CHAPTER 4.1: MODELING OF DAMAGE IN FIBER AND PARTICLE REINFORCED COMPOSITES. Wolfgang Lutz, Ming Dong, Ke Zhu, and Siegfried Schmauder

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1 325 CHAPTER 4.1: MODELING OF DAMAGE IN FIBER AND PARTICLE REINFORCED COMPOSITES Wolfgang Lutz, Ming Dong, Ke Zhu, and Siegfried Schmauder CONTENTS 1 Introduction Damage Phenomena in Short Fiber reinforced Composites Overview of Damage Modeling in Composites Modeling Self-Consistent Model (SCM) Combined Cell Model (CCM) Statistical Combined Cell Models Static Loading Conditions Quasi-Static Cyclic Loading Conditions Plastic-Damage Model Modifications of the Plastic- Damage Model Results and Application Metal Matrix Composites (MMC) Material Results: Self-Consistent Model Results: Combined Cell Model Results: Statistical Combined Cell Model Conclusions Polymer Matrix Composites (PMCs) Material Results: Combined Cell Model (CCM) Results: Statistical Combined Cell Model Conclusions Gypsum Fiber Composites Material Results: Plastic-Damage Model Conclusions Summary INTRODUCTION Composites such as Metal Matrix Composites (MMCs) and Polymer Matrix Composites (PMCs) are frequently reinforced with strong continuous or short fibers. In the case of short fiber reinforced MMCs and PMCs, random arrangements of fibers are observed. Their mechanical properties are highly dependent on their composition, on the matrix properties as well as on the type and volume fraction of reinforcements. The complexity of such affecting parameters makes a complete theoretical description of the behavior and the failure properties of reinforced composite with ductile and brittle matrix difficult. In this respect, a micromechanical analysis of the local composite failure process opens a possibility to predict the failure properties of these composites [1 3]. Micromechanical models shown in this Chapter can be applied to describe most technical relevant composites varying from simple inclusion type and interpenetrating microstructures to functionally graded materials. In this Chapter, models to simulate damage in fiber and particle reinforced composites with ductile (metal or polymer matrix) or brittle (gypsum) matrix are summarized on the basis of the works of Dong et al. [4 6], Zhu et al. [7, 8], Kabir et al. [9], Lutz et al. [10] and Rahman et al. [11]. They are the Self-consistent Model, the Combined Cell Model, the Statistical Combined Cell Model and the plastic-damage model. The embedded cell of the self-consistent model represents a composite where, instead of using fixed or symmetric boundary conditions around the fiber-matrix or particlematrix cell, the inclusion-matrix cell is embedded in

2 326 DAMAGE SIMULATION an equivalent composite material with the mechanical behavior to be determined iteratively in a selfconsistent manner. Using the Combined Cell Model in conjunction with the finite element method, the mechanical behavior of composites with a certain orientation of the fibers can be simulated numerically by averaging results from different 2D and 3D cell models representing a single fiber in three principal orthogonal planes in the composite. Applying an appropriate integration of the results of all fiber orientations, stress-strain curves in tension and compression of the global material can be simulated including the effects of residual stresses. As an advancement, the Statistical Combined Cell Model has been developed to consider fiber-cracks and fibermatrix debonding using a Weibull statistical approach and the rule of mixture. The parameters of the Weibull damage law have been determined using inverse modeling by comparing simulation and experiment. The Statistical Combined Cell Model was applied to static and cyclic loading conditions. The plastic-damage model includes stiffness degradation due to damage in the plasticity part by two independent scalar damage parameters, for tension and compression respectively. The effect of damage is introduced by replacing all stress definitions (true stress) by the reduced effective stress. Further on, the plastic-damage model is based on a stiffness recovery scheme to simulate the effect of micro-crack opening and closing. After introducing the models and the methods to simulate damage evolution, the presented models will be applied to different composites: MMCs, PMCs, and cellulose fiber reinforced gypsum composites. There, investigated materials will be explained. Moreover, an approach is presented to apply the Combined Cell Model to realistic microstructures of an injection molded PMC sample. 2. DAMAGE PHENOMENA IN SHORT FIBER REINFORCED COMPOSITES Failure of fiber or particle reinforced composites is generally preceded by an accumulation of different types of internal damages. During the damage process in composite materials, the following phenomena are known: formation and growth of microcracks or voids, clustering, coalescence, formation and growth of initial cracks, and propagation of one of the cracks up to the failure of the specimen [12]. These steps depend on the type of reinforcing fibers or particles (inclusion), and on the interface between inclusion and matrix. They vary with the type of loading. For instance, in an early stage of loading debonding appears if there is a weak inclusionmatrix interface. Stress transfer from the matrix to the inclusion in a composite takes place by shear at the inclusion-matrix interface. An important aspect is the loading direction relative to the orientation of the inclusion. For example, in a PMC with a glass fiber, debonding occurs preferentially if loading is applied perpendicular to the fiber orientation. If loading is applied in fiber direction, fibers will fail or will be pulled out after reaching a certain critical load. Further on, in fiber reinforced materials, fibers can exhibit a stitching action on the microcracks preventing them from propagating. This intervention retards the creation of micro-cracks leading to an overall improvement of the fracture resistance [13]. Strong interfaces result in high strength and stiffness, but low fracture toughness. On the other hand, weak interfaces promote deflection of matrix cracks along the interface and lead to high fracture toughness, but low strength and stiffness of composites. The process of transfer of load between fibers and matrix in the neighborhood of a fiber break or a matrix crack depends on the strength of the interface. Although fiber and matrix can be characterized by conducting simple tests, interface properties are most difficult to determine. Interfacial shear strength is an important parameter that controls the inclusion-matrix debonding process [14]. Mechanical fatigue is the most common type of failure of structures. It is defined as the failure of a component under the repeated application of a stress smaller than that required to cause failure in a single application. In fatigue, a crack is initiated and slowly grows under the action of the fluctuating stress until, eventually, failure occurs in a catastrophic manner with no great distortion preceding the event. To understand fatigue damage of fiber reinforced composites, a simple unidirectional composite loaded in tension parallel to the fibers is discussed in the following. If fiber breakage occurs when the local stress exceeds the strength of the weakest fiber, this causes shear stresses concentration at the fiber-matrix interface near the broken fiber tip. The interface area acts as a stress concentrator for the longitudinal tensile stress, which may exceed the fracture stress of the matrix, leading to transverse cracks in the matrix. These cracks can be randomly distributed [15]. With the development of the fatigue process, the local strains exceed a certain threshold, resulting eventually in fiber breakage and propagation of matrix cracks. During matrix crack propagation, the fiber-matrix interface will also fail due to severe shear stress at the crack tip. The final failure occurs when a sufficiently large crack has developed. The lower strain limit for the

