The present paper concerns the general solution for finite plane strain pure bending of incompressible, orthotropic sheets. In contrast to available solutions, the new solution is valid for inhomogeneous distributions of plastic properties. The solution is semi-analytic. A numerical treatment is only necessary for solving transcendent equations and evaluating ordinary integrals. The solution’s starting point is a transformation between Eulerian and Lagrangian coordinates that is valid for a wide class of constitutive equations. The symmetric distribution relative to the center line of the sheet is separately treated where it is advantageous. It is shown that this type of symmetry simplifies the solution. Hill’s quadratic yield criterion is adopted. Both elastic/plastic and rigid/plastic solutions are derived. Elastic unloading is also considered, and it is shown that reverse plastic yielding occurs at a relatively large inside radius. An illustrative example uses real experimental data. The distribution of plastic properties is symmetric in this example. It is shown that the difference between the elastic/plastic and rigid/plastic solutions is negligible, except at the very beginning of the process. However, the rigid/plastic solution is much simpler and, therefore, can be recommended for practical use at large strains, including calculating the residual stresses.

中文翻译：

---

非均质各向异性板的有限纯平面应变弯曲

本文涉及不可压缩正交各向异性板有限平面应变纯弯曲的一般解决方案。与现有解决方案相比，新解决方案适用于塑性特性的不均匀分布。解决方案是半解析的。数值处理仅对于求解超越方程和评估普通积分是必需的。解决方案的出发点是欧拉坐标和拉格朗日坐标之间的转换，该转换对多种本构方程均有效。在有利的情况下，相对于片的中心线的对称分布被分开处理。结果表明，这种对称性简化了求解过程。采用希尔的二次屈服准则。导出了弹性/塑性和刚性/塑性解决方案。还考虑了弹性卸载 并且显示出反向塑性屈服在相对大的内半径处发生。一个说明性的例子使用真实的实验数据。在此示例中，塑性特性的分布是对称的。结果表明，除了刚开始时，弹性/塑性和刚性/塑性解决方案之间的差异可以忽略不计。但是，刚性/塑性解决方案要简单得多，因此，建议将其实际应用在大应变下，包括计算残余应力。除了过程的开始。但是，刚性/塑性解决方案要简单得多，因此，建议将其实际应用在大应变下，包括计算残余应力。除了过程的开始。但是，刚性/塑性解决方案要简单得多，因此，建议将其实际应用在大应变下，包括计算残余应力。