Stability of harmonic materials in plane strain
Stability of harmonic materials in plane strain
复制标题
谐波材料在平面应变中的稳定性
DOI：
--
复制
发表时间：
1988
期刊：
影响因子：
--
通讯作者：
A. Pipkin
中科院分区：
文献类型：
--
作者：
D. Steigmann;A. Pipkin
其中和A2是主要的延伸。具有这种形式W的材料称为谐波。W的调和形式已被许多研究者[1-4]用于获得平衡方程的显式解析解。本文讨论了简谐材料平衡态的稳定性。考虑的问题是严格的二维，我们只考虑与平面替代品的稳定性。稳定性问题的一半由Graves [5]的定理解决，该定理意味着只有当变形中涉及的每个应变处的应变能是秩一凸的时，变形才是局部稳定的。我们证明了一个限制形式的匡威。对于简谐材料，以及没有体力的位移边值问题，如果W在每个应变处都是秩一凸的，则平衡状态是稳定的。此外，每个局部稳定的状态都是全局稳定的（第7节）。基本稳定性定理也可以用Wq表示，Wq是W的拟凸化。对于所考虑的问题，平衡态是稳定的当且仅当在变形体的每一点W = Wq。我们在第5节和第6节中明确地确定了Wq。对于I = Xi + A2和J = A1 A2，它具有以下形式：
where and A 2 are the principal stretches. A material with this form of W is called harmonic. The harmonic form of W has been used by a number of investigators [1-4] to obtain explicit analytical solutions of the equations of equilibrium. In the present paper we discuss the stability of equilibrium for harmonic materials. The problems considered are strictly two-dimensional, and we consider stability versus plane alternatives only. Half of the problem of stability is solved by a theorem of Graves [5] which implies that a deformation is locally stable only if the strain energy is rank-one convex at each strain involved in the deformation. We prove a restricted form of the converse. For harmonic materials, and for displacement boundary value problems with no body force, an equilibrium state is stable if W is rank-one convex at each strain involved. Moreover, every locally stable state is globally stable (Section 7). The basic stability theorem can also be stated in terms of Wq, the quasiconvexification of W. For the problems considered, an equilibrium state is stable if and only if W = Wq at each point in the deformed body. We determine Wq explicitly in Sections 5 and 6. With I = Xi + A2 and J = A1A2, it has the form