Triangles and tetrahedra in explicit dynamic codes for solids

Abstract

Explicit dynamic codes which are used currently for the study of plastic deformations in impact, or with some modification for metal forming, suffer two serious limitations.First, only quadrilateral or hexahedral linear elements can be used thus limiting the possibilities of adaptive refinement and adaptive meshing.Second, even with the use of such elements, special devices such as reduced integration must be introduced to avoid locking and reduce costs. These necessitate complex hour glass control, mending-type procedures.The main difficulties are those due to the need of treatment of (almost) incompressible deformation modes. Recently, similar difficulties have been overcome in the context of fluid dynamics and soil dynamics\ and we show here how the processes introduced there can be adopted effectively to the present problem, thus allowing an almost unrestricted choice of element interpolations.1998 John Wiley & Sons, Ltd.

Finite calculus formulation for incompressible solids using linear triangles and tetrahedra

“…It is interesting to note that the form of Equation (36) is very similar to that obtained using the CBS method [7,8,11]. The FIC and CBS forms can be made in fact identical if the time increment in the CBS method is chosen to be coincident with the intrinsic time parameter.…”

“…Equations (5) and (6) are completed with the boundary conditions on the displacements u i u i −ū i = 0 on u (7) and the equilibrium of surface tractions…”

“…It is well known that in these cases the use of equal order interpolations for displacements and pressure leads to locking solutions unless some precautions are taken. A stabilized finite element formulation based on the CBS method allowing for linear triangles and tetrahedra for transient dynamic analysis of quasi-incompressible solids was reported by the authors in References [7,11]. A similar formulation based on the FIC approach which does not require the split process is described next.…”

“…The pressure values also coincide reasonably well in the compression zone, although some differences are found in the peak pressure values induced by tensile (negative) stresses. These differences also occur for solutions obtained with linear triangles and the CBS method [7,11].…”

“…A similar method was derived for quasi-incompressible solids by Zienkiewicz and Taylor [1]. Zienkiewicz et al [7] have proposed a stabilization technique which eliminates volumetric locking in incompressible solids based on a mixed formulation and a characteristic based split (CBS) algorithm initially developed for fluids [8][9][10] where a split of the pressure is introduced when solving the transient dynamic equations in time. Extensions of the CBS algorithm to solve bulk metal forming problems have been recently reported by Rojek et al [11].…”