多晶体中马氏体微观结构的相容性
Compatibility of Martensitic Microstructures in Polycrystals
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中文总结 AI 辅助
研究多晶体中马氏体微观结构在晶界处的相容性，通过计算机辅助计算等方法，分析不同转变情形下的相容性条件，定义并研究泰勒集，证明其新上界，解释高阶层合板现象并给出相关结果证明。
中文摘要 AI 辅助
本文研究多晶体中的马氏体微观结构，重点关注其在晶界处的相容性。在简化为平面晶界的情况后，考虑晶界分隔两个零能量常数梯度的情形。结果表明，对于立方到四方的转变，当相对晶粒旋转不在立方群中时会出现这种构型。接着考虑晶界分隔两个零能量简单层合板的情形，通过计算机辅助符号计算表明，在立方到四方的情况下，相容性仅在晶界法线和相对晶粒旋转流形中的一个零测度闭集上才可能，对于立方到正交晶系的转变也有类似稍弱的结果。这些结果解释了为何在这种转变中常观察到高阶层合板。定义并研究了变形梯度的泰勒集，该集具有这样的性质：任何其梯度属于它的变形都对应于与晶粒几何形状和晶粒旋转无关的多晶体零能量微观结构。证明了立方到四方和立方到正交晶系转变的泰勒集的新上界，推广了Bhattacharya和Kohn使用几何线性化理论得到的结果。这些界特别暗示了Peigney关于三个四方阱拟凸包中正对角矩阵的一个结果，我们给出了该结果的一个简单独立证明。
英文摘要
The paper studies martensitic microstructures in polycrystals, focussing on their compatibility across grain boundaries. After a reduction to the case of a planar grain boundary, the case when the grain boundary separates two constant gradients of zero energy is considered. It is shown that for cubic-to tetragonal transformations such a configuration can occur when the relative grain rotation is not in the cubic group. Then the case when the grain boundary separates two simple laminates of zero energy is considered, it being shown using a computer-assisted symbolic calculation that in the cubic-to-tetragonal case compatibility is only possible for a closed set of measure zero in the manifold of grain boundary normals and relative grain rotations, and that a similar slightly weaker result holds for cubic-to-orthorhombic transformations. The results suggest why higher-order laminates are often observed for such transformations. The Taylor set of deformation gradients is defined and studied, this set having the property that any deformation whose gradient belongs to it corresponds to a zero-energy microstructure for the polycrystal independent of grain geometry and grain rotations. New upper bounds for the Taylor set are proved for cubic-to-tetragonal and cubic-to-orthorhombic transformations, generalizing those of Bhattacharya and Kohn using the geometrically linearized theory. We give a simple proof of a related result of Peigney characterizing the positive diagonal matrices in the quasiconvex hull of the energy wells for cubic-to-tetragonal transformations.