3 4.1 Modeling of Damage in Fiber and Particle Reinforced Composites 327 matrix is the threshold strain below which the matrix cracks remain arrested by the fibers. This strain is observed to be approximately the fatigue strain limit of the unreinforced matrix material. The upper limit is given by the strain to failure of the composite, which is the strain to failure of the reinforcing fibers. The progressive damage mechanism is matrix cracking with associated interfacial shear failure and this governs the fatigue life. When fibers are arranged perpendicular to the loading direction, damage mechanisms are slightly different (similar to the static case). Here, the lowest failure limit is given by transverse fiber-matrix debonding which is strongly connected to the fiber orientation angle θ. This is reflected in the fatigue limit strain [16]. A study of the fatigue damage mechanisms gives indications of the weakest microstructural element, which is a useful information in the selection of materials for improvement in service properties. Generally, PMCs possess weak interfaces, and fatigue failure occurs by distributed debonding or longitudinal matrix cracking followed by further fiber breakage. Macroscopically, the weak interface composites show shorter fatigue lives and more rapid fatigue degradation. Fatigue damage can be studied on a macroscopic and microscopic scale. There are several differences between the fatigue behavior of metals and of fiber reinforced composites. In metals, the stage of gradual and invisible deterioration spans nearly the complete lifetime during service conditions. No significant reduction of stiffness is observed during the fatigue process. The final stage of the process starts with the formation of small cracks, which are the only form of macroscopically observable damage. Gradual growth and coalescence of these cracks quickly produce large cracks and final failure of the structural component. As the stiffness of a metal remains quasi unaffected, the linear relation between stress and strain remains valid, and the fatigue process can be simulated in most common cases by a linear elastic analysis and linear fracture mechanics. In a fiber reinforced composite, damage starts very early and the extent of the damage zones grows steadily, while the damage type in these zones change (e.g., small matrix cracks leading to large size delaminations). The radial deterioration of a fiber reinforced composite with a loss of stiffness in the damaged zones leads to a continuous redistribution of stresses and to a reduction of stress concentrations inside a structural component. As a consequence, an estimation of the actual state or a prediction of the final state (when and where final failure is to be expected) requires the simulation of the complete path of successive damage states [16]. 3. OVERVIEW OF DAMAGE MODELING IN COMPOSITES Many of the established models only consider one or two of the above mentioned damage mechanisms. Using finite element method (FEM) to model damage, requires specific approaches to solve the discrepancies between the quasi-continuum statement of a problem and the random and discontinuous nature of crack growth [12]. The unit cell approach is often used to simulate the initiation of damage. Bao [17] uses a three phase damage cell model taking into account the failure of particles and particle-matrix debonding to simulate strength and creep resistance of metals such as Al and Ti reinforced with Al 2 O 3. The deformation of particle and whisker reinforced MMCs was investigated by Llorca et al. applying cylindrical unit cells to obtain the overall stress-strain behavior [18]. Axisymmetric unit cells were used by Walter to model damage initiation in fiber reinforced composites by cohesive elements [19]. The stress triaxiality and the shape of voids were taken into account by Brocks et al. to simulate effective stress vs. strain curves [20]. Thereafter, the relevant parameters of the Gurson- Tvergaard-Needleman damage model were determined. 3D hexagonal cells were used by Sun et al. to model the influence of micro-crack densities on the creep behavior of ferritic steels [21]. Also, weak interfaces of polymer specimens were simulated by cylindric unit cells containing a rigid particle [22, 23]. It is possible to include the effect of particle and fiber failure, the particle/fiber-matrix debonding by unit cell models. One basic assumption of many (non self-consistent) unit cells is the uniform distribution of the inclusions and, therefore, of the damage [12]. After a general introduction of unit cell models, the Self-consistent Model, the (Statistical) Combined Cell Model and the plastic-damage model will be presented to simulate the mechanical behavior of different fiber and particle reinforced composites. Initially, the mechanical behavior of a unidirectionally continuous fiber reinforced composite with fibers of circular cross-section was studied by Adams [24] adopting finite element cell models under plane strain conditions: a simple geometrical cell composed of matrix and inclusion material is repeated by appropriate boundary conditions to represent a composite with a periodic microstructure. The influence of different regular fiber arrangements on the strength of transversely loaded boron fiber reinforced Aluminum was analyzed in [25 27]. It was found that the square arrangement

4 328 DAMAGE SIMULATION of fibers represents two extremes of the strengthening: high strength levels are achieved if the composite is loaded in a 0 direction of nearest neighbors while the 45 loading direction is found to be very weak for the same fiber arrangement. A regular hexagonal fiber arrangement lies between these limits [25 29]. The transverse mechanical behavior of a realistic fiber reinforced composite containing about thirty randomly arranged fibers was found to be best described but significantly underestimated by the hexagonal fiber model [26]. Dietrich [30] found a transversely isotropic square fiber reinforced Ag/Ni composite material using fibers of different diameters. A systematic study in which the fiber volume fraction and the fiber arrangement effects have been investigated, was founded into a simple model in [29]. The influence of fiber shape and clustering was numerically examined by Llorca et al. [18], Dietrich [30], and Sautter [31]. It was observed that facetted fiber cross-sections lead to higher strengths compared to circular cross-sections except for fibers which possess predominantly facets with an angle of 45 with respect to the loading axis in close agreement with findings in particle reinforced MMCs [32]. Thus, hindering of shear band formation within the matrix was found to be responsible for strengthening with respect to fiber arrangement and fiber shape [18]. In [29, 33 35] local distributions of stresses and strains within the microstructure have been identified to be also strongly influenced by the arrangement of fibers. However, no agreement was found between the mechanical behavior of composites based on cell models with differently arranged fibers and experiments with randomly arranged fibers loaded in transverse direction. The overall mechanical behavior of a particle reinforced composites was studied with axisymmetric finite element cell models by Bao et al. [36] to represent a uniform particle distribution within an elasticplastic matrix. Tvergaard [37] introduced a modified cylindrical unit cell containing one half of a single fiber to model the axial performance of a periodic square arrangement of staggered short fibers. Hom [38] and Weissenbek [39] have used threedimensional finite elements to model different regular arrangements of short fibers and spherical as well as cylindrical particles with relatively small volume fractions ( f < 0.2). It was generally found that the arrangement of fibers strongly influences the different overall behavior of composites. When short fibers are arranged in a side-by-side manner, they constrain the plastic flow in the matrix and the computed stress-strain response of the composite in the fiber direction is stiffer than that observed in experiments. If the fibers in the model are overlapping, between neighboring fibers strong plastic shearing can develop in the ligament and the predicted load carrying capacity of the composite is closer to the experimental measurements. The influence of thermal residual stresses in fiber reinforced MMCs under transverse tension is studied in [27] and found to lead to significant strengthening elevations in contrast to findings in particulate reinforced MMCs where strength reductions were calculated [40]. A limited study on the overall limit flow stress for composites with randomly oriented disk-like or needle-like particles arranged in a packet-like morphology is reported by Bao et al. [36]. In [41, 42] a modified Oldroyd model has been proposed to investigate analytically-numerically the overall behavior of MMCs with randomly arranged brittle particles. Duva [43] has introduced an analytical self-consistent model to represent a random distribution of non-interacting rigid spherical particles perfectly bonded in a power law matrix in the dilute regime of volume fractions of f < 0.2. Composites with randomly arranged inclusions can be modeled by a self-consistent procedure with embedded cell models. This method of surrounding a simulation cell by additional equivalent composite material was introduced in [30, 33] for structures which are periodical in loading direction, and recently extended to non-periodic two-dimensional [4, 44, 45] and three-dimensional composites [31, 42]. One reason for the discrepancy between experiments and calculations based on simple cell models is believed to be the unnatural constraint governing the matrix material between inclusion and simulation cell border [26, 34, 36 38, 46 48] resulting in an unrealistic strength increase. Embedded cell models are known to remove the unrealistic constraints of the simple models described above. An initial comparison of two- and three-dimensional embedded cell models in case of perfectly plastic matrix material depicts elevated strength levels for the three-dimensional case [42] as it happened for composites with regularly arranged fibers [29]. For aligned short fiber reinforced composites, some approaches to determine the mechanical behavior have been introduced in [49, 50] by considering two geometrical aspects: cross-section along fiber and cross-section in transverse plane, which lead to different cell models for arrays of end-to-end aligned short fibers, axially clustered short fibers, transversely clustered arrays of short fibers or misaligned short fibers. In case of short fibers with small aspect ratio, periodic cell models can be used [49, 50] to follow morphological effects, especially to describe in-plane misaligned short ceramic fibers (SiCwhisker) with small misorientation angles in a single

5 4.1 Modeling of Damage in Fiber and Particle Reinforced Composites 329 cell model. A further analysis is presented in [51], where the Duva s model is applied to calculate the overall flow behavior of short fiber reinforced composites. The effects of fiber orientation, which influences the mechanical deformation behavior and the fiber damage behavior, have also been discussed in [51]. Unit cell models have been applied in [52] to analyze residual stress effects on uniaxial deformation of whisker reinforced Metal Matrix Composites, where models with different fiber aspect ratios are employed to predict the overall flow behavior of short fiber reinforced MMCs. Comparisons between experimental and numerical results on two composites demonstrate that it is not possible to use a single unit cell to predict the mechanical behavior of randomly oriented short fiber reinforced MMCs. A representative volume with different fiber orientations is described in a two-dimensional model in [53] to determine the mechanical behavior of composites with randomly oriented fibers, where the fibers are again very short (aspect ratio 2/1). A micromechanical model has been introduced in [54, 55] for short fiber reinforced aluminum alloys. There, three elementary microstructural mathematical processes were taken into account to investigate the creep behavior of these MMCs, without considering the morphological aspect of the fibers. 4. MODELING In this section several models will be presented, to simulate the mechanical behavior of composites. The self-consistent and the (Statistical) Combined Cell Models are indicative of unit cell approaches. As a last example the plastic-damage model is introduced, which can describe the homogenized constitutive behavior of fiber reinforced quasi-brittle materials. In Sec. 5, these four models will be then applied to fiber and particle reinforced composites with metal or polymer matrix and a cellulose fiber gypsum composite. There, also the potential and limitations of these models to simulate the specific mechanical behavior will be discussed Self-Consistent Model (SCM) For composites reinforced with aligned continuous fibers or with spherical particles, simple unit cells with single inclusions can be taken from a representative cross-section in a transverse plane [4, 24, 26, 30, 34, 56] or from a cross-section along the loading axis [5, 36, 40]. In the present section, 2D and 3D self-consistent embedded cell models will be applied to model the mechanical behavior of composites with random continuous fiber and particle arrangements. Fig. 1 describes schematically a typical plane strain (2D) embedded cell model with a volume fraction f =(r/r) 2 or axisymmetric (3D) embedded cell model with a volume fraction f =(r/r) 3. Here, instead of using fixed or symme- Figure 1. Embedded cell model with finite element mesh [4]. try boundary conditions around the fiber-matrix or particle-matrix cell, the inclusion-matrix cell is embedded in an equivalent composite material with the mechanical behavior to be determined iteratively in a self-consistent manner. If the dimension of the embedding composite is sufficiently large compared to that of the embedded cell, e.g., L/R = 5, the external geometrical boundary conditions introduced for the embedding composite are almost without influence on the composite behavior of the inner embedded cell. A typical FE mesh and corresponding symmetry and boundary conditions are given in Fig. 1, where a circular fiber or a spherical particle is surrounded by a circular (for 2D) or spherical (for 3D) shaped matrix, which is again embedded in the composite material with the mechanical behavior to be determined. The flow stresses for transverse loading of Metal Matrix Composites reinforced with continuous fibers and for uniaxial loading of spherical particle reinforced metal-matrix, composites were investigated in previous studies using embedded cell models [4, 5]. These works describe a fiber or a spherical particle as surrounded by a metal-matrix, which is again embedded in the composite material

