytical solution used by Tang (1977) employs an extension of a boundary layer theory developed by Reiss (1961) for isotropic elastic materials. The results of these solutions are compared with stresses calculated using the methods described in the previous Chapter. The results of Raju and Crews are compared first, since they present both interlaminar normal and shear stress distributions. The finite element mesh used in their formulation is increasingly finer closer to the free-edge at the ply interface for which the stresses are calculated. This is considered to be an added refinement that has greatly improved the accuracy of their calculations. In addition, the results of Raju and Crews are especially of interest, in that along with their 3-D solution they present a reduced 2 - D numerical solution which is the numerical method equivalent to the present approach. It must be pointed out that although the accuracy of finite element stress analysis can be effectively increased by element mesh refinement, it is not possible to achieve this at the very edge of a ply interface, where interlaminar stresses appear to display a singular nature. The presence of interlaminar stress singularities in multilayered composites have rigorously been proven by Wang and Choi (1982). Such singularities make stress calculations only tend toward accuracy, without convergence. From a practical point of view, this makes realistic estimates of interlaminar stresses at the free edge somewhat difficult, and any attempts to improve the solution accuracy through element mesh refinement (or other means) are superfluous. While the issue of convergence to the classical elasticity solution for a laminated structure remains unresolved, efforts to develop realistic descriptions of interlaminar stress fields are often being made. The present work is an attempt to estimate the relative magnitudes of interlaminar stresses and predict the general nature of the stress distribution around a circular hole. As such, the comparisons are made on the basis of relative changes that take place in the magnitudes and signs of interlaminar stresses as a function of angle around the hole. Comparisons with Literature 62 4.1.1 Raju and Crews (1982):-Interlaminar stress distributions have been calculated near a circular hole in [90/0] s and [0/90] s graphite/epoxy laminates by Raju and Crews, using a three-dimensional finite element analysis, based on a displacement formulation. We consider first the [90/0] s laminate subjected to a gross applied stress of o\u00E2\u0080\u009E with the elastic properties as used by Raju and Crews (see Table II - page 125). Plotted in Figures 4.1, 4.2, 4.3 and 4.4 are the distributions of interlaminar normal stress o Jo _ for the z = h plane from the point, average, z 6 modified point and modified average stress methods. A fixed boundary layer width of one laminate thickness was used for the results shown in these Figures. For each method, the effect of assuming different characteristic lengths / is shown. The effect of varying the characteristic length, while keeping the boundary layer width d at one laminate thickness is similar in all four cases. Since the thickness of the laminate considered by Raju and Crews is 0.2 times the hole radius a, the ratio of d/a remains at 0.2. With l/a = 0, that is using the stresses on the hole boundary only, all four methods reduce to the same solution, and we are considering only the effect of the laminate circumferencial stress. As we allow the characteristic length to increase (from l/a = 0 to 1), the increasing laminate shear stress and changing laminate circumferential stress cause the distribution of o z to shift to the smaller angles and increase in magnitude. Al l four methods give essentially similar values except that the modified point and modified average approximations predict a sign change for 6 > 75\u00C2\u00B0 . Plotted in Figures 4.5, 4.6, 4.7 and 4.8 are the distributions of interlaminar normal stress ( o z / o ) for the same laminate using a boundary layer Comparisons with Literature 63 width d equal to the characteristic length /. The effects of varying the characteristic length together with the boundary layer width is shown in these Figures for each different method in contrast to the case for fixed boundary layer width. As the characteristic length is increased (from lla = 0.1 to 0.5) oz is seen to decrease in magnitude. The contribution of the increasing laminate shear stress and the changing laminate circumferential stress is apparently offset by the effects of increased boundary layer width; the net result being lower o along the free-edge. The results are similar in all four methods except for the sign change behaviour predicted by the modified point and modified average stress methods. Comparison of the results of oz discussed so far with the finite element solution by Raju and Crews shows that the general characteristics of the stress distribution is best predicted when both / and d are equal to one laminate thickness. Thus, in calculating interlaminar stresses, the condition I = d = t where t is the thickness of the laminate is always satisfied (unless otherwise noted) throughout the rest of the work. In Fig. 4.9, results from the four different approximations for / = d = 0.2a are compared with the finite element solution. The value of 0.2a corresponds to one laminate thickness for the Raju and Crews geometry. Agreement is fairly good over most of the quadrant, though both the point and average methods predict compressive stresses throughout the region. The modified methods of approximation on the other hand do predict the sign change that takes place at around 80\u00C2\u00B0. It must be emphasized that the z = h plane is an interface between the 0\u00C2\u00B0 and 90\u00C2\u00B0 plies, and that the Raju and Crews solution indicates a singularity at this interface. Although the present solution cannot cope with a singularity, it appears to predict the same general shape, though with consistently lower magnitudes. Comparisons with Literature 64 Raju and Crews also present the distribution of oz for a [0/90] s laminate. These results are very similar to their [90/0] s values, both in magnitude and sign. The solution of the present work for a [0/90] s would be o equal in magnitude to the [90/0] s solution, but opposite in sign. However close inspection of the results in Raju and Crews shows that in the case of the [0/90] s laminate, not only does the interlaminar normal stress distribution through the thickness of the laminate become singular as it approaches the z = h interface but it also changes sign over a very short distance before the interface. The [90/0] s laminate does not show this very abrupt reversal, and therefore allows comparison. The [0/90] s behaviour appears not to be a 3 -D effect, but a result of having a very fine mesh in the region of interest. This is shown by the fact that their finite element 2-D results agree very well with their 3 -D results. It appears that it is possible, even in the 2 -D case, to have out-of-plane stresses at the free-edge different in sign to what an equlibrium argument would suggest Similar results exist in the straight free-edge literature. For example results from Wang and Crossman (1977) for cross-ply laminates show the same sign change behaviour at the z - h interface, but not at the z = 0 interface, where there is no discontinuity in material properties. Similarly, the Raju and Crews data show that at the z = 0 interface the [0/90] s laminate has an interlaminar stress of opposite sign to the [90/0] s laminate over most of the boundary. However, the mesh is very coarse at the z = 0 interface, and no detailed results are presented by them. Thus it appears that the problem lies not in reducing the 3-D problem to a 2 - D approximation, but in the 2 -D equilibrium argument However, there is a host of experimental evidence to back the equilibrium argument in the 2 - D case, and a great deal of use is made of i t Returning to the solution of oz for the [90/0] s laminate, we find that it is the modified average stress results which exhibit the best agreement Comparisons with Literature 65 when both / and d are equal to one laminate thickness. It has been reported by many, that the distance over which interlaminar effects occur is in the order of one laminate thickness.t Henceforth we will use a value of one laminate thickness for / and d in all comparisons. In order to better compare the shapes of the distributions, the stresses are sometimes normalized with respect to their maximum values, as shown in Fig. 4.10 for the [90/0] s laminate. This shows clearly that the essential characteristics of the distribution (eg., the location of the maximum stress and the range over which they are tensile) are predicted reasonably well by the modified average stress aproximation. It is however important to note that the absolute value of maximum oz for this lay-up given by Raju and Crews is approximately 3 times that obtained by the present approach. Raju and Crews also present interlaminar shear stress T Q Z distributions around the hole for both [0/90] s and [90/0] s laminates. These are compared with the results of the modified average stress calculation in Figs. 4.11 and 4.12. Shear stresses normalized with respect to their maxima are plotted in these Figures. Except for different signs, the T Q Z distributions of the finite element solution are identical for the two laminates, which is also true of the results obtained by the modified average stress method. The finite element solution predicts the maximum stress at about 75\u00C2\u00B0 while the present solution predicts it near 6 = 67\u00C2\u00B0. The angle at which the sign of the shear stress changes is predicted within a degree or two from that predicted by the finite element solution. The present solution, using the modified average stress approximation, thus appears to predict the same general shape of the distribution, though the absolute value of the maximum stress given by finite element solution is approximately 5.3 times that of t As noted by Pagano and Pipes, this is also in agreement with a (loose) interpretation of Saint Venant's principle, since the L.T. stresses on any plane y = const, and extending throughout the entire thickness dimension are self-equilibrating. Comparisons with Literature 66 the present solution. The increase in this ratio of absolute magnitudes over that for the normal stress may partly be due to the stronger stress singularity observed by Raju and Crews for r ^ r 4.1.2 Rybicki and Schmueser (1978):-Using a three-dimensional finite element program, the distribution of interlaminar normal stress, o z , around a hole at the laminate midplane was studied by Rybicki and Schmueser for a series of Graphite /epoxy laminates of the type [02/\u00C2\u00B18/^8]s, [\u00C2\u00B18/+8/02]s, [\u00C2\u00B18/3-8/%2]s and [902/\u00C2\u00B18/+'6]s where 8 is 30\u00C2\u00B0, 45\u00C2\u00B0 and 60\u00C2\u00B0 . t The material properties used in the analysis are given in Table II (page 125). All results are for the laminate midplane where there is no discontinuity in material properties, and thus no singularity is expected. Since a fairly coarse mesh was used by Rybicki and Schmueser, there is no indication of any such effects. Figures 4.13 and 4.14 show the results for the [ 0 2 / \u00C2\u00B1 3 0 / T 3 0 ] S laminate. Interlaminar normal stress is calculated using the present approximate methods with / = d = 1.2a which corresponds to one laminate thickness for all of the above lay-ups considered by Rybicki and Schmueser. Fig. 4.13 compares the computed results of o Jo _ of the four different methods with that of Rybicki and Schmueser. Agreement is reasonable with all four approximate methods predicting a sign change for oz at 8 > 60\u00C2\u00B0. The modified average approximation shows the best agreement It is also clear that we consistently predict larger magnitudes than the numerical results. Figure 4.14 compares the modified average stress results normalized by the maximum stress with numerical results, t Rybicki and Schmueser modeled the (0$ and (902) plies as one material, and the (\u00C2\u00B18/^-8) plies as a single material witn effective modulus properties. Comparisons with Literature 67 which are also normalized. As evident from this Figure the general shapes of o z distribution around the hole are in rough agreement Rybicki and Schmueser also present the distribution for a [ \u00C2\u00B1 3 0 / : f 30/02] s lay-up, and though the sign of the distribution is reversed, it is not an exact mirror image as predicted by the present solution. They predict magnitudes for the [ \u00C2\u00B1 3 0 / ? 