ABSTRACT
In the analysis of the filling problem, somewhat more complicated constitutive model is used. The relation between the mean normal stress, p, and the volume change, εvol, is assumed to be viscoelastic. If the granular material is non-cohesive, the relation can be expressed as follows:
where p0 and e0 denote pressure and void ratio for the state with respect to which the volume change is measured, κ is the elastic coefficient, µ the bulk viscosity ratio, the dot denotes the time partial derivative, and the brackets, ( )+, the positive part of an expression, (x)+ = max(0, x). The elastic part of pressure p (the first component of the right hand side in Equation (5)) follows from the relation between the void ratio and the pressure, well-known in soil mechanics, e.g. (Schofield & Wroth 1968):
The relation between the deviatoric parts of the rateof-deformation and rate-of-stress tensors is defined by Equations (2)–(4). The only difference is that the shear modulus depends on the pressure and volume change
3 MATERIAL POINT SOLUTION
The considered problem of silo discharge is solved by the use of the material point method, e.g. (Sulsky
et al. 1994; Sulsky & Schreyer 1996), where – as in the standard finite element method – the principle of virtual work is the starting point for the formulation of the method. The equation of virtual work has the following form:
where V0 denotes the space of kinematically admissible fields of displacements, σ the part of boundary where tractions are given, and c the part of the boundary where the frictional contact problem is to be solved. Let us introduce a division of the region occupied initially by the analyzed body into a set of subregions – each of them represented by one of its points called a material point. We assume that the mass density field satisfies the equation
where MP and XP denote the mass and the position of the P-th material point, δ(x) is the Dirac δ-function. Besides this space discretization (of Lagrangian type), another one – an Eulerian finite element mesh (called a computational mesh) covering the virtual position of the analyzed body – is also used. This mesh can be changed arbitrarily during calculations or remain constant. After substituting expression in Equation (7) to the equation of virtual work (6) and expressing the field of acceleration, ai, and the weight functions, wi, by the shape functions and nodal parameters, defined on the computational mesh, as in the finite element method, we obtain the following system of dynamic equations:
where M is the mass matrix, a the vector of nodal accelerations, F, Fc and R are the vectors of external, contact and internal nodal forces, respectively. The main difference between the finite element and material point methods is based on the fact that the state variables are traced at the material points, defined independently of the computational mesh in MPM, and at integration points connected with elements in FEM. The system of dynamic equations (8) is solved by means of the explicit time integration procedure.
