弹性张量的M-特征值与强椭圆性条件
On the M-eigenvalues of elasticity tensor and the strong ellipticity condition
摘要：
强椭圆性是弹性力学理论中的一个重要性质。在2009年，M-特征值被引入弹性张量。证明了弹性张量的M-特征值在坐标系选择下是不变的，并且弹性张量的强椭圆性条件成立当且仅当弹性张量的所有M-特征值都为正.因此，M-本征值是弹性张量的一些内禀参数。本文证明了弹性张量的M-特征值与体模量、剪切模量、Lamé第一参数、P波模量等弹性模量密切相关，并且M-特征值的正性对应于某些特殊情况下强椭圆性的存在条件，如各向同性情况、立方情况、极性各向异性的情况和四极异性的情况。我们还提出了正交各向异性情形的强椭圆性的新的充分条件。这在一定意义上进一步揭示了M-本征值的物理意义。
英文摘要：
Strong ellipticity is an important property in the elasticity theory. In 2009, M-eigenvalues were introduced for the elasticity tensor. It was shown that M-eigenvalues are invariant under coordinate system choices, and the strong ellipticity condition holds if and only if all the M-eigenvalues of the elasticity tensor are positive. Thus, M-eigenvalues are some intrinsic parameters of the elasticity tensor. In this paper, we show that the M-eigenvalues of the elasticity tensor are closely related with some elastic moduli, such as the bulk modulus, the shear modulus, Lamé's first parameter, the P-wave modulus, etc, and the positiveness of the M-eigenvalues are corresponding to some existing conditions for strong ellipticity in some special cases, such as the isotropic case, the cubic case, the polar anisotropic case and the tetragonal case. We also present new sufficient conditions for the strong ellipticity of the orthotropic case. These, in a certain sense, further reveal the physical meanings of M-eigenvalues.
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