30/02] s laminate that are roughly double those of the [ 0 2 / \u00C2\u00B1 3 0 / T 3 0 ] s lay-up. Since the modified average stress method seems to predict the stress distribution better than the other three methods, this method will be used to calculate the interlaminar stresses in future, unless otherwise noted. It was thought initially (Goonetilleke, Poursatip and Teghtsoonian, 1985) that the point and average stress methods constituted the most satisfactory approximations. The modified stress methods at first were difficult to explain in terms of physical reasoning. However, it was found later that a sound physical argument can be presented to explain these methods also. It is based on the fact that the phenomenon of free-edge effect is found to occur only in multilayered laminates where there is discontinuity in material properties, and not in homogeneous solids in general. The different ply k k stresses o r and T ^ predicted by the combined laminated plate theory (LPT) and exact plane stress solution at a given point within a laminate result from the material discontinuity in the thickness direction. If the material were truly homogeneous through the thickness, as assumed by the plane stress solution, these stresses would simply be the laminate stresses and respectively. The r rv difference between these values can therefore be considered as the source of interlaminar stresses observed in composite laminates. The modified stress methods use these deviatoric ply stresses to calculate the interlaminar stresses. It is however important to note that though the modified average stress method shows the best Comparisons with Literature 68 agreement with numerical results, the other three methods are not always worse. With some laminates, the distribution is predicted nearly as well by one or more of the other three methods. Figures 4.15 through 4.19 show the results of the modified average stress calculations for the laminates of the type [02/5 6/^6]$ where 6 is 45\u00C2\u00B0 and 60\u00C2\u00B0,and also for the [\u00C2\u00B16/36/902]$ type of laminates where 6 is 30\u00C2\u00B0, 45\u00C2\u00B0 and 60\u00C2\u00B0. The results of the present calculation for the laminates of the type [\u00C2\u00B16/3-6/02]$ ^ [902/56/+'6]s are not presented here, since they are the exact mirror images of the results shown in these Figures. Although of opposite sign, the results of the finite element solution for these lay-ups do not exhibit such mirror images, but rather the same general shapes with different stress magnitudes. Specific reference will be made to these results in the following paragraphs when comparing the curves in Figs. 4.15 to 4.19. Figure 4.15 compares the results for the [ f ^ / i : 4 5 / ^ 4 5 ^ laminate. Except for the hump observed at the centre, the present solution compares reasonbly well with the general character of the finite element stress distribution. Stresses are tensile (or compressive) within the same angular ranges as that predicted by the finite element solution. The absolute values are of the same order of magnitudes, although the present result is approximately 4 times that of the finite element solution at 6 = 35\u00C2\u00B0, where the deviation is found to be largest For the [ \u00C2\u00B1 4 5 / + 4 5 / 0 2 ] s laminate, this deviation is much less since the finite element stresses are approximately twice those for the [02/\u00C2\u00B145/3- 45] s . Fig. 4.16 shows the results for the [ 0 2 / \u00C2\u00B1 6 0 / T 6 0 ] s laminate. As it is a quasi-isotropic laminate, inplane stresses are calculated using the isotropic solution. The comparison between the calculated results and the finite element result is very much similar to that of [ 0 2 / \u00C2\u00B1 4 5 / T 4 5 ] s which is described in the previous Comparisons with Literature 69 paragraph. As before, the comparison is better, especially in terms of absolute magnitudes, for the [ \u00C2\u00B1 6 0 / T 6 0 / 0 2 ] s laminate in which the previous stacking sequence is reversed. The results shown in Fig. 4.17 for the [\u00C2\u00B130/ : F-30/902]s laminate were also obtained using the isotropic solution. For this lay-up the shape of the stress distribution resulting from the Finite element analysis is predicted well by the present solution, o z remains tensile all around the hole and is minimum at 0 \u00C2\u00B0 . Although the magnitudes of the present solution are consistently higher than that of the Finite element, it is not more than twice at any angle. For the [ 9 0 2 / \u00C2\u00B1 3 0 / T 3 0 ] s laminate, the present solution would predict the exact minor image of the [ \u00C2\u00B1 3 0 / T 3 0 / 9 0 2 ] s laminate stresses, whereas the solution by Rybicki and Schmueser, though of opposite sign, shows much less variation in magnitude as function of angle. Figures 4.18 and 4.19 show the results for the [ + 45/^45/902]s and [ \u00C2\u00B1 6 0 / + 60/ 9023 s laminates. Although of difTerent magnitudes, the variation of o around the hole predicted by the Finite element solution is essentially similar to that for the [ \u00C2\u00B1 3 0 / T 3 0 / 9 0 2 ] s laminate. This is also true of the present results, which compare reasonbly well with the finite element results. For the [ \u00C2\u00B1 4 5 / T 4 5 / 9 0 2 ] s laminate a good agreement was observed. At 0\u00C2\u00B0 , where the stresses are smallest, the value predicted by the present approach is only four times the numerical result and at 90\u00C2\u00B0, where they are largest, it is less than 1.5 times the latter. The Finite element solution would predict the interlaminar normal stress at 90\u00C2\u00B0 to be approximately 65% of the applied stress, whereas the present solution estimates this at 92%. For the [902^ 45/:P45]s laminate with the reverse stacking order, the difference between the results becomes larger as the Comparisons with Literature 70 angle increases. Although the same general shape is observed, the finite element result exhibits a much less variation in magnitude as function of angle. For the [ \u00C2\u00B1 6 0 / T 6 0 / 9 0 2 ] S laminate shown in Fig. 4.19 the difference between the results remains approximately constant throughout Although the finite element result would predict compressive oz at laminate midplane for angles less than 28\u00C2\u00B0, the present result indicates tensile oz for this region. However, compressive stresses are predicted for all angles, by both solutions, for the reverse stacking order in [ 9 0 2 / \u00C2\u00B1 6 0 / ^ 6 0 ] s laminate. While the present solution yields an exact mirror image of the result shown in Fig. 4.19, the finite element solution predicts a distribution that increases less rapidly than its counterpart In all twelve lay-ups given by Rybicki and Schmueser a good qualitative agreement was observed, demonstrating the possibility of evaluating interlaminar stresses around holes through equilibrium considerations. The method seems to have some success in predicting the approximate shape and sign of the stress distribution. The results reported by them are only midplane o z . No comparison could therefore be made with o z distribution at other interfaces or in the thickness direction. 4.1.3 Whitcomb (1981):-In an experimental and analytical study of fatigue damage in graphite/epoxy laminates Whitcomb has analysed interlaminar stress distribution around holes in two different stacking sequences; namely [45/90/-45/0] s and [90/\u00C2\u00B1 45/0] s . The elastic properties of the zero deg plies of these laminates are given in Table II (page 125). Using a conventional three-dimensional finite element analysis he has analysed the region around the hole and compared the delamination locations with the stress distributions. The analytical results reported in his work do Comparisons with Literature 71 not include a complete description of interlaminar stresses around the hole, but show the distribution of oz and T Q Z through the thickness at three angular locations. These results are compared with stresses calculated using the modified average stress method for the two stacking sequences in Figs. 4.20 and 4.21. The calculated values of a z through the thickness at different interfaces are shown in Fig. 4.20 for the [45/90/-45/0] s laminate. The results of Whitcomb's finite element analysis are shown by the solid curves in these diagrams. Comparison is good at angles 90\u00C2\u00B0 and 175\u00C2\u00B0 from the loading direction. The sign and relative magnitude of o z is predicted reasonably well at each interface, locating correctly the interfaces with maximum stress. At 120\u00C2\u00B0 however, the agreement is poor especially with regard to absolute magnitudes. The sign of o z at the second interface from the outer layer does not agree with that resulting from the finite element analysis, although the basic shapes of the distributions (through the thickness) roughly agree. A similar result was observed for T @ Z distribution. The agreement was better at 90\u00C2\u00B0 and 175\u00C2\u00B0 than at 120\u00C2\u00B0. At 90\u00C2\u00B0 , the high interlaminar shear stresses obtained by Whitcomb along the First and second interfaces are closely predicted by the present approach. At the third interface the stress changes sign and becomes negative, although the present result indicates only a slightly negative value, at 120\u00C2\u00B0 however, the signs are in poor agreement for the first two interfaces from outside. Nevertheless, the stresses calculated by the present method for these two interfaces are close to zero. The maximum shear stress at the third interface given by the finite element solution is predicted reasonably closely by the present calculations. Finally at 175\u00C2\u00B0, a good agreement was observed in the sign and relative magnitude of interlaminar shear stress through the thickness. Comparisons with Literature 72 For the [90/ \u00C2\u00B1 45/0] s laminate shown in Fig.4.21 a qualitatively good agreement was observed. For the angles 90\u00C2\u00B0 and 120\u00C2\u00B0 from the loading direction the present calculation overestimates the magnitude of o z near the midplane, but estimates accurately the stresses at every interface for 160\u00C2\u00B0. The signs and shapes of the distribution are predicted reasonably well for all three angles considered. The agreement between the computed results and the finite element solution is relatively good for the interlaminar shear stress TQZ distribution through the laminate thickness. Except for the two outermost interfaces at 120\u00C2\u00B0 from the loading direction, the signs are correctly predicted all throughout Even at these two interfaces the stresses calculated by the present method are nearly zero. The high interlaminar shear stresses predicted for the second interface at 90\u00C2\u00B0 and for the third interface at 120\u00C2\u00B0 are in good agreement, at 175\u00C2\u00B0 from the loading direction, the shear stresses become vanishingly small through the entire laminate thickness as predicted by both solutions. 4.1.4 Tang (1977):-In this work an extension of a boundary-layer theory, developed by Reiss (1961) for isotropic elastic plates, is used to obtain an analytical solution for the interlaminar stresses in laminated composites. The approach is based on a stress formulation. Results are given for a [0/90] s laminate made of boron/epoxy and for a [ \u00C2\u00B1 4 5 ] s laminate of graphite/epoxy. The elastic properties of the material used in the analysis are listed in Table II (page 125). The interlaminar normal stress o z is calculated at the midplane of the above laminate constructions, while the two shear stress components Comparisons with Literature 73 T Q Z and T R Z are derived at 0/90 and 45/-45 interfaces. It is not clear what the actual dimensions of the plate geometry used in the computations are, but a parallel study on the effect of t/a ratio on boundary-layer effects reported here assumes a range of values from 0.01 to 0.03 for t/a, the plate thickness-to-hole radius ratio. For the [0/90] s laminate containing a circular hole, the interlaminar stresses calculated by the modified average stress approximation are compared with those reported by Tang in Figs. 4.22 (a)-(c). The results of the present calculation seem to predict the same general shape of the distribution obtained by Tang for all stress components. For oz shown in Fig. 4.22 (a) the signs are predicted correctly. The maximum interlaminar normal stress is obtained almost at the same angular location as that given by Tang. A secondary peak in o z , analogous to that observed by Tang approximately 5\u00C2\u00B0 off the loading direction, is also obtained by the present calculation, but, at least 12\u00C2\u00B0 - 15\u00C2\u00B0 away from the loading direction. At 0\u00C2\u00B0 and 90\u00C2\u00B0 the absolute values of oz are almost equal, though at other angles lying in between they seem to differ by varying amounts. The results of the present calculation were particularly good with regard to the general shape of the interlaminar shear stress T Q Z distribution as shown in Fig. 4.22 (b). The sign and relative magnitude of T Q Z calculated by the present method is in good agreement with that given by Tang along the entire hole boundary. The absolute values of the present solution and the boundary-layer solution for the hole boundary are at a constant ratio of about 1:3. At 0 \u00C2\u00B0 , 24\u00C2\u00B0 and 90\u00C2\u00B0 from the loading direction T Q Z becomes zero, as predicted by both solutions. In fact, this is expected along the 6=0\u00C2\u00B0 and 90\u00C2\u00B0 for a [0/90] s k lay-up, since the inplane. ply shear stress r ^ is zero at these two locations and T Q Z is a direct product due to the matching of the inplane ply shear stresses at Comparisons with Literature 74 the free edge. The comparison of the interlaminar shear stress T N distributions of the [0/90] s laminate in Fig. 4.22 (c) shows a similarity in shape and magnitude. Within one-quater of the hole boundary the distribution of T RZ exhibits two maxima, one near 0\u00C2\u00B0 and the other one close to 90\u00C2\u00B0 . This results from the present approximate solution as well as Tang's boundary-layer solution, though the exact angular location of each maximum predicted by the two solutions differ by about 5 \u00C2\u00B0 . Further, the magnitudes of the two maxima resulting from these solutions are at the same ratio. The absolute values of T RZ are roughly equal around most of the boundary, although the solution by Tang predicts negative stresses in the range 20\u00C2\u00B0 to 45\u00C2\u00B0 . However, it is important to note that within this region T RZ is predicted to be less than 1% of the gross applied stress. It is also interesting to note that the magnitude of the maximum T R Z is approximately an order of magnitude less than the maximum T \u00C2\u00A7 Z shown in Fig. 4.22 (b). Thus, T RZ is found to be an insignificant interlaminar stress component for the [0/90] s laminate. As will be seen later, this is true for many other laminates of practical interest The comparison of interlaminar normal stress results for the [ + 45] s graphite/epoxy laminate shows poor agreement The results of the present calculation and the distribution of o z obtained by Tang are shown in Fig. 4.23 (a). Nevertheless, a good agreement was observed in the distribution of T Q Z as seen in Fig. 4.23 (b). Except for the difference in magnitude, the present solution agrees quite well with the boundary-layer solution given by Tang. The magnitudes are at a ratio of approximately 1:3 along the entire hole boundary. This is exactly the same ratio that was found to exist between the magnitudes of T Q Z in the [0/90] s laminate, (see Fig. 4.22 (b).) The stress becomes slightly negative in the Comparisons with Literature 75 neighbourhood of 6 = 4 0 \u00C2\u00B0 . Unlike the result for [0/90] s laminate, high values of T Q Z is observed at 8 = 0\u00C2\u00B0 and 90\u00C2\u00B0 for the [ \u00C2\u00B1 4 5 ] s configuration, since there are k high inplane ply shear stresses r ^ at these two locations. Figure 4.23 (c) presents the results of the interlaminar shear stress component r r z . The solution given by Tang is also reproduced here. A certain ambiguity exists in these results, t and therefore, a direct comparison with the results of the present analysis is not attempted. Instead, the results of the present work are shown separately in this figure. Here the stresses are negative in the region marked by the minus (-) sign. The overall magnitudes of the stresses seem to compare quite well. The results would agree well if the stresses obtained by Tang in the region indicated by the plus( + ) sign are in fact negative, and the loading direction is parallel to the horizontal axis. The direction of loading in the present work is parallel to the vertical axes of the plots. 