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.

中文翻译：

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非线性弹性的混合变分原理使用嘉当运动坐标系并用有限元外微积分实现

本文使用移动嘉当框架为固体力学提供了一个新视角，特别讨论了非线性弹性中的混合变分原理。我们将在参考和变形配置的同切丛上定义的数量视为额外的未知数。这样的处理会引起场与基空间（身体的配置）的兼容性，从而可以将配置实现为欧几里得空间的子集。我们首先使用微分形式重写度量和连接，它们进一步用于根据框架和共框架场编写变形梯度和柯西-格林变形张量。应力的几何理解为共向量值 2 形式，完全符合我们的整体程序。我们证明，对于超弹性固体，对于应力的共向量部分，可以写出类似于 Doyle-Erciksen 公式的方程。使用这些，我们根据微分形式编写了一个混合能量泛函，其极值导致变形、本构规则和平衡方程的兼容性。然后利用有限元外部微积分为变分原理中出现的微分形式构造有限维近似。然后使用这些近似来构造离散泛函，然后将其进行数值极值化。这种离散化导致了一种混合方法，因为它对与应力和变形梯度相关的微分形式使用独立近似。然后混合变分原理专门用于二维情况，其离散近似应用于非线性弹性问题。我们的 FE 技术的一个重要特征是缺乏额外的稳定性。从数值研究中发现，目前的离散化也没有受到形式锁定和相关收敛问题的影响。