Three Dimensional Cosserat Continuum Model And Its Numerical Application

In the Cosserat continuum theory, the local rotation is different from media’s global rotation at each material point. Based on this, three independent rotational degrees-of-freedom are introduced into this theory except three translational degrees-of-freedom. Accordingly, the couple-stresses energetically conjugate to the micro-curvatures, are introduced into the Cosserat theory, and also, as a kind of regularization mechanism, the concept of internal length scale which can be treated as material’s "characteristic dimension" is introduced into material’s constitutive equation. For these features, Cosserat continuum theory is mainly used in three aspects. Firstlly, because of the introduction of internal length scale, Cosserat theory can effectively explain and simulate the difference of mechanical properties of the material with micro-strcture under macroscopic or microcosmic circumstance, this phenomenon is also called material’s size effect. Secondly, by introducing the internal length scale into Cosserat theory as a type of regularization mechanism, the FE governing equation’s property can be preserved as elliptic type when using Cosserat theory to deal with soil’s strain localization problem, and this method will not only ensures the convergence and well-posedness of numerical solutions, but also avoid pathologically mesh-dependent solutions of the localization problem. Finally, by introducing independent rotational degrees-of-freedom which can be used to describe the micro-rotation of material point, the couple-stresses energetically conjugate to the micro-curvatures are introducing into the Cosserat theory naturely, this makes it able to consider the buckling effect of layerd structures, such as layered rock, and simulate the failure progress of layered structure caused by external force effectively. Bsed on the Cosserat theory, this paper has finished several works followed in three dimensional conditions:(1) The basic kinematics laws of Cosserat continuum are derived firstly, the components of stresses, couple-stresses, strains and micro-curvanatures are presented also, then, the equilibrium governing equations, geometric differential equations and constitutive equations are obtained in three dimensional elastic Cosserat continuum. Based on these works, an elastic FE program of three dimensional Cosserat model is developed by the UEL provided by ABAQUS, and numerical results of the deflections of a micro-cantilever beam and the relative angle of a micro-rod illustrate that numerical method based on Cosserat theory is able to reflect the size effect of specified materials, however, this is beyond the classical theroy’s capacity.(2) Based on the two-dimensional Cosserat theory, an elastoplastic Cosserat continuum model for three dimensional pressure-dependent geotechnical material is presented, the consistent algorithm of the pressure-dependent elastoplastic model, including the closed form of the consistent elastoplastic tangent modulus matrix, is derived also. This avoids inverse operation of consistent elastoplastic tangent modulus matrix, and ensures the convergence and well-posedness of numerical solutions. Similarly, the elastoplastic FE program of the pressure-dependent Cosserat continuum model is developed by the UEL. A series of numerial examples are applied to analyse the strain localization probems of geotechnical material and calculate the ouccurence and development of shear band. Through comparing the results calculated by classical numerical model and the developed Cosserat model, all of them demonstrate that Cossrat theory can more effectively simulate the progressive failure caused by strain localization in geotechnical material. This model excellently sovles the pathologically mesh-dependent problem of numerical results in localization problem, which frequently occurs in the classical model.(3) Taking the anisotropic properties of geotechnical matrial into consideration, a transversely isotropic Cosserat continuum model is developed based on the Cosserat continuum theory and transversely isotropic elastic theory in three-dimensional condition; also, an equivalent Cosserat continuum model for layered rock is developed based on former researches. The FE programs of these two models are both developed by the UEL, several numerical examples are analysed by them. The results show that these models can even better reflect anisotropic properties of geotechnical material, and, to specific geotechnical engineering problems, using appropriate model will be helpful to improve the precision of numerial results.