Necessary conditions for a minimum at a radial cavitating singularity in nonlinear elasticity
Necessary conditions for a minimum at a radial cavitating singularity in nonlinear elasticity
复制标题
非线性弹性中径向空化奇点处最小值的必要条件
DOI：
10.1016/j.anihpc.2006.11.013
复制
发表时间：
2008
影响因子：
1.9
通讯作者：
S. Spector
中科院分区：
文献类型：
--
作者：
J. Sivaloganathan;S. Spector
Ball [J.M.非线性弹性力学中的不连续平衡解和空穴。横河长索克A 306（1982）557-611]（以及随后许多其他人）是任何物理上合理的非线性弹性能量的全局极小值。因此，我们认为在本文中的相关问题，获得必要条件，这些径向的解决方案是最小的非径向扰动。一个标准的爆破参数适用于内部或外部的变化产生一个明显的新的不等式，对于大多数本构关系，尚未得到验证。然而，在可压缩neo-Hookean材料的特殊情况下，W（F）=μ2| F| 2+h（detF），我们证明了由外变分产生的不等式明显成立，而由内变分产生的不等式是一个著名的不等式（首先由Brezis，科龙和Lieb [H.布雷济斯M.科龙，E.H. Lieb，带有缺陷的谐波映射，Comm. Math. Phys. 107（1986）649-705]），它出现在向列型液晶理论中：对于所有n∈W1，2（B，B）（因此|n| =1 a.e.）满足n（x）=x/|X|其中B是R3中的单位球。
It is still not known if the radial cavitating minimizers obtained by Ball [J.M. Ball, Discontinuous equilibrium solutions and cavitation in nonlinear elasticity, Phil. Trans. R. Soc. Lond. A 306 (1982) 557–611] (and subsequently by many others) are global minimizers of any physically reasonable nonlinearly elastic energy. We therefore consider in this paper the related problem of obtaining necessary conditions for these radial solutions to be minimizers with respect to nonradial perturbations. A standard blowup argument applied to either an inner or an outer variation yields an apparently new inequality that, for most constitutive relations, has yet to be verified. However, in the special case of a compressible neo-Hookean material, W(F)=μ2|F|2+h(detF), we show that the inequality produced by an outer variation clearly holds whilst that produced by an inner variation is a well-known inequality (first proven by Brezis, Coron, and Lieb [H. Brezis, J.-M. Coron, E.H. Lieb, Harmonic maps with defects, Comm. Math. Phys. 107 (1986) 649–705] ) which arises in the theory of nematic liquid crystals: for all n∈W1,2(B,∂B) (so that |n|=1 a.e.) that satisfy n(x)=x/|x| on ∂B, where B is the unit ball in R3.