Modelling principal stress rotation effects with multilaminate type constitutive models for clay

Isotropic constitutive models where the yield surface is expressed in terms of stress invariants are not capable to predict the effects of principal stress axes rotation. The modelling of principal stress rotation could be achieved through the description of a translational or rotational rule for the yield surface, but the formulation may be complicated and very often has no physical meaning. Another method is to use the multilaminate framework where simple constitutive relations between rates of stresses and strains are used on spatially distributed planes. Overall behaviour is obtained by integrating inelastic contributions from all planes. Two different versions of multilaminate models are briefly introduced and used to simulate experimental results from the literature concerning principal stress axes rotation for clay. In both models volumetric hardening is applied by using a cap surface. The first model introduces also a shear hardening yield surface whereas the second model considers viscoplastic effects. Simulation results show good qualitative agreement with the experimental data. Inspiration for the present paper was the experimental work of Akagi & Saitoh (1994) and Akagi & Yamamoto (1997) where undisturbed and reconstituted clay samples were tested under pure principal stress rotation in hollow cylinder apparatus. These test results are used for evaluation of two versions of multilaminate models. 2 MULTILAMINATE CONSTITUTIVE MODELS 2.1 Multilaminate concept Taylor (1938) suggested to describe the stress-strain relationship for metal crystals independently on planes of various orientations in the material assuming that either the stresses or the strains on the planes are the decomposed components of the macroscopic stress or strain tensor, respectively. Calladine (1971) adopted this idea for clay. The physical model behind this suggestion is a solid block of homogeneous isotropic linear elastic material, that is intersected by an infinite number of randomly oriented planes, see Figure 2. It is assumed, that the deformation behaviour of such a material block is obtainable by a description of the sliding phenomenon under a current effective normal and shear stress component on the planes (σn, τ) and the opening/closing of the inter-boundary gap between two contact planes. Moreover it is supposed that all planes have the same properties. The infinite number of planes yields to an homogeneous material behaviour of the block that is no longer linear elastic. This so-called multilaminate framework was introduced by Zienkiewicz & Pande (1977), Pande & Sharma (1983) and Pietruszczak & Pande (1987) for rocks and soil. Within the multilaminate framework the local stress vector i σ on contact plane i is derived from the global stress vector σ by stress transformation, see Equation 1. 1 [ ] σ T σ ⋅ = i i (1) The stress-strain relations are formulated locally on the planes (microscopic level), except the elastic part, which is calculated on the global or macroscopic level. Figure 2. Isotropic material block intersected by randomly oriented planes. Figure 3. View of the 33 planes per hemisphere in space. The global behaviour is obtained by integration of the inelastic contributions from each contact plane and the global elastic part. The integration over an infinite number of planes is numerically replaced by an integration rule, which is a summation over n defined planes,