A Homogenization Procedure for the Numerical Simulation of Mechanical Behavior of CFCC by Considering the Overall and Local Anisotropic Damage

: An anisotropic damage constitutive model was presented to characterize mechanical behavior of continuous fiber-reinforced ceramic matrix composites (CFCC) with two-scale damage. An overall fourth-rank damage effect tensor was introduced to account for the overall damage of the composite system. In addition, two local (matrix and fiber) fourth-rank damage effect tensors were introduced to account for the local effects of damage experienced by both the matrix and fibers. The overall and local damage tensors were correlated together using homogenization procedure. In terms of the homogenization methods, the effective elastic properties were obtained, and the stress and strain concentration factors were derived for damaged composites. The model was applied in detail to the unidirectional laminate that was subjected to uni-axial tension. The results were compared well with experimental data. The effects of important parameters such as the fiber volume fraction and the damage material parameters on the nonlinear behavior of the composites were investigated. The model provided a useful tool for understanding the overall dependence of stress-strain behavior on all the underlying constituent material properties. (CFCC), model


INTRODUCTION
Continuous fiber-reinforced ceramic matrix composite materials (CFCC) play an important role in the industry today through the design and manufacture of advanced materials capable of attaining higher stiffnessdensity and strength-density ratios. Since the constituent materials are both brittle, and the non-linear response associated with ceramic-matrix composites is a direct result of damage interactions between its two linearelastic constituents, fiber and matrix. Micro-structural interactions associated with the constituents of the composite prevent catastrophic failure and promote toughness through load transfer and energy dissipation, and it is important to analyze and capture these phenomena in CFCC plates. Although the literatures were rich in the developments of the CFCC technology [1][2][3][4][5], most of them were focus on the experimental work, and the theoretical analyses were limited on classical analysis, such as shear-lag model or modified shear-lag model. These models attempt to estimate the material response by assuming simplified damage configurations (such as uniformly-spaced, infinitely-long matrix cracks, regular array of fibers, etc.) within the composites. The stochastic nature of brittle failure, however, has forced these models to rely on empirical data, thereby reducing the utility of the numerical models.
The overall deformation of the composite depends on the microscopic flaws in the matrix and fibers. Micromechanics is widely employed to relate the overall properties of strongly heterogeneous media with the properties of the constituents and the microstructure. In this context, the homogenization method usually qua-lifies the passage from the micro-to macro-scale. The advantage of homogenization procedure is that the physical details contained on the smaller scale are not lost, whereas they are not present at all when the phenomenological is utilized [6][7][8]. Furthermore, the macro-scale model may in fact be easier to be constructed by this method since the macro-scale properties are usually dependent on simpler micro-scale properties. This method has been applied in a variety of applications, for example, in studying mechanical behavior of a heterogeneous lamina [9][10] and laminates structures for armored vehicles [11], in predicting the effective elastic properties of composites [12], and in analyzing the mechanical behavior of elastic-perfectly plastic and strain hardening fiber-reinforced composites [13][14]. The number of literatures covering micro-mechanics applied to CFCC is rather limited. Hild et al. [15] employed continuum damage mechanics to derive the macroscopic stress-strain law for the damaged composites: matrix cracking and interface debonding are described through micro-scopic damage variables. Zhang et al. [16] focused their attention upon the evaluation of stress intensity factors associated with cracks in the matrix and at the fiber-matrix interface, assuming the un-cracked matrix to be linear elastic. It is noticeable that most of the studies have assumed that the fibers do not fail during matrix cracking, and there is little work on the application of the method to CFCC with anisotropic damage in the matrix and fiber simultaneously.
The importance of damage in the mechanical behavior depends upon a balance between the respective elastic properties of the matrix and fibers. The factors to determine the mechanical responses are critical for the Numerical Simulation of CFCCs with Multi-scale Damage development of optimized composite systems and the design of structural components using composites. By means of suitable selection and combination of single components, the properties of the composite can be varied over a wide range. In this paper, the asymptotic homogenization method is first used to determine the effective elastic properties of the undamaged CFCC with different fiber volume fractions. The results are verified by the results of the rule of mixture and the average method. Following the obtained effective elastic properties, the continuum damage mechanics is used with a micro-mechanical composite model to analyze two-scale damage in CFCC. Both overall and local damage variables are introduced to model the overall and local damage effects. The local damage relations are linked to the overall response by a homogenization procedure, and the stress and strain concentration factors are derived for the damaged composites. The subject of this paper is to predict the stress-strain and failure behavior of unidirectional CFCC as a function of the matrix and fiber damages, and to investigate the influence of important parameters on the mechanical behavior of composites. The effects of important parameters such as the fiber volume fraction and damage material parameters on the nonlinear mechanical behavior of the CFCC are examined by modeling the macro-structure.

