This paper aims to investigate the comprehensive influence of three microstructure parameters (fiber cross-section shape, fiber volume fraction, and fiber off-axis orientation) and strain rate on the macroscopic property of a polymer matrix composite. During the analysis, AS4 fibers are considered as elastic solids, while the surrounding PEEK resin matrix exhibiting rate sensitivities are described using the modified Ramaswamy-Stouffer viscoplastic state variable model. The micromechanical method based on generalized model of cells has been used to analyze the representative volume element of composites. An acceptable agreement is observed between the model predictions and experimental results found in the literature. The research results show that the stress-strain curves are sensitive to the strain rate and the microstructure parameters play an important role in the behavior of polymer matrix.
In the last few decades, polymer matrix composite materials (PMCs) have been developed rapidly to meet the demands for better materials with higher standards of performance and reliability in structures and machines [
Polymers are known to have a strain rate dependent deformation response that is nonlinear above 1 or 2% strain [
On the other hand, there are also many macromechanical and micromechanical models to predict the behavior of composite materials subjected to different strain rates [
Compared with macromechanical model, which considered composites as anisotropic medium with homogeneous distribution, the micromechanical model only needs to test the ingredient properties of composites, while macromechanical model needs to do repetitive experiments for composites [
In this paper, the rest outline is as follows. Section
The two-dimensional generalized method of cells is a micromechanical model developed originally by Paley and Aboudi [
When a micromechanical approach is used to model the mechanical response of fiber reinforced composites with periodic microstructures, a proper RVE is required to represent the microstructures of the materials such that the overall composites responses can be predicted directly from the representative volume element. In this study, three kinds of fiber cross-section shapes, such as square, circular, and elliptical, were considered as shown in Figure
Three kinds of fiber shapes.
Square
Circular
Elliptical
In the GMC analysis, the representative volume element is usually divided into
A typical RVE divided into
Based on the displacement continuity on the interface of the adjacent subcells in conjunction with the periodicity condition of the RVE, the relation between overall strain and the subcell strain is expressed as
At the same time, the interfacial traction continuity conditions can be expressed as
For each subcell of composites, the constitutive relationship of each subcell can be written as
Substituting (
Based on the homogenization theory, the overall stress of the RVE can be written as
Substituting (
It should be noted that the elements of matrixes
The matrix viscoplastic constitutive model is based on the modified Ramaswamy-Stouffer viscoplastic state variable model. The Ramaswamy-Stouffer viscoplastic state variable model [
The elastic strain rate can be obtained according to the time derivative of Hook’s law. The inelastic strain rate is defined in the following form:
The relation between the internal stress rate,
The term
In the modified Ramaswamy-Stouffer model, in order to consider the effect of hydrostatic stresses, (
The normal terms in the above expression are the same as the original definition while the shear terms are modified and can be written as
In (
Through the above introduction of the modified Ramaswamy-Stouffer model, it can be seen that the model does not depend on the yield rule and the inelastic strains are assumed to be present at all values of stress. Therefore, there is no need to judge whether the material is in elastic or plastic stage.
To verify the ability of the micromechanics model and the viscoplastic constitutive model in the prediction of rate effects of composites several examples are considered and discussed in this section. The material considered here is a composite composed of carbon AS4 fibers in a PEEK thermoplastic matrix. For the AS4 fibers, the longitudinal elastic modulus is 214 GPa, the transverse and in-plane shear modulus is 14 GPa, the longitudinal Poisson’s ratio is 0.2, and the transverse Poisson’s ratio is 0.25 [
Material properties of PEEK resin [
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4.0 | 0.4 | 104 | 0.7 | 630 | 310 | 52 | 0.40 |
Stress-strain response of AS4/PEEK [15°] laminate at strain rate of 0.1/sec and 10−5/sec.
Stress-strain response of AS4/PEEK [30°] laminate at strain rate of 0.1/sec and 10−5/sec.
Stress-strain response of AS4/PEEK [45°] laminate at strain rate of 0.1/sec and 10−5/sec.
Figure
Off-axis responses of AS4/PEEK laminate (
Increasing the fiber volume fraction further accentuates the differences in the composite’s transverse response due to the fiber’s cross-sectional shape. Figure
Off-axis responses of AS4/PEEK laminate (
Figure
Off-axis responses of AS4/PEEK laminate (
Figure
Off-axis responses of AS4/PEEK laminate (
Figure
Stress-strain response of AS4/PEEK [
A viscoplastic constitutive model has been employed in the micromechanical method based on generalized model of cells to analyze the inelastic, rate dependent stress-strain response of fiber-reinforced polymer matrix composites with three different microstructures at different fiber off-axis angles condition. The acceptable agreement between the model predictions and experimental results shows that the proposed model can well predict the behaviors of AS4/PEEK composite. At the same time, from the predicted results, the following conclusions are obtained. The AS4/PEEK composite is a kind of rate dependent material. When the strain rate changes from 10−5/sec to 0.1/sec, the composites provide an effective increase in the flow stress while the elastic behavior almost remain unchanged. The effects of fiber cross-sectional shape on the behavior of AS4/PEEK composite are related to the fiber volume fraction and fiber off-axis orientation. When the fiber volume fraction is smaller than 0.15, it can be seen that the composites response is hardly affected by the fiber cross-section shape; with the increasing of fiber volume fraction and fiber off-axis orientation, the effects of fiber cross-sectional shape become more obvious. Among the three kinds of fiber shapes, the stiffest response is obtained for the composites with the square fibers and the most compliant response for the composites with the circular fibers. The increasing of fiber volume fraction can improve the stiffness of AS4/PEEK composite. However, for the elliptical fiber, the maximum allowable fiber volume fraction is 0.59 in the case of fibers with an aspect ratio of 4/3, so it should be noted that the elliptical fiber may not be chosen when the fiber volume fraction needed is big. The influence of fiber off-axis orientation on the stress-strain curves of AS4/PEEK composite is very large. The response of composites decreases obviously when the off-axis orientation changes from 15° to 45° and then increases from 60° to 90°. So when the composites have been chosen to bear the load, the fiber off-axis orientation should be paid attention to.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China (nos. 51175401 and 51335006), the Research Fund for the Doctoral Program of Higher Education of China (no. 20120201110028), and the Program for Changjiang Scholars and Innovative Research Team in University.