Chapter 2:Multiscale Modeling of Tensile Failure 47 d+2r 9---0 (a) (b) Fig.2.5.(a)3D model with ID fibers and load transfer calculation in shear-lag and Green's function models and(b)the nodes around ith fiber matrix yielding as well as evolving fiber damage in a single,compact framework.A schematic view of the composite with a hexagonal fiber array and relevant notation is shown in Fig.2.5.The SLM assumes that the local matrix shear stress is governed by the smaller of(1)the elastic shear stress associated with the neighboring fiber displacements ()=Glu,()-u(]/d, (2.1) where un is the displacement of the nth near-neighboring fiber to fiber i (see Fig.2.5)and d is the fiber spacing or (2)=y,where ty is the yield strength for an elastic/perfectly plastic matrix or the debonded interfacial shear stress for a sliding interface.Within this framework,force equilibrium on the ith fiber in a hexagonal array with an elastic/plastic matrix is,when discretized by a uniform mesh size A=&given by 3AE a(u,(3H)-4,(e》-a.-(u,(s)-4(e-i》 (2+a+a-)82 (2.2) +[产e,）e+,0-6小-0
Fig. 2.5. (a) 3D model with 1D fibers and load transfer calculation in shear-lag and Green’s function models and (b) the nodes around ith fiber matrix yielding as well as evolving fiber damage in a single, compact framework. A schematic view of the composite with a hexagonal fiber array and relevant notation is shown in Fig. 2.5. The SLM assumes that the local matrix shear stress is governed by the smaller of (1) the elastic shear stress associated with the neighboring fiber displacements n m ( ) [ ( ) ( )]/ , n i τ z = − G u z uz d (2.1) where un is the displacement of the nth near-neighboring fiber to fiber i (see Fig. 2.5) and d is the fiber spacing or (2) n y τ =τ , where τy is the yield strength for an elastic/perfectly plastic matrix or the debonded interfacial shear stress for a sliding interface. Within this framework, force equilibrium on the ith fiber in a hexagonal array with an elastic/plastic matrix is, when discretized by a uniform mesh size ∆z = δ, given by , 1 ,1 1 f 2 , ,1 6 m y 1 ( ( ) ( )) ( ( ) ( )) 3 (2 ) ( ( ) ( )) (1 ) 0, ij i j i j ij i j i j ij ij nj ij n n n a uz uz a uz uz AE a a G h u z uz b b d δ τ + −− − = ⎡ ⎤ −− − ⎢ ⎥ + + ⎣ ⎦ ⎡ ⎤ + − +− = ⎢ ⎥ ⎣ ⎦ ∑ (2.2) Chapter 2: Multiscale Modeling of Tensile Failure 47
48 Z.Xia and W.A.Curtin where u)is the displacement of the jth node of the ith fiber located at longitudinal position with h=/3 and A=m2.In (2.2),ai,are damage parameters:a=0 if the element of fiber i between and is broken,and a=1 if unbroken;similarly,a=0 if the element between and is broken,and aij=1 if unbroken.The b (n=1-6)in (2.2)are yield indicator parameters,with b=1 if l is less than ty and b=0 otherwise.Periodic boundary conditions are used on the lateral edges of the composite,so that all fibers have six neighbors.The boundary con- ditions for uniaxial loading are zero displacement at z=0,u(0)=0,and a constant applied displacement U at =L,u(L)=U.The stresses in the unbroken fiber elements follow from Hooke's law as: o(ej)=E[4(3)-4,(3-]16 (2.3) The SLM predicts the stress concentration and recovery around an arbitrary collection of broken fibers,but the assumptions in the model are not always appropriate.Comparison of the SLM predictions against full finite element modeling for the exact same problem shows that the standard SLM can accurately predict the stress recovery length along a broken fiber for a wide range of fiber/matrix stiffness ratios [37].How- ever,the SCFs are only accurate for systems with a high fiber/matrix stiff- ness ratio and high fiber volume fraction;practically,this corresponds to polymer matrix composites with high fiber fraction.For other material systems,in particular metal matrix composites,factors such as the neglect of shear across the finite fiber dimensions in the SLM,the matrix load- carrying capability,and/or the loading history,makes the SLM less accurate for the stress transfer [37].The stress transfer is more diffuse in the SLM than in the full FEM studies,making the SLM less conservative in predictions of local damage evolution.Thus,care must be taken in using the SLM to model composite deformation and failure,although applications to polymer composites with stiff fibers and high fiber volume fractions should be accurate and realistic. Green's function model The Green's functional model(GFM)[36]uses the in-plane SCFs around a single fiber break as obtained from any detailed numerical model as a Green's function,makes a simple approximation for the SCFs along the length of the unbroken fibers,and then computes the 3D damage evolution due to multiple,interacting fiber breaks. Specifically,the data for in-plane SCFs,as shown in Fig.2.4b,define a Green's function G for stress transfer from broken fiber j to surrounding
