We study some mathematical properties of a novel implicit constitutive relation wherein the stress and the linearized strain appear linearly that has been recently put into place to describe elastic response of porous metals as well as materials such as rocks and concrete. In the corresponding mixed variational formulation the displacement, the deviatoric and spherical stress are three independent fields. To treat well-posedness of the quasi-linear elliptic problem, we rely on the one-parameter dependence, regularization of the linear-fractional singularity by thresholding, and applying the Browder–Minty existence theorem for the regularized problem. An analytical solution to the nonlinear problem under constant compression/extension is presented.

我们研究了一种新的隐式本构关系的一些数学特性，其中应力和线性化应变呈线性，最近已被用于描述多孔金属以及岩石和混凝土等材料的弹性响应。在相应的混合变分公式中，位移、偏应力和球面应力是三个独立的场。为了处理拟线性椭圆问题的适定性，我们依赖于单参数依赖，通过阈值对线性分数奇点进行正则化，并对正则化问题应用 Browder-Minty 存在定理。提出了恒定压缩/拉伸下非线性问题的解析解。