# 分数阶热弹理论下二维纤维增强弹性体的动态响应问题The Problem of Dynamic Response of Two Dimensional Fiber Reinforced Elastomer under Fractional Thermoelastic Theory

Abstract: Based on the Ezzat type fractional generalized thermoelastic coupling theory, the thermoelastic problem of a two-dimensional fiber-reinforced elastic body subjected to a linear mode I crack in a semi infinite space is studied. In this paper, the governing equations under the fractional genera-lized thermoelastic theory are given, and the regularized modal method is used to solve the go-verning equations. The distribution of dimensionless temperature, displacement, stress and other physical quantities in the half space infinite fiber reinforced elastomer is obtained. The influence of fractional order parameters and rotation effect on the physical quantities is mainly studied. The results show that the thermoelastic coupling effect occurs in the fiber reinforced elastomer due to the external load, and the fractional order parameters and rotation effect significantly affect the distribution of physical quantities.

1. 引言

2. 基本方程

Belfed等提出了纤维增强各向异性弹性介质的本构方程：

$\begin{array}{l}{\sigma }_{ij}=\lambda {e}_{kk}{\delta }_{ij}+2{\mu }_{T}{e}_{ij}+\xi \left({a}_{k}{a}_{m}{e}_{km}{\delta }_{ij}+{a}_{i}+{a}_{j}+{e}_{kk}\right)+\zeta {a}_{k}{a}_{m}{e}_{km}{a}_{i}{a}_{j}\\ \text{}+2\left({\mu }_{L}-{\mu }_{T}\right)\left({a}_{i}{a}_{k}{e}_{kj}+{a}_{j}{a}_{k}{e}_{ki}\right)-\gamma \left(T-{T}_{0}\right){\delta }_{ij}\end{array}$ (1)

${e}_{ij}=\frac{1}{2}\left({u}_{i,j}+{u}_{j,i}\right)$ (2)

${\sigma }_{xx}={A}_{1}{u}_{,x}+{A}_{2}{v}_{,y}-\gamma \left(T-{T}_{0}\right)$ (3)

${\sigma }_{yy}={A}_{3}{v}_{,y}+{A}_{2}{u}_{,x}-\gamma \left(T-{T}_{0}\right)$ (4)

${\sigma }_{zz}=\lambda {v}_{,y}+{A}_{2}{u}_{,x}-\gamma \left(T-{T}_{0}\right)$ (5)

${\sigma }_{xy}={\mu }_{L}\left({u}_{,y}+{v}_{,x}\right),{\sigma }_{zx}={\sigma }_{zy}=0$ (6)

$\kappa {\theta }_{,ij}=\left(1+\frac{{\tau }_{0}^{\alpha }}{\Gamma \left(\alpha +1\right)}\frac{{\partial }^{\alpha }}{\partial {t}^{\alpha }}\right)\left(\rho Ce\frac{\partial \theta }{\partial t}+{T}_{0}{\gamma }_{ij}{\stackrel{˙}{u}}_{i,j}\right)$ (7)

3. 热弹性问题描述

$\rho \left[{\stackrel{¨}{u}}_{i}+{\left(\Omega ×\left(\Omega ×u\right)+2\left(\Omega ×\stackrel{˙}{u}\right)\right)}_{i}\right]={\sigma }_{ij,j},\left(i,j=1,2,3\right)$ (8)

${A}_{1}\frac{{\partial }^{2}u}{\partial {x}^{2}}+{A}_{\text{4}}\frac{{\partial }^{2}v}{\partial x\partial y}+{\mu }_{L}\frac{{\partial }^{2}u}{\partial {y}^{2}}-\gamma \frac{\partial T}{\partial x}=\rho \left(\frac{{\partial }^{2}u}{\partial {t}^{2}}-{\Omega }^{2}u-2\Omega \frac{\partial v}{\partial t}\right)$ (9)

${\mu }_{L}\frac{{\partial }^{2}v}{\partial {x}^{2}}+{A}_{4}\frac{{\partial }^{2}u}{\partial x\partial y}+{A}_{3}\frac{{\partial }^{2}v}{\partial {y}^{2}}-\gamma \frac{\partial T}{\partial y}=\rho \left(\frac{{\partial }^{2}v}{\partial {t}^{2}}-{\Omega }^{2}v+2\Omega \frac{\partial u}{\partial t}\right)$ (10)

