论文标题
非线性弹性中Hu-Washizu变化原理的几何方法
A geometric approach to Hu-Washizu variational principle in nonlinear elasticity
论文作者
论文摘要
我们从几何的角度讨论了Hu-Washizu（HW）变化原理。当前方法的支柱是处理在参考捆绑包和变形构型束上定义的数量，为原始。这种治疗邀请兼容方程式，因此可以将基本空间（固体的构型）实现为欧几里得空间的子集。 Cartan的移动框架和关联的结构方程的方法建立了此兼容性。此外，它们允许我们使用1型编写指标和连接。使用可微分歧管提供的数学机械，我们将变形梯度和Cauchy-Green变形张量在框架和共框字段方面重写。对压力作为共同向量的几何理解与我们的整体计划完全符合2形式。我们还表明，对于超弹性固体，与Doyle-cerciksen公式相似的方程可能是为应力2形式的共同向量部分编写的。使用这种动力学和运动学理解，我们根据框架和差异形式重写了HW功能。最后，我们表明变形，本构规则和平衡方程的兼容性可作为HW功能的Euler-Lagrange方程，而随着牵引力1形式，变形1形和变形。这种涉及运动学闭合概念的新观点精确地阐明了对变异原理的必要几何限制，没有这些限制，不可能将变形的身体视为欧几里得空间的子集。它还提供了指向如何在非欧盟人环境中调整这些限制的指针。
We discuss the Hu-Washizu (HW) variational principle from a geometric standpoint. The mainstay of the present approach is to treat quantities defined on the co-tangent bundles of reference and deformed configurations as primal. Such a treatment invites compatibility equations so that the base space (configurations of the solid body) could be realised as a subset of an Euclidean space. Cartan's method of moving frames and the associated structure equations establish this compatibility. Moreover, they permit us to write the metric and connection using 1-forms. With the mathematical machinery provided by differentiable manifolds, we rewrite 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 also show that for a hyperelastic solid, an equation similar to the Doyle-Erciksen formula may be written for the co-vector part of the stress 2-form. Using this kinetic and kinematic understanding, we rewrite the HW functional in terms of frames and differential forms. Finally, we show that the compatibility of deformation, constitutive rules and equations of equilibrium are obtainable as Euler-Lagrange equations of the HW functional when varied with respect to traction 1-forms, deformation 1-forms and the deformation. This new perspective that involves the notion of kinematic closure precisely explicates the necessary geometrical restrictions on the variational principle, without which the deformed body may not be realized as a subset of the Euclidean space. It also provides a pointer to how these restrictions could be adjusted within a non-Euclidean setting.