6 330 DAMAGE SIMULATION with the mechanical behavior to be determined iteratively in a self-consistent manner. It has been verified in [4, 5] that such a self-consistent embedded cell method is appropriate to represent Metal Matrix Composites with randomly arranged particles or aligned continuous fibers. The inclusion behaves elastically and its stiffness is much higher than that of the matrix. In addition, the continuous fibers of circular cross-section and spherical particles are assumed to be well bonded to the matrix so that debonding or sliding at the inclusion-matrix interface is not permitted. The uniaxial matrix stressstrain behavior is described by a Ramberg-Osgood type of power law [5]. The global mechanical response of the composite under external loading is characterized by the overall stress σ as a function of the overall strain ε. Moreover, to describe the results in a consistent way, the reference axial yield stress σ 0 and yield strain ε 0 of the matrix (as defined in Eq. 1 for the 3D case) will be taken to normalize the overall stress and strain of the composite, respectively. Following Bao et al. [36], the composite containing hard inclusions will necessarily harden with the same strain hardening exponent N, as the matrix for the case of hard inclusions when strains are in the regime of fully developed plastic flow. At sufficiently large strains the composite behavior is then described by Figure 2. Composite strengthening [5]. Fig. 3. An initially assumed stress-strain curve (iteration 0 in Fig. 3) is first assigned to the embedding composite, in order to perform the first iteration step. An improved stress-strain curve of the composite (iteration 1) will be obtained by analyzing the average mechanical response of the embedded cell. This procedure is repeated until the calculated stress-strain curve from the embedded cell is almost identical to that of the previous iteration. The convergence of the iteration occurs typically at the fifth iteration step, as illustrated in Fig. 3. It has been σ = σ N [ ε ε 0 ] N, (1) where σ N is the asymptotic reference stress of the composite, which can be determined by normalizing the composite stress by the stress in the matrix at the same overall strain ε, as indicated in Eq. 2 and Fig. 2: [ ] σ( ε) σ N = σ 0 for ε>>ε 0. (2) σ( ε) For a matrix of strain hardening capability N, the limit value σ N / σ0 is defined as composite strengthening level, which is an important value to describe the mechanical behavior of composites. This value depends only on fiber and particle arrangement, inclusion volume fraction and matrix strain-hardening exponent. Under axial deformation at the external boundary, the overall response of the inner embedded cell can be obtained by averaging the stresses and strains at the boundary between the embedded cell and the surrounding volume. The embedding method is a self-consistent procedure, which requires several iterations, as shown in Figure 3. Iterative modeling procedure: stressstrain curves for different iteration steps[5]. found from systematic studies that convergence of the iteration to the final stress-strain curve of the composite is independent of the initial mechanical behavior of the embedding composite (iteration 0).

7 4.1 Modeling of Damage in Fiber and Particle Reinforced Composites Combined Cell Model (CCM) In this section, CCMs are presented to simulate the overall flow behavior of composites reinforced with discontinuous short fibers. These cell models involve two 2D models and two 3D models representing a single fiber in three principal orthogonal planes of the local system in a composite. The overall flow behavior of the composites will be predicted with in-plane randomly oriented and 3D randomly arranged short fibers by an appropriate integration over all fiber orientations. Two different kinds of fiber orientations are modeled: in-plane random (2D random) and 3D random. In Sec. 5.2 also composites with aligned and layered fiber orientations are considered. In the case of inplane random orientation the fibers are distributed in preferred parallel planes as illustrated schematically in Fig. 4a, whereas, in the other case, the fibers lie randomly in all directions of space without any preferred direction and plane, as shown in Fig. 4b. In Figure 5. (a) Global and local coordinate systems for a fiber with an orientation angle θ; (b) three principal orthogonal cross-section planes for a fiber in local coordinate system [6]. described by only one characteristic parameter θ, which defines the orientation of the fiber with respect to the external loading direction. Because of the high aspect ratio and the different orientations of the fibers, it is difficult to use a conventional FE unit cell model approach to represent the deformation and plastic flow behavior of composites. In order to get an appropriate cell model approach, a single fiber of an orientation angle θ with respect to the applied loading axis is considered in three principal orthogonal cross-sections (plane A, B and C in Figs. 5b and 6) in the local xyz-system of a composite. These three planes are parallel and per- Figure 4. Schematics of the composites with a) inplane 2D random and b) 3D random fiber orientations [6]. Fig. 5a, a global (X,Y,Z) and a local (x,y,z) coordinate system are introduced. The Z and z-axis are oriented in the loading direction. The local (x,y,z) coordinate system for each single short fiber in the composites is defined in such a way that the fiber of length L lies in the yz-plane. The orientation of the fiber is then defined as the angle θ between the fiber direction and the applied loading direction. The location of a fiber can then be described as a vector L in both coordinate systems: L = L(X,Y,Z)=(Lsinϕsinθ, Lcosϕsinθ, Lcosθ) (3) L = L(x,y,z) =(0, Lsinθ, Lcosθ). In the local coordinate system, the fiber orientation with respect to the loading direction is simply Figure 6. Construction of cell models from the projections of a fiber with an orientation angle θ on three principal orthogonal cross-section planes A, B and C [6]. pendicular to the loading direction and in the local xyz-system, so they can characterize the fiber orientation in a simple and direct way. On the plane A in Fig. 6, which is built up by the fiber direction and the applied loading direction, the orientation and the geometrical size (length L and diameter d) of the fiber can be represented by a rectangle (L xd) with an orientation angle θ. In the planes B and C in Fig. 6, the fiber is represented by its cuts of