4.2 EXPERIMENTAL COMPARISONS :-It is generally accepted that the presence of high interlaminar stresses at laminate free edges cause delamination along ply interfaces. The literature on straight free-edge problem (Foye and Baker, 1970; Whitney and Browning, 1972; Soni and Kim, 1986) clearly indicates the importance of either or both interlaminar normal and shear stresses in predicting delamination. But, the exact form of the correlation that exists between interlaminar stress components and delamination is still not known precisely. The necessary conditions or failure criteria for delamination initiation and propagation are not, however, clearly established due to number of reasons. Among them are the difficulties associated with experimental detection and quantifiable assesment of delamination, the use t An attempt to contact the author and clarify the results was unsuccessfid. Comparisons with Literature 76 of a large number of specimen and loading geometries -making it a formidable task to conduct complete three-dimensional analysis of inplane and interlaminar behaviour-, and the presence of a number of different interactive failure modes which may be operative simultaneously. Related studies on the hole problem are rare and limited because of the greater complexity associated with theoretical analysis. Delamination prediction based on such analysis is even more limited since any type of damage initiated at the hole boundary may change the original stress field significantly. Nevertheless, delamination initiated failure modes under uniaxial tension of composite plates with circular holes are studied by a number of authors. Daniel, Rowlands, and Whiteside (1974), for example, investigated the influence of ply stacking sequence on the strength of laminated plates with circular holes and attributed the differences in strength to the differences in the state of interlaminar stresses near the boundary. Stacking sequences associated with tensile interlaminar normal stresses or high interlaminar shear stresses calculated at the straight free-edge resulted in laminates weaker by 10 to 20 percent than the corresponding alternate stacking sequence. Whitcomb (1981) studied fatigue damage development around holes in graphite/epoxy laminates by examining fatigue loaded specimens for damage type and location using light microscopy, ultrasonic C-scans and X-ray radiography. Delamination and ply cracking were found to be the dominant types of fatigue damage. Comparison of observed delamination with finite element stress analysis indicated that both interlaminar normal and shear stresses must be considered to explain the observed delamination. In a similar study by Kress and Stinchcomb (1985) X-ray radiography and a deply technique were used to determine the distribution of damage in each ply around the hole in two quasi-isotropic grphite/epoxy laminates subjected to tension fatigue. Their observations on the locations of initial delaminations in these two laminates agreed with the interlaminar stress analysis by O'Brien and Raju (1984). Comparisons with Literature 77 In the following sections the experimental results mentioned above are compared with the stresses calculated using the present approach. The observed delamination damage is compared with the results of the present approach which considers delamination as the only damage mode. The presence of other interactive damage modes may, however, alter the stress distribution to the extent which makes such comparisons only qualitative. The altered stress distribution, accompanied by delamination and other damage growth, can possibly initiate delamination at new locations or change the direction of delamination propagation. 4.2.1 Whitcomb (1981):-Damage development around holes in two orthotropic and two quasi-isotropic laminates under both tension and compression fatigue was investigated by Whitcomb. Delamination and ply cracking were found to be the primary modes of damage that took place. C-scan records of typical delamination locations have been made for the specimens after 10 tension or compression fatigue cycles. Comparisons of the C-scan records for the two orthotropic laminates [ 0 / \u00C2\u00B1 4 5 / 0 ] s and [45/0/-45/0] s reveals that the difference in stacking sequence of these two laminates affected the delamination growth ( Fig. 4.24 ). It has been noted by Whitcomb that under tensile fatigue loading the [ 0 / \u00C2\u00B1 4 5 / 0 ] s specimen delaminated preferentially at 0\u00C2\u00B0 and 180 \u00C2\u00B0 from the loading direction, but the [45/0/-45/0] s specimen delaminated uniformly around the hole. For the [ 0 / \u00C2\u00B1 4 5 / 0 ] s specimen, the interlaminar normal stress calculated using the present approximate technique is found to be tensile for almost all interfaces above and below the hole, as shown in Fig.4.25. The curves which lie outside the base circle of this polar plot indicate (positive) tensile stresses, while those which fall inside the circle represent compressive o r In contrast the results of the present calculation for the Comparisons with Literature 78 [45/0/-45/0] s laminate (Fig. 4.26) predict tensile oz within a small angular region on either side of the hole and compressive oz in regions above and below the hole. Since the interlaminar shear stress T Q Z and T R Z distributions are essentially similar for the two stacking sequences at each interface, the difference in delamination observation can be attributed to the difference in o distribution. Thus, more delamination can be expected to occur along the loading direction, above and below the hole, for the [ 0 / \u00C2\u00B1 4 5 / 0 ] s laminate than for the [45/0/-45/0] s laminate. It has also been observed by Whitcomb that the sign of the loading also affected delamination growth. In particular, the [45/0/-45/0] s specimen delaminated much more extensively under compression than tension. This can be expected since the compressive a stresses (shown inside the base circle in Fig. 4.26) would become tensile under compressive fatigue loading. The highest tensile stresses would be obtained at an angle equivalent to that at which extensive delamination is observed in the C-scans. Specimens of quasi-isotropic laminates were sectioned and examined for delamination locations in the thickness direction. Micrographs of these sections are given by Whitcomb at angles 90\u00C2\u00B0 , 120\u00C2\u00B0 and 175\u00C2\u00B0 for the [45/90/-45/0] s laminate subjected to tension fatigue ( Fig 4.27 ), and at angles 90\u00C2\u00B0 , 120\u00C2\u00B0 and 160\u00C2\u00B0 for the [90 /\u00C2\u00B1 45/0] s laminate subjected to compression fatigue (Fig 4.28). For the [45/90/-45/0] s laminate at 90\u00C2\u00B0, delaminations were observed at 90/-45 interfaces where the interlaminar shear stress T Q Z calculated by the present approach is found to be maximum. The stresses shown are those calculated at the edge of the hole and normalized with respect to the absolute value of gross axial stress (see Fig. 4.20). T Q Z has the same maximum value at neighbouring 45/90 interface too, but the interlaminar normal stress calculated for this interface is found to be slightly lower. At 120\u00C2\u00B0, delaminations were observed along -45/0 interfaces, Comparisons with Literature 79 where T Q Z is found to be maximum and oz relatively high. However, no delamination was observed by Whitcomb at the edge of the hole at about 175\u00C2\u00B0 from the loading direction. The results of the present calculation for this angular location predict low interlaminar shear stresses and compressive oz through the entire laminate thickness. It is however interesting to note the delamination of -45/0 interfaces away from the edge in the corresponding micrograph. The form of the o z distribution along an interface, infact, causes the compressive interlaminar normal stresses calculated at this angle to change sign and become tensile away from the free edge. Also, the through thickness interlaminar shear stress T RZ distribution resulting from the present analysis is found to have its maximum at this interface between -45-deg and zero-deg plies. In the [90/\u00C2\u00B1 45/0] s laminate subjected to compression fatigue ( Fig 4.28 ) delamination was observed at 90\u00C2\u00B0 from the loading direction between the 45-deg plies. These delaminations can be associated with coincidental peaks in the shear stress T Q Z and high tensile normal stress oz calculated by the present approach and shown in Fig. 4.21. At 120\u00C2\u00B0, the location of the shear ( T Q Z ) stress peaks shifted to the adjoining interface between -45-deg and zero-deg plies causing delamination at that interface. Here the magnitude of the tensile interlaminar normal stress too remained high and near its maximum which occured at the mid-plane. At 160\u00C2\u00B0 from the loading direction delamination was observed at the midplane between the zero-deg plies. At this angle, the tensile interlaminar normal stress was found to be maximum at the mid-plane while the shear stresses remained low in magnitude through the entire thickness. The delamination locations obtained by section studies compare well with the stress distributions calculated by the present approach. Comparisons have been made at three angular locations for each quasi-isotropic laminate Comparisons with Literature 80 presented. Results of microscopic section studies are not presented by Whitcomb for the orthotropic laminates. However, the examination of C-scan records of these laminates show that the calculated interlaminar stress can still be related to the observed delamination. 4.2.2 Kress and Stinchcomb (1985):-In a study on the fatigue response of two quasi-isotropic graphite/epoxy laminates Kress and Stinchcomb investigated damage development around circular holes during cyclic tensile loading. Non-destructive inspection of damage using zinc iodide enhanced X-radiography provided information on the continuous damage process during the fatigue life. The study is of special interest here because in addition to the information on damage growth during fatigue life it provides information on damage initiation. Data on early stages of damage development have been obtained by Kress and Stinchcomb by sequential loading of test specimens to progressively higher loads. Zinc-iodide enhanced X-radiographs of the specimens made during each unloading and reloading process produced information on damage initiation. Matrix cracks were the first to appear at lower stress levels, before any delaminations were detected at slightly higher stress levels, in both laminate types. Kress and Stinchcomb report that in the [ 0 / 9 0 / \u00C2\u00B1 4 5 ] s laminate delaminations were first detected in the 90/45 interface after loading to 80% of its mean tensile strength. Radiographs of the zinc-iodide infiltrated hole region for this laminate made after 90% and 105% stress loadings t are reproduced in Fig. 4.29. Delamination at the hole boundary appears at an angle to the loading t The stress is expressed as a percentage of the mean tensile strength of the laminate which is determined independently using several test specimens. Comparisons with Literature 81 direction on four locations around the hole, symmetrical with respect to the horizontal and vertical axes. No delamination appears to have initiated on either side of the hole perpendicular to the loading direction, or, above or below the hole parallel to the loading direction. The stress solution for this laminate using the present approximate technique yields compressive interlaminar normal stresses within a small angular region on either side of the hole and tensile stresses around the rest of the hole. Figures 4.30 (a)-(c) show the stress solutions for this laminate obtained by the present method. High interlaminar shear T Q Z stresses are obtained at 0/90 and 90/45 interfaces at roughly the same angle on four locations around the hole where delaminations are observed. It is nearly zero on either side across the horizontal diameter and above and below the hole. The high interlaminar shear stresses at 90/45 interfaces are now supplemented by relatively high interlaminar normal stresses in the vicinity of the delaminated regions. The apparent delamination of the interfaces between 90-deg and 45-deg plies thus appears to be governed by high interlaminar normal and (T Q Z) shear stresses. The r r z component of interlaminar shear is almost uniformly distributed around the hole thus having little influence on the location of delamination initiation. Delamination initiation in the [45/90/-45/0] s laminate has been observed by Kress and Stinchcomb on radiographs made after loading the specimens to 60% of their mean tensile strength. They report that delaminations were first detected in the 45/90 and 90/-45 interfaces. The radiographs provided in the published work ( made after 90% and 130% stress loadings as shown in Fig. 4.31 ) clearly indicates severe delaminations on opposite sides of the hole, perpendicular to the loading direction, with no visible delamination on top or bottom of the hole. These observations agree with the results of the present stress calculations, shown in Figs. 4.32 (a)-(c). The interlaminar normal stress is tensile only within a small angular region perpendicular to the loading direction, and is Comparisons with Literature 82 largest at 45/90 and 90/-45 interfaces. The interlaminar shear T Q Z in Fig. 4.32 (b) is also high for these interfaces, but nearly zero for other interfaces at this loacation. The same is true of T RZ distribution shown in Fig. 4.32 (c). Thus good correlation between theoretical and experimental results are observed for this laminate. The absence of any visible delamination above and below the hole parallel to the loading direction can be seen as a result of having low interlaminar shear