1. Introduction
In the last few decades, nonlinear micromechanical models of multiphase composite materials have been the subject of interest for many investigations. The aim of these models is to study the effect of volume fraction and the interphase effects on generating the nonlinear effective response of the composite materials. A detailed review of this class of nonlinear micromechanical methods can be found in [
1,
2,
3], among several others. In addition, the prediction of the local deformation process of periodic alternate multiphase materials requires the detailed description of the spacial description of the soft and hard phases subject to different kinds of loading [
4]. Hence, the formation of constitutive equations that govern the large deformations of composite materials, which consist of elasto-viscoplastic phases, is necessary for the modeling and analysis of their deformation behavior. Eutectic Cu-Ag composites form an example of a layered two-phase composite structure that consists of alternate layers of Cu and Ag lamellae inside the grains.
Semi-analytical homogenization such as a generalized method of cells (GMC) has been developed by [
1]. This method enables the modeling of complex unit-cell morphologies and is able to study the mechanical response of heterogeneous materials. The generalized method of cells (GMC) is used to predict the thermoelastic behavior of multiphase composites by including the thermal tangent tensor in the constitutive law. The effective tangent and thermal tensors are functions of the mechanical and thermal concentration tensor, respectively. The concentration tensor indicates the relationships between the local and overall behavior. Aboudi [
3] also proposed an expanded form of the micromechanical approach to predict the hyperelastic response of viscoelastic multiphase composites. This approach uses viscoelastic, as well as mechanical concentration tensors to form relationships between the local and effective deformation. Moghaddam et al. [
5] apply the GMC for determining the elastoplastic behavior by employing the single-crystal plasticity constitutive model. Aboudi proposes a method that can predict the viscoplastic, as well as thermoelastic behavior of multiphase composites. The proposed high fidelity generalized method of cells (HFGMC) includes second order displacement expansion in the multi-cell configuration of the repeating unit-cell (RUC). The HFGMC has been developed in order to predict the local distribution of stresses and strains in a refined manner that previously could not be generated by the GMC [
6].
Numerical micromechanical models for two-phase repeating unit-cell (RUC) can be achieved using phenomenological material models at lower length scales, such as finite element, viscoplastic self-consistent and Taylor models [
5,
7,
8]. Another alternative model directly specified for periodic composites is the high fidelity generalized method of cells (HFGMC). HFGMC can be considered as a higher order extension of the generalized method of cells (GMC) [
6]. In the HFGMC approach for two-phase composites, the RUC consists of different numbers of subcells. In this approach, the higher order displacement expansion is used in the subcells, utilizing both displacement and stress microvariables to satisfy on an average basis, the equilibrium and continuity (traction and displacement) equations and periodic conditions across the interface between the cells. In Aboudi et al. [
3], the HFGMC computational approach has been applied to predict the deformation behavior of various types of multiphase composite materials (elastic, viscoelastic, thermoviscoelastic and electroelastic). In the current work, the HFGMC was enhanced to include crystal plasticity material modeling for two-phase copper-silver composites.
This study presents general formulations for the HFGMC modeling framework implemented with the crystal plasticity material model for two-phase Cu-Ag composites undergoing large deformations. The behavior of composites with elasto-viscoplastic material constituents undergoing large deformations is allowed to deviate far away from equilibrium and is modeled by the HFGMC micromechanical theory. For a given constitutive model, both the micromechanical and macromechanical governing equations based on the homogenization technique for periodic composites are developed, wherein the instantaneous mechanical concentration tensors that relate the local induced deformation in the phase to the current externally-applied deformation gradient are given. Furthermore, the macroscopic constitutive equations of the two-phase composite in terms of its instantaneous stiffness tensors are also provided.
The structure of this paper is outlined as follows.
Section 2 deals with the experimental observations for both samples.
Section 3 introduces the constitutive equations of a finite elasto-viscoplastic material model based on the isomorphy concept of the elastic laws [
9].
