变形弹性材料中的非均匀“纵向”平面波
Inhomogeneous "longitudinal" plane waves in a deformed elastic material
摘要：
均匀各向同性可压缩阿达玛材料具有这样的性质：当材料保持在任意有限静态均匀变形状态时，无限小的纵向均匀平面波可以在每个方向上传播。这里，关于波，“均匀”意味着波的传播方向平行于最终衰减的方向，并且“纵向”意味着波在平行于传播方向的方向上线性偏振。换句话说，位移的形式为u = n cos k（n.x - ct），其中n是实向量。可以看出，阿达玛材料是最一般的一个，其中的“纵向”不均匀平面波也可以在任何方向上传播的预变形体。这里，“非均匀”意味着波在与传播方向不同的方向上衰减，而“纵向”意味着波在包含这两个方向的平面中椭圆偏振。换句话说，位移具有形式u = Re {S exp[i（ω S. x-ct）]}，其中S是复向量（或双向量）。然后引入了广义Hadamard材料。它是最普遍的均匀各向同性可压缩材料，允许传播无限小的“纵向”非均匀平面圆偏振波的所有选择的各向同性定向双矢量。最后，对于均匀变形的各向同性弹性材料，找到了响应函数的最一般形式，其中“纵向”非均匀平面波可以在B^n-椭球的中心圆形截面的两个平面中的每一个中以圆偏振传播，其中B是对应于原始纯均匀变形的左柯西-格林应变张量。
英文摘要：
A homogeneous isotropic compressible Hadamard material has the property that an infinitesimal longitudinal homogeneous plane wave may propagate in every direction when the material is maintained in a state of arbitrary finite static homogeneous deformation. Here, as regards the wave, 'homogeneous' means that the direction of propagation of the wave is parallel to the direction of eventual attenuation and 'longitudinal' means that the wave is linearly polarized in a direction parallel to the direction of propagation. In other words, the displacement is of the form u = n cos k(n.x - ct), where n is a real vector. It is seen that the Hadamard material is the most general one for which a 'longitudinal' inhomogeneous plane wave may also propagate in any direction of a predeformed body. Here, 'inhomogeneous' means that the wave is attenuated in a direction distinct from the direction of propagation and 'longitudinal' means that the wave is elliptically polarized in the plane containing these two directions. In other words, the displacement is of the form u = Re {S exp[i (omega S.x - ct)]}, where S is a complex vector (or bivector). Then a Generalized Hadamard material is introduced. It is the most general homogeneous isotropic compressible material which allows the propagation of infinitesimal 'longitudinal' inhomogeneous plane circularly polarized waves for all choices of the isotropic directional bivector. Finally, the most general forms of response functions are found for homogeneously deformed isotropic elastic materials in which 'longitudinal' inhomogeneous plane waves may propagate with a circular polarization in each of the two planes of central circular section of the B^n-ellipsoid, where B is the left Cauchy-Green strain tensor corresponding to the primary pure homogeneous deformation.
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