stresses and compressive interlaminar normal stresses in this region. The progressive damage development in the two laminate types under constant amplitude tension-tension fatigue is shown by Kress and Stinchcomb through a series of radiographs taken at different times in the loading history. Matrix cracking and delamination seemed to be the dominant modes of damage that occured. Delamination at early stages of fatigue life is very much similar to that observed in sequential static loading, conforming to the predictions based on the present stress calculations. Successive delamination, though appears to be in agreement with analytical results, is much more widespread and influenced by the matrix cracks. In order to determine the shape and size of the delaminated region on a given interface the authors have used the deply technique. The essential features of the delamination zone for each deplied layer determined for the two laminate types are shown in Figs.4.33 and 4.34. In spite of the possible stress alteration associated with these delaminations ( and ply cracking ), the damage on each interface can be compared qualitatively with the stresses shown in Figs. 4.30 and 4.32. The damage on 0/90, 90/45 interfaces of the [0 /90/\u00C2\u00B1 45] s laminate ( Fig. 4.33 ) appears to have initiated at an angle to the loading direction and propagated towards the upper and lower edges of the hole. This can Comparisons with Literature 83 be expected on the basis of high interlaminar shear and normal stresses found at an angle to the loading direction, and, relatively high normal and T RZ shear stresses which continue across the upper and lower edges of the hole. On the 45/-45 interface delaminations surround the hole completely perhaps due to the fact that compressive o z near 90\u00C2\u00B0 from the loading direction is offset by high interlaminar shear stresses in the region. Over the remaining upper and lower parts of the hole boundary o z is found to be highly tensile. The delaminated regions in the [45/90/-45/0] s laminate are generally smaller than in the above [0 /90 /\u00C2\u00B1 45] s laminate which can be attributed to largely compressive interlaminar normal stresses around the hole. The overall interlaminar shear stress distributions can be considered as equal for the two stacking sequences. Nevertheless, on 45/90, 90/-45 interfaces of the [45/90/-45/0] s laminate delaminations appear at diametrically opposite positions ( Fig. 4.34 ) slightly inclined to the horizontal axis. A careful examination of the interlaminar normal and T Q Z shear stress distributions of Figs. 4.32 (a) and (b) reveals that the maximum stresses occur at positions diametrically opposite to each other, aligned closely with the horizontal axis as the observed delaminations. T RZ shear stress too is high on either side at 90\u00C2\u00B0 from the loading direction, intensifying the influence of oz and T Q Z in the near region. Damage on -45/0 interface on the other hand appears at four locations around the hole. Although the interlaminar normal stress is compressive, each of the two shear stress components exhibits peak values almost at same locations around the hole. The observations on the location of initial delaminations in the two laminate types agree with the present stress analysis. Delamination growth at early stages in fatigue life also appear very much similar to that observed in sequential static loading. Despite the changes in stress distribution due to existing Comparisons with Literature 84 damage, interlaminar stresses of the present analysis correlate well with the general form of delamination observed at different interfaces. Comparisons with Literature 85 POINT STRESS METHOD Angle, degrees 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 1 I,, ,.< I I I l _ _ \" l/a = 0.00 X \" ^ ^ l/a = 0 * 0 5 ^ . /i/'*' / \ l/a = 0 . 2 0 \ N / / l/a = 1.00'N^ / Fig. 4.1. Effect of l/a on oz ax i = h in a [90/0] s laminate. M E T H O D = Point Stress. B O U N D A R Y L A Y E R WIDTH = laminate thickness. Comparisons with Literature 86 AVERAGE STRESS METHOD Angle, degrees 20.0 30.0 40.0 \u00E2\u0080\u0094 \u00E2\u0080\u0094 J 1 \u00E2\u0080\u0094 50.0 60.0 70.0 80.0 \u00E2\u0080\u0094 L _ 1 L I/a = 0.00 \ l/a = 0.05^\" \ \ l/o = 100 l/a = 0.20 / Fig. 4.2. Effect of l/a on oz at z = h in a [90/0] s laminate. M E T H O D = Average Stress. B O U N D A R Y L A Y E R WIDTH = laminate thickness. Comparisons with Literature 87 MODIFIED POINT STRESS METHOD Angle, degrees 10.0 20.0 30.0 40.0 50.0 60.0 70.0 \u00E2\u0080\u0094I t... \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2J ....I 1 1 y ^ - i - - - - -^ l/a = 0.05 j l /a = 0.00 \ l/a = 0.20 ^ _ \ 7 l /a = 1.00 v . / Fig. 4.3. Effect of l/a on oz at z = h in a [90/0] s laminate. M E T H O D = Modified Point Stress. B O U N D A R Y L A Y E R WIDTH = laminate thickness. Comparisons with Literature 88 MODIFIED AVERAGE STRESS METHOD Angle, degrees Fig. 4.4. Effect of l/a on o z at z = h in a [90/0] s laminate. M E T H O D = Modified Average stress. B O U N D A R Y L A Y E R WIDTH = laminate thickness. Comparisons with Literature 89 POINT STRESS METHOD Angle, degrees 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 \u00E2\u0080\u00A2 J L ^ I I I I I I _ J ^ l /a = 0.50 l /a = 0.20 l /a = 0.10 \ Fig. 4.5. Effect of l/a on oz at z = h in a [90/0] s laminate. M E T H O D = Point Stress. B O U N D A R Y L A Y E R WIDTH = Characteristic length. Comparisons with Literature 90 AVERAGE STRESS METHOD Angle, degrees 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 I 1 1 1 1 l/a = 0.50 * - \ ^ -l/a = 0.20 l/a = 0.10 \ Fig. 4.6. Effect of l/a on oz at i = h in a [90/0] s laminate. M E T H O D = Average Stress. B O U N D A R Y L A Y E R WIDTH - Characteristic length. Comparisons with Literature 91 MODIFIED POINT STRESS METHOD Angle, degrees l /a = 0.20 l/a = 0.10 \ Fig. 4.7. Effect of l/a on a z at z = h in a [90/0] s laminate. M E T H O D = Modified Point Stress. B O U N D A R Y L A Y E R WIDTH = Characteristic length. Comparisons with Literature 92 MODIFIED AVERAGE STRESS METHOD Angle, degrees 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 _ J I I L_ I I I , J , r l/a = 0.50 \u00E2\u0080\u0094 \u00E2\u0080\u0094 \u00E2\u0080\u0094 ^ l/a = 0.20 l/a = 0.10 \ Fig. 4.8. Effect of IIa on oz at z = h in a [90/0] s laminate. M E T H O D = Modified Average stress. B O U N D A R Y L A Y E R WIDTH = Characteristic length. Comparisons with Literature 93 Point Stress Average Stress Modified Point Stress Modified Average Stress Raju and Crews 1982 Fig. 4.9. Present results compared with numerical solution for [90/0] s laminate at i = h. Comparisons with Literature 95 Fig. 4.11. Present results of TQZ distribution at z = h in a [0/90] s laminate compared with the solution of Raju and Crews (1982). Comparisons with Literature % Fig. 4.12. Present results of TQZ distribution at z = h in a [90/0] s laminate compared with the solution of Raju and Crews (1982). Comparisons with Literature 97 Point Stress Average Stress Modified Point Stress Modified Average Stress Rybicki and Schmueser 1978 Fig. 4.13. Present results compared with numerical solution for a [02/\u00C2\u00B110/130]s laminate at i = 0. **** Fig. 4.14. Results of Modified Average Stress method compared with U 3 numerical solution for a [ 0 2 / \u00C2\u00B1 3 0 / T 3 0 ] s laminate at z = 0. ? az is normalized with respect to az(maxy vii* Lile-mure 99 \u00E2\u0080\u00A2 i \u00E2\u0080\u00A2 lfj'ri|Ml||\u00C2\u00BB|l III fO.f C.v,v~ X , -\u00E2\u0080\u00A2 fj j Comparisons with Literature 100 Fig. 4.16. Present results of interlaminar normal stress distribution at midpane in a [ 0 2 f i60 /3-60] s laminate compared with the solution of Rybicki and Schmueser (1977). Comparisons with Literature . 101 Fig. 4.17. Present results of interlaminar normal stress distribution at midpane in a [ \u00C2\u00B1 3 0 / : f 3 0 / 9 0 2 ] s laminate compared with the solution of Rybicki and Schmueser (1977). Comparisons with Literature 102 Fig. 4.18. Present results \u00E2\u0080\u00A2 of interlarninar \u00E2\u0080\u00A2 normal stress distribution at midpane in a [ \u00C2\u00B1 4 5 / ^ 4 5 / 9 0 2 l s laminate compared with the solution of Rybicki and Schmueser (1977). Comparisons with Literature - 103 Fig. 4.19. Present results of interlaminar normal stress distribution at midpane in a [ \u00C2\u00B1 6 0 / ^ 60/902] s laminate compared with the solution of Rybicki and Schmueser (1977). Comparisons with Literature 104 T82/\u00C2\u00B0g -0.6 -0.3 0.0 0.3 0.6-06 -0.3 0 0 0.3 0.6 90 1 1 V > 1 1 \u00C2\u00BB t 1 1 9 120 175* i J -- S r -y i 1 i f \u00E2\u0080\u0094 J 1 4 1 -0.6 -0.3 0.0 0.3 0.6-0.6 -0.3 00 0.3 0 6 Whitcomb 1981 Present Solution! Fig. 4.20. Present results compared with numerical solutions (Whitcomb, 1981) of interlaminar stress distributions across laminate thickness in [45/90/-45/0] s specimen, (in tension) Comparisons with Literature 105 \u00C2\u00B0z/\u00C2\u00B09 ' r6z/\u00C2\u00B0g \ -0.6 -0.3 0.0 0.3 0.6-0.6 -0.3 0.0 0.3 0.6 -0.6 -0.3 0.0 0.3 0.6-0.6 -0-3 0-0 0.3 0 6 Whitcomb 1981 Present Solution Fig. 4.21. Present results compared with numerical solutions (Whitcomb, 1981) of interlaminar stress . distributions across laminate thickness in [90/\u00C2\u00B1 45/0] s specimen, (in compression) Comparisons with Literature Fig. 4.22 (a). Present results compared with Tang's (1977) solution of interlaminar normal stress oz at z = 0 in a [0/90] s laminate. Comparisons with Literature 107 CO 6 H Fig. 4.22 (b). Present results compared with Tang's (1977) solution of interlaminar shear stress T g z a t z = h i n a [0/90] s laminate. Comparisons with Literature 108 Fig. 4.22 (c). Present results compared with Tang's (1977) solution of interlaminar shear stress T RZ at z = h in a [0/90] s laminate. Comparisons with Literature 109 Fig. 4.23 (a). Present results compared with Tang's (1977) solution of interlaminar normal stress oz at z = 0 in a [ \u00C2\u00B1 4 5 ] s laminate. Comparisons with Literature 110 Fig. 4.23 (b). Present results compared with Tang's (1977) solution of interlaminar shear stress T Q Z at z = h in a [ \u00C2\u00B1 4 5 ] s laminate. Comparisons with Literature Fig. 4.23 (c). Present results compared with Tang's (1977) solution of interlaminar shear stress r rz at z = h in a [ \u00C2\u00B1 4 5 ] s laminate. Comparisons with Literature 112 T E N S I O N C O M P R l I ON O R T H O T R O P I C ( 0/\u00C2\u00AB4>'0>, (45/0/-45/OI la) O U A S I - I S O T R O P I C I0/s45/0l (c) (45/0/ 4VGi (0 I Fig. 4.24. C-scan records of various notched laminates after 10 7 tensile or compressive fatigue cycles. (Whitcomb, 1981). Fig. 4.25. Present solution of o z distribution in a [ 0 / \u00C2\u00B1 4 5 / 0 ] s laminate. (Whitcomb, 1981). Comparisons with Literature 114 Fig. 4.26. Present solution of o z distribution in a [45/0/-45/0]s laminate. (Whitcomb, 1981). Comparisons with Literature 115 Fig. 4.27. Delamination location for [45/90/-45/0] s specimen subjected to tension fatigue. (Whitcomb, 1981). Fig. 4.28. Delamination location for [90/\u00C2\u00B1 45/0] s specimen subjected to compression fatigue. (Whitcomb, 1981). 4.29. Radiographs of damage in sequential loading to (a). 0.9 Kress and Stinchcomb (1985) a [0 /90 /\u00C2\u00B1 45] s laminate after ^ and (b). 1.05 o ^ . Comparisons with Literature 117 Fig. 4.30 (a). Interlaminar normal stress oz distribution in a [0 /90 /\u00C2\u00B145] laminate. (Kress and Stinchcomb, 1985). Comparisons with Literature 118 Fig. 4.30 (b). Interlaminar shear stress T Q Z distribution in a [0/90/\u00C2\u00B145] s laminate. (Kress and Stinchcomb, 1985). Comparisons with Literature 119 Fig. 4.30 (c). Interlaminar shear stress r r z distribution, in a [0 /90 /\u00C2\u00B1 45] s laminate. (Kress and Stinchcomb, 1985). Fig. 4.31. Radiographs of damage in a [45/90/-45/0] s laminate after sequential loading to (a). 0.9 a ^ and (b). 1.3 o^. Kress and Stinchcomb (1985) Comparisons with Literature 121 Fig. 4.32 (a). Interlaminar normal stress o z distribution in a [45/90/-45/0] s laminate. (Kress and Stinchcomb, 1985). Fig. 4.32 (b). Interlaminar shear stress T Q Z distribution in a [45/90/-45/0] s laminate. (Kress and Stinchcomb, 1985). Comparisons with Literature 123 Fig. 4.32 (c). Interlaminar shear stress r r z distribution in a [45/90/-45/0] s laminate. (Kress and Stinchcomb, 1985). Comparisons with Literature 124 + Fig. 4.33. Damage on 0/90, 90/45 and 45/-45 interfaces of a [0/90/\u00C2\u00B145] s laminate. Kress and Stinchcomb (1985) X ** Fig. 4.34. Damage on 45/90, 90/-45 and -45/0 interfaces of a [45/90/-45/0]s laminate. Kress and Stinchcomb (1985) 125 T A B L E II. In-plane ply calculations:-A U T H O R Raju and Crews (1982) Rybicki and Schmueser (1977) Whitcomb (1981) Tang (1977) elastic properties used by M A T E R I A L Graphite/epoxy Graphite/epoxy Graphite/epoxy Boron/epoxy Graphite/epoxy different authors in stress ELASTIC PROPERTIES Ell \u00E2\u0080\u0094 138.0 GPa E22 = 14.5 GPa G12 = 5.86 GPa v12 = 0.21 Ell = 151.60 GPa E22 11.00 GPa G12 = 6.89 GPa v12 = 0.25 Ell = 140.0 GPa -E22 = 14.0 GPa G12 5.9 GPa V12 - 0.21 Ell - 211.68 GPa E22 = 19.93 GPa G12 - 4.99 GPa v12 - 0.21 Ell 137.90 GPa E22 : 14.48 GPa G12 - 5.86 GPa V12 = 0.21 126 C H A P T E R V E X P E R I M E N T A L OBSERVATIONS 5.1 INTRODUCTION:-An experimental programme was undertaken to prepare a series of composite laminates and examine them for delamination damage around circular holes under quasi-static loading. The main objectives were to (i) ascertain the nature of damage induced at hole boundaries by static loading (ii) correlate delamination damage with interlaminar stresses calculated using the approximate methods described in Chapter III. Composite laminates of differing ply orientation and stacking sequence, to include angle ply, cross ply and quasi-isotropic laminates, were used in the investigation; Al l laminates were constructed at U.B.C. except for the quasi-isotropic XAS/914 graphite/epoxy laminates (Poursartip, 1984). The laminates were layed up using either the 305 mm (12-in) wide Magnamite AS/3501-6 graphite/epoxy prepreg tapes or the same width 3M Scotchply-type 1003 glass/epoxy prepreg tapes to form panels of dimension 280- by 190-mm (11 by 7.5 in). The subsequent tests were carried out almost entirely on laminates made out of graphite/epoxy material system. Due to experimental difficulties associated with non-destructive detection of delamination damage, the use of Scothply glass/epoxy material system was later abandoned. Typical material properties of Experimental Observations 127 the prepreg tapes and the manufacturer's recommended cure cycles are found in Radford (1982). The in-plane ply elastic properties, used to calculate the interlaminar stresses at hole boundaries, are listed in Table III (page 214) for each material system. The angle ply laminates investigated are the 12-ply [Ojl^Ojl \u00C2\u00B1 30] s and [ \u00C2\u00B1 3 0 / 9 0 2 / 0 2 ] s laminates and 8-ply [ 0 2 / \u00C2\u00B1 4 5 ] s and [ \u00C2\u00B1 4 5 / 0 2 ] s laminates of AS/3501-6 graphite/epoxy material. In addition, a glass/epoxy laminate of [ 0 2 / \u00C2\u00B1 3 0 ] s construction was also included in the investigation. The crossply and quasi-isotropic layups investigated are respectively the 4-ply [0/90] s and [90/0] s AS/3501-6 graphite/epoxy laminates and 8-ply [45/0/-45/90] s construction of XAS/914 graphite/epoxy system. The experimental procedure described in Sec. 5.2 briefly outlines the fabrication of laminates from prepreg material and the preparation of test specimens from these panels. It also provides a description of the test programme employed. The results are reported and analysed in Sec. 5.3 for each individual laminate separately. The experimental observations are compared with analytical results to determine the validity of the present approximate technique. 