International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP) (12th : 2015)

The role of fiber volume fraction in tensile strength of fibrous composites Rypl, Rostislav; Vořechovský, Miroslav; Chudoba, Rostislav Jul 31, 2015

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12th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015The Role of Fiber Volume Fraction in Tensile Strength of FibrousCompositesRostislav RyplResearch assistant, Dept. of Mechanics, Czech Tech. Univ. in Prague, Czech RepublicMiroslav VorˇechovskýProfessor, Dept. of Struct. Mechanics, Brno Univ. of Technology, Brno, Czech RepublicRostislav ChudobaSenior researcher, Inst. of Structural Concrete, RWTH Aachen Univ., Aachen, GermanyABSTRACT: It has been proved experimentally that a finer crack pattern in brittle matrix compositeswith heterogeneous fibrous reinforcement increases the reinforcement efficiency in terms of fiber strengthby up to 100 %. In the present paper, we simulate this phenomenon by a semi-analytical model of acomposite crack bridge based on probabilistic fiber bundle models. The model is able to quantify thereinforcement efficiency increase with finer crack spacing given the information on the reinforcementheterogeneity. Possible sources of heterogeneity are variabilities in fiber diameter, modulus of elasticityor bond quality. With finer crack spacing, the heterogeneous stress state of the reinforcement is homog-enized which leads to a more efficient load bearing behavior. Since the crack spacing is (within thepractical range of values) a monotonic function of the fiber volume fraction and fiber diameter, thesequantities should be taken into account in structural analysis and design of composites with heteroge-neous reinforcement.1. INTRODUCTIONThe combination of brittle matrix (ceramic, ce-mentitious) with fibrous reinforcement provides thepossibility to design composites with tuned proper-ties, in particular with a favorable quasi-ductile be-havior and high load bearing capacity, see Phoenix(1993); Curtin (1993). The high tensile stiffnessand strength of micro-fibers that create the loadbearing component in brittle matrix composites canonly be utilized if cracks form in the matrix. Fibersthen stretch and transmit tensile load between thecrack planes providing the composite with highductility and stress redistribution capacity (Evansand Zok (1994)). The debonded lengths of fibers,and thus the compliance of a crack bridge, growwith the applied load and fiber radius, and decreaseswith the bond strength Cox (1952); Marshall andEvans (1985); Aveston et al. (1971).When loaded in tension, well designed brittle ma-trix composites exhibit multiple cracks developingin the matrix perpendicularly to the loading direc-tion over a range of applied stresses up to a state ofcrack saturation and ultimate failure Aveston et al.(1971); Hui et al. (1995); Curtin (1991). Duringthis process, which is accompanied by damage evo-lution and significant stress redistribution, fibersdebond in all crack bridges. Starting from the crackplanes, the debonding process advances until thedebonded zones meet between two adjacent matrixcracks. From that point on fibers behave like fixedbetween the cracks and the compliance of crackbridges upon further crack opening remains con-stant (if not affected by growing fiber damage). Thequalitative and quantitative characteristics of thewhole stress-strain response of composites dependon the mechanical, geometrical (size effect) andstatistical properties of the constituents and theirinterface Ibnabdeljalil and Curtin (1997); Phoenixand Raj (1992).The present modeling framework is based on a112th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015special class of mechanical models – probabilis-tic models. These models use probabilistic meth-ods for the evaluation of representative mechani-cal responses of composite materials Curtin (1993);Hui et al. (1995); Thouless and Evans (1988);Smith (1982) and provide a fully probabilistic out-put in terms of statistical distributions of the ana-lyzed measures (e.g. strength, stiffness, toughness).There are good reasons for the use of probabilisticmethods to model the mechanical behavior of com-posites: a) the random nature of fiber failure andfiber properties in general; b) the large number offibers (of the order of 104-108) in the composite.If one incorporated these features in deterministicmodels, computational limits would be exceededvery fast Chudoba et al. (2006).Even though the models referenced so far havecontributed important insights and are, in general,methodologically sound, the set of assumptionsthey use represents