Three-dimensional isogeometric solutions to general boundary value problems of Toupin’s gradient elasticity theory at finite strains
Three-dimensional isogeometric solutions to general boundary value problems of Toupin’s gradient elasticity theory at finite strains
复制标题
有限应变下Toupin梯度弹性理论一般边值问题的三维等几何解
DOI：
10.1016/j.cma.2014.06.015
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发表时间：
2014
影响因子：
7.2
通讯作者：
K. Garikipati
中科院分区：
文献类型：
--
作者：
S. Rudraraju;Anton Van der Ven;K. Garikipati
我们目前，据我们所知，第一个完整的三维数值解的范围广泛的边界值问题的一般理论有限应变梯度弹性。我们选择了Toupin理论（Toupin，1962）--应变梯度弹性的一个更一般的公式。我们的框架有三个关键组成部分：第一个是等几何分析（Hughes等人，2005），我们已经采用了它的直接和强大的C1-连续性表示。第二个是一个较弱的高阶狄利克雷边界条件的制定，控制应变梯度的发展在解决方案中的治疗。第三个要素是算法（自动）微分，它消除了需要线性化的“手”的相当复杂的几何和材料的非线性梯度弹性在有限的应变。我们提出了一些数值解决方案，以证明该框架是适用于任意边值问题的三维。我们讨论的长度尺度效应，高阶边界条件的作用，也许最重要的是，相关的框架与弹性自由能密度函数，是非凸的应变空间的问题。
We present, to the best of our knowledge, the first complete three-dimensional numerical solutions to a broad range of boundary value problems for a general theory of finite strain gradient elasticity. We have chosen for our work, Toupin’s theory (Toupin, 1962)–one of the more general formulations of strain gradient elasticity. Our framework has three crucial ingredients: The first is isogeometric analysis (Hughes et al., 2005), which we have adopted for its straightforward and robust representation of C 1-continuity. The second is a weak treatment of the higher-order Dirichlet boundary conditions in the formulation, which control the development of strain gradients in the solution. The third ingredient is algorithmic (automatic) differentiation, which eliminates the need for linearization “by hand” of the rather complicated geometric and material nonlinearities in gradient elasticity at finite strains. We present a number of numerical solutions to demonstrate that the framework is applicable to arbitrary boundary value problems in three dimensions. We discuss length scale effects, the role of higher-order boundary conditions, and perhaps most importantly, the relevance of the framework to problems with elastic free energy density functions that are non-convex in strain space.