5.2 EXPERIMENTAL P R O C E D U R E : -Uni-directional prepreg tapes available in roll form were used to prepare the laminates described above. A standard autoclave was used to cure the prepregged material. After the completion of the cure cycle the completed laminates were removed from the autoclave. In order to avoid any edge effects due to higher resin content, at least half an inch wide strips were cut off from all four sides of the plate. Later analysis showed a high void content in most of the laminates. This is considered to be due to moisture absorption by the prepreg material during storage. However, since we are interested only in damage initiation and not in the failure, these laminates were Experimental Observations 128 considered to be satisfactory. 5.2.1 Specimen Preparation :-Specimens were cut from the composite plates using a diamond cutting wheeel. Except for the XAS/914 graphite/epoxy specimens, all others were cut to a nominal width of 50 mm and a length of 280 mm. A central circular hole of 12.7 mm (0.5 in) diameter was drilled in each specimen using a high speed diamond drill. When drilling the holes, the specimens were clamped tightly between two glass sheets to prevent any delamination at hole edges. The drill travel through the thickness was also set at a very low speed. Water coolant was used to prevent any overheating of the drill bit. The holes thus produced were of good quality, possessing damage free smooth edges. This was later confirmed by zinc iodide enhanced X-ray radiography prior to loading. The straight free edges on either side of the specimen width were smootheried by mechanical polishing on wheels upto 400 grit-Specimens which were examined during the early part of the experimental programme were tabbed at the ends to prevent any possible grip failure. This was later considered to be unnecessary, because most of the specimens were loaded only up to a fraction of the failure load. Even those which were loaded up to failure had damage confined mostly to the central hole region. Experimental Observations 129 5.2.2 Testing and Observations :-After checking for any initial delamination introduced by the machining process, the specimens were tested quasi-statically for damage development around the holes. Using a pair of 2-in wide tensile friction grips the specimens were loaded in an lnstron testing machine. The specimens without tabs were mounted using medium grade sand paper held between the grips. The free span between the grips was approximately 170 mm (6.75 in) for the 50 mm wide specimens. A cross head speed of 0.5 mm/min (0.02 in/min) was used in both loading and unloading. The specimens were loaded sequentially to higher loads until delamination was introduced at hole boundaries. After loading a specimen to a particular load value it was immediately unloaded to half that value. While keeping the load on the specimen constant at this value, the hole was covered with scotch tapes on both sides, and X-ray opaque zinc iodide solution was injected into the cavity formed by the tape and the circular free edge. The specimen was left under load for at least half an hour allowing enough time for zinc iodide to penetrate into cracks and delaminations. It was then removed from the machine and cleaned thoroughly of residual zinc iodide on the surface, before making the radiographs. The X-ray radiographs of the central hole region were made by exposing the specimens to an X-ray beam, for a length of time depending on the specimen thickness. For a typical 8-ply graphite/epoxy laminate this was about 45 seconds at 30 kV and 10 mA X-ray current A Polaroid type 55 positive-negative film was placed behind the specimen in the path of the X-ray beam. Once the radiography was completed, the specimen was reloaded in the Experimental Observations 130 Instron to a higher load, and the above process repeated. At selected times in the loading sequence, the specimens were also sectioned at the hole in order to determine on which interfaces the delaminations and matrix cracking appeared. A slow diamond cutter was used to cut sections along radial planes around the hole, at different angles to the loading direction. A typical set of cuts is shown in Fig. 5.1. These sections were then polished to a finish of 6 microns for microscopic examination, and later to 1 micron finish for potographic and replication work. Replication of the sections was found to be of great use in further establishing microscopic evidence of delamination and ply cracking. The replicas were prepared by making surface impressions of the polished sections on .125 mm (.005 in) thick pieces of cellulose acetate sheets, moistened with acetone. After applying a mild pressure on the sample and allowing sufficient time for hardening, the acetate sheet was gently peeled off from the replicating surface. When viewed under the microscope it displayed a negative form of the actual surface, manifesting delaminations in the form of thin walls of acetate. Attempts to replicate the circular free edge, while the specimen was still under load, met with little success. One reason for employing a wide specimen geometry was to have a hole big enough for easy access in replicating the circular edge. Yet, a successful replication technique could not be developed due to difficulties involved in polishing the curved surface to an acceptable fine finish, wetting and making an imprint of the surface on cellulose acetate quickly and evenly. Also, the hardened acetate sheet could not be straightened out for subsequent microscopic work, without inducing much damage to the imprint Experimental Observations 131 5.3 RESULTS:-The results of the experimental programme are presented separately for laminates with different ply orientations. The results for laminates with the same ply orientations but different stacking sequences are discussed together. The results of the angle ply laminates are presented first, following which those of the crossply and quasi-isotropic laminates are given. In each case, comparisons have been made with the corresponding stress calculations. Although the stresses calculated can only predict the onset of delamination, comparisons have been made between the extent of initial delamination and the stresses, even after significant damage growth. 5.3.1 [02 /902/\u00C2\u00B130] s and [\u00C2\u00B130 /90 2 / 02 J S laminates:-The radiographs of damage around the hole in a [O2/9O2/ \u00C2\u00B1 3 0 ] s laminate under sequential static loading are shown in Fig. 5.2. The first damage in the specimen appeared after loading to about 50 MPa. Two matrix cracks of the zero-deg plies, extending outward from the hole boundary, appeared about 25\u00C2\u00B0 from the loading direction above and below the hole. These cracks were on the same side of the hole with respect to the specimen vertical centre line. A third matrix crack of the zero-deg ply was also visible, on the other side of the hole about 55\u00C2\u00B0 from the loading direction. There were strong indications that these cracks resulted from a possible weakness in the zero-deg ply, inherited from a defective prepreg material or from poor laminate construction or specimen preparation. These cracks grew slowly but continuously as the stress level was increased. Experimental Observations 132 After repeated loading to higher stress levels, cracks emanating from the hole boundary appeared in the 90-deg plies. Several of these cracks were seen on each side of the hole in the radiograph made after loading to 280 MPa stress. At the same time, zero-deg ply cracks tangent to each side of the hole were also seen to emerge. They grew in length and became more visible as the load was increased, while 90-deg ply cracks increased in length and number spreading over a wider segment of the hole boundary on either side. Signs of delamination initiation were first observed in the radiograph made after loading to 370 MPa stress level. Dark regions representing delamination were visible in the radiograph at ah angle of at least 70\u00C2\u00B0 from the loading direction. With increasing load these regions extended in size, spreading over the hole boundary on either side, covering approximately the region between 50\u00C2\u00B0 and 130\u00C2\u00B0. The specimen was sectioned at the hole after loading to 470 MPa, when the gross damage around the hole was as shown in the final radiograph of Fig. 5.2. The distribution of interlaminar stresses around a hole for the [ O 2 / 9 O 2 / \u00C2\u00B1 3 0 ] s laminate are shown in Fig. 5.3. The stresses shown are those calculated for the particular specimen geometry resulting from an applied stress of 470 MPa and the residual thermal stresses, normalized with respect to the applied stress. High tensile interlaminar normal stresses are obtained around 50\u00C2\u00B0 and 130\u00C2\u00B0 from the loading direction, which become compressive at 90\u00C2\u00B0, on both sides of the hole. For many interfaces the interlaminar shear TQZ is found to be high around 70\u00C2\u00B0 and 110\u00C2\u00B0, while r r z is found to be high around 55\u00C2\u00B0, 90\u00C2\u00B0 and 125\u00C2\u00B0 from the loading direction. Thus high interlaminar stresses are obtained within the same angular region in which much of the damage and delamination took place. The compressive interlaminar normal stresses on opposite sides of the hole are perhaps counteracted by high TRZ and TQZ found between the 30-deg and -30-deg plies. Experimental Observations 133 The specimen was sectioned at angles of 0 \u00C2\u00B0 , 45\u00C2\u00B0, 70\u00C2\u00B0 . 90\u00C2\u00B0 , 110\u00C2\u00B0, 135\u00C2\u00B0 and 180\u00C2\u00B0 from the loading direction. Fig. 5.4 shows the microscopic sections and replicas for each angle. For the angles 0\u00C2\u00B0 and 180\u00C2\u00B0 these micrographs appeared very much similar to each other, with little or no delamination. This can be expected on the basis of low interlaminar normal stresses and negligibly small shear stresses found at these angles. (see Fig. 5.3.) Only those micrographs of the 0\u00C2\u00B0 are included in Fig. 5.4. At 45\u00C2\u00B0 from the loading direction delamination was observed at fourth and fifth interfaces from the surface, corresponding to 90/30 and 30/-30 interfaces respectively. Comparatively little delamination was seen at the second interface between the zero-deg and 90-deg plies. The calculated stresses at 45\u00C2\u00B0 show higher interlaminar normal stresses at fourth and fifth interfaces than at other interfaces, with the exception of the midplane. Although a is maximum the shear stress components are both zero at the midplane, while finite values of T \u00C2\u00A7 Z and T RZ are obtained at fourth and fifth interfaces. Thus delamination can be expected at 90/30 and 30/-30 interfaces prior to any other place on the basis of the present stress calculations. The traces of delamination observed at the second interface are also explained by the existence of maximum TQZ and r r z at this interface despite the presence of relatively low tensile o z . At 70\u00C2\u00B0, significant delamination was observed at the second interface between zero-deg and 90-deg plies, and a little at the fourth between 90- and 30-deg plies. The zero-deg ply cracking appeared previously in the radiographs can also be seen on these micrographs, close to the hole boundary on both sides of the laminate. The interlaminar shear stresses calculated at 70\u00C2\u00B0 are both maximum at the second interface from outside making it a favourable location for delamination. Note that the normal stresses calculated at this angle are nearly Experimental Observations 134 equal for all interfaces. At the fourth interface the shear stress components are both high, making it the next favourable location for delamination. Delaminations at 90\u00C2\u00B0 from the loading direction were observed at fourth and fifth interfaces from outside, and found to be connected through 30-deg ply cracking. The interlaminar normal stress distribution in Fig. 5.3 predicts compressive oz at 90\u00C2\u00B0 for the entire laminate, increasing in magnitude towards the laminate midplane. However, at this angle a high, non-zero TQZ is obtained at the fifth interface between the 30-deg and -30-deg plies. TQZ is zero for all other interfaces. The interlaminar shear r r z , on the other hand, has its maximum at the fourth interface, and shows high values at the fifth and third interfaces. The delamination at the second 0/90 interface at 110\u00C2\u00B0 from the loading direction can be related to interlaminar shear stresses, both of which show their maximum at this interface. The normal stress is again nearly equal for all interfaces. Slight delamination is observed at fourth and fifth interfaces corresponding to 90/30 and 30/-30 respectively. At 135\u00C2\u00B0, delamination at the fifth 30/-30 interface can be related to the peak in o z , and that at the second 0/90 to the peaks in TQZ and TRZ stresses. Slight delamination was also observed at the fourth interface where the interlaminar normal stress is nearly as high as that at the