8 332 DAMAGE SIMULATION two ellipses, one with a minor axis of d (diameter of the fiber) and major axis of d/sinθ on plane B, and another one with a minor axis of d and major axis of d/cosθ on plane C. The major axes vary with the fiber orientation in the local xyz-system. The rectangular- and ellipse-shaped cross-section of discontinuous fibers in MMCs can be seen, e.g., in the optical micrograph of a polished section. For composites reinforced with in-plane randomly oriented fibers (Fig. 4a), the global XYZ-system is identical to the local xyz-system. In this case, all the ellipses of fibers in cross-sections B and C possess the same orientation (all major axes are parallel, see Fig. 4a), whereas in the case of 3D random fiber orientations they depict different directions of major axes, see Fig. 4b, in the global cross-section. Due to different fiber orientations, which lead to different shapes of rectangles and ellipses in the crosssections, as seen in Fig. 4, a single unit cell is not sufficient to represent the complicated geometrical situation and the mechanical behavior of the short fiber reinforced MMCs. More computational cells should be taken into account in order to obtain the mechanical behavior of the composites by simple cell models. On the basis of the geometrical description outlined above, it is possible to define an approach which uses simple cell models to calculate and predict the mechanical deformation and flow behavior. From the geometrical shape of the single fiber on the three local principal orthogonal cross-sections (plane A, B and C) four unit cell models (see Fig. 6) can be constructed, which represent the local stress state of a single fiber with an orientation θ. The cross-section of the single fiber on the local plane A is a rectangle with the orientation angle θ. Unit cells can only be applied for the orientation of θ = 0 or θ = 90, as shown in Fig. 6. The oriented fiber is then separated into two essential parts, i.e. model A2 and A3, where A2 is a 2D model for fibers under transverse loading and A3 is an axisymmetrical model for fibers under axial loading. The cross-sections of the single fiber on the local planes B and C are two ellipses with the minor axis d and the major axes d/sinθ and d/cosθ, where d is the diameter of the fiber. The representative approaches to these cross-sections are given by constructing two cell models, one is B2 and the other is C3. B2 is a two-dimensional model with an elliptical inclusion of a minor axis d and a major axis d/ sinθ, whereas C3 is a three-dimensional model with an axisymmetrical ellipsoidal inclusion of the same minor and major axes as those of B2. There exist four unit models, i.e. A2, A3, B2 and C3 for each fiber orientation θ. Forθ = 0 and θ = 90 one model, i.e. A3 and A2, respectively, is sufficient. From the fiber volume fraction f, the fiber aspect ratio L/d and the fiber orientation θ, the geometric size of four cell models can be determined and the local stress-strain curves σ A2 (ε,θ), σ A3 (ε,θ), σ B2 (ε,θ) and σ C3 (ε,θ) can be calculated from the cell models with the conventional unit cell technique [36, 56]. Four stress-strain curves, σ A3 (ε,θ), σ A2 (ε,θ), σ B2 (ε,θ), and σ C3 (ε,θ) from four unit cells for a given single fiber orientation θ, must be connected in an appropriate way to get the stress-strain curve of a composite with fibers oriented in a direction θ. The following heuristic procedure has been used to establish the desired connection: at first, the stressstrain curve σ A (ε,θ) can be calculated by averaging the stresses σ A3 (ε,θ) and σ A2 (ε,θ) with the help of the volume relationship between the two separate parts A2 and A3 of the cross-section on the plane A, V A2 = V sinθ and V A3 = V cosθ: σ A (ε,θ)= σa2 (ε,θ) V A2 + σ A3 (ε,θ) V A3 V A2 +V A3 (4) = σa2 (ε,θ)sinθ + σ A3 (ε,θ)cosθ. sinθ + cosθ On the cross-sections of the plane B and C there exists the same volume relationship between the two models B2 and C3, so that an average stress σ BC (ε,θ), associated with σ B2 (ε,θ) and σ C3 (ε,θ) can be written as σ BC (ε,θ)= σb2 (ε,θ)sinθ + σ C3 (ε,θ)cosθ. (5) sinθ + cosθ The three stress-strain curves, σ A (ε,θ), σ B (ε,θ) and σ C (ε,θ) must be averaged to get an overall flow behavior of fiber reinforced composites with a single fiber orientation, in such a way that each of the three stresses σ A, σ B and σ C contributes to the overall flow behavior σ(ε, θ): σ(ε,θ)= [ σ A (ε,θ)+2σ BC (ε,θ) ]/ 3. (6) It is assumed that Eq. 6 provides the mechanical behavior of a single fiber as a function of the fiber orientation θ with respect to the applied loading in a composite. To obtain the overall flow behavior of short fiber reinforced composites, a weighted integration of stress-strain curves for all the fiber orientations, as defined in Eq. 7, can be carried out by introducing a weighting function f (θ), which describes the distri-

9 4.1 Modeling of Damage in Fiber and Particle Reinforced Composites 333 bution density of short fibers in a composite: σ(ε)= π/2 0 σ(ε,θ) f (θ)dθ π/2 0 f (θ)dθ. (7) If the short fibers are randomly distributed in MMCs in a plane (2D random), they possess the same distribution density in all directions of the plane, as schematically illustrated in Fig. 4a. In this case, the weighting function f (θ) has the constant value 1. If the short fibers are distributed randomly in MMCs in all directions of space (3D random), the distribution density changes with the orientation angle in the same way as the change of the latitude in a spherical coordinate, which is considered by introducing a weighting function f (θ)=sin θ. For composites with preferred fiber orientation the weighting function f (θ) must be determined in correspondence with the preferred fiber orientations. Weibull s law which corresponds to the scatter of the fiber breaking in the composite. σ 0F is a scale parameter and equivalent to the mean value of the fiber strength, which gives a cumulative breaking probability of 63% and it corresponds to fraction of broken fibers for a given fiber reinforced composite. This parameter is strongly related to the reinforcement material. In this manner, we can obtain the mechanical behavior of composites for the unit cells A3, A2, B and C with Eqs. 4 and 5. The damage behavior of composites can be calculated according to Eqs. 8 and 9: [ σ brk ( ε,θ)= 1 P brk ( σ F)] σ UD( ε, brk θ) +P brk ( σ F) σ brk FD ( ε,θ). (9) A further principal source of damage is the failure 4.3. Statistical Combined Cell Models Static Loading Conditions Statistical Combined Cell Models (SCCMs) for short fiber reinforced composites with different fiber volume fractions have been developed on the basis of the Combined Cell Models of the previous section [6, 57, 58] and a Weibull statistical approach [59], originally developed for fiber fracture in composites. The SCCM takes into consideration fiber-cracks and fiber-matrix debonding. This allows to calculate the two types of unit cells separately, i.e. unit cells with unbroken and with broken fibers. Then, the global mechanical behavior of composites reinforced with short fibers is calculated on the basis of the rule of mixture. When loading is parallel to the fiber orientation or if no debonding occurs between fiber and matrix, it is found that fiber failure is the main source of damage in the composite (Fig. 7). The fracture probability of each fiber is a function of its volume and of the maximum principal stress σ U F in the fiber. Therefore, the Weibull law, from Eq. 8, can be written in terms of fiber failure as follows: P brk ( σ F )=1 exp [ ( σ U F σ 0F ) mf ]. (8) In this equation, P brk (σ F ) is the failure probability of fiber fracture and m F is the shape parameter of Figure 7. Schematics of the Statistical Combined Cell Models (SCCM) with fracture of brittle fibers in short fiber strengthened composites. of the fiber-matrix interface (Fig. 8). This failure is governed by a local criterion that is dominated by interfacial normal stress. Because the interfacial damage is distributed statistically as a function of the spatial distribution of the microstructure, the local interface failure criterion must be written in a statistical form following Weibull s law: ( exp σ U L σ 0L P deb ( σ L )=1 (10) ) m L 2, ) 2 ( τ U + L τ 0L where P deb (σ L ) denotes the fiber-matrix interfacial debonding probability relative to a given interfacial state σ U L, which is a function of the microscopic stress σ L, σ 0L denotes the interfacial stress, and m L

10 334 DAMAGE SIMULATION is the statistical parameter. The parameter τ U L denotes the interfacial shear stress and τ 0L is the characteristic shear stress. If the fiber is perpendicular to the loading direction (90 ), there is no significant influence of shear stresses and the equation can be written as [ ) σ P deb U ml ] ( σ L )=1 exp ( L. (11) σ 0L The stress state of a cell can be predicted by the mixing rule [7, 60] in which undebonding stresses and debonding stresses are taken into account: [ ] σ( ε)= 1 P deb ( σ L ) σ ud ( ε)+ P deb ( σ L ) σ db ( ε), (12) where σ ud ( ε) is the stress in an undamaged unit cell and σ db ( ε) is the stress in a damaged unit cell due to fiber-matrix interfacial debonding. In this equation, σ( ε) is the stress behavior of a composite cell with fiber perpendicular to the loading direction that includes the debonding damage behavior. The first term on the right-hand side indicates the stress behavior of the undamaged interface (ud) and the second term indicates the stress behavior of damaged interface (db) in a composite. Thus, the arithmetic sum in Eq. 12 implies the stress behavior of a composite cell with debonding failure. The mechanical loading direction. In this case there was damage at the boundary layer between fibers and matrix, but no fiber fractures took place. When using the CCM, we have in this case θ = 0, so that a description of the model through the model part A3 is sufficient (Fig. 8). As a first application of the SCCM, the parameters in Eqs. 11 and 11 can be calculated by a comparison between the computation and the experiment. From the experiments it can be seen that in case of parallel loading there is a combined effect of fiber breaking and debonding on the composite failure. Both effects can be combined in a composite unit cell using the mixing rule [7, 60] [ ] σ( ε)= 1 P deb ( σ L ) P brk ( σ L ) σ ud ( ε)+ P deb ( σ L ) σ db ( ε)+p brk ( σ L ) σ brk ( ε), (13) where σ brk ( ε) is the stress in a damaged unit cell due to broken fibers. The two Weibull parameters for interface failure and fiber failure are numerically identified by using the data from micromechanical models and the calculated finite element results to compare them with the experimental curves Quasi-Static Cyclic Loading Conditions The micromechanical fatigue damage model in this section is based on a statistical microscopic damage law. Predictions of these types of failure have been applied to determine damages in each loading cycle. By comparing the simulation with the experimental stress-strain curves for tension, the Weibull damage parameters are determined. Using these damage parameters a mesoscopic model (Sec ) including the effect of fiber-clusters is developed and the damage during cyclic loading is predicted. Figure 8. Schematics of the Statistical Combined Cell Models with damage in the boundary layer between fibers and matrix. behavior derived from the unit cells A3, A2, B and C with consideration of the damage between fibers and matrix follows in an analogous manner. For the numerical investigation with consideration of the fiber-matrix adhesion effect, fibers were arranged in the tensile specimen perpendicularly to the To study the behavior of fiber reinforced composites, 3D unit cell models are used to analyze the microscopic failure. The statistical analysis of fiber breaking and fiber-matrix interfacial debonding will be predicted by Weibull s law [61, 62] as described above. It is based on the assumption that the composite fails as a result of accumulation of statistically distributed fiber flaws. The equations of Weibull s damage law for fiber failure [7, 63] were taken from Eqs. 8 and 13. The fiber failure can be supplemented by fiber-matrix interfacial debonding. Evolution of damage in a composite under cyclic loading is calculated on the basis of the statistical

