Localizations of plastic deformation such as shear band and necking have been investigated during past several decades. From experimental observations, microstructural details inside shear band are related to combination of re-crystallization, phase transformation, and high temperature rising. Because it happens in such a short time, e.g., a few nanoseconds, that initiation of localization can hardly be captured by current experimental methods. On the other hand, theoretical explanations of initiation of localized plastic deformation have attempted in many ways. Early investigations are based on corotational constitutive models and singular yielding surface theory. Recently different viscoplastic constitutive models are applied to understand localization. But a clear picture is still not obtained because of complex mathematical structures and phenomenological aspect of current constitutive models and yielding theories. For example, it is not clear that initiation of localized plastic deformation is from dislocation softening or thermal softening. In this paper, a new proposed physical model from the perspective of energy transmission is used to explain initiation of localization of plastic deformation. It is shown in this mathematical framework that a transition from homogenous plastic deformation to localized plastic deformation is mathematically related to a transition from hyperbolic differential system to elliptical differential system. In this transition, wave equation is changed to become Laplace equation. Initial value problem is changed to boundary value problem. Heat generated due to plastic energy is calculated by mechanical version of Joule’s law. The trigger of localization is not only initial imperfections but also can be hydrostatic pressure and grain size changes.

在过去的几十年中，已经研究了塑性变形的局部性，例如剪切带和颈缩。从实验观察，剪切带内部的微观结构细节与重结晶，相变和高温上升的组合有关。因为它发生在这么短的时间内（例如几纳秒），所以当前的实验方法几乎无法捕获定位的开始。另一方面，以多种方式尝试了对局部塑性变形的引发的理论解释。早期的研究是基于规范的本构模型和奇异屈服面理论。最近，使用了不同的粘塑性本构模型来理解定位。但是，由于当前本构模型和屈服理论的复杂数学结构以及现象学方面，仍然无法获得清晰的图像。例如，不清楚局部塑性变形是由位错软化或热软化引起的。本文从能量传递的角度提出了一种新的物理模型，用以解释塑性变形局部化的开始。在该数学框架中表明，从均匀塑性变形到局部塑性变形的转变在数学上与从双曲微分系统到椭圆微分系统的转变有关。在此转变中，波动方程式变为拉普拉斯方程式。初始值问题变为边值问题。由塑性能产生的热量是通过焦耳定律的机械形式计算的。定位的触发因素不仅是初始缺陷，还可能是静水压力和晶粒尺寸变化。