the material structure with ahigh level of idealization. All material and interfaceproperties are deemed to be perfectly homogeneouswith fiber strength as the only considered sourceof randomness and the fibers do not interact in anyway. Probabilistic models with these idealizationshave a great ability to predict qualitative tendencies,such as tough-to-brittle transitions Curtin (1993)and size effects Phoenix and Raj (1992). However,when reinforcement is far from being perfectly ho-mogeneous, e.g. due to variability in interface qual-ity, fiber diameter or fiber stiffness, the composite’sresponse changes dramatically, rendering the pre-dictions of common probabilistic models inaccurateRypl et al. (2013). Experimental observations re-garding textile reinforced concrete (TRC) yield insome aspects reversed tendencies than predicted byexisting models for composites with homogeneousreinforcement Weichold and Hojczyk (2009); Ryplet al. (2013).2. COMPOSITE CRACK BRIDGE MODEL2.1. Assumptions and notationA unidirectional composite with constant cross-sectional area containing fibers of volume fractionVf is considered. The fibers exhibit linear elastic be-havior with the modulus of elasticity Ef and brittlefailure upon reaching their breaking strain ξ . Thefiber cross-section is assumed circular with radiusr. Elastic deformations of the matrix are neglectedso that it is assumed to be rigid. This is justifiedfor cross-sections with much higher matrix stiff-ness compared to the stiffness of the reinforcementEm(1−Vf) EfVf, where Em denotes the matrixmodulus of elasticity. Matrix cracks in a compos-ite subjected to tensile load are assumed to be pla-nar and perpendicular to the loading direction. Anyresidual force carried by the matrix crack planes isneglected so that the force is transmitted solely bythe fibers. When the tensile load is increased, fibersdebond at the bond strength τ and slide against aconstant frictional stress τ at the fiber-matrix inter-face along the debonded length a (Fig. 2).Although detailed analyses of stress profiles withina fiber cross-section have been performed in thepast Nairn (1997); Xia et al. (2002), the stress con-centrations at the fiber perimeter close to the ma-trix crack plane are assumed to have a minor ef-fect (see also Curtin (1993)). Therefore, the funda-mental assumption of shear-lag models Cox (1952);Nairn (1997) of constant fiber stress over the cross-section can be anticipated. Nevertheless, the stressis variable for individual fibers due to the parame-ters which affect the fiber-matrix bond and whichare assumed to be of random nature. The mechan-ical idealization of the composite can thus be de-scribed as a parallel set of independent 1D springsrepresenting the fibers coupled to a rigid body rep-resenting the matrix through a (possibly random)frictional bond.2.2. Homogenized composite responseThe (quasi-static) matrix crack width w is chosenas control variable of the composite loaded in ten-sion because it enables a model formulation withrandom properties of fibers and fiber-matrix inter-face. Also, this way the composite response canbe tracked along the complete descending branch.Note that the far field composite stress taken as con-trol variable would result in an unstable (dynamic)damage process if monotonically increased beyondthe peak stress. The task of the present model isto evaluate the far field composite stress σc givena value of w. The composite stress σc,X is definedas the sum of (random) fiber forces ff,i(w,Xi), i ∈212th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Figure 1: Typical fiber crack bridge function (a) andmean composite crack bridge function (b).1,2 . . .nf transmitted by the nf fibers within a crackplane at a given nonnegative crack opening w yield-ing the total transmitted force, which is divided bythe composite cross-sectional area Acσc,X(w,X) =1Acnf∑i=1ff,i(w,Xi), w≥ 0. (1)Here, Xi is a sampling point from the X ∈ Rn sam-pling space of the n considered random variableswith the joint distribution function GX. Hence, thesampling points Xi are random n-dimensional vec-tors containing the fiber and bond properties. Theforce of a single fiber, ff,i(w,Xi), maps the vector Xion a nonnegative scalar – the fiber force – as a func-tion of the crack opening w. The σc,X(w,X) func-tion then sums the random fiber contributions and istherefore itself a random variable sharing the samesampling domain as the fibers X ∈Rn. One realiza-tion of the random variable σc,X(w,X) is thus thesum of randomly chosen samples (fiber forces) di-vided by Ac. These realizations have unique globalmaxima σ?c,X(X) in the w dimension at some