Strashnov Stanislav, Alexandrov Sergei, Lang Lihui
General Education Courses Department, Рeoples' Friendship University of Russia (RUDN University), 6 Miklukho-Maklaya Street, 117198 Moscow, Russia.
School of Mechanical Engineering and Automation, Beihang University, 37 Xueyuan Road, Beijing 100191, China.
Materials (Basel). 2021 Mar 2;14(5):1166. doi: 10.3390/ma14051166.
The present paper provides a semianalytic solution for finite plane strain bending under tension of an incompressible elastic/plastic sheet using a material model that combines isotropic and kinematic hardening. A numerical treatment is only necessary to solve transcendental equations and evaluate ordinary integrals. An arbitrary function of the equivalent plastic strain controls isotropic hardening, and Prager's law describes kinematic hardening. In general, the sheet consists of one elastic and two plastic regions. The solution is valid if the size of each plastic region increases. Parameters involved in the constitutive equations determine which of the plastic regions reaches its maximum size. The thickness of the elastic region is quite narrow when the present solution breaks down. Elastic unloading is also considered. A numerical example illustrates the general solution assuming that the tensile force is given, including pure bending as a particular case. This numerical solution demonstrates a significant effect of the parameter involved in Prager's law on the bending moment and the distribution of stresses at loading, but a small effect on the distribution of residual stresses after unloading. This parameter also affects the range of validity of the solution that predicts purely elastic unloading.
本文针对不可压缩弹塑性薄板在拉伸作用下的有限平面应变弯曲问题，采用结合各向同性强化和随动强化的材料模型给出了一种半解析解。仅需通过数值方法求解超越方程并计算普通积分。等效塑性应变的任意函数控制各向同性强化，而普拉格定律描述随动强化。一般而言，薄板由一个弹性区域和两个塑性区域组成。如果每个塑性区域的尺寸增大，则该解是有效的。本构方程中涉及的参数决定了哪个塑性区域达到其最大尺寸。当本解失效时，弹性区域的厚度非常窄。文中还考虑了弹性卸载。一个数值例子说明了在给定拉力情况下的一般解，其中包括纯弯曲这一特殊情况。该数值解表明，普拉格定律中涉及的参数对加载时的弯矩和应力分布有显著影响，但对卸载后的残余应力分布影响较小。该参数还会影响预测纯弹性卸载的解的有效范围。
