Keywords: meshfree methods, RKPM, FEM, computational plasticity, cap model, pressure-sensitive material.

Although the finite element method has been the most popular and widely used method in engineering computations, its procedure is not always advantageous. Since the FEM interpolation functions are built upon a mesh, the numerical compatibility condition is not the same as the physical compatibility condition of a continuum. For instance, mesh distortion can either end the computation or result in drastic deterioration of accuracy. In addition, FEM often requires a very fine mesh in problems with high gradients or a distinct local character, which can be computationally expensive [1]. Therefore, it would be computationally efficacious to discretize a continuum by only a set of nodal points or particles, without mesh constraints. This is the leitmotif of contemporary meshfree Galerkin methods [2].

According to computational modelling, particle methods may be categorized into two different types; those serving as approximations of strong forms of Partial Differential Equations, and those serving as approximations of weak forms of PDEs. The second class of particle methods is used with various Galerkin weak formulations. Reproducing Kernel Particle Method (RKPM) is an example of this type. The RKPM introduces a correction function to the Smooth Particle Hydrodynamics (SPH) kernel function to correct the boundary errors in it. In fact, the basic idea of the RKPM is to formulate the discrete consistency that is lacking in SPH. Due to the correction function the accuracy enhances near, or on the boundary domain and the RKPM kernel function obtains the consistency conditions throughout the problem domain. A corrected collocation method is used to enforce essential boundary conditions within this meshfree method and conditions are enforced exactly at a discrete set of boundary nodes [3]. In this study, the reproducing kernel particle method is applied to the analysis of two-dimensional problems involving pressure-sensitive materials.

The yielding of granular materials, such as: geomaterials, rock, and concrete is pressure-sensitive. The yield criterion for these materials should include the influence of hydrostatic pressure. Development of constitutive stress-strain models that can provide adequate physical representation of observed mechanical behaviour in such materials, is a challenging problem. Cap models provide a flexible tool for representing the many diverse aspects of the observed dynamic stress-strain behaviour of pressure-sensitive materials [4].

In order to describe the constitutive model of the non-linear behaviour of pressure-sensitive materials, a cone-cap plasticity model is applied based on a hardening rule to define the dependence of the yield surface on the degree of plastic straining [5]. The main features of the cap model include a failure surface and an elliptical yield cap which closes the open space between the failure surface and the hydrostatic axis. The yield cap expands in the stress space according to a specified hardening rule. The model is based on a generalized plasticity theory, which consists of three different yield surfaces. The first is used to confine the tension limit. Second yield surface is the fixed one which depends on the material characteristics. The third surface is the moving cap surface, which can move in stress space with increase of volumetric plastic strain and model the hardening behavior of material. The role of the exponential failure envelop is to limit the level of shear stress that the material can support without failure. The moving cap surface has an elliptical form with a constant ratio of major to minor radius that intersects the failure envelope in a nonsmooth fashion. The tension cutoff surface is a fixed surface that determines the ultimate tensile strength of the material.

In the present paper, an application of a two invariant cap model for pressure- sensitive materials is presented. The meshfree algorithms based on the reproducing kernel particle method is employed to the analysis of two-dimensional problems involving pressure-sensitive materials. Several numerical examples are solved to demonstrate the applicability of the RKPM algorithm in modelling of pressure- sensitive material.

1

A.R. Khoei and R.W. Lewis, "Adaptive finite element remeshing in a large deformation analysis of metal powder forming", Int. Jou. Num. Meth. Eng., 45, 801-820, 1999. doi:10.1002/(SICI)1097-0207(19990710)45:7<801::AID-NME604>3.3.CO;2-R

2

Sh. Li, W.K. Liu, "Meshfree and particle methods and their applications", Applied Mechanics Review, 55(1), 2002. doi:10.1115/1.1431547

3

G.J. Wagner, W.K. Liu, "Application of essential boundary conditions in meshfree methods: a corrected collocation method", International Journal for Numerical Methods in Engineering, 47(8), 1367-1379, 2000. doi:10.1002/(SICI)1097-0207(20000320)47:8<1367::AID-NME822>3.0.CO;2-Y

4

G. Hofstetter, J.C. Simo, R.L. Taylor, "A modified cap model: closest point solution algorithms", Computers & Structures, 46(2), 203-214, 1993. doi:10.1016/0045-7949(93)90185-G

5

R.W. Lewis and A.R. Khoei, "A plasticity model for metal Powder forming processes", Int. Jou. Plasticity, 12, 1659-1692, 2001. doi:10.1016/S0749-6419(00)00096-6

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