Section 4 describes the computational methodology for HFGMC formulation.
Section 5 presents results in terms of mechanical behavior, and finally, conclusions are given in
Section 6.
Notation: We use the symbolic notation given in the continuum mechanics text book of Bertram [
9]. Scalars, vectors, second-order and fourth-order tensors are denoted by
a,
,
and
, respectively. The scalar, dyadic and Rayleigh product are given by ·, ⊗ and
, respectively, where
,
,
. : denotes the double contraction between tensors, i.e.,
.
,
and
denote the transpose, the inverse and the material time derivative of a second-order tensor
. The linear mapping of a second-order tensor
by a fourth-order tensor
is written as
.
2. Experimental Methodology
The two cold-drawn Cu-Ag rods having diameters of 12.42 mm (
) and 6.73 mm (
) are produced by die casting. The volume fraction of each phase in the polycrystals is given as
,
. Both samples (
,
) exhibit a crystallographic texture with different strengths, respectively. X-ray analysis is used to determine the bulk texture of two different cold-drawn Cu-Ag rods. X-ray diffraction (XRD) measurements are performed by using an X-ray tube with a chromium-anode in point focus mode and a 1D detector with secondary
filtering. The measured texture from both samples (
,
) has been approximated by 100 grains as a compromise between the precision and computational costs. More details about the initial texture and approximated texture (100 grains) are given in [
10,
11]. The compression tests are carried out for both cylindrical samples on a universal testing machine. These tests are operated at room temperature for a constant strain rate of
s
. The results are reported in terms of true stress (
) and compressive strain (
).
4. Computational Methodology
The HFGMC micromechanical modeling theoretical framework with periodic microstructure is shown in
Figure 1. The composite with a periodic microstructure is shown in
Figure 1a with respect to the initial global coordinates (
,
,
). The repeating unit-cell (RUC) geometry of a two-phase periodic composite is divided into an orthogonal number of subcells, defined with respect to initial local coordinates (
,
,
) (see
Figure 1b). The RUC of the two-phase composite is separated into
,
and
subcells in the
,
and
directions, respectively. Each subcell contains a different homogeneous elasto-viscoplastic material. The dimensions of subcell (
) along the 1, 2 and 3 axes are denoted by
,
and
, respectively. In each subcell, a local coordinate system (
,
,
) is introduced, and the origin is located at the center of the subcell (see
Figure 1c).
In the framework of HFGMC micromechanical formulation, the increment of the displacement vector is expanded to second-order as follows:
where
consists of the externally-applied mechanical loading, and the unknown coefficients
are determined, as shown in the following, by implementing the constitutive relations, equilibrium equations together with the interfacial and periodic conditions in the average (integral) sense.
The increments of the components of the deformation gradient tensor are:
After lengthy manipulations [
3] in which the equilibrium equations in the subcell, interfacial conditions between the subcells and periodicity conditions are imposed in the average sense, the following equation is obtained:
where
is a fourth-order concentration tensor, which relates the local increment of the deformation gradient in the subcell
to the externally-applied one
. Consequently, once the increment of the deformation gradient in the subcell has been established, the increment of the first Piola–Kirchhoff stress (
) in the subcell can be readily determined. Thus, Equation (
17) can be rewritten as follows:
where
is the fourth-order tensor of the material within subcell
.
In the numerical implementation of HFGMC, the analytical derivation of the HFGMC model [
20] is implemented and coupled with the crystal plasticity material model for two-phase Cu-Ag composites. Each subcell contains either Cu or Ag material constituents. Each subcell in the RUC is labeled by the indices (
) with
,
and
. The material parameters for both samples (
,
) are listed in
Table 1 [
11]. The repeating unit-cell (RUC) is subject to uniaxial compression for two different textured Cu-Ag samples (
,
) to study the deformation behavior.