where ui(zj) is the displacement of the jth node of the ith fiber located at longitudinal position zj, with h = πr/3 and A = πr 2 . In (2.2), ai,j are damage parameters: , 1 0 i j a − = if the element of fiber i between j 1 z − and zj is broken, and , 1 1 i j a − = if unbroken; similarly, ai,j = 0 if the element between zj and zj+1 is broken, and ai,j = 1 if unbroken. The bn (n = 1–6) in (2.2) are yield indicator parameters, with bn = 1 if |τn| is less than τy and bn = 0 otherwise. Periodic boundary conditions are used on the lateral edges of the composite, so that all fibers have six neighbors. The boundary conditions for uniaxial loading are zero displacement at z = 0, ui(0) = 0, and a constant applied displacement U at z = L, ui(L) = U. The stresses in the unbroken fiber elements follow from Hooke’s law as: f 1 ( ) [ ( ) ( )]/ . ij ij ij σ z Euz uz δ = − − (2.3) The SLM predicts the stress concentration and recovery around an arbitrary collection of broken fibers, but the assumptions in the model are not always appropriate. Comparison of the SLM predictions against full finite element modeling for the exact same problem shows that the standard SLM can accurately predict the stress recovery length along a ever, the SCFs are only accurate for systems with a high fiber/matrix stiffness ratio and high fiber volume fraction; practically, this corresponds systems, in particular metal matrix composites, factors such as the neglect of shear across the finite fiber dimensions in the SLM, the matrix loadcarrying capability, and/or the loading history, makes the SLM less accurate predictions of local damage evolution. Thus, care must be taken in using the SLM to model composite deformation and failure, although applications to polymer composites with stiff fibers and high fiber volume fractions should be accurate and realistic. Green’s function model The Green’s functional model (GFM) [36] uses the in-plane SCFs around a single fiber break as obtained from any detailed numerical model as a Green’s function, makes a simple approximation for the SCFs along the length of the unbroken fibers, and then computes the 3D damage evolution due to multiple, interacting fiber breaks. Specifically, the data for in-plane SCFs, as shown in Fig. 2.4b, define a Green’s function Gij for stress transfer from broken fiber j to surrounding 48 Z. Xia and W.A. Curtin broken fiber for a wide range of fiber/matrix stiffness ratios [37]. Howthan in the full FEM studies, making the SLM less conservative in to polymer matrix composites with high fiber fraction. For other material for the stress transfer [37]. The stress transfer is more diffuse in the SLM
Chapter 2:Multiscale Modeling of Tensile Failure 49 fiber i.The stress distribution around a broken fiber is then modeled by the simple constantSLM.In other words,for fiber m broken at position the stress along the broken fiber is approximated as a"(=)=21-/r,a(=)sa(=) (2.4) as shown in Fig.2.4a,where o()is the axial stress in the fiber existing before the break,so that (2.4)is operative only within the slip length around the fiber break.The stress lost by the broken fiber at position z, Pm(=)=Goppm (=m)-2t=-zhIr Pnm(2)≥0m(a）(2.5) is transferred to the surrounding fibers using the Green's function computed in the plane of the break.With these two features,the total stress z)on unbroken fiber i in plane z due to broken fibers {m)is approxi- mated as ,()=Opp ()+[G(1-G)Gi ]p(), (2.6) where app)is the applied stress on fiber i at position z and there is an implied sum over the repeated indices k,I,m.Equations(2.4)-(2.6)predict that the stress transferred to surrounding fibers decreases linearly with distance from the fiber break until the slip region ends.This approximation is shown in Fig.2.4c,from which it is evident that the model captures the basic features of the deformation but misses the subtle details associated with bending and compatibility that arise in the full FEM and also in the SLM.By construction,however,the GFM always satisfies equilibrium of the axial load,i.e.,the