Definitions and Assumptions
Considering a body of CFCC in the initial undeformed and un-damaged configuration C0. Let C be the configuration of the body that is both damaged and deformed after a set of external loading act on it, and C is the state of the body after it had deformed without damage. Assume that the representative volume element (RVE) in C0 is statistically homogeneous, and is free of voids and cracks initially. Assume also that the composite material is loaded by an overall stress or strain field which is followed by increments of loading. The overall stress and strain fields are assumed to be uniform. The effective overall stress is defined in the configuration C as the stress in a perfectly-bonded twophase composite free of cracks or voids.
The composite material is assumed to consist of elastic fibers and an elastic matrix. The fibers are continuous, aligned and equally spaced. In the following, quantities are defined in the configuration C of the overall composite system. Barred quantities are defined in the configuration C of the overall composite system. Quantities with a superscript M or F refer to matrix or

The Mechanical Behavior of Composites
In this section, the relations between the local (matrix and fiber) and overall (composites) relations are presented in the configuration C [17], In the configuration C, the effective stress tensor where B,nijkl is a fourth-rank tensor indicating the elastic phase stress concentration factor. As a result of volume integration and averaging of the local stress fields, the following relation is obtained between the local (matrix and fiber) stresses and the overall stress in C: where VM and VF are the matrix and fiber volume fractions, respectively, given by: where V is the total volume of the RVE.
As the same method, the local-overall relationships C are presented. Upon volume integrating and averaging the local stress fields [18], the following local-overall relation is obtained for the effective spatial strain tensor: where the appropriate relations for the effective matrix and fiber strain tensors are used as follows: where Anijkl is a fourth-rank tensor denoting the elastic phase strain concentration factor.

Anisotropic
Damage Analysis There are two steps that can be followed in order to develop a continuum damage model for a composite system consisting of fibers and a matrix. First, one considers that damage in the overall composite system is whole continuous.
At this step, the model will reflect various types of damage mechanisms such as void growth and coalescence in the matrix, fiber fracture, debonding and delamination, etc. It should be noted that at this step, no distinction is made between these types of damage as they are all reflected through the fourth-rank overall damage effect tensor Mijkl. In the second step, one considers the damage that the matrix and fibers undergo separately such as nucleation and growth of voids and void coalescence for the matrix and fracture for the fibers. In this case, two fourth-rank matrix and fiber damage effect tensors MMijkl and MFijkl are introduced that reflect all types of damage that the matrix and fibers undergo.
Subsequently, the local-overall relations are used to transform these local damage effects to the whole composite system.
Following the first step outlined above and utilized an overall damage effect tensor M for the whole composite system, the overall effective Cauchy stress Dongmei Luo and Shigeo TAzoNo Substituting Eq. (6) into Eq. (1), we can obtain: Following the second step discussed at the beginning of this section, we can establish the following local transformation equation: Comparing Eqs. (7) and (8), the following relation between the phase stress tensor and the overall stress tensor can be obtained: where stress concentration factor that includes geometrical and damage related effects as can be seen from Eq. (10).
Similar relations are exist for strain concentration factor: where Substituting Eq, (8) into Eq. (2) and simplifying, the required relationship between the local damage effect tensors MM, MF and the overall damage effect tensor M becomes: The above equation is an explicit relation between the effective local concentration factors and the overall damage effect tensor. It is clear that once the local (matrix and fiber) damage mechanisms have been described through the tensors MM and MF, then the overall damage in the composite system can be described which includes the matrix and fiber related damage as well as the damage resulting from the interaction of the two phases such as debonding.
The local (matrix and fiber) damage tensors are given by Voyiadjis and Kattan as follows [19]:

Evolution of Damage
The criterion for damage evolution used here is that proposed by Lee et al [20] and is given by the function g(y, B) defined by: The generalized thermodynamic force conjugate to the kinematic damage variable [21] is: Combining the energy dissipation inequality (22), and the definition of damage evolution (25) with Kuhn-Tucker relations, the damage evolution conditions become: 3. HOMOGENIZATION METHOD AND ITS APPLICATION

The Theory of Homogenization Method
The homogenization procedure has already been fully derived in Pellegrino et al. [6], therefore we only briefly recall it here. We take into consideration a periodic composite material, which is a material made of a large number of regularly distributed and equal unit cells with linear boundary conditions on them, and only a unit cell need be considered for microscopic analysis.
An asymptotic expansion of the displacement field is: With consideration of the indirect differentiation rule, the strain field can be expanded as: The stress field: The stresses and displacements fields satisfy the following equations on macroscopic domain: boundary portion (as seen in Fig. 1 (a)). The subscript pairs with parentheses denote the symmetric gradients defined as: Substituting Eqs. (31), (32) into Eqs. (33) and (34), we have: Due to the linearity of Eq. (36), the following characteristic function is introduced: as a polarization function.
The weak form boundary value problem defined by Eq. (39) can be solved by finite element method. This process has been implemented into the ANSYS program through a user-supplied material model subroutine [23].
The elastic homogenized stiffness is The reliability of the adopted numerical model is first checked by the calculation of elastic properties for different fiber volume fractions in the linear elastic field. The considering composite materials are Nicalon/CAS composites.
In Fig.2 the macroscopic elastic properties are plotted vs. the fiber volume fraction, VF. The data plotted in Fig.2 include the numerical results given by the present model, the theoretical prediction by the rule of mixture and average method in [24]. The "rule of mixture" model has a low-estimation, and the average method has an over-estimation to the elastic properties of composites. The homogenization method gives a moderate result.
With the increase of fiber volume fraction, the agreement between "rule of mixture" predictions and the homogenization method is definitely encouraging. However, the "rule of mixture" predictions are only limited on linear-elastic composites, and the results obtained from average method are actually the average relation between the stress and strain, and they are not the real material properties. Homogenization method is a rigorous approach to determine the effective material properties of composites with linear and non-linear mechanical behavior.

The Influence of Fiber Volume Fraction
The influence of fiber volume fraction on mechanical behavior of damaged composites is studied here. Figure 3 shows the dependence of the tensile strength on the fiber volume fractions. Four stages can be divided for Fig. 3. The laminate behaves linearly up to VF=30%, and it is controlled by the matrix strength, and the macroscopic brittle behaviors are obvious shown in Fig.4, in which the macroscopic stress-strain curves show a strain-softening behavior. In this stage matrix cracking is the main damage. With the increase of fiber volume fraction, the stress increases gradually, and saturation of matrix damage tendency by shown by a "tough" stable stage of the curves.
For VF= 30-40%, the stress where the stress nearly keeps constant or even has a little decrease, and the stiffness (secant modulus) has a sudden drop. Then the stress continues to increase till fiber failure, which represents the stress of fiber experienced. For VF=40-60%, the stresses are controlled mainly by the fiber, and the stress-strain curve experiences another linear stage, which represent the fiber linear elastic strength increases very slowly and the failure strength almost keeps constant. It means that it has no any meaning to improve the strength of composites by only increasing fiber volume fraction after the fiber volume fraction is big enough. Figure  3 shows that the stress-strain curves are very sensitive to fiber volume fractions, and the maximum stress of composites increases with the increase of fiber volume fraction. The loading sharing between the fibers and the matrix is modified with the change of fiber volume fraction, and the fiber volume fractions are optimized as VF=30-40% by considering the strength changes of CFCC in parameter studied. This is consistent with the choice of typical specimen in the experiment.