$\begin{array}{l}{x}^{\prime }=\frac{{\eta }_{0}}{{c}_{0}}x,{y}^{\prime }=\frac{{\eta }_{0}}{{c}_{0}}y,{u}^{\prime }=\frac{\rho {c}_{0}{\eta }_{0}}{\gamma {T}_{0}}u,{v}^{\prime }=\frac{\rho {c}_{0}{\eta }_{0}}{\gamma {T}_{0}}v,{t}^{\prime }={\eta }_{0}t,{\tau }_{0}={\eta }_{0}{\tau }_{0}\\ {\Omega }^{\prime }=\frac{\Omega }{{\eta }_{0}},{\theta }^{\prime }=\frac{\theta }{{T}_{0}},{{\sigma }^{\prime }}_{ij}=\frac{{\sigma }_{i}{}_{j}}{\gamma {T}_{0}},\left(i,j=1,2\right)\end{array}$ (11)

${h}_{11}\frac{{\partial }^{2}u}{\partial {x}^{2}}+{h}_{22}\frac{{\partial }^{2}v}{\partial x\partial y}+{h}_{23}\frac{{\partial }^{2}u}{\partial {y}^{2}}-\frac{\partial \theta }{\partial x}=\frac{{\partial }^{2}u}{\partial {t}^{2}}-{\Omega }^{2}u-2\Omega \frac{\partial v}{\partial t}$ (12)

${h}_{23}\frac{{\partial }^{2}v}{\partial {x}^{2}}+{h}_{22}\frac{{\partial }^{2}u}{\partial x\partial y}+\frac{{\partial }^{2}v}{\partial {y}^{2}}-\frac{\partial \theta }{\partial y}=\frac{{\partial }^{2}v}{\partial {t}^{2}}-{\Omega }^{2}v+2\Omega \frac{\partial u}{\partial t}$ (13)

$\frac{{\partial }^{2}\theta }{\partial {x}^{2}}+\frac{{\partial }^{2}\theta }{\partial {y}^{2}}=\left(1+\frac{{\tau }_{0}^{\alpha }}{\Gamma \left(\alpha +1\right)}\frac{{\partial }^{\alpha }}{\partial {t}^{\alpha }}\right)\frac{\partial }{\partial t}\left[{h}_{12}\theta +{h}_{13}\left(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}\right)\right],\left(0<\alpha \le 1\right)$ (14)

${\sigma }_{xx}={h}_{11}{u}_{,x}+\frac{{A}_{2}}{{A}_{3}}{v}_{,y}-\theta$ (15)

${\sigma }_{yy}={v}_{,y}+\frac{{A}_{2}}{{A}_{3}}{u}_{,x}-\theta$ (16)

${\sigma }_{zz}=\frac{\lambda }{{A}_{3}}{v}_{,y}+\frac{{A}_{2}}{{A}_{3}}{u}_{,x}-\theta$ (17)

${\sigma }_{xy}={h}_{23}\left({u}_{,y}+{v}_{,x}\right),{\sigma }_{zx}={\sigma }_{zy}=0$ (18)

4. 正则模态分析

$\left[u,v,\theta \right]\left(x,y,t\right)=\left[{u}^{\ast }\left(x\right),{v}^{\ast }\left(x\right),{\theta }^{\ast }\left(x\right)\right]{\text{e}}^{\omega t+iay}$ (19)

$\left({h}_{11}{D}^{2}-{A}_{5}\right){u}^{*}+\left(2\Omega \omega +ia{h}_{22}D\right){v}^{*}-D{\theta }^{*}=0$ (20)

$\left({h}_{23}{D}^{2}-{A}_{6}\right){v}^{*}+\left(ia{h}_{22}D-2\Omega \omega \right){u}^{*}-ia{\theta }^{*}=0$ (21)

${A}_{7}D{u}^{*}+{A}_{8}{v}^{*}+\left({A}_{\text{9}}-{D}^{2}\right){\theta }^{\text{*}}=0$ (22)

$\begin{array}{l}{A}_{5}={\omega }^{2}-{\Omega }^{2}+{h}_{23}{a}^{2},{A}_{6}={\omega }^{2}-{\Omega }^{2}+{a}^{2},{A}_{7}={h}_{13}\omega \left(1+\frac{{\tau }_{0}^{\alpha }}{\Gamma \left(\alpha +1\right)}{\omega }^{\alpha }\right)\\ {A}_{8}=ia{A}_{7},{A}_{9}=\frac{{h}_{12}}{{h}_{13}}{A}_{7}+{a}^{2}\end{array}$

$\left[{D}^{6}+{B}_{2}{D}^{4}+{B}_{1}{D}^{2}+{B}_{0}\right]\left\{{u}^{*}\left(x\right),{v}^{*}\left(x\right),{\theta }^{*}\left(x\right)\right\}=0$ (23)

${B}_{2}=-\frac{1}{{h}_{11}{h}_{23}}\left({A}_{6}{h}_{11}-{a}^{2}{h}_{22}^{2}+{A}_{5}{h}_{23}+{A}_{7}{h}_{23}+{A}_{9}{h}_{11}{h}_{23}\right)$

