Finite difference method

Abstract: Engineering materials are generally non-homogeneous, yet standard continuum descriptions of such materials are admissible, provided that the size of the non-homogeneities is much smaller than the characteristic length of the deformation pattern. If this is not the case, either the individual non-homogeneities have to be described explicitly or the range of applicability of the continuum concept is extended by including additional variables or degrees of freedom. In the paper the discrete nature of granular materials is modelled in the simplest possible way by means of finite-difference equations. The difference equations may be homogenised in two ways: the simplest approach is to replace the finite differences by the corresponding Taylor expansions. This leads to a Cosserat continuum theory. A more sophisticated strategy is to homogenise the equations by means of a discrete Fourier transformation. The result is a Kunin-type non-local theory. In the following these theories are analysed by considering a model consisting of independent periodic 1D chauns of solid spheres connected by shear translational and rotational springs. It is found that the Cosserat theory offers a healthy balance between accuracy and simplicity. Kunin’s integral homogenisation theory leads to a non-local Cosserat continuum description that yields an exact solution, but does not offer any real simplification in the solution of the model equations as compared to the original discrete system. The rotational degree of freedom affects the phenomenology of wave propagation considerably. When the rotation is suppressed, only one type of wave, viz. a shear wave, exists. When the restriction on particle rotation is relaxed, the velocity of this wave decreases and another, high velocity wave arises.