non-negative crack opening w?. Such a maximum iscalled ’composite strength’.Assuming a large number of fibers, the term∑nfi=1 ff,i(w,Xi) in Eq. (1) can be approximated byexpected value stating thatnf∑i=1ff,i(w,Xi)≈ nfE[ ff,X(w,X)] (2)where ff,X(w,X) is the fiber force as a continuousfunction spanning the Rn+1 space (n random vari-ables + the crack opening w). The formula can beinterpreted as stating that the sum of random fiberforces is asymptotically equal to the mean fiberforce multiplied by the total number of fibers. Sim-ilarly, Ac can be for large nf substituted byAc ≈ nfE[Af]Vf, (3)where Af = pir2 is the single fiber cross-sectionalarea. It is assumed that for a nonnegative w thefibers exhibit linear elastic behavior, i.e.ff,X(w,X) = AfEf εf0,X(w,X) (4)with εf0,X(w,X)∈Rn+1 standing for the fiber strainat the matrix crack derived below. Then, the sub-stitution of Eqns. (2) and (3) into Eq. (1) is the ex-pected value of σc,X denoted as µσc,X and referredto as the ’mean composite crack bridge function’.With the dependencies on w and X omitted, it isderived asσc,X ≈ µσc,X =VfE[ ff,X]E[Af]=VfE[AfEf εf0,X]E[Af]= EfVfE[AfE[Af]εf0,X].(5)The fraction in the square brackets in Eq. (5) is de-fined as the dimensionless fiber cross-sectionνf(r) =AfE[Af]=r2E[r2], (6)312th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015so that the general form of the mean compositecrack bridge function readsE[σc,X(w,X)]= EfVf E[νf(r)εf0,X(w,X)], w≥ 0(7)which we will use the notion µσc,X(w) for in furthertext. The expectation in the formula (referred toas the ’mean composite crack bridge function’) isevaluated asµσc,X(w) = EfVf∫Xνf(r)εf0,X(w,X)gX(X)dX (8)with gX being the joint probability density functionof the random variables. Since the expectation op-erator E[·] maps the Rn sampling space of the ran-dom variables onto a scalar (the mean value), theresult, µσc,X(w) is defined in R – the dimension ofthe control variable w. The maximum of the meancomposite crack bridge function will be referred toas the ’mean composite strength’ and is defined asµ?σc,X = sup{µσc,X(w); w≥ 0}. (9)In order to evaluate Eqns. (7) and (9), the unknownfiber strain εf0,X(w,X), which shall be referred toas ’fiber crack bridge function’, has to be resolved.The formulation of εf0,X(w,X) is considered in thenext section.2.3. Fiber crack bridge functionIndividual fibers in a composite with rigid matrixare mechanically independent so that their straincan be defined regardless of the strain state ofneighboring fibers. When a matrix crack opens, thebridging fibers debond and transmit an amount offorce that is linearly proportional to their debondedlengths a and the bond strength τ at the fiber-matrixinterface. The debonded length is a function of therandom variables from the X sampling space andof the crack opening w, i.e. a = f (w,X). Followingdifferential equilibrium condition for the debondedfibers at the longitudinal distance z from the matrixcrack is statedEfε ′f,X(z)+Tz(z,X) = 0, (10)where εf,X(z) is introduced as the longitudinal fiberstrain, ε ′f,X(z) as its derivative with respect to z andFigure 2: Multi-scale modeling approach diagram.Tz(X,z) as the bond intensity. The function Tz(X,z)is defined as the interface shear flow 2τpir actingon the fiber cross-section pir2 within the debondedlength a (Fig. 2) with the corresponding sign de-pending on the longitudinal distance z from the ma-trix crackTz(z,X) = T (X) · sign(z) (11)withT =2pirτpir2 =2τr. (12)Since the sampling domain X and the distance vari-able z are included in Tz, Eq. (10) has the dimensionRn+1. The fiber strain derivative for the debondedrange of a fiber with respect to the longitudinal po-sition is derived from Eq. (10) asε ′f,X(z) =−Tz(z,X)Ef. (13)Analyzing the fiber strain εf,X along z, the maxi-mum is found at the crack plane z = 0 and with412th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015growing distance from the crack the function de-cays linearly with the slope −T/Ef until it reacheszero at z = ±a (Fig. 2a). An explicit expressionfor the fiber strain can be obtained by integratingEq. (13). For the complete z domain, the fiber strainthen yieldsεf,X(w,z,X)=Ta−Tz(z,X)zEf: |z|< a0 : |z|> a.(14)Note that these formulas involve the debondedlength a which is a function of w and X. The di-mension of εf,X is thus Rn+2 corresponding to X,z and w. The maximum fiber strain εf0,X(w,X) =εf,X(w,z = 0,X) then reduces the z dimension. Athis point, we refer to Rypl et al. (2013) for detailedderivation and give the fiber crack bridge functionin the final form asεf0,X(w,X) = ε intactf0,X (w,X)+ εbrokenf0,X (w,X). (15)The two parts of the fiber crack bridge function arethe contributions of the intact and broken fibers, re-spectively (see Fig. 1). The first term has the formε intactf0,X (w,X) = εf0,rτ(w,X) ·H(ξ − εf0,rτ(w,X))(16)whereεf0,rτ(w,X) =√TwEf(17)and H(·) denotes the Heaviside step function de-fined asH(x) ={0 : x < 01 : x≥ 0.