11 4.1 Modeling of Damage in Fiber and Particle Reinforced Composites 335 evolution of damage in the fiber-matrix interfaces and in the broken fibers. Debonding failure and failure due to broken fibers are considered mutually dependent on each other. That means that, if debonding occurs around the fiber-matrix interface, fiber failure will not occur. On the other hand, where the fibers break, there is a negligible influence of debonding failure. The damage stress for each cycle is calculated according to the total failure probability due to fiber failure and interface debonding. The effect of damage is embedded in the model by replacing the stress of the previous cycle (the true stress in the first cycle) with the effective stress in the present cycle. Any strain constitutive equation for the damaged material is derived in the same way as for the virgin material, except that the true stress is replaced by the current effective stress [64]. Accordingly, material properties are changed during the cycle due to fiber failure and interface debonding. Applying the mixing rule, the stress after the k-th loading cycle can be expressed as follows: σ k+1 ij (ε ij )=σ k ij(ε ij ) P brk (σ k ij,unbr (ε ij) σ k ij,brk (ε ij)) P deb (σ k ij,unde (ε ij) σ k ij,deb (ε ij)), (14) where i is the Element index, and j is the loading step index in one cycle. Therefore, the new material properties of the composite are calculated for each loading cycle, which is then included into the ABAQUS input file [65] for the next loading cycle calculation Plastic-Damage Model In this section, the homogenized constitutive fracture behavior of materials will be described for static and quasi-static cyclic loading with a plasticdamage model proposed by Lubliner et al. [66] and Lee and Fenves [67]. In this model, stiffness degradation due to damage is embedded in the plasticity part of the model. Damage is represented by two independent scalar damage parameters, one for tension (d t ) and another one for compression (d c ). This is necessary because many materials show different damage mechanisms in tension and compression. In tension, the damage is associated with cracking, while in compression, it is associated with crushing. The initial undamaged state and complete damaged state of the material under tension and compression are indicated by d t, d c = 0 and d t, d c = 1, respectively. Apart from this, a stiffness recovery scheme is used for simulating the effect of microcrack opening and closing. The effect of damage is embedded in the plasticity theory and all stress definitions (true stress) are reduced to the effective stress [64]. This enables the decoupling of the constitutive relations for the elastic-plastic response from stiffness degradation (damage) response. In the following equations, underlined symbols indicate vector or tensor quantities, overlined stress expressions indicate effective stresses. Symbols without underline are to be understood as scalar quantities. All strain symbols with a tilde are equivalent strains. In Eq. 15 Macaulay brackets have been used, which are defined as x = x if x > 0, otherwise x = 0. For the plasticity part, a nonassociated plasticity scheme is used. The yield surface proposed by Lubliner et al. [66] is based on modifications of the classical Mohr-Coulomb plasticity (Eq. 15): F( σ, ε pl )= 1 1 α ( q 3α p + β ( εpl ) ˆ σ max γ ˆ σ max ) σc ( ε c pl ), (15) where σ corresponds to the stress tensor, σ c is the uniaxial compressive stress, p corresponds to the effective hydrostatic pressure, α and γ are material constants, q corresponds to the equivalent effective deviatoric stress, ˆ σ max is the max. principal stress, and ε pl corresponds to the equivalent plastic strain. A separate flow potential is used to determine the direction of plastic flow in the principal stress space. The flow potential chosen for this model is the Drucker-Prager hyperbolic function G (Eq. 22 in Sec ). At high confining pressure stress, the function asymptotically approaches the linear Drucker-Prager flow potential in the deviatoric plane and intersects the hydrostatic pressure axis at 90 [68]. In Fig. 9 the yield surface and the flow potential function are illustrated in the 2D principal stress space. The material modeling has been performed based on an existing implementation of the plastic-damage model in ABAQUS. The details of the mathematical formulation of the model are given in [66, 68 70] Modifications of the Plastic-Damage Model The simulation results of the static behavior of the material presented in Sec , obtained by using the implemented plastic-damage model in ABAQUS, is close to that of the experiment (see Sec ). However, when applying the implemented model to cyclic loading, considerable model limitations are observed. The implemented model

12 336 DAMAGE SIMULATION damage parameter d t as d = d t s, (16) s = 1 w, (17) where s is the stiffness recovery factor and w is the weight factor that controls the stiffness recovery. w = 1 means complete stiffness recovery corresponding to d = 0, whereas w = 0 means no stiffness recovery corresponding to d = d t. On the yield surface, d is obtained from Eq. 16 and 17. The evolution of yield stress, tension damage, d t and weight factor, w are functions of plastic strain in tension, ε p t. The material subroutine UMAT requires the evolution information as strain softening, damage evolution and stiffness recovery curves. During unloading and reloading in the elastic domain, d is redefined in the material subroutine UMAT based on rules derived by observing the unloading/reloading slope (E) in the uniaxial quasi-static cyclic stressstrain curves. The corresponding damage parameter d is obtained from the varying slope (E) and the initial stiffness (E 0 )as d = 1 E E 0. (18) The determination process of the rules controlling d in the elastic domain and strain softening, damage evolution and stiffness recovery curves are discussed in Sec Figure 9. Illustration of (a) yield surface, and flow potentials, (b) dilation angle [11]. reaches up to the point where unloading starts. Beyond this point the material behavior is complex, showing different stiffnesses at different stages of unloading and reloading. It is not possible to handle the varying unloading and reloading stiffnesses with the available stiffness recovery effects implemented in the plastic-damage model provided by ABAQUS. Therefore, the plastic-damage model has been re-implemented with the necessary modifications in a user defined material subroutine UMAT in ABAQUS to improve the simulation of the quasistatic cyclic experiments. If the yield point in compression is not reached, compression damage is absent (d c = 0). Then, damage occurs only due to tensile loading, which is represented by the scalar tension damage parameter d t. The total damage parameter d in the modified plastic-damage model is correlated with tension 5. RESULTS AND APPLICATION After introduction of the investigated materials, in this section the above presented models will be applied to different composites which are MMCs, PMCs, and cellulose fiber reinforced gypsum materials. In the case of MMCs the self-consistent and the (Statistical) Combined Cell Models are used. The PMCs are studied applying the Statistical and Combined Cell Model. Finally, the plastic-damage model suit especially for quasi-brittle materials such as the investigated cellulose fiber reinforced gypsum composite Metal Matrix Composites (MMC) As a first example the self-consistent and the (Statistical) Combined Cell Model will be applied to Metal Matrix Composites. MMCs with strong inclusions are a relatively new class of materials (compare Chapter 2.1.4). Due to high strength and light weight, they are potentially valuable in aerospace and transportation applications [71].

