Energy minimization and the formation of microstructure in dynamic anti-plane shear
Energy minimization and the formation of microstructure in dynamic anti-plane shear
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动态反平面剪切中的能量最小化和微观结构的形成
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发表时间：
1992
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通讯作者：
P. Holmes
中科院分区：
文献类型：
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作者：
P. Swart;P. Holmes
我们研究了连续介质模型的行为，该模型旨在深入了解某些材料在位移相变过程中观察到的微观结构的动态发展。该模型是在非线性粘弹性的框架内提出的，并且作为一个强耗散的无限维动力系统的例子也很有趣，该动力系统的前向轨道不需要位于有限维吸引集上，并且它对初始条件的依赖性与经典的有限维“混沌”有很大的不同。（二维）线性粘弹性阻尼反平面剪切。在非线性超弹性的框架内，我们考虑各向同性和各向异性的本构关系，可以允许不同的阶段，我们表征他们的能力，提供最小化和最小化序列的存储的弹性能量（定理2.3）。利用Rybka的变换，我们将问题转化为半线性退化抛物系统，从而允许应用半群理论建立Lp空间中解的存在性、唯一性和正则性（定理3.1）。我们还讨论了能量最小化和传播的应变不连续性的问题。我们评论的困难，试图利用特定的本构关系的几何特性。特别是，我们无法得到类似的缺乏极小和非传播的应变不连续性所发现的球，霍姆斯，詹姆斯，Pego和Swart [1991]的一维模型问题。看来，一个绝对极小的情况下，可以防止能量最小化，从而提供了一个动力学机制，以限制所观察到的微观结构的精细度，已经证明在一维的情况下。类似地，粘弹性阻尼似乎可以防止应变不连续性的传播。在精细结构的极其缓慢的发展过程中，观察到的解决方案，以显示局部细化，努力克服与边界和初始条件的不兼容性，与所得到的更精细的尺度显示对初始条件的微妙依赖的分布和形状。
We investigate the behavior of a continuum model designed to provide insight into the dynamical development of microstructures observed during displacive phase transformations in certain materials. The model is presented within the framework of nonlinear viscoelasticity and is also of interest as an example of a strongly dissipative infinite-dimensional dynamical system whose forward orbits need not lie on a finite-dimensional attracting set, and which can display a subtle dependence on initial conditions quite different from that of classical finite-dimensional “chaos”.We study the problem of dynamical (two-dimensional) anti-plane shear with linear viscoelastic damping. Within the framework of nonlinear hyperelasticity, we consider both isotropic and anisotropic constitutive laws which can allow different phases and we characterize their ability to deliver minimizers and minimizing sequences of the stored elastic energy (Theorem 2.3). Using a transformation due to Rybka, we recast the problem as a semilinear degenerate parabolic system, thereby allowing the application of semigroup theory to establish existence, uniqueness and regularity of solutions in Lpspaces (Theorem 3.1). We also discuss the issues of energy minimization and propagation of strain discontinuities. We comment on the difficulties encountered in trying to exploit the geometrical properties of specific constitutive laws. In particular, we are unable to obtain analogues of the absence of minimizers and of the non-propagation of strain discontinuities found by Ball, Holmes, James, Pego & Swart [1991] for a one-dimensional model problem.Several numerical experiments are presented, which prompt the following conclusions. It appears that the absence of an absolute minimizer may prevent energy minimization, thereby providing a dynamical mechanism to limit the fineness of observed microstructure, as has been proved in the one-dimensional case. Similarly, viscoelastic damping appears to prevent the propagation of strain discontinuities. During the extremely slow development of fine structure, solutions are observed to display local refinement in an effort to overcome incompatibility with boundary and initial conditions, with the distribution and shape of the resulting finer scales displaying a subtle dependence on initial conditions.