Nonlinear Mechanical Behavior of Glass Fiber/ Epoxy Resin Composite Under Medium and Low Strain Rates Loading

Abstract

The necessity to design composite building structures that are both safe and reliable has prompted the academic community to delve into the investigation of the bearing capacity of composite materials and forecast their mechanical behavior. Since most deformation of composite structures under impact is in the range of low to medium strain rate ($\dot{\varepsilon }\le 100\hspace{0.33em}{s}^{-1}$), this paper conducted experimental study and finite element analysis (FEA) on the nonlinear mechanical behavior of glass fiber reinforced plastic (GFRP) before damage under medium and low strain rates loading. A strain rate dependent elastic-viscoplastic constitutive equation considering the tension and compression strength-difference effect was proposed based on a nonlinear elastic–plastic constitutive model. The mechanical behaviors of GFRP laminate at medium and low strain rates were obtained by writing the explicit user-defined material subroutine (VUMAT). The prediction results of FEA are in good agreement with the experimental findings. Thus, the constitutive model can be used to predict the mechanical behaviors of the GFRP building structures at medium and low strain rates.

Appendices

Appendix A

1.1 Explanation of the Validation of Experimental Tests

According to “GB/T 3354–2014” and “GB/T 1446–2005”, there should be no less than 5 valid samples in each group. For each set of tests, the arithmetic mean, standard deviation and the dispersion coefficient of each measurement performance are calculated.

Performance values for each sample: X₁, X₂,…, X_n. If necessary, the damage to each sample should be described.

The arithmetic mean $\overline X$ is calculated to three significant numbers.

$$\overline{X}=\frac{\sum\limits_{i=1}^n X_i}{n}$$

(25)

where X_i is the performance value of each sample and n is the number of samples.

The standard deviation S is calculated to two significant numbers.

$$S=\sqrt{\frac{{\sum\limits_{i=1}^n}\left(X_i-\overline X\right)^2}{n-1}}$$

(26)

where the symbols are the same as those in Eq. 25.

The dispersion coefficient C_v is calculated to two significant numbers.

$$C_v=\frac S{\overline X}$$

(27)

where the symbols are the same as those in Eq. 25, 26.

The dispersion coefficients of the experimental data selected in this paper are all less than 1%, thus these data are valid.

Appendix B

2.1 Parameter Acquisition Method in Nonlinear Constitutive Model

Fig. 11

Off-axis loading diagram of unidirectional laminate specimen

In the off-axis loading experiment shown in Fig. 11, the rectangular coordinate system for the direction of the main axis of the unidirectional composite laminates is (O-1–2-3), the global rectangular coordinate system is (o-x–y-z), and the off-axis loading Angle θ is the Angle between the loading direction and the fiber laying direction. In off-axis loading, the stress in the direction of specimen thickness is not considered. Through the coordinate transformation matrix, the stress and strain in the global coordinate system o-x–y-z can be expressed under the axis coordinate O-1–2-3:

$$\begin{array}{c}\sigma_{11}=\sigma_x\;\cos^2\;\theta\\\sigma_{22}=\sigma_{x\;}\sin^2\;\theta\\\sigma_{12}=-\sigma_x\;\sin\;\theta\;\cos\;\theta\end{array}$$

(28)

$$d\varepsilon_x^{\mathrm{vp}}=d\varepsilon_{11}^{\mathrm{vp}}\;\cos^2\;\theta+d\varepsilon_{22}^{\mathrm{vp}}\;\sin^2\;\theta-d\gamma_{12}^{vp}\;\sin\;\theta\;\cos\;\theta$$

(29)

By substituting Eq. 28 into the equivalent stress function and combining Eq. 14, 15, 16 and 29, the equivalent stress function and equivalent viscoplastic strain can be expressed through the stress and strain in the global coordinate system:

$$\begin{array}{c}\sigma_{eq}=h\left(\theta\right)\cdot\left|\sigma_x\right|\\\varepsilon_{\mathrm{eq}}^{\mathrm{vp}}=\frac1{h\left(\theta\right)}\\h\left(\theta\right)=\sqrt{\frac32\kappa_1\;\sin^4\;\theta+3\alpha_{66}\kappa_2\;\sin^2\;\theta\;\cos^2\;\theta}+\alpha_{11}\kappa_3\end{array}$$

(30)

where Ex is the modulus of the specimen along the loading direction, which can be obtained from the initial linear elastic section of the stress–strain curve in the off-axis loading test. The main fitting process of constitutive model parameters is as follows:

• (a) Select the off-axis loading test results at reference strain rate and fit them with Eq. 30 to obtain the equivalent stress-equivalent viscoplastic strain master curve of composite laminae under tension and compression, and obtain $\alpha_{11},\;\alpha_{66},\;\Gamma_1,\;\Gamma_2,\;\Gamma_3$ in Eq. 9 of equal effect force function and material constant R_oi, A_1i, B_1i, A_2i, B_2i, A_3i, B_3i (i = T, C) in Eq. 17 of isotropic hardening function at reference strain rate.

• (b) Based on the experimental results of off-axis loading under other strain rate conditions, the hardening functions corresponding to tension and compression under various strain rate conditions are obtained by fitting Eq. 30, and then the hardening functions under tension and compression under reference strain rate conditions are respectively referred to, and parameter Q_i, M_i, N_i(i = T, C) in the overstress function is obtained by fitting Eq. 22. Select the reference strain rate 10^–4 s^-1 and obtain Fig. 12 according to the above parameter acquisition steps.

Fig. 12

The equivalent stress-equivalent viscoplastic strain master curve of composite laminae. (a) Tension conditions, (b) Compression conditions

According to the equivalent stress-equivalent viscoplastic strain master curve shown in Fig. 12, the parameters of the constitutive model obtained by fitting are shown in Table 5. Where, a44 is taken from the recommended value of Sun and Chen [26]. The fitting process reveals that the coefficient of the hydrostatic pressure term $\alpha_{11},\;\Gamma_3$ is insignificantly small, exerting negligible influence on the overall nonlinear deformation of the material. Therefore, it can be disregarded and assumed to have a value of zero in this study. The material constant in the equivalent stress function remains unchanged when the principal curve is fitted under different strain rates.

When fitting the parameters of isotropic hardening function model, the equivalent strain rate ${\dot{\upvarepsilon }}_{{\text{eq}}}$ in dynamic yield function is replaced by the axial loading strain rate ${\dot{\upvarepsilon }}_{{\text{x}}}$. The equivalent stress-equivalent viscoplastic strain principal curve with a reference strain rate of 10–4 s-1 was selected as the reference, and the rate correlation of the principal curve at different strain rates was analyzed in combination with Eq. 22, and the rate-dependent isotropic hardening function of the following form was formed:

$${\text{R}}\left({\upvarepsilon }_{{\text{eq}}}^{{\text{vp}}},{\dot{\upvarepsilon }}_{{\text{x}}}\right)={{\text{R}}}^{*}\left({\upvarepsilon }_{{\text{eq}}}^{{\text{vp}}},{\dot{\upvarepsilon }}_{{\text{x}}}^{*}\right)\cdot {\left({{\text{Q}}}_{{\text{i}}}+{{\text{M}}}_{{\text{i}}}\cdot {\text{ln}}\frac{{\dot{\upvarepsilon }}_{{\text{x}}}}{{\dot{\upvarepsilon }}_{{\text{x}}}^{*}}\right)}^{{{\text{N}}}_{{\text{i}}}}$$

(31)

where i = T, C. The fitting constants are shown in Table 6.