${B}_{1}=\frac{1}{{h}_{11}{h}_{23}}\left({A}_{8}a{h}_{22}i-{A}_{7}{a}^{2}{h}_{22}-{A}_{9}{a}^{2}{h}_{22}^{2}-{A}_{8}a{h}_{11}i+4{\Omega }^{2}{\omega }^{2}+{A}_{5}{A}_{6}+{A}_{6}{A}_{7}+{A}_{6}{A}_{9}{h}_{11}+{A}_{5}{A}_{9}{h}_{23}\right)$

${B}_{0}=\frac{1}{{h}_{11}{h}_{23}}\left({A}_{5}{A}_{8}ai-{A}_{5}{A}_{6}{A}_{\text{9}}-4{A}_{\text{9}}{\Omega }^{2}{\omega }^{2}\right)$

${m}^{6}+{B}_{2}{m}^{4}+{B}_{1}{m}^{2}+{B}_{0}=0$ (24)

$\left\{{\theta }^{*}\left(x\right),{u}^{*}\left(x\right),{v}^{*}\left(x\right)\right\}=\underset{j=1}{\overset{3}{\sum }}\left\{{C}_{j}^{1},{C}_{j}^{2},{C}_{j}^{3}\right\}{\text{e}}^{-{m}_{j}x}$ (25)

$\left\{{C}_{j}^{2},{C}_{j}^{3}\right\}=\left\{{\beta }_{j}^{2},{\beta }_{j}^{3}\right\}{C}_{j}^{1},\left(j=1,2,3\right)$ (26)

$\left\{\begin{array}{c}{\beta }_{j}^{2}\\ {\beta }_{j}^{3}\end{array}\right\}={\left[\begin{array}{cc}{h}_{11}{m}_{j}^{2}-{A}_{\text{5}}& 2\Omega \omega -ia{h}_{22}{m}_{j}\\ -2\Omega \omega -ia{h}_{22}{m}_{j}& {h}_{23}{m}_{j}^{2}-{A}_{\text{6}}\end{array}\right]}^{-1}\left\{\begin{array}{c}-{m}_{j}\\ ia\end{array}\right\}$ (27)

$\left\{\theta ,u,v\right\}\left(x,y,t\right)=\underset{j=1}{\overset{3}{\sum }}\left\{1,{\beta }_{j}^{2},{\beta }_{j}^{3}\right\}{C}_{j}^{1}{\text{e}}^{iay-{m}_{j}x+\omega t}$ (28)

${\sigma }_{xx}=\underset{j=1}{\overset{3}{\sum }}\left[-{h}_{11}{m}_{j}{\beta }_{j}^{2}+\frac{{A}_{2}}{{A}_{3}}ia{\beta }_{j}^{3}-1\right]{C}_{j}^{1}{\text{e}}^{iay-{m}_{j}x+\omega t}$ (29)

${\sigma }_{yy}=\underset{j=1}{\overset{3}{\sum }}\left[-\frac{{A}_{2}}{{A}_{3}}{m}_{j}{\beta }_{j}^{2}+ia{\beta }_{j}^{3}-1\right]{C}_{j}^{1}{\text{e}}^{iay-{m}_{j}x+\omega t}$ (30)

${\sigma }_{zz}=\underset{j=1}{\overset{3}{\sum }}\left[-\frac{{A}_{2}}{{A}_{3}}{m}_{j}{\beta }_{j}^{2}+\frac{\lambda }{{A}_{3}}ia{\beta }_{j}^{3}-1\right]{C}_{j}^{1}{\text{e}}^{iay-{m}_{j}x+\omega t}$ (31)

${\sigma }_{xy}={h}_{23}\underset{j=1}{\overset{3}{\sum }}\left[ia{\beta }_{j}^{2}-{m}_{j}{\beta }_{j}^{3}\right]{C}_{j}^{1}{\text{e}}^{iay-{m}_{j}x+\omega t}$ (32)

5. 边界条件

${\sigma }_{xy}\left(0,y,t\right)=0,{\sigma }_{yy}\left(0,y,t\right)=-p{\text{e}}^{iay+\omega t}$ (33)

Figure 1. Mode I crack model

$\theta \left(0,y,t\right)=f{\text{e}}^{iay+\omega t}$ (34)

$\left\{\begin{array}{c}{C}_{1}^{1}\\ {C}_{2}^{1}\\ {C}_{3}^{1}\end{array}\right\}={\left[\begin{array}{ccc}{R}_{1}& {R}_{2}& {R}_{3}\\ {S}_{1}& {S}_{2}& {S}_{3}\\ 1& 1& 1\end{array}\right]}^{-1}\left\{\begin{array}{c}0\\ -p\\ f\end{array}\right\}$ (35)