The Influence of Damage Material Parameters
Damage material parameter is another important factor to influence the mechanical behavior of composites. The different stresses of matrix crack initiation can be observed in Fig.5. The maximum stress of composites decreases with decreasing fiber damage parameter.     When the fiber damage parameter is equal to matrix damage parameter, the fiber damage becomes so big that its influence on mechanical behavior of composites cannot be ignored, and this induces the brittleness of composites.
When the fiber damage parameter is big sufficiently, the stress-strain curves tend to stable, and fiber damage is ignorable. From Fig. 6, we can find that the damage is most sensitive to the fiber damage parameter between kFd=0.15-10.0. Figure 7 shows that matrix damage parameter has no influence on the trends of stress-strain response, and it influences only the stress values in stable stage.

Stress-strain Curves for Unidirectional CFCC
For unidirectional composites with VF=35%, we suppose that the damage parameters are selected as kFD=50.0, and kMD=0.15, respectively. The elastic constants of the constituents are listed in Table 1. The typical stress-strain curves are shown in Fig. 8. Both the longitudinal and transverse strain responses are plotted. The numerical results are in good agreement with that in S. W. Wang's experiment [4]. They summarize important trends in the mechanical behavior of brittle-matrix composites. The first deviation from linearity in the curve of stress versus strain is at a longitudinal strain of 0.2%. Towards the higher stress end, the stress-strain curves show upward convexity. The non-linear stressstrain relations show that the CFCC is damage sensitive, and the curved domain of deformation results essentially from transverse cracking in the matrix. The cracks are arrested by the fibers and they are deflected at the fiber/matrix interfaces causing fiber debonding [25]. Table 1. Experimental values of material properties [4].

The Damage Analysis
The basic damage phenomena in unidirectional CFCC involve multiple micro-cracks or crack is formed in the matrix, perpendicularly to the loading direction. These cracks are arrested by the fibers and deflected at the interface between the fiber and the matrix [25]. The main idea in this paper is to describe the local damage evolution in matrix and fiber, then the relationship between the overall damage and the local damage is established by using the stress concentration factors. Comparing the evolution of damage in the matrix and fibers in Fig. 9, we can fmd that the damages of composites are mainly controlled by the matrix damage. This is consistent with the experimental results in [4].
The experimental results also show that no fiber fracture and no clear indication of fiber/matrix debonding in unidirectional composites. Figure 10 is the relationship between overall (local) damages and strain. The overall damage redistribution is clearly indicated in Fig. 10, in which the fiber/matrix debonding can be described by D12. The variations of the damaged elastic properties are shown in Fig. 11. There are obvious reductions in the elastic properties of the unidirectional composites as a consequence of damage. With the increase of the fiber volume fractions, the failure strains decrease, and the brittleness of composites is strengthened. Fig. 10. The relationship between the overall (local) damage and strain.

CONCLUSIONS
The micro-mechanical constitutive model was established to predict the mechanical behavior of CFCC with anisotropic damage in the matrix and fiber. An overall damage variable was introduced to model damage in the composite system while two local damage variables were used to model damage in the matrix and fibers. The overall and local damage variables were then related to the matrix and fiber volume fractions and stress concentration factors. The homogenization method was used to derive the elastic material properties, and the stress and strain concentration factors of the undamaged and damaged composites. The main conclusions were summarized as follows: (1) The homogenization method is a rigorous approach to determine the effective elastic properties and the stress and strain concentration factors of undamaged and damaged composites.
(2) The general influence of important model parameter such as the fiber volume fraction and damage material parameters on the mechanical behavior of the CFCC was studied. The optimization of parameters was analyzed by considering the strength variation of CFCC. (3) The continuum damage mechanics was used with a micro-mechanical composite model to analyze damage in composite materials. Both overall and local damage variables were used to model the overall and local damage effects. The continuous damage evolution in matrix and fiber can be captured, and damage redistribution can be found from the comparison between the overall damage and local damage.