Nonlinear Meshless Methods For Contact-Impact Problems With Applications In Metal Forming Analysis

The analysis of both two-dimensional and three-dimensional contact-impact problems by using the Reproducing Kernel Particle Method (RKPM) is developed. Prior to the treatment of the contact-impact interfaces, a general benchmark test and lower integration rules are introduced.A general benchmark test is developed for the verifications of the formulations and the implementation of the numerical memod in computer programs. Like in the patch test, in the benchmark test procedure the displacement field is prescribed on the entire boundary of the domain. Linear as well as higher order displacement fields can be prescribed. The major innovation in the benchmark test is that the body forces, which are imposed to be zero in the standard patch, are taken into account. In fact usmg higher order displacement field, the strain and the stress fields are no longer constant through the domain. As consequence, to satisfy the linear momentum conservation law or the equilibrium equation, the body forces are to be added. When linear displacement field is prescribed in the benchmark test, the body forces vanish because the strain and the stress fields are constant; therefore the patch test is a special case of the present benchmark test Through the domain, the values of the fields variables (displacements, strains, stresses ...) computed using the analytical mapping of the deformation are compared to those obtained by the execution of the RKPM-codes (2D and 3D).The meshless methods are characterized by their high computational cost compared to the conventional finite element method. The support domain of the shape function covers a large number of particles and integration points. Therefore the calculation of the stress/strain involves several loops over mis large number of integration points. Reducing the number of integration points without affecting the accuracy of the result will obviously improve the computation efficiency. For this purpose, lower integration schemes are proposed to reduce the computational cost using the RKPM. In two-dimensional, instead of the traditional 2×2 Gauss quadrature rule, two Gauss quadrature points are set in the integration cell while in three-dimensional, instead of thecommonly used 2×2×2 Gauss quadrature rule, four points are set in the integration cell. The correctness and the effectiveness of the reduced integration schemes are proven through the general benchmark test procedure. Both two-dimensional and' three-dimensional cases have been successfully investigated.The simulation of the contact-impact is a challenging task. The contact problems are characterized by the so-called impenetrability condition which does not allow overlapping between contacting bodies. The constraints methods are used to enforce this condition. But prior to the determination of the contact constraints the contacting boundaries are to be found. For mis purpose a particle to segment contact algorithm is developed. The algorithm is originally developed for the simulation of the contact between a deformable body and several rigid bodies. The boundaries of the rigid bodies are represented by flat portions (segments) of surface. The deformable body is discretized by a set of particles from which those located on the boundary may contact with the rigid surfaces. After determination of the contacting boundaries, the penalty method is used to enforce the impenetrability condition by application of penalty forces. This contact algorithm is found to be very suitable for metal forming simulation. In fact in metal forming simulation the tools are usually assumed to be rigid, and the workpiece/blank is deformable. A series of problems involving contact or impact; ranging from rod impact, bulk metal forming, sheet metal forming to the penetration simulations is treated to illustrate the performance of the algorithm using the new integration schemes as well as the traditional 2×2 and 2×2×2 Gauss quadrature rules. The advantage of RKPM over the finite element method in large deformation analysis is confirmed.