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This article presents a numerical technique for computing the biaxial yield surface of polymermatrix composites with a given microstructure. Generalized Method of Cells in combination with an Improved BodnerPartom Viscoplastic model is used to compute the inelastic deformation. The validation of presented model is proved by a fiber Bragg gratings (FBGs) strain test system through uniaxial testing under two different strain rate conditions. On this basis, the manufacturing process thermal residual stress and strain rate effect on the biaxial yield surface of composites are considered. The results show that the effect of thermal residual stress on the biaxial yield response is closely dependent on loading conditions. Moreover, biaxial yield strength tends to increase with the increasing strain rate.
Due to their remarkable mechanical characteristics and wide range of potential applications, composites have attracted extensive attention of researchers. Composites present evident plastic behavior, which is primarily characterized by yielding and rate sensitivity in service. Because of the complexity of composite materials, experimental methods require considerable financial and human resources. Furthermore, compared with uniaxial loading conditions, it is difficult to acquire yield strength of composites through macroscopic experimental methods under complex stress conditions. Therefore, more and more investigators rely to theoretical research on biaxial yield responses of composites.
Two main methods, namely the analytical micromechanical method and the finite element method, have been used to study yield the behaviors of composites under complex stress conditions. Azizi
In addition, it should be noted that traditional strain gauges can hardly capture dynamic strain changes exactly under highrate loading conditions due to the sensitivity to electromagnetic interference and low speed response. In this paper, repeatability and sensitivity of FBGs sensor are validated by a cantilever system. Meanwhile, the prediction results under uniaxial tensile conditions are validated by experimental data of a FBGs strain test system. On this basis, the effects of thermal residual stress and strain rate on the yield surfaces of composites with different fiber offaxis angles are investigated.
In the most micromechanical models, it is supposed that inclusions or fibers present periodic configuration in the composites, as shown in
Generalized Method of Cells (GMC), one of the most important micromechanical models, has been used in predicting effective elastic constants, mechanical properties of composites [10, 11 and 12]. For fiberreinforced composites, the representative volume element (RVE) is extracted from the cross section which is perpendicular to the fiber direction. The RVE is divided into
According to the homogenization theory, the relationship between macroscopic average stress
In order to satisfy displacement continuity conditions between adjacent subcells and axial deformation constraint conditions, the relationship between subcell average strain and macro strain can be expressed as follows:
According to the stress continuity conditions between subcells and the constitutive equation of the subcells, the relationship among average plastic strain components
Combining
Substituting
Through comparing
Due to their small dimensions and accurate measurements, as well as resistance to corrosion and electromagnetism, FBGs sensors have been used in cardiac ablation [
In acquiring strain signals, the next two methods are always used [
In order to validate the strain test of FBGs sensor, a cantilever beam structure is used as shown in
Measurement data are linear fitted by the leastsquares method.
The average wavelengths of four repeated tests are used to fit the strain sensitivity coefficients by the leastsquares method under loading and unloading conditions. The sensitivity experimental results and corresponding error analysis can be seen in
In order to describe the nonlinear behaviors of polymer matrix composites, an Improved BodnerPartom (IBP) model is incorporated into the GMC model. Supposing that the fiber is linearly elastic, the polymer matrix is viscoplastic. The flow law for the viscoplastic strain rate components of the IBP model is formulated as follows [
In the above, the overhead dot of the variables indicates the differentiation with respect to time
Good correlation has been found between the micromechanical model and the experimental data. On this basis, the biaxial yield responses of polymermatrix composites are studied.
With the increase of loading, material deformation is translated from the elastic to the plastic state. This procedure is called yield. A yield surface is defined as the locus of points in a stress or strain space when a specified yield criterion is satisfied. Stress state of materials can be easily discerned by the stress yield surface. Generally speaking, the relationship between stress
Under the condition of ignoring parameters
Many researchers have studied the yield surface of materials by experimental methods. The results show that the yield surface presents different shapes. Ishikawa [
In order to build a fiberreinforced composites yield surface, equivalent plastic strain
As mentioned in references [
For biaxial loading under constant strain rate conditions (
The region above the line AB is defined as region I, while the other one is defined as region II. From the
The effects of a strain rate range of 0.0001/s to 0.01/s on the biaxial yield strength of fiberreinforced composites with thermal residual stress under σ
The Generalized Method of Cells can be used to predict the nonlinear stressstrain of metalmatrix composites, polymermatrix composites and ceramicmatrix composites through incorporating different viscoplastic models. In this paper, the method has been used to investigate the thermal residual stress and strain rate influence on the biaxial yield responses of polymermatrix composites with different fiber offaxis angles. In the σ
This work was supported by the National Natural Science Foundation of China (No. 51175397), the key project of the National Natural Science Foundation of China (No. 51035007) and the First Aircraft Institute of Aviation Industry Corporation of China (HX0112040106).
Fiberreinforced composites with periodic array.
Discretization of the RVE.
The structure of FBGs sensor.
Test principle diagram.
Repeatability experiment of FBGs sensor: (
Wavelength absolute error of the FBGs sensor under loading and unloading conditions.
Sensitivity experiment of the FBGs sensor: (
Experimental system of the FBGs strain test system: (
Experimental research on polymermatrix composites: (
Thermal residual stress influence on σ
Thermal residual stress influence on the σ
Strain rate influence on σ
Strain rate influence on σ
Sensitivity and initial wavelength error analysis of FBGs sensor.
loading  1560.2390  1560.2403  0.0013  1.194  0.50%  0.9959 
unloading  1560.2390  1560.2413  0.0023  1.158  3.50%  0.9948 
Material parameters of the glass fiber and polymer matrix.



glass fiber  71.42  0.2                5 × 10^{−6} 
Polymer  3.3  0.22  0.63  0.104  0.184  0.391  0.803  10^{6}  168.5  25 × 10^{−6} 