Energy scaling and branched microstructures in a model for shape-memory alloys with SO(2) invariance
Energy scaling and branched microstructures in a model for shape-memory alloys with SO(2) invariance
复制标题
具有 SO(2) 不变性的形状记忆合金模型中的能量缩放和分支微观结构
DOI：
--
复制
发表时间：
2014
期刊：
影响因子：
--
通讯作者：
S. Conti
中科院分区：
文献类型：
--
作者：
Allan Chan;S. Conti
在许多材料微观结构的奇异摄动模型中，边界附近的区域分支都会出现，Kohn和Muller首先在描述形状记忆合金弹性行为的标量问题上证明了这一点。在这里，我们研究形状记忆合金的模型，该模型基于非线性弹性的全矢量问题，包括在旋转下的不变性，在二维两个井的情况下。结果表明，对于具有两个一阶连接的两个势垒，能量与表面能的2/3次方成正比，符合标量模型。在只存在一个一阶连接的情况下，我们证明了能量表现出不同的行为，与表面能量的4/5次方成正比。这种较低的能量是通过变形的两个分量的适当相互作用实现的，因此不能用标量模型来再现。通过显式构造和匹配下界证明了这两种标度。
Domain branching near the boundary appears in many singularly perturbed models for microstructure in materials and was first demonstrated mathematically by Kohn and Muller for a scalar problem representing the elastic behavior of shape-memory alloys. We study here a model for shape-memory alloys based on the full vectorial problem of nonlinear elasticity, including invariance under rotations, in the case of two wells in two dimensions. We show that, for two wells with two rank-one connections, the energy is proportional to the power 2/3 of the surface energy, in agreement with the scalar model. In a case where only one rank-one connection is present, we show that the energy exhibits a different behavior, proportional to the power 4/5 of the surface energy. This lower energy is achieved by a suitable interaction of the two components of the deformations and hence cannot be reproduced by the scalar model. Both scalings are proven by explicit constructions and matching lower bounds.
影响因子：
3
作者：
H. Knüpfer;R. V. Kohn;F. Otto
通讯作者：
F. Otto