A unifying treatise on variational principles for gradient and micromorphic continua
A unifying treatise on variational principles for gradient and micromorphic continua
复制标题
关于梯度和微态连续体变分原理的统一论文
DOI：
10.1080/14786430500362421
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发表时间：
2005
影响因子：
1.6
通讯作者：
P. Steinmann
中科院分区：
文献类型：
--
作者：
N. Kirchner;P. Steinmann
大多数材料的有效宏观性能在很大程度上取决于其微观结构。在最近的几十年里，相当多的努力被用于发展解释材料固有微观结构的连续统理论。然而，目前还没有一个统一的理论。可靠和有效地利用微结构材料将受益于前者，这也是任何后续数值分析的基础。以两种微结构材料（梯度连续体和微形态连续体）为研究对象，尝试在一般理论中确定它们在数值实现方面的作用和潜力。在基于狄利克雷原理的各种变分公式框架下对这两个连续体的分析表明，它们是密切相关的（这在直观上并不明显）：例如，梯度连续体可以通过将混合变分原理应用于微态连续体来建模。这对数值实现有相当大的影响：欧拉-拉格朗日方程描述了看似更复杂的微形态连续体，在FEM代码中实现时只需要连续近似，而相反，为梯度连续体推导的欧拉-拉格朗日方程需要涉及的未知量的可微性。
The effective macroscopic behaviour of most materials depends significantly on the structure exhibited at the microscopic level. In recent decades, considerable effort has thus been directed towards the development of continuum theories accounting for the inherent microstructure of materials. A unified theory is, however, not yet available. A reliable and efficient use of microstructured materials would benefit from the former, which is also the foundation of any subsequent numerical analysis. Focusing on two kinds of microstructured materials (gradient continua, micromorphic continua), an attempt is made to identify their role and their potential with regard to numerical implementations in a generic theory. An analysis of both continua in the framework of various variational formulations based on the Dirichlet principle shows that they are closely related (which is intuitively not obvious): a gradient continuum can, for example, be modelled by applying a mixed variational principle to a micromorphic continuum. This has considerable consequences for the numerical implementation: the Euler–Lagrange equations describing the seemingly more intricate micromorphic continuum require only continuous approximations when implemented in a FEM code, while, in contrast, the Euler–Lagrange equations deduced for the gradient continuum require differentiability of the involved unknowns.