Abstract

Conditions for the strong ellipticity of equilibrium equations are formulated within strain gradient elasticity under finite deformations. In this model, the strain energy density is a function of the first and second gradients of the position vector (deformation gradient). Ellipticity imposes certain constraints on the tangent elastic moduli. It is also closely related to infinitesimal stability, which is defined as the positive definiteness of the second variation of the potential-energy functional. The work considers the first boundary-value problem (with Dirichlet boundary conditions). For a 1D deformation, necessary and sufficient conditions for infinitesimal stability are determined, which are two inequalities for elastic moduli.

中文翻译：

---

关于应变梯度弹性和无穷小稳定性静态方程的椭圆率

摘要

平衡方程的强椭圆性条件是在有限变形下的应变梯度弹性内制定的。在此模型中，应变能密度是位置矢量（变形梯度）的第一和第二梯度的函数。椭圆率对切线弹性模量施加了某些限制。它还与无穷小稳定性密切相关，无穷小稳定性被定义为势能泛函二次变体的正定性。这项工作考虑了第一个边界值问题（具有 Dirichlet 边界条件）。对于一维变形，确定无穷小稳定性的充分必要条件，即弹性模量的两个不等式。