H-B. Muhlhaus

A variational principle for gradient plasticity

Abstract We elaborate on a generalized plasticity model which belongs to the class of gradient models suggested earlier by Aifantis and co-workers. The generalization of the conventional theory of plasticity has been accomplished by the inclusion of higher-order spatial gradients of the equivalent plastic strain in the yield condition. First it is shown how these gradients affect the critical condition for the onset of localization and allow for a wavelength selection analysis leading to estimates for the width and or spacing of shear bands. Due to the presence of higher-order gradients, additional boundary conditions for the equivalent plastic strain are required. This question and also the associated problem of the formulation and solution of general boundary value problems were left open in the previous work. We demonstrate here that upon assuming a certain type of additional boundary conditions, the structural symmetries of the gradient-dependent constitutive model are such that there exists a variational principle for the displacement rates and the rate of the equivalent plastic strain. The variational principle can serve as a basis for the numerical solution of boundary value problems in the sense of the finite element method. Explicit expressions for the tangent stillness matrix and the generalized nodal point forces are given.

Wave propagation, localization and dispersion in a gradient-dependent medium

A continuum model that incorporates a dependence upon the Laplacian of the inelastic strain is used to regularize the initial value problem that results from the introduction of strain softening or non-associated flow. It is shown that the introduction of this gradient dependence preserves well-posedness of the initial value problem and that wave propagation in the enhanced continuum is dispersive. An analysis of the dispersive wave propagation reveals the existence of an internal length scale. Numerical analyses of one-dimensional and two-dimensional problems confirm that this internal length scale sets the localization zone and show that the results are insensitive to the fineness of the discretization and to the direction of the grid lines. This holds true with respect to the strain profiles, the energy dissipation and the extent of wave reflection.

Dispersion and wave propagation in discrete and continuous models for granular materials

Abstract A generalised continuum model for granular media is derived by direct homogenisation of the discrete equations of motion. In contrast to previous works on this topic, continuum concepts such as stress and moment stress are introduced after homogenisation. First, a very simple one-dimensional model is considered and the continuum version for this model is derived by replacing the difference quotients of the discrete model by differential quotients. The dispersion relations of the discrete and the continuous model are derived and compared. Variational boundary conditions for the continuous model are deduced from the stationarity of the corresponding Lagrangian. The three-dimensional case is treated in an essentially similar fashion. The resulting continuum theory is a combination of a Cosserat Continuum and a higher-order deformation gradient continuum. The salient features of the theory are illustrated by means of the dispersion relations for planar wave propagation.

A numerical study of pore‐fluid, thermal and mass flow in fluid‐saturated porous rock basins

We present a numerical methodology for the study of convective pore‐fluid, thermal and mass flow in fluid‐saturated porous rock basins. In particular, we investigate the occurrence and distribution pattern of temperature gradient driven convective pore‐fluid flow and hydrocarbon transport in the Australian North West Shelf basin. The related numerical results have demonstrated that: (1) The finite element method combined with the progressive asymptotic approach procedure is a useful tool for dealing with temperature gradient driven pore‐fluid flow and mass transport in fluid‐saturated hydrothermal basins; (2) Convective pore‐fluid flow generally becomes focused in more permeable layers, especially when the layers are thick enough to accommodate the appropriate convective cells; (3) Large dislocation of strata has a significant influence on the distribution patterns of convective pore‐fluid flow, thermal flow and hydrocarbon transport in the North West Shelf basin; (4) As a direct consequence of the formation of convective pore‐fluid cells, the hydrocarbon concentration is highly localized in the range bounded by two major faults in the basin.

Path independent integrals for cosserat continua and application to crack problems

We generalise Noethers theorem to include Cosserat Continua and derive corresponding conservation laws and path independent boundary integrals. From translational invariance of the deformation energy we optain the Cosserat generalisation of the J-integral. In a Cosserat Continuum, additional integrals follow from rotational invariance.

Fractures and Defects in Cosserat Continua Modelling Layered Materials

This paper considers fracture of layered materials consisting of many layers that are thin compared to the characteristic size of the loading (eg, wave length). In this case the explicit modelling of fracturing of every layer becomes cumbersome. Instead, we represent the material as an equivalent continuum that provides a large-scale (average) description of the material response to loading. This is achieved by introducing volume elements that are much greater than the layer thickness and, at the same time, much smaller than the characteristic length of stress variations. It should be noted that in this approach no lengths smaller than the volume element size can be distinguished.