固-固相变的双阱线性化
Two-well linearization for solid-solid phase transitions
摘要：
在本文中，我们考虑非线性弹性，框架无关，和奇摄动两井模型的材料进行固-固相变在任何空间尺寸，我们同时进行通过尖锐的界面和小应变的限制。通过适当的Caccioppoli分区分解成分段恒定的刚性运动和适当的重新缩放的位移的总和的变形与等界能量的序列。这些分别收敛到极限分区、变形和位移。极限变形是梯度只有两个值的简单层合板，而极限位移则属于有界变差的特殊函数。后者的特点弹性的贡献，测量简单的层压板的距离，以及跳跃相关的两个连续的相变具有消失的距离，因此不会被检测到的限制变形。通过伽玛收敛，我们确定了一个有效的限制模型的总和的二次线性化的弹性能量的位移以及两个表面项。第一个与由极限变形梯度中的跳跃产生的界面的总长度成比例。第二个是成比例的两倍的总长度的界面所产生的跳跃在极限位移，以及由限制分区的边界。我们分析的一个主要工具是一个新的双阱刚度估计，它已经在[32]中推导出了一个具有各向异性二阶扰动的模型。
英文摘要：
In this paper we consider nonlinearly elastic, frame-indifferent, and singularly perturbed two-well models for materials undergoing solid-solid phase transitions in any space dimensions, and we perform a simultaneous passage to sharp-interface and small-strain limits. Sequences of deformations with equibounded energies are decomposed via suitable Caccioppoli partitions into the sum of piecewise constant rigid movements and suitably rescaled displacements. These converge to limiting partitions, deformations, and displacements, respectively. Whereas limiting deformations are simple laminates whose gradients only attain two values, the limiting displacements belong to the class of special functions with bounded variation (SBV). The latter feature elastic contributions measuring the distance to simple laminates, as well as jumps associated to two consecutive phase transitions having vanishing distance, and thus not being detected by the limiting deformations. By Gamma-convergence we identify an effective limiting model given by the sum of a quadratic linearized elastic energy in terms of displacements along with two surface terms. The first one is proportional to the total length of interfaces created by jumps in the gradient of the limiting deformation. The second one is proportional to twice the total length of interfaces created by jumps in the limiting displacement, as well as by the boundaries of limiting partitions. A main tool of our analysis is a novel two-well rigidity estimate which has been derived in [32] for a model with anisotropic second-order perturbation.
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