6. 数值结果及讨论

$\begin{array}{l}\lambda =7.59×{10}^{9}\text{N}/{\text{m}}^{\text{2}},{\mu }_{T}=1.89×{10}^{9}\text{N}/{\text{m}}^{\text{2}},{\mu }_{L}=2.45×{10}^{9}\text{N}/{\text{m}}^{\text{2}},\\ \xi =-1.28×{10}^{9}\text{N}/{\text{m}}^{\text{2}},\zeta =0.32×{10}^{9}\text{N}/{\text{m}}^{\text{2}},\rho =7800\text{\hspace{0.17em}}\text{kg}/{\text{m}}^{\text{2}},\\ {\alpha }_{t}=1.78×{10}^{-5}\text{N}/{\text{m}}^{\text{2}},k=386,Ce=383.1,{\tau }_{0}=0.02,a=1,\\ {T}_{0}=293\text{\hspace{0.17em}}\text{K},p=2,f=1,\omega ={\omega }_{0}+i\varsigma ,{\omega }_{0}=2,\varsigma =1,\mu =3.86×{10}^{10}\text{kg}/\text{m}\cdot {\text{s}}^{\text{2}}\end{array}$

Figure 2. Distribution of dimensionless displacement u on x axis when $\alpha$ takes different values

Figure 3. Distribution of dimensionless displacement v on x axis when $\alpha$ takes different values

Figure 4. Distribution of dimensionless temperature $\theta$ on x axis when $\alpha$ takes different values

Figure 5. Distribution of dimensionless stress ${\sigma }_{xx}$ on x axis with different values of $\alpha$

Figure 6. Distribution of dimensionless stress ${\sigma }_{yy}$ on x axis with different values of $\alpha$

Figure 7. Distribution of dimensionless stress ${\sigma }_{xy}$ on x axis with different values of $\alpha$

Figure 8. Distribution of dimensionless displacement u on x axis when $\Omega$ takes different values

Figure 9. Distribution of dimensionless displacement v on x axis when $\Omega$ takes different values

Figure 10. Distribution of dimensionless temperature $\theta$ on x axis when $\Omega$ takes different values

Figure 11. Distribution of dimensionless stress ${\sigma }_{xx}$ on x axis with different values of $\Omega$

Figure 12. Distribution of dimensionless stress ${\sigma }_{yy}$ on x axis with different values of $\Omega$

Figure 13. Distribution of dimensionless stress ${\sigma }_{xy}$ on x axis with different values of $\Omega$

Figure 14. Distribution of dimensionless displacement u on x axis when y takes different values

Figure 15. Distribution of dimensionless displacement v on x axis when y takes different values

Figure 16. Distribution of dimensionless temperature $\theta$ on x axis when y takes different values

Figure 17. Distribution of dimensionless stress ${\sigma }_{xx}$ on x axis with different values of y

Figure 18. Distribution of dimensionless stress ${\sigma }_{yy}$ on x axis with different values of y

Figure 19. Distribution of dimensionless stress ${\sigma }_{xy}$ on x axis with different values of y

$\Omega =0,\alpha =0.5$，在x轴固定位置( $x=1$$x=1.4$ )，给出无量纲位移u、v和应力 ${\sigma }_{xx}$ 随时间t的变化情况，如图20~22所示。

Figure 20. When x takes different values, the dimensionless displacement u changes with time t

Figure 21. When x takes different values, the dimensionless displacement v changes with time t

Figure 22. The change of dimensionless stress ${\sigma }_{xx}$ with time t when x is different

7. 结论

1) 分数阶参数的不同导致预测的结果不同，因此不同分数阶参数下，对弹性体内物理量的分布预测结果不同。例如，不同分数阶参数下，位移v、温度、应力的分布不同；

2) 由于转速影响物理量的振幅，大多数物理量随着转速的增加，振幅减小。因此可得旋转效应的存在该模型中具有重要意义；

3) 不同分数阶参数下，无量纲位移、温度、应力的大小、分布不同，在相同分数阶参数下，各物理量的分布随着坐标值得增加而变化。因此，各物理量的大小及分布，不仅取决于分数阶参数，还受时间和空间变量的影响；

4) 各物理量的大小随着距离远离边界都趋向于零，表明在半无限大弹性体中，外力冲击仅在一定范围内形成影响，并不能无限传播，这与材料中物理量传播的事实是一致的，同时所有表示物理量的函数都是连续的，表明各物理量在传播过程中不会产生突变，这也符合纤维增强材料的材料特性。

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