This article offers a new perspective for the mechanics of solids using moving Cartan's frame, specifically discussing a mixed variational principle in non-linear elasticity. We treat quantities defined on the co-tangent bundles of reference and deformed configurations as additional unknowns. Such a treatment invites compatibility of the fields with base-space (configurations of the body), so that the configuration can be realised as a subset of the Euclidean space. We first rewrite the metric and connection using differential forms, which are further utilised to write the deformation gradient and Cauchy-Green deformation tensor in terms of frame and co-frame fields. The geometric understanding of stress as a co-vector valued 2-form fits squarely within our overall program. We show that, for a hyperelastic solid, an equation similar to the Doyle-Erciksen formula may be written for the co-vector part of stress. Using these, we write a mixed energy functional in terms of differential forms, whose extremum leads to the compatibility of deformation, constitutive rules and equations of equilibrium. Finite element exterior calculus is then utilised to construct a finite dimensional approximation for the differential forms appearing in the variational principle. These approximations are then used to construct a discrete functional which is then numerically extremised. This discertization leads to a mixed method as it uses independent approximations for differential forms related to stress and deformation gradient. The mixed variational principle is then specialized for 2D case, whose discrete approximation is applied to problems in nonlinear elasticity. An important feature of our FE technique is the lack of additional stabilization. From the numerical study, it is found that the present discretization also does not suffer form locking and related convergence issues.
翻译:此文章为使用移动 Cartan 框架的固态机械化提供了一个新视角, 特别是讨论非线性弹性的混杂变异原则。 我们将共同切分的参照包和变形配置中确定的数量作为额外的未知。 这样的处理可以使字段与基空间( 体体的配置) 兼容, 这样配置可以作为 Euclidean 空间的一个子组实现。 我们首先使用不同格式重写度和连接, 这些格式进一步用于写写变形梯度和 Causy- 绿色变异变异在框架和共同框架字段中的变异原则。 我们将压力的几何度理解作为共同变异值, 值值为值 2 的值变异性, 我们的整个程序。 我们显示, 对于超弹性, 一个类似于 Doyle- irciksen 公式的方程式, 可以用共同压力部分来写成。 我们使用这些变异变异形式来写一种混合的能量功能功能, 这些变异变式的外形形式, 其外形也会导致变异性、 变异性规则和正数原则的变异性变异性, 的变变化是用来变变变变变法, 。 这些变变变变法的变的变法, 的变的变的变的变的变的变法在的变的变的变式, 变式在的变变的变的变的变的变的变的变的变式, 的变的变的变的变的变法, 的变法在的变法在的变的变的变式在的变的变的变的变的变式, 的变式, 的变的变的变式, 的变的变的变的变的变的变的变的变式, 的变的变的变式, 的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变的变