A multi-field incremental variational framework for gradient-extended standard dissipative solids
A multi-field incremental variational framework for gradient-extended standard dissipative solids
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DOI：
10.1016/j.jmps.2010.11.001
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发表时间：
2011-04
影响因子：
5.3
通讯作者：
C. Miehé
中科院分区：
文献类型：
--
作者：
C. Miehé
本文提出了一个基于紧致变分理论的耗散微结构固体本构框架。它开发了一类梯度型耗散材料的增量最小化和鞍点原理，该材料包含微结构场（微位移，序参数或广义内变量），其梯度进入能量存储和耗散函数。与传统的基于局部演化内部变量的非弹性固体局部连续方法不同，这些全局微结构场由包括微结构边界条件在内的附加平衡方程控制。它们描述了材料的子结构的变化，这些变化相对于整个材料而演变。典型的例子是相场演化理论、梯度损伤理论或应变梯度塑性理论。这种模型结合了基于长度尺度的非局部效应，其反映了材料微观结构的性质。我们概述了一个统一的框架，广泛的一类一阶梯度型标准耗散固体。特别强调的是放在替代多场表示，其中既有微观结构变量本身，以及它的双重驱动力。这些三场设置适用于在驱动力空间中制定阈值或屈服函数的模型。结果表明，耦合的宏观和微观平衡遵循自然的方式作为欧拉方程的最小化和鞍点原则，这是基于适当定义的增量潜力。这些多场势泛函概述了在连续速率配方和时空离散增量设置。所提出的多场制剂的固有对称性是一个有吸引力的功能，就其数值实现。该框架的统一特征是由一系列模型问题所证明的，这些问题包括相场模型和梯度损伤与塑性的公式。
The paper presents a constitutive framework for solids with dissipative micro-structures based on compact variational statements. It develops incremental minimization and saddle point principles for a class of gradient-type dissipative materials which incorporate micro-structural fields (micro-displacements, order parameters, or generalized internal variables), whose gradients enter the energy storage and dissipation functions. In contrast to classical local continuum approaches to inelastic solids based on locally evolving internal variables, these global micro-structural fields are governed by additional balance equations including micro-structural boundary conditions. They describe changes of the substructure of the material which evolve relatively to the material as a whole. Typical examples are theories of phase field evolution, gradient damage, or strain gradient plasticity. Such models incorporate non-local effects based on length scales, which reflect properties of the material micro-structure. We outline a unified framework for the broad class of first-order gradient-type standard dissipative solids. Particular emphasis is put on alternative multi-field representations, where both the microstructural variable itself as well as its dual driving force are present. These three-field settings are suitable for models with threshold- or yield-functions formulated in the space of the driving forces. It is shown that the coupled macro- and micro-balances follow in a natural way as the Euler equations of minimization and saddle point principles, which are based on properly defined incremental potentials. These multi-field potential functionals are outlined in both a continuous rate formulation and a time–space-discrete incremental setting. The inherent symmetry of the proposed multi-field formulations is an attractive feature with regard to their numerical implementation. The unified character of the framework is demonstrated by a spectrum of model problems, which covers phase field models and formulations of gradient damage and plasticity.