On Strain Gradient Approach for Detefmination of Sheet Metal Forming Limits

In this paper, the sheet metal forming limits are presented in strain space. The computations are based on strain gradient theory of plasticity in conjunction with M-K model. This approach introduces an internal length scale into conventional constitutive equations and takes into account the effects of deformation inhomogeneity and material softening. The nonlinear differential equations of the thickness variation is the second order ordinary type. The shooting method has been used with different solvers such as Backward Differentiation Formula (BDFs or Gear Method), explicit Runge-Kutta (the Dormand-Prine pair), explicit Runge-Kutta (the Bogacki- Shampine pair), Adams-Bashforth-Moulton and Rosenbrock formula and Trapezoidal rule. The numerical results of these techiniques are almost the same. However, major differences can be observed in the speed of calculations. It was found that the explicit Runge-kutta (the Dormand-Prince pair) initial value problem solver has the best speed of calculation. The sheet metal forming limits obtained theoretically in this work have been compared with some published experimental data.

Keywords:

Forming Limit Diagram , Strain Gradient , Shooting Method