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SUMMARYMany finite elements exhibit the so-called 'volumetric locking' in the analysis of incompressible or quasi-incompressible problems. In this paper, a new approach is taken to overcome this undesirable effect. The starting point is a new setting of the governing differential equations using a finite calculus (FIC) formulation. The basis of the FIC method is the satisfaction of the standard equations for balance of momentum (equilibrium of forces) and mass conservation in a domain of finite size and retaining higher order terms in the Taylor expansions used to express the different terms of the differential equations over the balance domain. The modified differential equations contain additional terms which introduce the necessary stability in the equations to overcome the volumetric locking problem. The FIC approach has been successfully used for deriving stabilized finite element and meshless methods for a wide range of advective-diffusive and fluid flow problems. The same ideas are applied in this paper to derive a stabilized formulation for static and dynamic finite element analysis of incompressible solids using linear triangles and tetrahedra. Examples of application of the new stabilized formulation to linear static problems as well as to the semi-implicit and explicit 2D and 3D non-linear transient dynamic analysis of an impact problem and a bulk forming process are presented.

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 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.

In this paper, a residual correction method based upon an extension of the finite calculus concept is presented. The method is described and applied to the solution of a scalar convection‐diffusion problem and the problem of elasticity at the incompressible or quasi‐incompressible limit. The formulation permits the use of equal interpolation for displacements and pressure on linear triangles and tetrahedra as well as any low order element type. To add additional stability in the solution, pressure gradient corrections are introduced as suggested from developments of sub‐scale methods. Numerical examples are included to demonstrate the performance of the method when applied to typical test problems.

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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.

Summary This work presents a new original formulation of the discrete element method (DEM) with deformable cylindrical particles. Uniform stress and strain fields are assumed to be induced in the particles under the action of contact forces. Particle deformation obtained by strain integration is taken into account in the evaluation of interparticle contact forces. The deformability of a particle yields a nonlocal contact model, it leads to the formation of new contacts, it changes the distribution of contact forces in the particle assembly, and it affects the macroscopic response of the particulate material. A numerical algorithm for the deformable DEM (DDEM) has been developed and implemented in the DEM program DEMPack. The new formulation implies only small modifications of the standard DEM algorithm. The DDEM algorithm has been verified on simple examples of an unconfined uniaxial compression of a rectangular specimen discretized with regularly spaced equal bonded particles and a square specimen represented with an irregular configuration of nonuniform‐sized bonded particles. The numerical results have been verified by a comparison with equivalent finite element method results and available analytical solutions. The micro‐macro relationships for elastic parameters have been obtained. The results have proved to have enhanced the modeling capabilities of the DDEM with respect to the standard DEM.