(18)Broken fibers contribute with the strainεbrokenf0,X =ξm+1·H(εf0,rτ −ξ ), (19)with m being the Weibull modulus of the fiberstrength distribution, see Rypl et al. (2013).2.4. Strength of multiply cracked compositesWith an a priori known (or approximately pre-dicted) periodic crack spacing `CS, which is amonotonic function of the fiber volume fraction Vf,the composite crack bridge model can be adapted toreflect the periodic stress field of a multiply crackedcomposite. Once the debonded lengths reach thevalue `CS/2 (see Fig. 3b), fibers can be assumed asfixed to the matrix at the distance `CS/2 from thematrix crack. For these fibers, further debondingis not possible so that they only stretch elasticallywith the composite stiffness EfVf/`CS resulting in alinear response upon crack opening (see Fig. 3b).To include this constraint, the fiber crack bridgefunction for intact fibers with infinite strength hasto be modified to take on the linear formεMCf0,rτ(w) =w`CS+T `CS4Ef, for(a > `CS/2) (20)where the superscript MC denotes the multiplecracking state. The derivation of Eq. (20) isstraightforward and details can be found in Ryplet al. (2013).For the mean pullout length of broken fibers ina multiply cracked composite, the approxima-tion µ` ≈ aξ/2, which is derived and justified inPhoenix (1993), can be applied. The variable aξdenotes the debonded length of the fiber at the in-stant of its rupture. This assumption becomes accu-rate as the composite approaches its ultimate statewhere the matrix crack spacing can be assumed nar-row and the fiber strains high. The contribution ofbroken fibers to the fiber crack bridge function canbe written asεMC,brokenf0,X (w,X) =ξ2·H(εMCf0,rτ(w)−ξ ). (21)To remain consistent with the structure of the fibercrack bridge function (Eq. 15), the fiber crackbridge function for a multiply cracked compositeis written asεMCf0,X(w,X) = εMC,intactf0,X (w,X)+ εMC,brokenf0,X (w,X),(22)where εMC,intactf0,X is obtained asεMC,intactf0,X (w,X) = εMCf0,rτ(w) ·H(ξ − εMCf0,rτ(w)).(23)2.4.1. Conclusions and discussionExisting models predict a strength reduction ofmultiply cracked composites when compared to acomposite with a single crack Phoenix and Raj512th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Figure 3: Comparison of single fiber crack bridge with free debonding (single crack bridge) and boundary condi-tions (multiple cracking): (a) fiber strain profiles along z; (b) fiber crack bridge functions.(1992); Phoenix (1993); Curtin (1993). This can beexplained by the higher average fiber strain withinthe length ±a from a matrix crack as compared tothe case of a single matrix crack (Fig. 3a). Thissource of strength reduction can be implementedin the fiber breaking strain distribution Gξ , seePhoenix and Raj (1992); Rypl et al. (2013), and forcomposites with homogeneous reinforcement, it isthe only source of interaction of strength with crackdensity.However, experimental investigations of textile re-inforced concrete involving a single crack bridgeand multiple cracks show the opposite effect, seeRypl et al. (2013). The strengths of multiplycracked specimens were up to 1.7 times higher thanthe strengths of specimens with a single matrixcrack. Textile reinforced concrete is known for itspronounced heterogeneity of bond quality. There-fore, the strength increase for the multiple crackingstate observed with textile reinforced concrete spec-imens has to be connected with the reinforcementheterogeneity. In the following paragraphs, theeffect of boundary conditions (crack spacing) onthe strength of composites with heterogeneous re-inforcement is explained mathematically and phe-nomenologically.Using the presented model, a stress-homogenizingeffect of the periodic boundary conditions on fibersdue to multiple cracking can be observed. Themore uniform is the stress in the reinforcement,the higher load it can transmit – this is a generalprinciple of materials mechanics. The variance offiber strain in a crack bridge can thus be consideredas a measure of the crack bridge’s performance inthe sense that a high variance denotes low strength.For the two respective cases – single and multiplecracking – the variance operator, D[·], is applied onEqs. (17) and (20) (assuming randomness in T andomitting the effect of fiber rupture) as followsD[εf0,rτ ] = D[√TwEf]=wEfD[√T](24)andD[εMCf0,rτ ] = D[w`CS+T `CS4Ef]=`2CS16E2fD [T ] . (25)Analyzing these formulas, it is apparent that thevariability of strains in the single crack bridge casegrows linearly with w while the variability of fibersbridging cracks in a multiply cracked composite isindependent of w and decreases quadratically withdecreasing crack spacing `CS. When `CS → 0, thevariance completely vanishes and the strain in allfibers is uniform. Therefore, the growing crackdensity can be said to cause strain homogenizationin the fibers despite the scatter in the bond inten-sity T and therefore increases the overall compositestrength.612th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015Figure 4: Effect of boundary conditions on the mean composite strength.Comparing the strength of composite specimenswith a single crack (in the sense of a few isolatedcracks corresponding to a low Vf) and multiple in-teracting cracks, an unambiguous conclusion can-not be drawn. This is because the crack spacinginfluences the composite strength in two oppositeways. It reduces the strength because the aver-age fiber strain along the specimen grows but atthe same time, variability in fiber strains is reducedwhich increases the composite strength. Generally,it depends on the ratio of variability in T to the vari-ability in ξ which of the two effects of the crackspacing will take the upper hand. Will it be the ho-mogenizing effect, the multiply cracked specimenwill have higher strength than a single crack bridge.If the more severe stress state effect is stronger, themultiply cracked specimen will be weaker.The interaction of these effects is depicted in Fig. 4,which shows the ratio between the single and mul-tiple cracking strength (assuming 1 mm crack spac-ing). The bond strength and the fiber strength inthe study are assumed to follow the two parame-ter Weibull distribution with shape parameters mτand mξ , respectively. The variable µCBσc,X is themean composite crack bridge function of a singlecrack and µMCσc,X the mean composite crack bridge712th International Conference on Applications of Statistics and Probability in Civil Engineering, ICASP12Vancouver, Canada, July 12-15, 2015function in a multiply cracked composite. It isworth noting that for homogeneous reinforcement,the multiple cracking strength approaches the sin-gle crack bridge strength when the fiber breakingstrength is a deterministic value (mξ → ∞, bolddashed curves in Fig. 4). For the studied mate-rial, textile reinforced concrete, the scatter of bondstrength is very high and therefore the homogeniza-tion due to increasing fiber volume fraction whichincreases the crack density is likely do dominate.3. ACKNOWLEDGMENTThis publication was supported by the Euro-pean social fund within the framework of re-alizing the project „Support of inter-sectoralmobility and quality enhancement of researchteams at Czech Technical University in Prague“,CZ.1.07/2.3.00/30.0034 and by the Czech ScienceFoundation project No. 13-19416J. The support isgratefully acknowledged.4. REFERENCESAveston, J., Cooper, G., and Kelly, A. (1971). “Sin-gle and multiple fracture, the properties of fibre com-posites.” Proc. Conf. National Physical Laboratories,IPC Science and Technology Press Ltd. London, 15–24.Chudoba, R., Vorˇechovský, M., and Konrad, M. (2006).“Stochastic modeling of multifilament yarns. I. ran-dom properties within the cross-section and size ef-fect.” International Journal of Solids and Structures,43(3-4), 413–434.Cox, H. (1952). “The elasticity and strength of paperand other fibrous materials.” British Journal of Ap-plied Physics, 3, 72–79.Curtin, W. 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(1985). “Failure mechanismsin ceramic-fiber ceramic-matrix composites.” J. Am.Ceram. Soc., 68(5).Nairn, J. (1997). “On the use of shear-lag methods foranalysis of stress transfer in unidirectional compos-ites.” Mechanics of Materials, 26(2), 63–80.Phoenix, S. (1993). “Statistical issues in the fractureof brittle-matrix fibrous composites.” Composites Sci-ence and Technology, 48(1-4), 65–80.Phoenix, S. and Raj, R. (1992). “Scalings in frac-ture probabilities for a brittle matrix fiber composite.”Acta Metallurgica et Materialia, 40(11), 2813–2828.Rypl, R., Chudoba, R., Scholzen, A., and Vorˇechovský,M. (2013). “Brittle matrix composites with hetero-geneous reinforcement: Multi-scale model of a crackbridge with rigid matrix.” Composites Science andTechnology, 89, 98–109.Smith, R. (1982). “The asymptotic distribution of thestrength of a series-parallel system with equal load-sharing.” The Annals of Probability, 10(1), 137–171.Thouless, M. and Evans, A. 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