fifth interface. We now consider the [ \u00C2\u00B1 30/902 ^ 2^s laminate and study the damage development under sequential static loading. The minute amount of damage visible in the radiographs of early stages of loading in Fig. 5.5 is a result of inherent material defects or poor specimen construction. These include matrix cracking on either side of the hole about 50\u00C2\u00B0 from the loading direction and a zero-deg ply crack at 170\u00C2\u00B0 on the right side. Damage due to loading was first Experimental Observations 135 seen in the radiograph taken after 185 MPa of gross applied stress. These were 90-deg ply cracking on the right side of the hole, which also started on the left side at a higher stress level. At 275 MPa, zero-deg ply cracks appeared on each side tangent to the hole boundary. With increasing load they extended in length in either direction, while the 90-deg ply cracks increased in length as well as in number. The first delamination appeared in the radiograph taken after 365 MPa of applied stress. Delamination was observed on each side of the hole at 90\u00C2\u00B0 from the loading direction. Subsequent loading to higher stress levels caused the delaminated regions to spread over a wide area. The specimen was sectioned after reaching 550 MPa applied stress and after making the final radiograph of Fig. 5.5. The interlaminar stresses around the hole in [\u00C2\u00B130/902^2^% laminate due to an applied stress of 550 MPa and residual thermal stresses are shown in Fig. 5.6. High tensile normal stresses are obtained in regions centred around 90\u00C2\u00B0 from the loading direction. As in the case of [O2/9O2/\u00C2\u00B130] s laminate the interlaminar shear TQZ is found to be high for many interfaces around 70\u00C2\u00B0 and 110\u00C2\u00B0. Similarly, rrz is also found to be high around 55\u00C2\u00B0, 90\u00C2\u00B0 and 125\u00C2\u00B0 from the loading direction. Thus one significant difference between the two stacking sequences is that the [\u00C2\u00B130/902^2^% lay-up generates tensile oz on each side around 90\u00C2\u00B0 from the loading direction, whereas [O2/9O2/\u00C2\u00B130] s generates compressive o z in this region. This may have caused the delamination to initiate at 90\u00C2\u00B0 from the loading direction in the [ \u00C2\u00B1 30/902 ^ 2^s laminate. The distribution of shear stresses, though appears to be similar in general for both laminates, is different for any given interface. The specimen was sectioned at angles of 0 \u00C2\u00B0 , 45\u00C2\u00B0, 70\u00C2\u00B0, 90\u00C2\u00B0, 110\u00C2\u00B0, 135\u00C2\u00B0 and 180\u00C2\u00B0 as before, and the micrographs of polished sections and replicas at these angles are shown in Fig. 5.7. At 0 \u00C2\u00B0 , no delamination was Experimental Observations 136 apparent The interlaminar stresses were either compressive or negligibly small. At 45\u00C2\u00B0, delamination was observed at the first and fourth interfaces, and perhaps a little at the second. It is interesting to note that much of this delamination has taken place away from the free edge and that the interlaminar normal stresses at this angle were compressive for all interfaces, being nearly zero at the first and highest at the midplane. Delamination can therefore be expected to occur on interfaces closer to the outside surface if supplemented by shear stresses. The high interlaminar shear TQZ at the first 30/-30 interface and TRZ at the second -30/90 interface can thus be considered as the cause of observed delamination. On the other hand, both TQZ and T r z become maximum at the fourth interface between 90 and zero-deg plies causing it to delaminate, despite the presence of relatively high compressive o z at this location. At 70\u00C2\u00B0 from the loading direction, severe delamination was observed at the fourth interface from outside surface. Relatively little delamination was observed at the second between -30-deg and 90-deg plies. The zero-deg ply cracking observed in the radiographs can also be seen clearly on polished sections. The crack seen here runs through the four centre plies, close to the hole boundary, between the delaminated surfaces. Some 90-deg ply cracks are also visible in the photographs. Comparing these observations with the results of the present stress analysis (Fig. 5.6), good correlation can be found between the analytical and experimental results. While the distribution of o z is nearly zero throughout the laminate TQZ and r r z are both found to be very high and maximum at the fourth interface between 90 and zero-deg plies, causing it to delaminate more than any other interface. These two components of interlaminar shear were also high at the second interface making it the next possible location for delamination initiation. Experimental Observations 137 At 90\u00C2\u00B0 from the loading direction, significant delamination at the first interface and a little at the second were observed. Except for the first interface, TQZ is found to be zero for all others, r r z on the other hand was highest for the second interface and nearly half that for the first and third interfaces, with neglegibly small values everywhere else through the laminate thickness, o z was tensile all throughout, increasing progressively in magnitude from outer surfaces to midplane. Delamination at the first and second interfaces can thus be related to the presence of high interlaminar shear and tensile interlaminar normal stresses at these locations. The high o z derived for those interfaces near the midplane are not sufficiently aided by additional interlaminar shear stresses to cause any delamination near the midplane. As at 70\u00C2\u00B0 , severe delamination was observed at the fourth interface, 110\u00C2\u00B0 from the loading direction. The interlaminar shear stresses TQZ and T r z are both very high and maximum at this location. Although of relatively small magnitude a z is also slightly tensile at this interface. At 135\u00C2\u00B0, delamination away from the free-edge can be seen on the first and fourth interfaces. Here again, the shear stress components are both maximum at the fourth, and high at the first interfaces. However, the interlaminar normal stress at the hole boundary is compressive for all interfaces increasing in magnitude towards the laminate midplane. This may have prevented initiation of delamination at the free-edge, though away from the edge, delamination may have been assisted by o z which changes sign and become tensile in this region. At 180\u00C2\u00B0 from the loading direction no delamination was observed. This is in agreement with neglegibly small shear stresses and compressive o, obtained for all interfaces across the entire laminate thickness. Experimental Observations 138 The comparison between the experimental observations of delamination and analytical solutions of interlaminar stresses made so far, have all been descriptive. In order to present them in a direct, concise manner the results are tabulated in tables IV (page 215) and V (page 217) for the two laminates [ O 2 / 9 O 2 / \u00C2\u00B1 3 0 ] s and [\u00C2\u00B130 /9 f J2 /02 ] s respectively. These tables show the relative extent of delamination observed at different interfaces, at each angle of sectioning. The amount of delamination is estimated for a given interface, on a scale of 0 to 10, zero being no delamination and 10 being the complete separation of the plies which form that interface. For convenience, the interfaces are numbered beginning from the outmost, such that the interface nearest the outside surface is designated one (1) and the midplane of a 12-ply laminate six (6). The interlaminar stresses shown in the tables are the stresses at the free-edge, normalized with respect to the gross applied stress. The values given for TQZ and T are the absolute magnitudes of shear stresses, ignoring the sign, while for o z both positive and negative values are given indicating tensile and compressive normal stresses. The stresses calculated by the present approximate technique are those for the undamaged hole that exist at the free-edge. Strictly, these stresses can only be used to predict delamination initiation, since the original stress state may change with initiation and propagation of damage at the hole. Nevertheless, comparisons have been made after the initiation and propagation of finite amounts of delamination achieveing good agreement between the results. 5.3.2 [ 0 2 / \u00C2\u00B1 4 5 ] s and [ \u00C2\u00B1 4 5 / 0 2 ] s laminates:-The radiographs taken of the above laminates, before loading, indicated the existence of fabrication defects in every specimen. These included voids and ply cracking that appeared around the hole. A typical radiograph with such Experimental Observations. 139 defects is shown in Fig. 5.8 for a specimen of [ \u00C2\u00B1 45 / 02 ] s configuration. The voids in the specimens were visible in polished sections prepared after sequential loading. In spite of the defects observed in those laminates, tests were carried out to investigate the influence of interlaminar stresses on delamination at hole boundaries. Fig. 5.9 shows a series of radiographs of a [02/ i\"45] s specimen, made after repeated loading to increasingly higher in-plane stresses. A large number of cracks in the 45-deg plies appeared throughout the specimen, while only two zero-deg ply cracks were seen on each side tangent to the hole boundary. A significant amount of delamination was observed above and below the hole, in the region bounded by the zero-deg ply cracks. The extent of the delaminated region and the lengths of the zero-deg ply cracks were seen to grow rapidly with increasing load. In contrast, the radiographs of [ \u00C2\u00B1 4 5 / 0 2 ] s specimens exhibited much less damage around the hole, as shown in Fig. 5.10 for a specimen loaded to same stress levels. Though fewer in number, matrix cracks of the zero-deg and 45-deg plies still appeared in the radiographs. Delamination was also observed above and below the hole as before, within the same angular region bounded by two zero-deg ply cracks tangent to the hole. However, the gross amount of delamination (and ply cracking) observed in the [ \u00C2\u00B1 4 5 / 0 2 ] $ specimen at a given stress was much less than that observed in the [ 0 2 / \u00C2\u00B1 4 5 ] s specimen. This was so, even in spite of the fact that a given stress level represented a higher fraction of the mean failure stress foT [ \u00C2\u00B1 4 5 / 0 2 ] s laminate than for the [ 0 2 / \u00C2\u00B1 4 5 ] s laminate. The mean (far- field) failure stress for each laminate type was determined by monotonic tension tests of three specimens of each type. The maximum stress applied before making each radiograph is given in Figs. 5.9 and 5.10 as a percentage of the mean failure stress. Experimental Observations 140 In order to compare the above observations with interlaminar stresses, the stress distributions at the hole boundary obtained for each laminate type are shown in Figs. 5.11 and 5.12. Except within a narrow angular region on each side of the hole perpendicular to the loading direction, the interlaminar normal stresses are found to be tensile for the [ 0 2 / \u00C2\u00B1 4 5 ] s laminate and compressive for the [ \u00C2\u00B1 4 5 / 0 2 J S laminate. The magnitude of this normal stress at any interface around the hole is seen to increase with the distance from the surface to that interface. On the other hand, the shear stress distributions in the two laminates are found to be similar for any interface between similar plies. A close inspection of the shear stress distributions shown in Figs. 5.11 and 5.12 show that the distributions of both TQZ and rrz are essentially similar in view of the two-fold symmetry associated with specimen and loading geometry. Thus, the only difference between interlaminar stress distributions of the two laminate types is found to be that of a z , which is mostly tensile for [ 0 2 / i 4 5 ] s and compressive for [ \u00C2\u00B1 4 5 / 0 2 ] s configurations. The excessive delamination observed in [ 0 2 / \u00C2\u00B1 4 5 ] s laminate over the other, can therefore be attributed to this difference in o z at the hole boundary. Though delamination due to combined effects of interlaminar normal and shear stresses occurs in both laminates, the tensile normal stresses in [ 0 2 / \u00C2\u00B1 4 5 ] s laminate lead to extensive delamination at the hole boundary. Microscopic examination of the circular edge in delaminated specimens revealed that, in both laminates, much of the delamination took place at the second interface from the outside between zero-deg and +45-deg plies. Furthermore, in the [ 0 2 / \u00C2\u00B1 4 5 ] s specimen, narrow lips extending outward in the vertical direction from the hole boundary, consisting of surface zero-deg plies and bounded by the zero-deg ply cracks tangent to the hole, were seen. These lips formed above and below the hole when loaded to more than 80% of its mean failure stress. At failure, complete separation of these lips from the rest of the Experimental Observations 141 specimen occured at 0/45 interface near the hole region. These observations agree with the results of the present stress calculations. The interlaminar shear TQZ and TRZ are both maximum at the second interface between zero-deg and \u00C2\u00B1 45-deg plies, over most of the boundary in both stacking sequences. The appearence of fracture surfaces of the two laminate types are shown in Fig. 5.13. The long narrow lips of a fractured [ 0 2 / \u00C2\u00B1 4 5 ] s specimen are clearly visible in this figure, while failure across the hole through entire laminate thickness is seen for the [ \u00C2\u00B1 4 5 / 0 2 ] s specimen. Microscopic examination of polished sections revealed the existence of voids in the laminates. The presence of voids along ply interfaces made any comparison between through the thickness locations of delamination initiation and stress distributions difficult. This is evident from Fig. 5.14, which shows a series of replicas of polished sections prepared from a [ \u00C2\u00B1 4 5 / 0 2 ] s specimen, loaded to 265 MPa gross applied stress (42% of the mean failure stress). The sections are made at angles of 0 \u00C2\u00B0 , 60\u00C2\u00B0, and 90\u00C2\u00B0 from the loading direction. Although major delamination of the second interface from outside (between -45-deg and zero-deg plies) can be seen on 0\u00C2\u00B0 and 60\u00C2\u00B0 replicas, the presence of voids make it unclear