13 4.1 Modeling of Damage in Fiber and Particle Reinforced Composites Material The metal-matrix is an Al/12% vol. Si cast alloy (M124). The composite considered here has been produced by Mahle GmbH, Stuttgart, via pressure infiltration of a fiber preform with randomly oriented short Al 2 O 3 -fibers (Saffil). These fibers behave elastically (Young s modulus E = 300 GPa, and Poisson s Ratio ν = 0.23), the fiber content in the composite is 15% vol., and the fiber aspect ratio is approximately 200µm/3µm. Fig. 10 shows an optical micrograph of a polished section of the short fiber reinforced composite [72]. As a further example an Al/46% vol. B composite with random fiber packing taken from [26] has been selected to verify the embedded cell model. This MMC is a 6061-O aluminum alloy reinforced with unidirectional cylindrical boron fiber of 46% volume fraction. The room temperature elastic properties of the fibers are a Young s modulus of E (B) = 410 GPa, and a Poisson s Ratio of ν (B) = 0.2. The experimentally determined mechanical properties of the 6061-O aluminum matrix are Young s modulus, E (Al) = 69 GPa, Poisson s Ratio, ν (Al) = 0.33, 0.2% offset tensile yield strength, σ 0 = 43 MPa, and strain-hardening exponent N = 1/n = 1/3. Furthermore, the composite Ag/58% vol. Ni [42] with random particle arrangement (Young s modulus, E (Ni) = GPa,E (Ag) = 82.7 GPa, Poisson s Ratio, ν (Ni) = 0.312, ν (Ag) = 0.367, and yield strength σ (Ni) 0 = 193 MPa, σ (Ag) 0 = 64 MPa) has been investigated. Figure 10. Optical micrograph of a polished section of the discontinuous short fiber Al alloy/15% vol. Al 2 O 3 composite with 3D random fiber orientation [6] Results: Self-Consistent Model In this section, self-consistent embedded cell models, which are described in Sec. 4.1, are applied to simulate the transverse behavior of MMCs containing fibers in a regular square or hexagonal arrangement as well as the mechanical behavior of MMCs containing particles in a regular arrangement. Two aims are pursued: one is to investigate the mechanical behavior of MMCs reinforced with regular or random arranged continuous fibers under transverse loading and particles under uniaxial loading. The other one is to systematically study composite strengthening as a function of inclusion volume fraction and matrix hardening ability. The Finite Element Method (FEM) is employed to carry out the calculations. The overall response of MMCs is elastic-plastic. As regular fiber spacings are difficult to achieve in practice, most of the present fiber reinforced MMCs contain aligned but randomly arranged continuous fibers. The LARSTRAN finite element program [73] was employed using 8 noded plane strain elements (for 2D) as well as axisymmetric biquadrilateral elements (for 3D) generated with the help of the preand post-processing program PATRAN [74]. Fig. 11a shows a comparison of the stress-strain curves of the composite Al/46% vol. B under transverse loading from simulations of a real microstructure together with results from different cell models. The stress-strain curve from the embedded cell model employed in this Chapter shows close agreement with the curve from the calculated random fiber packing in the elastic and plastic regime, which lies between the curves from square unit cell modeling under 0 loading and hexagonal unit cell modeling. Furthermore, the stress-strain curve from another experiment [42] on the composite Ag/ 58% vol. Ni with random particle arrangement has been compared with that from the self-consistent embedded cell model (Fig. 11b). Close agreement in the regime of plastic response is obtained, although the Niparticles in the experiment were not perfectly spherical. These results indicate that the embedded cell model can be used to successfully simulate composites with random inclusion arrangements and to predict the elastic-plastic composite behavior. A comparison of the stress-strain curves for the composite Al/46% vol. B in Fig. 11a shows that the stressstrain curve from random fiber packing given in [26] lies also between the curves from square unit cell modeling under 0 loading and hexagonal unit cell modeling.

14 338 DAMAGE SIMULATION Figure 12. Embedded cell models: influence of (a) different matrix shapes on (b) stress-strain curves for an Al-46% vol. B (N=1/3, f=0.46) composite [5]. Figure 11. Comparison of the mechanical behavior of (a) an Al-46% vol. B fiber reinforced composite (N=1/3, f=0.46) under transverse loading from different models, and (b) an Ag-58% vol. Ni particulate composite from embedded cell model and experiment [5]. Geometrical shape of the embedded cell As mentioned above, different shapes of crosssection of the embedded cell model with a circular shaped fiber, as shown in Fig. 12a, are also taken into account to investigate the influence of the geometrical shape of the embedded cells on the overall behavior of the composite. The stress-strain curves of all embedded cell models with different geometrical shapes are plotted in Fig. 12b. With an exception of square - 45 embedded cell model, the stress-strain curves are very close for all embedded cell shapes, namely, square - 0, circular, rectangular-0, rectangular - 90, elliptic - 0 and elliptic -90. From the calculated results of the embedded cell models, localized flows have been found around the hard fiber with preferred yielding at 45. Because of the special geometry of the square - 45 embedded cell model with the cell boundary parallel to the preferred yielding at 45, the overall stresses of the composite with such a geometrical cell shape are therefore reduced, such that a relative lower stressstrain curve has been obtained from the modeling. The almost identical responses of the other embedded cell models indicate that, besides of the special shape of matrix with 45 cell boundaries, the predicted mechanical behavior of fiber reinforced composites under transverse loading is independent of the modeling shape of the embedded composite cell. Strengthening model The strength of MMCs reinforced by hard inclusions under external mechanical loading has been shown to increase with inclusion volume fraction and strain-hardening ability of the matrix for all inclusion arrangements investigated. From the presented numerical predictions, a strengthening model for aligned continuous fiber reinforced MMCs with random, square (0 ) and hexagonal arrangements

15 4.1 Modeling of Damage in Fiber and Particle Reinforced Composites 339 as well as for spherical particle reinforced MMCs with random, primitive cubic and hexagonal arrangements can be derived as a function of the inclusion volume fraction f, and the strain-hardening exponent N, of the matrix: ( σ N = σ 0 1 ) f (c2 N+c 3 ) c 1 (2 + N) σ 0 c 4 ( f + N 5 ), (19) where σ 0 is the matrix yield stress, and c 1, c 2, c 3 and c 4 are constants summarized in Tab. 1. Eq. 19 2D c 1 c 2 c 3 c 4 SCM SM (0 ) HM D SCAM AM PCM Table 1. Constants for strengthening models: selfconsistent embedded cell model with random fiber arrangement (SCM), square unit cell model (SM), hexagonal unit cell model (HM), Self-consistent axisymmetric embedded cell model (SCAM), axisymmetric unit cell model (AM), primitive cubic unit cell model (PCM) [5]. represents the closest approximation to the calculated composite strengthening values σ N for matrix strain-hardening exponents N in the limit of 0.0 < N < 0.5 for square 0, hexagonal and random fiber arrangements, (practical fiber volume fractions f in the range of 0.0 < f < 0.7), respectively. A comparison of this strengthening model (Eq. 19) for random fiber arrangements with the values calculated by using self-consistent embedded cell models shows close agreement with an average error of 1.25% and maximum error of 6.95%. Eq. 19 is also available for matrix strain-hardening exponents N in the limits of 0.0 < N < 0.5 for self-consistent axisymmetric embedded cell models (particle volume fraction f in the range of 0.05 < f < 0.65 with an average error of 1.59% and a maximum error of 6.68% for the extreme case f = 0.05, N = 0.5), axisymmetric unit cell models (particle volume fraction f in the range of 0.05 < f < 0.55 with an average error of 1.22% and a maximum error of 6.18% for the extreme case f = 0.55, N = 0.5) and for primitive cubic unit cell models (particle volume fraction f in the range of 0.05 < f < 0.45 with an average error of 1.43% and a maximum error of 6.38% for the extreme case f = 0.05, N = 0.5) Results: Combined Cell Model The purpose of the present section is to investigate the mechanical and thermo-mechanical behavior of MMCs (Al/15% vol. Al 2 O 3 aluminum matrix composite (Fig. 10) reinforced with randomly oriented short fibers by applying the Combined Cell Model described in Sec The fibers are well bonded to the matrix so that debonding or sliding at the fibermatrix interface is not permitted. The finite element method (FEM) is employed within the framework of continuum mechanics to carry out the calculations. The uniaxial matrix elasto-plastic stress-strain behavior measured from experiments at room temperature can be described by an exponential hardening law: σ = Eε ε ε 0, (20) σ = σ 0 [ ε ε 0 ] N ε>ε 0, where σ and ε are the uniaxial stress and strain of the matrix, respectively, σ 0 is the flow stress, the matrix yield strain is given as ε 0 = E/σ 0, E is Young s modulus, and N is the strain hardening exponent. J 2 flow theory of plasticity with isotropic hardening is employed with a von Mises yield criterion to characterize the rateindependent matrix material. The flow behavior is different in tension and compression and can be described using the following parameters: E = MPa, ν = 0.33, N = 0.2, and σ tension 0.2 = 225 MPa, σ compression 0.2 = 234 MPa. Figs. 13a and 13b show the numerically obtained stress-strain curves of the composite (M124/15% vol. Al 2 O 3 ) in uniaxial compression (a) and tension (b), respectively. The orientation angles considered here are 0,5,10,15,30,45,60 and 90. The experimental stress-strain curves of elastic fiber (Al 2 O 3 ) and elastic-plastic matrix (Al/12% vol. Sialloy) are also shown in these figures. Compositestrengthening increases with decreasing the fiber orientation angle from 90 to 0. From 90 to 30 the increase is very small, but it becomes larger and larger from 30 to 0. After the integration by applying Eq. 7 (Sec. 4.2) for both cases of 2D and 3D random orientations with weighting functions f (θ) =1 and f (θ) =sinθ, respectively, we obtain the two stress-strain curves (bold continuous and dashed lines in Fig. 13) for the overall flow behavior of these composites. For all the cases analyzed, the averaged stress-strain curves of composites with 2D random as well as 3D random fiber reinforcements lie in the neighboring of the stress-strain curve of composites with 30 and 60 fiber orientation, respectively. The fact that the stress-strain curves of

