On the Numerical Solution of Nonlinear Problems in Continuum Mechanics - Abo-Elkhier, Abdel-Ghany Mahmoud (My University, 1985)
A critical discussion of the formulation methods for the finite clement analysis of nonlinear problems is given, which includes the Lagrangian, the updated Lagrangian, and the Eulerian formulation. It is shown that each formulation is suitable for a specific class of nonlinear problems. In the literature many authors treat updated Lagrangian formulation as an Eulerian formulation. Therefore, the basic differences between the two formulations are critically discussed.
Consistent Lagrangian and updated Lagrangian formulation are derived from the virtual work principle expressed in current configuration, then transformed to the proper reference configuration. A detailed Eulerian formulation in the current configuration is derived by means of the virtual work principle. Explicit forms for the stiffness matrices contributing to the total nonlinear stiffness matrix, for the mass matrix, and for the load increments are presented in each case. Differences between the presented Lagrangian and the updated Lagrangian formulations and similar formulations in the literature are found to exist in the number of the stiffness matrices in the final incremental equilibrium equations as well as in the definition of the load increments. These differences as well as those between the existing formulations in the literature are assessed within the framework of the basic equations of the continuum mechanics. Specific forms of constitutive equations for elastic and elasto-plastic response of the materials are presented. A discussion on the use of the stress rates to derive acceptable constitutive equations is also given.
For the Lagrangian and the updated Lagrangian formulation two example problems have been solved to demonstrate the applicability of the presented formulations and the effect of the individual stiffness matrices as well as the definition of the follower load which results from the consistent formulation. These problems are elastic, large deformation static analysis of a cantilever under uniformly distributed load and elastic-perfectly plastic dynamic analysis of a pipe-whip problem.
To assess the presented Eulerian formulation and to show the effectiveness of the Eulerian finite element analysis using fixed mesh in space: a metal-extrusion problem has been solved. In this approach, the mesh is maintained fixed in space and the increment of stress tensors for a forward incremental step are added to a set of interpolated stress tensors. Then these stresses are interpolated back to obtain the state of stress of the body-points momentarily occupying the fixed integration points of the mesh.
Keywords: Mechanical Engineering; Mechanical Engineering