The role of dissipation and defect energy in variational formulations of problems in strain-gradient plasticity. Part 1: polycrystalline plasticity
The role of dissipation and defect energy in variational formulations of problems in strain-gradient plasticity. Part 1: polycrystalline plasticity
复制标题
耗散和缺陷能量在应变梯度塑性问题的变分公式中的作用。
DOI：
10.1007/s00161-011-0194-9
复制
发表时间：
2011
影响因子：
2.6
通讯作者：
B. Reddy
中科院分区：
文献类型：
--
作者：
B. Reddy
针对应变梯度塑性中的小变形速率无关问题，建立了一套通用的流动规律和相关的变分公式。该框架基于Gurtin和Anand提出的热力学一致性理论（J Mech Phys Solids 53:1624-1649, 2005），并包括一组具有能量和耗散分量的微应力作为变量。流动规律是关联型的。它被表示为关于一个凸的但在其他方面是任意的屈服函数的正态性律，或者等价地表示为对应的耗散函数。所研究的两种情况是，第一，经典的Hill-Mises或J2流动定律的扩展，第二，一种形式写成塑性应变和应变梯度的大小的线性和。后一种形式是由Evans和Hutchinson （Acta Mater, 57:1675-1688, 2009）以及Nix和Gao （J Mech Phys Solids, 46:41 - 425, 1998）的工作推动的，他们表明，至少对于特定类别的问题，这种形式与实验结果非常吻合。相应的屈服函数由对偶参数得到。变分问题是基于用耗散函数表示的流动规律，并将问题表述为位移、塑性应变和硬化参数的变分不等式。微应力的耗散分量是不确定的，在公式中不存在。研究了广义Hill-Mises和线性和耗散函数解的存在性和唯一性，以及缺陷能的各种组合。问题的适定性条件主要取决于耗散函数的选择，塑性应变或塑性应变梯度中是否存在缺陷能量，以及内变量硬化。
A general set of flow laws and associated variational formulations are constructed for small-deformation rate-independent problems in strain-gradient plasticity. The framework is based on the thermodynamically consistent theory due to Gurtin and Anand (J Mech Phys Solids 53:1624–1649, 2005), and includes as variables a set of microstresses which have both energetic and dissipative components. The flow law is of associative type. It is expressed as a normality law with respect to a convex but otherwise arbitrary yield function, or equivalently in terms of the corresponding dissipation function. Two cases studied are, first, an extension of the classical Hill-Mises or J2 flow law and second, a form written as a linear sum of the magnitudes of the plastic strain and strain gradient. This latter form is motivated by work of Evans and Hutchinson (Acta Mater 57:1675–1688, 2009) and Nix and Gao (J Mech Phys Solids 46:411–425, 1998), who show that it leads to superior correspondence with experimental results, at least for particular classes of problems. The corresponding yield function is obtained by a duality argument. The variational problem is based on the flow rule expressed in terms of the dissipation function, and the problem is formulated as a variational inequality in the displacement, plastic strain, and hardening parameter. Dissipative components of the microstresses, which are indeterminate, are absent from the formulation. Existence and uniqueness of solutions are investigated for the generalized Hill-Mises and linear-sum dissipation functions, and for various combinations of defect energy. The conditions for well-posedness of the problem depend critically on the choice of dissipation function, and on the presence or otherwise of a defect energy in the plastic strain or plastic strain gradient, and of internal-variable hardening.