各种对称类中三维弹性张量的秩一凸性、多凸性和极值性
Rank-one convexity, polyconvexity, and extremality for three-dimensional elasticity tensors in various symmetry classes
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中文总结 AI 辅助
本文研究\(N = n = 3\)维中弹性张量不同对称类下的极值拟凸二次型。证明正交各向异性张量中拟凸性蕴含多凸性，而三角对称时二者不等价，给出非多凸三角极射线及证明相关性质的一般准则，改进了已有结果。
中文摘要 AI 辅助
已知对于二次函数，拟凸性等于秩一凸性，但不一定等于多凸性。虽然二次能量的多凸性有明确的特征描述，但在\(n,N\geq3\)维中，拟凸性的类似特征尚不清楚。本文关注在\(N = n = 3\)维中，在弹性张量的各种对称类中寻找被视为线性弹性能量的极值拟凸二次型。特别地，我们证明对于具有正交各向异性对称的张量，拟凸性意味着多凸性，因此正交各向异性类不包含非平凡极值。我们表明这种等价性首先在三角对称时不成立，在一些陈述中，我们提供了一族非多凸的三角极射线。我们还提供了一些证明秩一凸性、多凸性和极值性的一般准则，改进了文献[\ref{bib: this http URL.}]中的一个结果。
英文摘要
It has been known that, for quadratic functions, quasiconvexity equals rank-one convexity but need not equal polyconvexity. While there is an explicit characterization of polyconvexity for quadratic energies, a similar characterization for quasiconvexity is not known in dimensions $n,N \geq 3.$ This paper is concerned with the search of extremal quasiconvex quadratic forms regarded as a linear elastic energy in dimensions $N=n=3$ in various symmetry classes of the elasticity tensor. In particular, we prove that for tensors with orthotropic symmetry, quasiconvexity implies polyconvexity, and thus the orthotropic class contains no nontrivial extremals. We show that this equivalence fails first at trigonal symmetry, where among some statemets, we provide a one-parameter family of non-polyconvex trigonal extreme rays. We also provide some general criteria for proving rank-one convexity, polyconvexity, and extremality, improving a result in [\ref{bib:Har.Hov.}].