sum of the forces over any cross-section of the fiber system is equal to the total force applied across the section.Equations(2.4) (2.6)are solved at a discrete set of points =along each fiber and,thus, provide the analog of the stresses emerging from the solution of(2.1)(2.3) in the SLM.Since the GFM takes the input directly from a more detailed calculation,it has a wider range of applicability than the SLM.However, for cases where the SLM is a good approximation,such as polymer matrix composites,the GFM contains some additional assumptions that could modify the predictions.A comparison of the GFM vs.the SLM,when the SCFs from the SLM are used as the input to the GFM,shows that the GFM predicts damage evolution and tensile strength in good agreement with the SLM for the systems considered [36],thus suggesting that the approxima- tions made in the GFM model are reasonable
fiber i. The stress distribution around a broken fiber is then modeled by the simple constant τ SLM. In other words, for fiber m broken at position b mz , the stress along the broken fiber is approximated as b bb () 2 /, () () m m mm σ τ σσ z =− ≤ zz r z z (2.4) as shown in Fig. 2.4a, where ( ) m σ z is the axial stress in the fiber existing before the break, so that (2.4) is operative only within the slip length around the fiber break. The stress lost by the broken fiber at position z, () () m m p z z ≥σ (2.5) is transferred to the surrounding fibers using the Green’s function computed in the plane of the break. With these two features, the total stress σi(z) on unbroken fiber i in plane z due to broken fibers {m} is approximated as 1 app, ( ) ( ) [ (1 ) ] ( ), i i ik kl lm m σ σ z z G GG pz − = +− (2.6) where σapp,i(z) is the applied stress on fiber i at position z and there is an implied sum over the repeated indices k, l, m. Equations (2.4)–(2.6) predict that the stress transferred to surrounding fibers decreases linearly with distance from the fiber break until the slip region ends. This approximation is shown in Fig. 2.4c, from which it is evident that the model captures the basic features of the deformation but misses the subtle details associated with bending and compatibility that arise in the full FEM and also in the SLM. By construction, however, the GFM always satisfies equilibrium of the axial load, i.e., the sum of the forces over any cross-section of the fiber system is equal to the total force applied across the section. Equations (2.4)– j provide the analog of the stresses emerging from the solution of (2.1)–(2.3) in the SLM. Since the GFM takes the input directly from a more detailed calculation, it has a wider range of applicability than the SLM. However, for cases where the SLM is a good approximation, such as polymer matrix composites, the GFM contains some additional assumptions that could modify the predictions. A comparison of the GFM vs. the SLM, when the SCFs from the SLM are used as the input to the GFM, shows that the GFM predicts damage evolution and tensile strength in good agreement with the , () ( ) 2 / b b pz z zz r m a = −− σ τ ppm m m Chapter 2: Multiscale Modeling of Tensile Failure 49 tions made in the GFM model are reasonable. (2.6) are solved at a discrete set of points z along each fiber and, thus, SLM for the systems considered [36], thus suggesting that the approxima-
50 Z.Xia and W.A.Curtin 2.2.4 Predictions of Tensile Strength in Small Samples The ultimate tensile strength of the composite is determined by two contributions.The first contribution is the fiber bundle strength o,which is determined via simulation of the evolution of fiber damage and stress transfer from broken to unbroken fibers using the shear-lag or Green's function method in a stochastic simulation model to be described below. The second contribution is the load-carrying capacity of the matrix.Since the fiber damage that drives ultimate failure is fairly localized in space,in both the longitudinal and transverse directions,most of the matrix is deform- ing as if in an undamaged composite.Thus,to a very good approximation, the average stress carried by the matrix is the axial stress in an undamaged composite at a stress equal to the composite strength.The ultimate strength can thus be expressed as Guts=Vioi+(1-Vi)om(Outs), (2.7) where om is the axial matrix stress and is a function of the applied stress The main goal is to compute the fiber bundle strength o. For any fixed state of