which other interfaces suffer delamination. Delamination of the first interface from outside between + 45-deg plies is clearly seen near the free-edge at 60\u00C2\u00B0. This is probably due to high TQZ shear stress and non-compressive oz obtained at this angle as shown in Fig. 5.12. At 90\u00C2\u00B0 from the loading direction, damage is seen at almost every interface across the laminate thickness. It is clearly difficult to differentiate between any separation due to voids formed along ply interfaces and delamination due to high interlaminar stresses. Experimental Observations 142 5.3.3 [ 0 2 / \u00C2\u00B1 3 0 ] s laminate:-Laminates of [ 0 2 / \u00C2\u00B1 3 0 ] s configuration were prepared using Scotchply-type 1003 glass/epoxy material system. Non-destructive detection of damage development using zinc-iodide enhanced X-ray radiography was not feasible with this laminate, due to high absorbancy of X-rays by glass fibres. However, the transparency properties of this material aided in visual detection of damage at the hole boundary. A source of light placed behind the specimen illuminated undamaged regions of the specimen leaving damaged regions less illuminated. Delamination at the circular edge could therefore be seen as dark areas when viewed from the front Specimens were loaded sequentially until damage was introduced at the hole boundary. Matrix cracking of the zero-deg ply, tangent to the hole, was observed prior to any delamination. Delamination was first detected within one of the 60\u00C2\u00B0 - 80\u00C2\u00B0 Regions measured from the loading direction. Dark regions of delaminations were then seen to extend gradually in size with increasing load. Careful examination of the circular edge revealed delaminations at the edge as very fine cracks running between the plies. Lines of cracks associated with these delaminations could be seen between zero-deg and 30-deg plies, within the same angular regions mentioned above. Micrographs of sections taken at 0 \u00C2\u00B0 , 70\u00C2\u00B0 and 90\u00C2\u00B0 from the loading direction are shown in Fig. 5.15, for a specimen loaded to 233 MPa of gross applied stress. The interlaminar stresses at the hole boundary for this laminate, resulting from both an applied stress of 233 MPa and residual thermal stresses, are shown in Fig. 5.16. The values plotted in these figures are the stresses normalized with respect to the applied stress. The normal stresses shown in Fig. 5.16 a are Experimental Observations 143 found to be tensile at every interface around the hole, except within a narrow angular region on each side of the hole. The interlaminar shear T Q Z on the other hand is high at second and third interfaces in the neighbourhood of 70\u00C2\u00B0 and 90\u00C2\u00B0 respectively. However, the other component of interlaminar shear, T is not so high but largest at the second interface between zero-deg and 30-deg plies for most of the boundary. The importance of T RZ in predicting delamination at any given interface is very little, since the distribution of r rz is nearly uniform around the hole. At 0\u00C2\u00B0 from the loading direction no delamination was observed. Although oz is high at this location, T Q Z is zero throughout, except at the third interface from outside where it is still relatively small, r r z is maximum at the second and zero at the mid plane, with intermediate values at others. At 70\u00C2\u00B0 , delamination was observed at the second interface between zero-deg and 30-deg plies. This delamination and the zero-deg ply cracks tangent to the hole fully separated parts of zero-deg plies from the specimen segment near the free-edge. While oz is only slightiy tensile for every interface at this angle, T Q Z is very high at the second interface between zero-deg and 30-deg plies. At the first interface T Q Z is nearly half that at the second and is- zero at others. Since the interlaminar shear T RZ is also small at this angle, the observed delamination can be related to the presence of high T Q Z at the boundary. A similar result was obtained at 90\u00C2\u00B0 from the loading direction. Delamination of the third interface .between \u00C2\u00B130-deg plies was observed at this angle, where T Q Z of the third interface is near its maximum. The magnitude of T Q Z is significantly high at this interface but zero at every other interface. The observed delamination of Fig. 5.15 can thus be related to high T Q Z which may also have been supplemented by T at the boundary. The net influence Experimental Observations 144 of these stresses in predicting delamination may also have offset any negative effects due to compressive oz. It is also interesting to note the high density of ply cracks near the free-edge in this micrograph. In predicting delamination at the hole boundary in [ 0 2 / \u00C2\u00B1 3 0 ] s glass/epoxy laminates the distribution of TQZ was found to be more significant than those of oz and TRZ. This may partly be due to the fact that the peaks in TQZ are higher than those of oz and TRZ. 5.3.4 [0790] s and [90/0] s laminates:-The radiographs of damage around the hole due to sequential loading in 4-ply [0/90] s and [90/0] s laminates are shown in Figs. 5.17 and 5.18. They show damage after the same stress loadings in the two laminate types. No damage is visible at the begining in either specimen. The first zero-deg ply crack appears in the [90/0] s specimen, after loading to 165 MPa stress. In the [0/90] s specimen zero-deg ply cracks are visible in the radiograph taken after 250 MPa stress loading. Here, the cracks are seen on each side of the hole tangent to the boundary, and found to be associated with many 90-deg ply cracks. However, these 90-deg ply cracks are not visible in the corresponding micrograph of the [90/0] s specimen. Subsequent loading to higher stress levels increased the lengths of the zero-deg ply cracks in both laminates. Delamination at the hole boundary was first observed in the [0/90] s specimen. Delaminations were seen at four locations around the hole boundary, symmetrical with respect to the longitudinal and transverse axes. As seen in Fig. 5.17, their location on the boundary makes an angle of approximately 60\u00C2\u00B0 -65\u00C2\u00B0 with the loading direction. Subsequent delamination growth is found to occur along the zero-deg ply cracks. Experimental Observations 145 The observed delamination in the [90/0] s specimen, shown in Fig. 5.18, is also similar to that just described. The locations of delamination initiation at the hole boundary and its growth appear very much similar to that in the [0/90] s specimen. However, at any given stress level, the amount of delamination seen in the [90/0] s specimen is less than that observed in the [0/90] s specimen. The interlaminar stresses at the hole boundary due to an applied stress of 420 MPa are shown in Figs. 5.19 and 5.20 for the two laminates. The stresses shown are those for the 0/90 interface and the mid-plane normalized with respect to the maximum applied stress. The contribution of laminate residual thermal stresses are also included. The interlaminar normal stress, o z , is found to be tensile for the [0/90] s laminate, both at the first ply interface and. the mid-plane. The distributions of o z at these interfaces remain tensile with their maxima occuring at 70\u00C2\u00B0 from the loading direction. They are also symmetrical with respect to the longitudinal and transverse axes. In contrast, the distributions of oz in the [90/0] s laminate are found to be compressive through the laminate thickness, and also around the hole. However, the basic shapes of the distributions and their magnitudes remain same for both laminate types. The interlaminar shear stress distributions, on the other hand, are same for both laminate types as shown in Figs. 5.19 and 5.20. Both TQZ and T R Z are finite at the first interface between the zero-deg and 90-deg plies. Their distributions exhibit maxima at around 70\u00C2\u00B0 from the loading direction, in both laminate types. Since the distributions are also symmetrical with respect to the longitudinal and transverse axes, the above mentioned maxima occur at four locations around the hole. Experimental Observations 146 The delaminations observed in the two laminate types show good qualitative agreement with the stress distributions described above. Although the general appearance of delamination is similar in both laminates, less delamination is observed in the [90/0] s type. The interlaminar stress distributions obtained for the two laminates are also similar except for the compressive o z obtained for the [90/0] s laminate. In the [0/90] s laminate all three stress components exhibit maxima at around 70\u00C2\u00B0 from the loading direction. In the [90/0] s laminate the two shear stress components exhibit their peaks at this location, ln agreement with these results the delaminations observed are also seen at an angle of 60\u00C2\u00B0-65\u00C2\u00B0 from the loading direction in both laminates. As with the stress distribution, the delaminations on the hole boundary are also found to be symmetrically located at four positions around the hole. In both laminates delamination was observed at the first interface from outside between the zero-deg and and 90-deg plies. Fig. 5.21 shows a typical micrograph of a polished section taken at an angle of 65\u00C2\u00B0 from the loading direction. The observed delamination of the first ply interface, rather than the laminate mid-plane, must be due to the presence of interlaminar shear stresses at this interface. 5.3.5 [45/0/-45/90] s laminate:-Quasi-isotropic XAS/914 graphite/epoxy laminates were used to prepare tensile specimens of 18 mm width and 100 mm length. The hole drilled at the centre of each specimen was 6.5 mm (0.25 in.) in diameter. The specimens were loaded in an Instron testing machine using a pair of 1-in. wide tensile friction grips. Experimental Observations 147 The radiographs of damage introduced at the hole boundary of a specimen due to sequential static loading is shown in Fig. 5.22. No damage is visible in the radiograph taken after loading to about 115 MPa of gross stress. Matrix cracking of the 90-deg ply and \u00C2\u00B145-deg plies are visible after 140 MPa stress. These cracks increase in length and number with increasing applied stress. Matrix cracking of the zero-deg ply begins to appear in the radiographs taken after loading to 200 MPa stress. Delamination at the hole boundary first appears in the radiograph taken after loading 185 MPa gross applied stress. Dark regions of delamination appears at four locations around the hole, each making an angle of nearly 75\u00C2\u00B0 with the loading direction. These regions become more prominent and clearer in the subsequent radiographs. The final radiograph of Fig. 5.22 was taken after loading the specimen to 235 MPa stress. the distribution of interlaminar stresses calculated for this specimen geometry is shown in Fig. 5.23. The distribution of interlaminar normal and shear stresses are calculated for a maximum applied stress of 235 MPa. The normalized stresses also include the effects due to residual thermal stresses. The interlaminar normal stress, az, distribution of Fig. 5.23 remains tensile over a significant part of the hole boundary. High o stresses are obtained in the region around 90\u00C2\u00B0 from the loading axis. Also the magnitude of oz at different interfaces increases with the distance to that interface from outside. The interlaminar shear TQZ distribution of Fig. 5.23 predicts high stresses only for the first two interfaces from outside, that between 45-deg and zero-deg plies and zero-deg and -45-deg plies. As seen in this figure, the peaks in the distribution of TQZ at these interfaces are obtained in regions near 75\u00C2\u00B0 from the loading direction. Four of these peaks are found around the hole, in different quadrants. Experimental Observations 148 each making approximately the same angle with the loading axis. The interlaminar shear T N distribution on the other hand predicts high stresses for regions near 45\u00C2\u00B0 and 90\u00C2\u00B0 from the loading direction, T TZ distribution at the second interface reaches its maximum at 45\u00C2\u00B0 , while that at the first and third interfaces reach their maxima at 90\u00C2\u00B0. The interlaminar stress distribution discussed above is sufficient to explain the experimental observations of delamination. The locations of observed delamination coincide with the locations of stress peaks in TQZ distribution. The distributions of TQZ shown in Fig. 5.23 reach high magnitudes at either the first or the second interface near the regions where delamination is observed (Fig. 5.22). Also, the distribution of both oz and T RZ seem favourable for the particular location of delamination observed, o z is tensile throughout the laminate thickness at this angular location, while T RZ is significantly high for the first two interfaces at this location. For the third interface between the 45-deg and 90-deg plies however, oz and TRZ are both found to be even higher though TQZ is predicted to be relatively small. Thus, delamination is likely to occur even at the third interface in regions near 90\u00C2\u00B0 from the loading direction. Microscopic sectioning of a specimen loaded to nearly 235 MPa stress was done in order to compare the results in the thickness direction. Micrographs of sections taken at angles of 0 \u00C2\u00B0 , 45\u00C2\u00B0 and 90\u00C2\u00B0 are shown in Fig. 5.24. At 0 \u00C2\u00B0 , no delamination was observed. This can be expected on the basis of slightly compressive oz and relatively small T RZ obtained for different interfaces through the laminate thickness. Relatively high TQZ shear stresses are obtained only for the first two interfaces at this location. At 45\u00C2\u00B0, delamination was observed at the second interface between zero-deg and -45-deg plies. This is in agreement with the stress Experimental Observations 149 distributions shown in Fig. 5.23. oz calculated for this interface is tensile and has a relatively high magnitude. T Q Z too is high for this interface at 45\u00C2\u00B0 from the loading direction. In addition, the highest value of r r z at this angular location is also obtained for the second interface where delamination is observed. At 90\u00C2\u00B0, delamination was observed along the third interface between -45-deg and 90-deg plies. Here oz is very high and smaller to only that at the mid-plane. The interlaminar shear r r z too has the highest value at this interface. However, the T Q Z component of interlaminar shear is found to be zero for the third interface at this angle. 