5. Numerical Results
In order to illustrate an application of these constitutive equations, we consider two different textured samples (
,
) whose material properties are given in
Table 1.
Figure 2A,B illustrates the uniaxial mechanical response calibrated from the proposed elasto-viscoplastic HFGMC model compared to the experimental and finite element simulation results under isothermal conditions.
In
Figure 2A, the Cauchy stress
variations with
are caused by the application of uniaxial compression in the Z direction of the large Cu-Ag sample
. In addition comparisons between the true stresses based on the experimental investigations and finite element simulations [
11] of the two-phase Cu-Ag sample
are shown. The slight difference in the yield stress predicted by the HFGMC method may be attributed to the fact that in the HFGMC model, the constitutive equations and boundary conditions are implemented in a point-wise manner and in the average sense, respectively. The correspondence of the HFGMC plot to the experimental data and finite element simulated curve is reasonable.
In
Figure 2B, the composite mechanical response to uniaxial compression loading of the two-phase Cu-Ag sample
is shown by the macroscopic mechanical response (
vs.
). Besides, the comparisons between the experimental and finite simulation results [
11] are shown. The HFGMC deformation behavior is in reasonable agreement with the experimental and finite element-simulated curve. However, there is a slight difference in the yield stress and strain hardening in the HFGMC approach. The difference may be due to the heterogeneity in the microstructure and the identified material parameters for sample
. In addition, the governing equations and boundary conditions are enforced in a point-wise manner and in the average sense, respectively. As seen in
Figure 2A,B, the mechanical response are identical in the elastic and elastoplastic region. It can be seen that there is a smooth transition from the elastic to elastoplastic region. The mechanical response however coincides with the corresponding experimental and finite element simulated results. Here, the nonlinearity of the elasto-viscoplastic material is well observed.
The distribution of the constituents (Cu/Ag), deformation gradient
and the first Piola–Kirchhoff stress
for sample
is examined in
Figure 3.
Figure 3a shows the distribution of the material in the plane
-
at
, i.e.,
on the examined cross-section. Here, Cu and Ag constituents with different color contours are shown in the figure (see
Figure 3a).
Figure 3b,c shows the axial deformation gradient
and the first Piola–Kirchhoff stress
, respectively. The results show that all subcells are under compression
, and the stresses are distributed accordingly for the hard (Cu) and soft (Ag) phase, respectively. The distribution of the
and
for the sample
in the plane
-
at
, i.e.,
, is demonstrated in
Figure 4. It can be clearly observed that subcells are under compression
(see
Figure 4a). The stress distribution shown in
Figure 4b presents that the HFGMC homogenization under a crystal plasticity material modeling predicts the expected distribution at the macroscale.
6. Conclusions
A new elasto-viscoplastic micromechanical formulation for the HFGMC is proposed using elasto-viscoplastic material for Cu and Ag constituents. A HFGMC micromechanical model is employed for the prediction of the mechanical response of two-phase Cu-Ag composites for two different samples (, ) in which each constituent (Cu or Ag) is considered as a rate-dependent elasto-viscoplastic material. The reliability of the micromechanical model prediction is investigated by comparison with the experimental results and finite element simulations, which is valid under a uniaxial compression loading. The stress-strain results for both samples under uniaxial compression show that the effect of plastic deformation behavior is significant.
This HFGMC homogenization model has the advantage of modeling a two-phase periodic composite by discretizing the RUC into quite a few subcells. For example, in the present investigation, 10 × 10 × 10 subcells are sufficient to provide good accuracy. Besides, the running time of the program is quite short (several minutes). In addition, due to the rectangular shape of the subcells, the discretization of the repeating unit-cell is quite simple.
The proposed HFGMC model is shown to be effective for two different Cu-Ag composites calibrated using available experimental data and finite element simulation results. The results of the deformation behavior of the two-phase composites from the proposed HFGMC show the applicability of the proposed model to the wide range of crystalline materials, as well as the multiphase composites.