damage,i.e.,spatial distribution of broken fibers, the SLM and the GFM compute the associated tensile stresses in all fibers in the system.Damage evolution then occurs by further failure of fibers due to the increasing stress concentrations.The progressive fiber damage occurs because the fibers have a statistical distribution of flaws within them,leading to a corresponding statistical distribution of strength on any set of fiber elements.Modeling of the damage evolution thus requires the appropriate fiber strength distribution as input.The cumulative probability of fiber failure P(,L)in a gauge length L at stress o is usually modeled as a Weibull distribution that accounts for the flaw-sensitive,weak-link nature of the brittle fiber failure.In a two-parameter Weibull model, P(o,L)is given by (2.8) where oo is a characteristic fiber strength for fibers of length Lo and m is the Weibull modulus describing the statistical spread in strengths.For most commercial fibers,the fiber strength properties are well characterized by the two-parameter Weibull strength model.The Weibull parameters oo and m are usually obtained from experiments in which a large number of fibers of length Lo are tested in tension prior to incorporation into the composite
2.2.4 Predictions of Tensile Strength in Small Samples The ultimate tensile strength of the composite is determined by two contributions. The first contribution is the fiber bundle strength * σ f , which is determined via simulation of the evolution of fiber damage and stress transfer from broken to unbroken fibers using the shear-lag or Green’s function method in a stochastic simulation model to be described below. The second contribution is the load-carrying capacity of the matrix. Since the fiber damage that drives ultimate failure is fairly localized in space, in both the longitudinal and transverse directions, most of the matrix is deforming as if in an undamaged composite. Thus, to a very good approximation, the average stress carried by the matrix is the axial stress in an undamaged composite at a stress equal to the composite strength. The ultimate strength can thus be expressed as * uts f f f m uts σ σ σσ = +− V V (1 ) ( ), (2.7) where σm is the axial matrix stress and is a function of the applied stress. The main goal is to compute the fiber bundle strength * σ f . For any fixed state of damage, i.e., spatial distribution of broken fibers, the SLM and the GFM compute the associated tensile stresses in all fibers in the system. Damage evolution then occurs by further failure of fibers due to the increasing stress concentrations. The progressive fiber damage occurs because the fibers have a statistical distribution of flaws within them, leading to a corresponding statistical distribution of strength on any set of fiber elements. Modeling of the damage evolution thus requires the appropriate fiber strength distribution as input. The cumulative probability of fiber failure f P L (,) σ in a gauge length L at stress σ is usually modeled as a Weibull distribution that accounts for the flaw-sensitive, weak-link nature of the brittle fiber failure. In a two-parameter Weibull model, f P (,) σ L is given by f 0 0 ( , ) 1 exp , m L P L L σ σ σ ⎡ ⎤ ⎛ ⎞ =− −⎢ ⎥ ⎜ ⎟ ⎢ ⎝ ⎠ ⎥ ⎣ ⎦ (2.8) where σ0 is a characteristic fiber strength for fibers of length L0 and m is the Weibull modulus describing the statistical spread in strengths. For most commercial fibers, the fiber strength properties are well characterized by the two-parameter Weibull strength model. The Weibull parameters σ0 and m are usually obtained from experiments in which a large number of fibers of length L0 are tested in tension prior to incorporation into the composite. 50 Z. Xia and W.A. Curtin
Chapter 2:Multiscale Modeling of Tensile Failure 51 However,composite processing can damage the fibers,modifying the in situ strength distribution compared to the initial ex situ distribution.To address this problem,fibers can sometimes be extracted from as-processed composites and then tested to obtain the appropriate strength parameters [11].Another approach is to examine the fracture mirrors on fibers protrud- ing from the fracture surface of a tested composite,from which the fiber strength statistics can be derived [7].In any case,simulations of composite tensile strength require accurate knowledge of the in situ fiber strength distribution. Within the constant r shear model for interface