16 340 DAMAGE SIMULATION (a) compression (a) compression (b) tension (b) tension Figure 13. Numerical results of (a) compression and (b) tension flow behavior of the M124/15% vol. Al 2 O 3 (3D random fiber orientation, fiber aspect ratio: 200µm/3µm) with different fiber orientations θ [6]. in-plane randomly oriented fiber reinforced composites coincides with those of approximately 30 oriented fiber reinforced composites, has been also reported in [51]. In Figs. 14a and 14b the numerical results obtained for 3D random fiber orientation are compared to the experimental data obtained by uniaxial compression and tension tests, respectively. In the case of compression loading close agreement exists between experiments and simulation in the elastic and plastic regimes. However, at strains above 1.5% the numerical simulation predicts higher strain hardening than observed in the experiments. In the case of tensile loading, close agreement between the experimental measurement and the numerical prediction is obtained only for the elastic regime (see Fig. 14b). The observed deviations between experimental and numerical results can be attributed to the onset of mi- Figure 14. Effects of residual stresses on the overall flow behavior of the M124/ 15% vol. Al 2 O 3 (3D random fiber orientation, fiber aspect ratio: 200mm/3mm) and comparison with experiments. [6] crodamage such as fracture of the brittle constituents of the composite. Such damaging processes have been observed both in metallographic studies and in acoustic emission measurements (see Chapter 3.1). The different deviation in tension and compression may be attributed to the fact that the damaging processes mentioned above are sensitive to the direction of loading [75]. These results indicate that the Combined Cell Model used in this study can be applied successfully to composites with random fiber orientation as long as effects from micro-damage can be neglected. In order to predict the macroscopic stress-strain curve of short fiber reinforced MMCs in tension, a more accurate model including microscopic damage events must be developed (see Sec. 4.3). In a second step, the effects of residual stresses have been estimated using the model. The internal stresses and strains that form during cooling from 400 C to room temperature were calculated for each cell under the simplifying assumptions that the thermal expansion coefficients of the constituents as

17 4.1 Modeling of Damage in Fiber and Particle Reinforced Composites 341 well as the flow behavior of the matrix alloy are identical in the whole temperature range. Figs. 14a and 14b show, besides the comparison with experiments, the comparisons between computer predictions of the overall flow behavior of MMC randomly reinforced with short fibers, with and without considering the initial thermal stresses. Significant influences of residual stresses on the overall mechanical behavior of short fiber reinforced MMCs were found: in both tension and compression the effective Young s moduli are found to be lower while the yield stresses are increased. Because of the plastic deformation in the local area near the fiber under thermal loading, in these areas the local stress states under mechanical loading are different compared to the case without thermal loading. Under mechanical loading, further flow in some local areas directly after thermal loading reduces the overall stress response at the small strain state. Because metal-matrix hardening takes place in some local areas under thermal loading and the hardening is isotropic, the material in this area is harder than it would be without undergoing thermal loading. With increasing the overall strain, i.e. when higher flow stresses in the local area are reached, the overall yield stresses of the composites will be higher compared to the case when thermal stresses are absent. The composite strengthening including thermal loading naturally depends on temperature change ΔT, difference of thermal expansion coefficients of matrix and fiber Δα and on the value of the yield stress of the matrix. The composite strengthening with thermal loading increases with increasing value of ΔT, Δα and the yield stress of the matrix Results: Statistical Combined Cell Model The SCCM model is applied to MMC materials in static and quasi-static cyclic loading conditions. These results are presented in the following two paragraphs. Static loading In [58, 59] it has been experimentally established that prior to the failure of the composite, fracture of brittle fibers takes place in short fiber reinforced metal matrix composite M124-Saffil under tensile stress. Fig. 15 shows calculated stress-strain curves of the fiber composite M124-Saffil (15% vol. Al 2 O 3 -Saffil fibers) under consideration of fiber failure in dependence of global strain. In Fig. 15a, the Weibull modulus m was varied from 1 to 3, and in Fig. 15b, the characteristic stress σ 0 of fibers, from 500 MPa to infinite. It can be observed that when (a) (b) Figure 15. Comparison of stress-strain curves, for a) different Weibull moduli m with σ 0 = 1000 MPa, and b) different characteristic stresses σ 0 with Weibull modulus m = 1 for metal matrix composite with 3D random short fibers [7]. the global strains are lower than 0.15%, the difference among the numerical and experimental results are very small, because in this area there is hardly any damage in fibers. Close agreement of the calculated stress with the experimental result is found for m = 1 and σ 0 = 1000 MPa. The Weibull modulus m is usually found between 3 and 8 [58]. However, the calculated stress-strain curve for m = 1 deviates from the experimental results. As reported in [58], strong fiber clusters exist in the analyzed fiber composite M124-Saffil (15% vol. Al 2 O 3 -Saffil fibers), but are not considered in the present model. To consider the influence of fiber clusters on the simulation results, a mesoscopic concept has to be established which takes into account accidental changes of fiber volume content and which allows to calculate statistical fiber failure in different fiber cluster areas (see Sec. 5.2). Quasi-static cycling loading The presented Statistical Combined Cell Model is based on the reduction of the effective Young s mod-

18 342 DAMAGE SIMULATION ulus and damage as introduced by the evolution of failure probability of fibers and fiber-matrix interfaces determined by the Weibull damage law. The predescribed procedures are performed on the Al- Al 2 O 3 short fiber composite and compared with test results from [63, 76, 77]. The calculations are performed up to 10 loading cycles and the evolution of damage on the Young s modulus is calculated after each loading cycle. FE simulation results and test results are plotted for comparison in Fig. 16. Close Figure 16. Comparison of experiment and simulation with the reduction of the effective Young s modulus in fatigue of an Al-Al 2 O 3 composite, for 0.15% strain [9]. agreement is found between the experiment and the simulation using the proposed damage model. It is seen that the simulated model shows a slightly lower decrease in the effective Young s modulus. It is assumed that, in reality, matrix cracks influence the composite failure, and hence reduce the effective Young s modulus. Taking also into account matrix cracks would minimize the differences in the reduction of the effective Young s modulus between experiment and calculation Conclusions The transverse elastic-plastic response of MMCs reinforced with unidirectional continuous fibers and the overall elastic-plastic response of Metal Matrix Composites reinforced with spherical particles have been shown to depend on the arrangement of reinforcing inclusions as well as on the inclusion volume fraction f, and the matrix strain-hardening exponent, N. Self-consistent axisymmetric embedded cell models have been employed to predict the overall mechanical behavior of Metal Matrix Composites reinforced with randomly arranged continuous fibers and spherical particles perfectly bonded in a power law matrix. Experimental findings on an aluminum matrix reinforced with aligned but randomly arranged boron fibers (Al/46% vol. B) as well as a silver matrix reinforced with randomly arranged nickel inclusions (Ag/58% vol. Ni) and the overall response of the same composites predicted by embedded cell models are found to be in close agreement. The strength of composites with aligned but randomly arranged fibers cannot be properly described by conventional fiber-matrix unit cell models, which simulate the strength of composites with regular fiber arrangements. Systematic studies were carried out for predicting composite limit flow stresses for a wide range of parameters f and N. The results for random 3D particle arrangements were then compared to regular 3D particle arrangements by using axisymmetric unit cell models as well as primitive cubic unit cell models. The strength of composites at low particle volume fractions were in close agreement except for the modified Oldroyd model. With increasing particle volume fractions f, and strain hardening of the matrix N, the strength of composites with randomly arranged particles cannot be properly described by conventional particle-matrix unit cell models, as those are only able to predict the strength of composites with regular particle arrangements. Finally, a strengthening model for randomly or regularly arranged continuous fibers and particle reinforced composites under axial loading is derived, providing a simple guidance for designing the mechanical properties of Metal Matrix Composites. For any required strength level, Eq. 19 will provide the possible combinations of particle volume fraction f, and matrix hardening ability, N. The flow behavior for Metal Matrix Composites reinforced with 2D (planar) and 3D randomly oriented short Al 2 O 3 -fibers is investigated by Combined Cell Models in conjunction with the FEM. The mechanical behavior of short fiber reinforced Metal Matrix Composites (MMCs) with a given fiber orientation can be simulated numerically by averaging results derived from different cell models. These cell models involve two 2D models and two 3D models representing a single fiber in three principal orthogonal planes in the composite. Stress-strain curves have been calculated for MMCs reinforced with 2D randomly planar and 3D randomly oriented short fibers by an appropriate integration of results of all fiber orientations. The numerical results are compared with experimental data of a fiber reinforced aluminum alloy composite obtained in uniaxial tension and compression tests. Close agreement is obtained between experimental results and the predictions of the model in the regimes where no microdamage is observed experimentally. Finally, the effects of residual stresses have been estimated using the model. Both in tension and in compres-

