Formulation and survey of ALE method in nonlinear solid mechanics
Although the application of the finite element method to many nonlinear problems has been successfully carried out, there are numerous areas of concern and investigation in that regard. One of such areas is the solution of large strain plasticity and metal-forming problems with pseudo-type boundary nonlinearities. A number of critical difficulties arise in the finite element analysis of such problems. Among these difficulties are: the proper formulation of the problem, mesh distortion during the deformation process, the proper modeling of the contact boundary conditions, incorporation of the plastic incompressibility condition and accounting for plastic anisotropy. Foundations of large strain analysis of elastioplastic solids may be traced back to the early work of Hill Ill. It took some time, until Hibbit et al.  introduced, however, the first finite element formulation for large strain problems. In their approach they used a total Lagrangian formulation (TLF). Later on, McMeeking and Rice  pioneered the use of updated Lagrangian formulation (ULF) in the same area of applications. The two formulation methods have been widely used for both steady and non-steady static large plastic strain problems. On the other hand, Eulerian formulation (EF) has been initially introduced for finite element applications in the fluid mechanics area. Several trials aiming at adapting this formulation to large strain and metal-forming problems were attempted [4-7]. Owing to the difficulties in obtaining material time derivatives in spatial reference frame, no generally accepted Eulerian formulation is available for such problems. Some developments, e.g., Gadala et al. , give the proper material time derivatives in a spatial reference frame but original geometric parameters are included in the final equilibrium equation. Therefore, such approach may not be strictly considered as purely Eulerian. Also, since the mesh is spatially fixed in EF, it is not easy to simulate non-steady static or dynamic behavior. Trials have been made to relate the moving material points (in terms of FE Gauss points) to the fixed spatial mesh [6, 7, 8]. Much work is still required, however, to refine and establish this approach. (1.136Mb)
Gadala, Mohamed S.
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This paper investigates the applicability and accuracy of existing formulation methods in general purpose finite element programs to the finite strain deformation problems. The basic shortcomings in using such programs in these applications are then pointed out and the need for a different type of formulation is discussed. An arbitrary Lagrangian-Eulerian (ALE) method is proposed and a concise survey of ALE formulation is given. A consistent and complete ALE formulation is derived from the virtual work equation transformed to arbitrary computational reference configurations. Differences between the proposed formulations and similar ones in the literature are discussed. The proposed formulation presents a general approach to ALE method. It includes load correction terms and is suitable for rate-dependent and rate-independent material constitutive law. The proposed formulation reduces to both updated Lagrangian and Eulerian formulations as special