sliding,an analytic model that ignores local stress concentrations,the so-called Global Load Sharing model,permits for the identification of a characteristic stress that embodies most of the major dependencies of composite behavior on fiber and interfacial characteristics [6,7].This characteristic stress oe is the char- acteristic fiber strength at a characteristic length P()=0.632, and these interrelated quantities are given by [6] 1m+H) r (2.9) T In a simulation model,it is often convenient to normalize all lengths by & and stresses by using an appropriate value of r to define the length Even if r is approximate,(2.9)condenses some of the major physical dependencies of the composite failure into two key parameters. With the above preliminaries,the computation of the fiber bundle strength o,is straightforward.The simulation algorithm proceeds as illus- trated in Fig.2.6.A simulation model contains a computationally tractable number of fibers (typically ~1,000)each of length L>26..Each fiber is discretized into a series of small elements of length .as illustrated in Fig.2.5.Each fiber element is then assigned a tensile strength at random from the Weibull distribution,i.e.,a random number R in the interval [0,1]is selected;and the strength of the element is assigned to be (/)m(-In(1-R)).An initial tensile load is applied to the fiber bundle,and fiber breaks are introduced into those fiber elements for which the applied stress exceeds the assigned element strength.After these fibers break,the stress redistribution is calculated with the shear-lag or Green's function model.Under the redistributed stress,some fiber elements may then exceed their assigned strengths and are broken;and the stress redistribution is computed again.This fiber break and stress redistribution
However, composite processing can damage the fibers, modifying the in situ strength distribution compared to the initial ex situ distribution. To address this problem, fibers can sometimes be extracted from as-processed composites and then tested to obtain the appropriate strength parameters [11]. Another approach is to examine the fracture mirrors on fibers protruding from the fracture surface of a tested composite, from which the fiber strength statistics can be derived [7]. In any case, simulations of composite tensile strength require accurate knowledge of the in situ fiber strength distribution. Within the constant τ shear model for interface sliding, an analytic model that ignores local stress concentrations, the so-called Global Load Sharing model, permits for the identification of a characteristic stress that embodies most of the major dependencies of composite behavior on fiber and interfacial characteristics [6, 7]. This characteristic stress σc is the characteristic fiber strength at a characteristic length δc, f cc P ( , ) 0.632 σ δ = , 1/( 1) 0 0 c c c , . m m L r r σ τ σ σ δ τ + ⎛ ⎞ = = ⎜ ⎟ ⎝ ⎠ (2.9) In a simulation model, it is often convenient to normalize all lengths by δc and stresses by σc, using an appropriate value of τ to define the length δc. Even if τ is approximate, (2.9) condenses some of the major physical dependencies of the composite failure into two key parameters. With the above preliminaries, the computation of the fiber bundle strength * σ f is straightforward. The simulation algorithm proceeds as illustrated in Fig. 2.6. A simulation model contains a computationally tractable number of fibers (typically ∼1,000) each of length c L ≥2δ . Each fiber is discretized into a series of small elements of length c δ δ , as illustrated in Fig. 2.5. Each fiber element is then assigned a tensile strength at random 1/ 1/ c c ( / ) ( ln(1 )) m m σδ δ − − R . An initial tensile load is applied to the fiber bundle, and fiber breaks are introduced into those fiber elements for which the applied stress exceeds the assigned element strength. After these fibers break, the stress redistribution is calculated with the shear-lag or Green’s function model. Under the redistributed stress, some fiber elements may then exceed their assigned strengths and are broken; and the stress redistribution is computed again. This fiber break and stress redistribution 51 from the Weibull distribution, i.e., a random number R in the interval [0, 1] is selected; and the strength of the element is assigned to be Chapter 2: Multiscale Modeling of Tensile Failure and these interrelated quantities are given by [6]