Finite Pure Plane Strain Bending of Inhomogeneous Anisotropic Sheets

Abstract

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.

Plastic Bending at Large Strain: A Review

“…Papers [46,47] extended the method above to sheets with continuously varying through-thickness properties. The material model adopted in [46] is isotropic.…”

“…The material model adopted in [46] is isotropic. Paper [47] generalized this solution to anisotropic materials. The solution in [46] is for elastic/plastic materials.…”

“…The solution in [46] is for elastic/plastic materials. Both elastic/plastic and rigid/plastic models were considered in [47].…”

“…According to this approach, the distribution of stresses found from a rigid/plastic solution at the end of the loading stage is used as the initial distribution for the stage of unloading, which allows one to calculate the distribution of residual stresses and strains. Paper [47] explored the accuracy of the approach proposed in [58] for the analysis of bending.…”

“…The method described in the previous section was used in [47] for obtaining elastic/plastic and rigid/plastic solutions for the pure bending of anisotropic sheets with a non-uniform through-thickness distribution of material properties. The quantitative results below are for the distribution of properties given in [59].…”