19 4.1 Modeling of Damage in Fiber and Particle Reinforced Composites 343 sion Young s modulus is found to be lower while the yield stresses are increased compared to the case when residual stresses are absent. Applying the Statistical Combined Cell Model, which includes damage effects in form of a statistical Weibull approach, also the quasi-static cyclic behavior of the MMC composite could be investigated Polymer Matrix Composites (PMCs) (a) Material PMCs are frequently reinforced with strong continuous or short fibers (Chapter 2.1.3). In the case of short fiber reinforced PMCs, a specific orientation distribution is observed. Their mechanical properties are highly dependent on their structure. The complexity of such affecting parameters impedes a complete theoretical description of the behavior and the failure properties of these composites. In this respect, a micromechanical analysis of the local failure process opens a possibility to predict the macroscopic failure property of composites [1 3]. In this section, on the basis of [7], the Combined and the Statistical Combined Cell Model [6] are applied to describe the overall flow behavior of composites reinforced with short fibers (polypropylene matrix with 8.1% vol. glass fibers) with good and sparse adhesive strength (Fig. 17). The failure of such composites with different fiber volume fractions is investigated using Statistical Combined Cell Models based on Combined Cell Models [6] and Weibull statistical approach [58, 59]. For this purpose, an injection molded polypropylene was used. Injection molded specimens usually show a complex layered morphology with fibers mainly oriented in processing direction at the skin layer and normal to it in the center of the specimen (core layer) due to shear and elongation flow. Applying a push-pull processing the melt can be pushed through the cavity several times forth and back using a two component injection molding machine. Push-pull processing leads to highly oriented fibers (Fig. 17a) also in the center of the specimen while the thickness of the core layer is considerably reduced. This fact is expressed in a high value of the effective Young s modulus of the composite in push-pull direction ( ) compared to the effective Young s modulus perpendicular ( ) to it [78]. The properties of the matrix and fiber as well as the composite are given in Tab. 2. (b) Figure 17. Micrograph of a polymer matrix with 8.1% vol. glass fibers with (a) good adhesive strength [7], and (b) sparse adhesive strength. Properties Value Young s modulus of matrix 1.9 GPa Young s modulus of fibers 72.0 GPa Aspect ratio of fibers 25 Diameter of fibers 10 µm Number of push-pull cycles 4 Fiber content 8.1% vol. Young s modulus (composite) 5.5 GPa Young s modulus (composite) 2.5 GPa Table 2. Properties of the matrix, the glass fibers and the push-pull processed composite [9, 78] Results: Combined Cell Model (CCM) In this Chapter, CCMs [6, 57, 58] are applied to describe the overall flow behavior of composites reinforced with short fibers and polymer matrix (Fig. 17). As described in Sec , the overall flow behavior of composites with a certain fiber orientation can be calculated by an appropriate integration over all fiber orientations. The numerical results are compared to experimental data of short fiber reinforced Polymer Matrix Composites under

20 344 DAMAGE SIMULATION tension. Close agreement has been obtained at small strains between experiments and numerical predictions by using these models. The larger the strain, the stronger the deviation between experiments and numerical predictions (Fig. 18). In order to predict Figure 20. Averaged fiber orientation over thickness of a PA6GF30-specimen (simulation). Figure 18. Comparison of experiments and FE predictions for polypropylene matrix composite with planar random short fibers [7]. termined via microwave anisotropy measurements (details of the method can be found in Chapter 1.2.3) has been measured in the region near the hole within a measurement field of 60 x 45 mm 2 and a raster of 1,25 x 1,25 mm 2. The experimental results are dis- the flow behavior of short fiber reinforced composites in tension to a higher accuracy, fiber cracking and fiber-matrix debonding can be taken into account [58], which is done in Sec Consideration of complex fiber orientations The injection molding process leads to a complex arrangement of the fibers in the cavity due to shear flow and elongational flow. The different orientations of the fibers result in anisotropy of the component properties. Using inserts to fabricate plates containing a hole, leads to the splitting of the melt front and finally to the formation of a weldline as a result of the joining of the two melt fronts (Fig. 19). Weldlines are known to be mechanically weak regions of the component (compare Chapter 2.1.3). In Figure 19. Successive patterns of filltime of the melt (polyamide 6 reinforced with 30 weight-% glassfibers - PA6GF30) at different stages of the process: 0.25 s, 0.96 s, 1.02 s and 1.17 s (end of filling). this region, the fiber orientations show great variations (Fig. 20). The fiber orientation distribution de- Figure 21. Experimental microwave orientation of a PA6GF30-specimen (see Chapter 2.1.3). played in Fig. 21 and compared to the simulation result in Fig. 20. Horizontal orientation in the left part of the microwave orientation image, the flow around the hole and the coalescence to the weldline with a horizontal orientation of the fibers can be identified. Since the measurement field is larger than the raster distance, artefacts appear near the free edges (here: hole). The comparison of the simulated fiber orientation (Fig. 20) with experimental results of microwave anisotropy investigations shows a good correlation (Fig. 21). The fiber orientation was simulated for several layers (Fig. 22) in each element with the use of an injection molding simulation software (Moldflow Plastic Insight). The linearelastic results were transferred (Fig. 23) to a strength analysis FE code (ABAQUS). The simulation procedure is shown in Fig. 23. As a result, the effect of fiber orientations on the local mechanical behavior as well as the macroscopic properties of a model plate containing a hole and a weldline were investigated by applying the CCM and compared to the results of the Tandon-Weng model [79]. Fig. 24 shows the result of the linear-elastic simulation. According

21 4.1 Modeling of Damage in Fiber and Particle Reinforced Composites 345 Figure 22. Multilayer model. Figure 25. σ/ε-graph of the PA6-component with isotropic and anisotropic properties (CCM model) compared to the Tandon-Weng model. Figure 23. Procedure of simulation. to the Tandon-Weng model, the lowest values of the stiffness E 11 appear near the injection point, behind the hole and at the flow end. The maximal stiffness is reported with 9.9 GPa (red regions). The resulting macroscopic stiffness of the global component is calculated to be 7.1 GPa. In a second step, the CCM is used to calculate the global mechanical tensile properties of the component (polyamide 6, 30% glass fibers) taking into account the local fiber orientation, as shown in Fig. 20. Elastic and elastic-plastic properties are considered. The results of these simulations are shown in Fig. 25. The solid line, which represents the results of the Combined Cell Model, is compared to the isotropic elastic-plastic properties (dashed line) and to the stiffness prediction of the Tandon-Weng model (straight line). The anisotropic simulation exhibits a stiffness of 6.9 GPa compared to 6.7 GPa of the isotropic model. These two results are in the same order of magnitude as the results of the Tandon-Weng model (7.1 GPa). The differences between isotropic and anisotropic simulations are attributed to the fact that the CCM model does not take into account fiber-matrix debonding and fiber failure Results: Statistical Combined Cell Model The two Weibull parameters, for interface and fiber failure for the polypropylene (Sec ), are numerically identified by using the data from micromechanical models and the calculated finite element results to fit the experimental curves. For this purpose, unit cell models with fibers reinforced composites and the stress-strain behavior due to debonding and fiber breaking are calculated. Tensile test data are taken from experimental tests at IKP, University of Stuttgart [78]. The simulated Weibull Figure 24. Local stiffness E 11 on the basis of the simulated fiber orientation distribution. Figure 26. The Weibull curve is compared to the experimental curve to determine Weibull parameters [9]. curves are calculated using Eqs. 12 and 13. Then,