5.4 DISCUSSION:-The comparisons of experimental observations of delamination with interlaminar stresses calculated using the present approximate methods show good qualitative agreement. Tne locations of delamination around the hole and in the thickness direction compare well with the interlaminar stresses calculated for each laminate. The relative amounts of delamination observed at different interfaces and at different angles in a given laminate are also in agreement with the stresses calculated. The agreement between theoretical calculations and experimental results appear to justify the use of approximate methods of stress calculations in predicting delamination. Yet, a number of related problems remain unresolved. First, the stresses calculated by the present methods are those that exist at the free-edge of an undamaged hole. Strictly, these stresses can only be used to predict delamination initiation at the undamaged hole. In practice, delamination at the hole boundary is frequently preceded by matrix cracking or splitting, which alters the original stress distribution, strictly, the stresses calculated for the undamaged hole can not then be used to predict Experimental Observations 150 delamination initiation. At present, no solution is available which allows for changes in the stress distribution due to initial damage. Only those solutions which incorporate the effects due to damage development would provide the necessary mathematical background for predicting the propagation of delamination and other forms of damage. Even here, the approximate methods of stress calculations, such as that presented in this work may prove to be useful. Second, a reliable mixed-mode failure criterion is required in order to compare, quantitatively or semi-quantitatively the stress distributions with observed delamination. A method is required to combine all three interlaminar stress components into a single failure criterion. Since delamination occurs at the ply interface, where the strengths are mainly governed by the bonding matrix material, a criterion that predicts failure in an isotropic medium may be used. As a first approximation, the Von Mises yield criterion can be modified to generate a stress combination function. Using all six stress components, Von Mises criterion can be written as / 2 ^ o e - o r ) 2 + (or-oz)2 + (o-oe)2 + H T ^ + T ^ + T ^ 2 ) ] \" 2 = k (5.1) At a given interface near the hole boundary, the in-plane radial and shear stress' components (viz; or and T ^ ) become zero. The in-plane stress O Q , which is finite at the ply interface, can also be neglected since the reinforcing plies on either side carry a greater percentage of the gross laminate tangential stress. The matrix material at the interface near the hole boundary is assumed to have tangential stresses that are small compared to the interlaminar stresses. Thus by equating all in-plane stress components to zero equation (5.1) can be reduced as; [ oz 2 + 1( r r z 2 + r 6 z 2 ) ] 1 / 2 = k (5.2) Experimental Observations 151 Delamination is expected to occur when the stress combination function on the left hand side of equation (5.2) reaches a critical value. It predicts that the sign of interlaminar normal stress has no influence on delamination. In order to include the detrimental effect of tensile interlaminar normal stresses on delamination equation (5.2) can be modified as; [ az\ oz | + 3( rrz 2 + r6z 2 ) ] 1 / 2 = k (5.3) This predicts that tensile interlaminar normal stresses contribute towards delamination initiation while compressive stresses oppose it The onset of delamination due to tensile interlaminar normal stresses has been observed by many investigators, (e.g., Whitney and Browning, 1972; Pagano and Pipes, 1973) However, according to equation (5.3), the net influence of interlaminar shear stress components on delamination seems to be higher than that of the normal stress component The stresss combination function suggested here can be evaluated at different interfaces and angles around the hole. It is also possible to compare the semi-quantitative observations of the extent of delamination with this criterion. Such a comparison is shown in Figs. 5.25 and 5.26 for the [ O 2 / 9 O 2 / \u00C2\u00B1 3 0 ] s and [ \u00C2\u00B1 30/902 ^2^s laminates described in section 5.3.1. The observations of the extent of delamination (Table IV and V, pages 215-218) are compared with the stress combination function of equation 5.3, evaluated for different interfaces and angles around the hole. For ease of comparison, the same scale is used in both plots. Although there is considerable scatter, the extent of delamination is seen to increase with the value of the above stress function. There are many overlapping points on the horizontal stress axis with zero delamination, but most of these correspond to low values of the stress function. The points on the two plots also overlap very closely. Experimental Observations 152 The points on Figs. 5.25 and 5.26 have been obtained by comparing the delamination observed at every interface through the laminate thickness with the corresponding value of the stress combination function. Strictly, the interlaminar stresses calculated for the undamaged hole can only be used to predict the onset of the first delamination. Delamination at any interface at a given angle will influence the subsequent interlaminar stress distribution at that angle significantly. A more discriminating comparison is to plot the length of the longest delamination at each angle as a function of the predicted stresses for that interface at that angle. Fig. 5.27 shows such a plot of the maximum delamination in both the [ 0 2 / 9 0 ^ / \u00C2\u00B1 3 0 ] s and [ \u00C2\u00B1 30/902/02] s laminates as a function of the stress combination function. A linear correlation between the extent of delamination and the magnitude of the stress function is clearly observed. Considering the approximations involved in formulating the suggested stress function its agreement with experimental results is very pleasing. Though one would not necessarily predict a linear relation in Fig. 5.27, the observed trend of an increasing calculated stress combination function (which is the driving force for initiation of delamination) leading to a larger measured delamination (which is truly a propagation effect) is to be expected. Experimental Observations 153 Polished Surface Fig. 5.1. Sectioning of a specimen at the hole. Experimental Observations 154 (c) 280 MPa Fig. 5.2. Radiographs of damage in a [O2/9O2/\u00C2\u00B130]s laminate after sequential loading to (a) 50 MPa (b) 185 MPa (c) 280 MPa (d) 370 MPa (e) 470 MPa Fig. 5.3 (a), interlaminar normal stress a z distribution in a [ 0 2 / 9 0 2 / \u00C2\u00B1 3 0 ] s laminate. Fig. 5.3 (b). interlaminar shear stress TBZ distribution in a [ 0 2 / 9 0 2 / \u00C2\u00B1 3 0 ] s laminate. Experimental Observations 158 Fig. 5.3 (c). interlaminar shear stress [ 0 2 / 9 0 2 / \u00C2\u00B1 3 0 ] s laminate. rr distribution in a Fig. 5.4 (a). Micrograph and replica showing delamination in a [ 0 2 / 9 0 2 / \u00C2\u00B1 3 0 ] s laminate at 0 \u00C2\u00B0 . Experimental Observations 160 Fig. 5.4 (c). Micrograph and replica showing delamination in a [ 0 2 / 9 0 2 / \u00C2\u00B1 3 0 ] s laminate at 70\u00C2\u00B0. Fig. 5.4 (d). Micrograph and replica showing delamination in a [ 0 2 / 9 0 2 / \u00C2\u00B1 3 0 ] s laminate at 90\u00C2\u00B0. Experimental Observations 163 Experimental Observations 164 Experimental Observations 165 (c) 275 MPa Fig. 5.5. Radiographs of damage in a [ \u00C2\u00B1 30/902 / 0 2 ] s laminate after sequential loading to (a) 50 MPa (b) 185 MPa (c) 275 MPa (d) 365 MPa (e) 550 MPa Experimental Observations 166 (e) 550 MPa Experimental Observations 167 Fig. 5.6 (a). Interlaminar normal stress o z distribution in a [ \u00C2\u00B1 30/902 / 0 2 J S laminate. Experimental Observations 168 Fig. 5.6 (b). Interlaminar shear stress TQZ distribution in a [\u00C2\u00B130/902/02]s laminate. Experimental Observations 169 Fig. 5.6 (c). Interlaminar shear stress r r z distribution in a [\u00C2\u00B1 30/902 /023s laminate. Experimental Observations Fig. 5.7 (a). Micrograph and replica showing delamination in [ \u00C2\u00B1 3 0 / 9 0 2 / 0 2 ] s laminate at 0 \u00C2\u00B0 . Fig. 5.7 (b). Micrograph and replica showing delamination in a [ \u00C2\u00B1 30/902 / 0 2 ] s laminate at 45\u00C2\u00B0. Fig. 5.7 (c). Micrograph and replica showing [ \u00C2\u00B1 3 0 / 9 0 2 / 0 2 ] s laminate at 70\u00C2\u00B0 . delamination in a Fig. 5.7 (d). Micrograph and replica showing delamination in a [ \u00C2\u00B1 30/90 2 /02]s laminate at 90\u00C2\u00B0 . Experimental Observations 174 Fig. 5.7 (e). Micrograph and replica showing delamination in a [ \u00C2\u00B1 3 0 / 9 0 2 / 0 2 ] s laminate at 110\u00C2\u00B0. Experimental Observations 175 Fig. 5.7 (f). Micrograph and replica showing [ \u00C2\u00B1 3 0 / 9 0 2 / 0 2 ] s laminate at 135\u00C2\u00B0. delamination in a Fig. 5.7 (g). Micrograph and replica showing delamination in a [ + 30/902/0 2]s laminate at 180\u00C2\u00B0. Experimental Observations 177 Fig. 5.8. Radiographs taken before loading a [ \u00C2\u00B1 4 5 / 0 2 ] s laminate. Experimented Observations 178 (0 2/\u00C2\u00B145) s 178 MPa 25% of uts IX 45\u00C2\u00B0 p lamjnatj nitiation 311 MPa 44% of uts x splitti 0\u00C2\u00B0|Ny Fig. 5.9. Radiographs of damage in a [ 0 2 / \u00C2\u00B1 4 5 ] s laminate. (\u00C2\u00B145/02), Experimental Observations 179 178 MPa 27.8% of uts Fig. 5.10. Radiographs of damage in a [ \u00C2\u00B1 4 5 / 0 2 ] s laminate. Fig. 5.11 (a). Interlaminar normal stress oz distribution in a [02/i~45] s laminate. Experimental Observations 181 Fig. 5.11 (b). Interlaminar shear stress rQz distribution in a [ 0 2 / \u00C2\u00B1 4 5 ] s laminate. Fig. 5.11 (c). Interlaminar normal stress r rz distribution in a [ 0 2 / \u00C2\u00B1 4 5 ] s laminate. Experimental Observations 183 Fig. 5.12 (a). Interlaminar normal stress oz distribution in a [\u00C2\u00B145/02J\"s laminate. Experimental Observations 184 Fig. 5.12 (b). Interlaminar shear stress TQZ distribution in a [ \u00C2\u00B1 4 5 / 0 2 ] s laminate. Experimental Observations 185 Fig. 5.12 (c). Interlaminar shear stress T RZ distribution in a [ \u00C2\u00B1 4 5 / 0 2 ] s laminate. Experimental Observations J86 Fig. 5.13. Fracture surfaces of [ 0 2 / \u00C2\u00B1 4 5 ] s and [ \u00C2\u00B1 45/C>2]s laminates. Replicas of sections showing delamination at different angular locations in a [ \u00C2\u00B1 4 5 / 0 2 ] s laminate (a), at 0\u00C2\u00B0 (b). at 60\u00C2\u00B0 (c). at 9 0 \u00C2\u00B0 . Experimental Observations Fig. 5.15. Micrographs of sections showing delamination at different angular locations in a [ 0 2 / \u00C2\u00B1 3 0 ] s laminate (a), at 0\u00C2\u00B0 (b). at 70\u00C2\u00B0 (c). at 90\u00C2\u00B0 (c). at 90\u00C2\u00B0 Fig. 5.16 (a). Interlaminar normal stress az distribution in a [ 0 2 / \u00C2\u00B1 3 0 j s laminate. Experimental Observations 191 Fig. 5.16 (b). Interlaminar shear stress TQZ distribution in a [ 0 2 / \u00C2\u00B1 3 0 ] s laminate. Fig. 5.16 (c). Interlaminar shear stress T RZ distribution in a [fJ2/\u00C2\u00B130] s laminate. Experimental Observations 193 (c). 335 MPa. Fig. 5.17. Radiographs of damage in a [0/90] s laminate after sequential loading to (a).165 MPa. (b). 250 MPa. (c). 335 MPa. (d). 375 MPa. (e). 420 MPa. Experimental Observations 194 (d). 375 MPa. (e). 420 MPa. Experimental Observations 195 (c). 335 MPa. Fig. 5.18. Radiographs of damage in a [90/0] s laminate after sequential loading to (a).165 MPa. (b). 250 MPa. (c). 335 MPa. (d). 375 MPa. (e). 420 MPa. Experimental Observations 196 (e). 420 MPa. Experimental Observations 197 Fig. 5.19 (a). Interlaminar normal stress a z distribution in a [0/90] s laminate. Experimental Observations 198 Fig. 5.19 (b). Interlaminar shear stress TQZ distribution in a [0/90] s laminate. Experimental Observations 199 Fig. 5.19 (c). Interlaminar shear stress r r z distribution in a [0/90] s laminate. Experimental Observations 200 Fig. 5.20 (a). Interlaminar normal stress o z distribution in a [90/0] s laminate. Experimental Observations 201 to Fig. 5.20 (b). Interlaminar shear stress TQZ distribution in a [90/0] s laminate. Experimental Observations 202 Fig. 5.20 (c). Interlaminar shear stress T RZ distribution in a [90/0] s laminate. Fig. 5.21. Micrographs of sections taken at 65\u00C2\u00B0 from the loading direction in [0/90] s laminate. Experimental Observations (c) 185 MPa Fig. 5.22. Radiographs of damage in a [45/0/-45/90] s laminate after sequential loading to (a) 115 MPa (b) 140 MPa (c) 185 MPa (d) 210 MPa (e) 235 MPa. Experimental Observations 205 (d) 210 MPa (e) 235 MPa. Experimental Observations 206 Fig. 5.23 (a). Interlaminar normal stress a z distribution in a [45/0/-45/90] s laminate. Experimental Observations 207 Fig. 5.23 (b). Interlaminar shear stress T Q Z distribution in a [45/0/-45/90] s laminate. Experimental Observations 208 Fig. 5.23 (c). Interlaminar shear stress [45/0/- 45/90] s laminate. r distribution in a Fig. 5.24. Micrographs of sections showing delamination at different angular locations in a [45/0/-45/90] s laminate, (a), at 0 \u00C2\u00B0 . (b). at 4 5 \u00C2\u00B0 . (c). at 9 0 \u00C2\u00B0 . Experimental Observations (c). at 90\u00C2\u00B0 Experimental Observations 211 p o 10 \u00E2\u0080\u00A2 p fO \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 o o r \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 p d tpODTJ \u00E2\u0080\u00A2 0.0 - B \u00E2\u0080\u0094 ^ \u00E2\u0080\u0094 B \u00E2\u0080\u0094 B - T 50.0 100.0 150.0 200.0 Stress Combination Function MPa \u00E2\u0080\u0094 i 250.0 Fig. 5.25. Comparison of delamination in [ 0 2 / 9 0 ^ / \u00C2\u00B1 3 0 ] s laminate with stress combination function. Experimental Observations 212 o O - i \u00E2\u0080\u00A2 p CO c o \u00E2\u0080\u00A2\u00E2\u0080\u0094 D c \u00E2\u0080\u00A2 \u00E2\u0080\u0094 E a \u00C2\u00A9 q U3 c O E < q q c>. -S\u00E2\u0080\u0094B-\u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2o \u00E2\u0080\u00A2 C D T T 1 250.0 0.0 50.0 100.0 150.0 200.0 Stress Combination Function MPa Fig. 5.26. Comparison of delamination in [ \u00C2\u00B1 307902/O^Js laminate with stress combination function. Experimental Observations 213 p d-i p CO c o 1 9 E \" o