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.
Highlights
Sheet metal forming processes usually include bending
An exact rigid perfectly plastic solution for finite pure plane strain bending and plane strain bending under tension of sheets has been found in [11]
If the distribution of material properties is symmetric relative to the surface ζ = −1/2, W(0) = W(−1) and it follows from the third case in (35) that 16se(se − ae) = 1
Summary
Sheet metal forming processes usually include bending. A brief review of typical sheet metal forming processes that incorporate bending is provided in [1]. A solution for pure plane strain bending of anisotropic sheets has been derived in [17]. A more detailed review of solutions for plane strain bending of anisotropic sheets is provided in [20]. Symmetry 2021, 13, 145 more detailed review of solutions for plane strain bending of anisotropic sheets is provided in [20]. According to the model proposed in [35], the principal anisotropy axes coincide with the ζ− and η− coordinate curves throughout the process of deformation. T is the shear yield stress with respect to the ζ− and η− coordinate curves, and c is expressed through the yield stresses in the principal anisotropy axes’ directions [11].
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