The theory of microstretch elastic bodies was first developed by Eringen (1971, 1990, 1999, 2004). This theory was developed by extending the theory of micropolar elastcity. Each material point in this theory has three deformable directors. The governing equations of a transversely isotropic microstretch material are specialized in x-z plane. Plane wave solutions of these governing equations results into a bi-quadratic velocity equation. The four roots of the velocity equation correspond to four coupled plane waves which are named as Coupled Longitudinal Displacement (CLD) wave, Coupled Longitudinal Microstretch (CLM) wave, Coupled Transverse Displacement (CTD) wave and Coupled Transverse Microrotational (CTM) wave. The reflection of Coupled Longitudinal Displacement (CLD) wave is considered at a stress-free surface of half-space of material. The appropriate displacement components, microrotation component and microstretch potential for incident and four reflected waves in half-space are formulated. These solutions for incident and reflected waves satisfy the boundary conditions at a stress free surface of half-space and we obtain a non-homogeneous system of four equations in four reflection coefficients (or amplitude ratios) along with Snell’s law for the present model. The speeds of plane waves are computed by Fortran program of bi-quadratic velocity equation for relevant physical constants of the material. The reflection coefficients of various reflected waves are also computed by Fortran program of Gauss elimination method. The speeds of plane waves are plotted against angle of propagation direction with vertical axis. The reflection coefficients of various reflected waves are plotted against the angle of incidence. These variations of speeds and reflection coefficients are also compared with those in absence of microstretch parameters. For a specific material, numerical simulation in presence as well as in absence of microstretch shows that the coupled longitudinal displacement (CLD) wave is fastest wave and the coupled transverse microrotational (CTM) is observed slowest wave. The coupled longitudinal microstretch (CLM) wave is an additional wave due to the presence of microstretch in the medium. The presence of microstretch in transversely isotropic micropolar elastic solid affects the speeds of plane waves and the amplitude ratios of various reflected waves. 74J

中文翻译：

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横观各向同性微拉伸弹性固体中的波传播

微拉伸弹性体的理论最早由Eringen（1971，1990，1999，2004）提出。该理论是通过扩展微极弹性理论而发展起来的。该理论中的每个实质点都有三个可变形的指向矢。横观各向同性微拉伸材料的控制方程式专门用于xz平面。这些控制方程的平面波解导致了一个双二次速度方程。速度方程的四个根对应于四个耦合平面波，分别被称为耦合纵向位移（CLD）波，耦合纵向微拉伸（CLM）波，耦合横向位移（CTD）波和耦合横向微旋转（CTM）波。纵向位移（CLD）波的反射被认为是材料半空间的无应力表面。制定了适当的位移分量，微旋转分量和微拉伸势，以用于半空间中的入射波和四个反射波。这些入射波和反射波的解决方案满足了半空间无应力表面的边界条件，并且针对本模型，我们获得了具有四个反射系数（或振幅比）的四个方程的非均匀系统。平面波的速度是通过双二次速度方程的Fortran程序来计算材料的相关物理常数的。各种反射波的反射系数也可以通过高斯消去法的Fortran程序来计算。相对于垂直轴的传播方向角度绘制平面波的速度。相对于入射角绘制了各种反射波的反射系数。还将速度和反射系数的这些变化与没有微拉伸参数的情况进行比较。对于特定的材料，存在和不存在微拉伸的数值模拟表明，耦合纵向位移（CLD）波是最快的波，而耦合横向微旋转（CTM）是观察到的最慢的波。由于介质中存在微拉伸，因此耦合纵向微拉伸（CLM）波是附加波。横向各向同性的微极性弹性固体中微拉伸的存在会影响平面波的速度和各种反射波的振幅比。74J 还将速度和反射系数的这些变化与不具有微拉伸参数的情况进行比较。对于特定的材料，存在和不存在微拉伸的数值模拟表明，耦合纵向位移（CLD）波是最快的波，而耦合横向微旋转（CTM）是观察到的最慢的波。由于介质中存在微拉伸，因此耦合纵向微拉伸（CLM）波是附加波。横向各向同性的微极性弹性固体中微拉伸的存在会影响平面波的速度和各种反射波的振幅比。74J 还将速度和反射系数的这些变化与不具有微拉伸参数的情况进行比较。对于特定的材料，存在和不存在微拉伸的数值模拟表明，耦合纵向位移（CLD）波是最快的波，而耦合横向微旋转（CTM）是观察到的最慢的波。由于介质中存在微拉伸，因此耦合纵向微拉伸（CLM）波是附加波。横向各向同性的微极性弹性固体中微拉伸的存在会影响平面波的速度和各种反射波的振幅比。74J 存在和不存在微拉伸的数值模拟表明，耦合纵向位移（CLD）波是最快的波，而耦合横向微旋转（CTM）是观察到的最慢的波。由于介质中存在微拉伸，因此耦合纵向微拉伸（CLM）波是附加波。横向各向同性的微极性弹性固体中微拉伸的存在会影响平面波的速度和各种反射波的振幅比。74J 存在和不存在微拉伸的数值模拟表明，耦合纵向位移（CLD）波是最快的波，而耦合横向微旋转（CTM）是观察到的最慢的波。由于介质中存在微拉伸，因此耦合纵向微拉伸（CLM）波是附加波。横向各向同性的微极性弹性固体中微拉伸的存在会影响平面波的速度和各种反射波的振幅比。74J 横向各向同性的微极性弹性固体中微拉伸的存在会影响平面波的速度和各种反射波的振幅比。74J 横向各向同性的微极性弹性固体中微拉伸的存在会影响平面波的速度和各种反射波的振幅比。74J