Finite Element Simulation Of Strain Localization Based On Cosserat Continuum Model

As strain softening constitutive behavior is incorporated into a computational model inthe frame of classical plastic continuum theories, the initial and boundary value problem ofthe model will become ill-posed, resulting in pathologically mesh-dependent solutions.Furthermore, the energy dissipated at strain softening is incorrectly predicted to be zero, andthe finite element solutions converge to incorrect, physically meaningless ones as the elementmesh is refined. To correctly simulate strain localization phenomena characterized byoccurrence and severe development of the deformation localized into narrow bands of intenseirreversible straining caused by strain softening, it is required to introduce some type ofregularization mechanism into the classical continuum model to preserve the well-posednessof the localization problem.One of the radical approaches to introduce the regularization mechanism into the modelis to utilize the Cosserat micro-polar continuum theory, in which high-order continuumstructures are introduced. As two dimensional problems are concerned, a rotational degree offreedom with the rotation axis orthogonal to the 2D plane, micro-curvatures as spatialderivatives of the rotational degree of freedom, coupled stresses energetically conjugate to themicro-curvatures and the material parameter defined as internal length scale are introduced inthe Cosserat continuum. In this paper, the Cosserat micro-polar continuum theory isintroduced into the FEM numerical model, which is used to simulate the strain localizationphenomena occuring in solid and saturated porous media.First, a pressure-dependent elastoplastic Cosserat continuum model is presented. Thenon-associated Drucker-Prager yield criterion is particularly considered. Splitting the scalarproduct of the stress rate and the strain rate into their deviatoric and spherical parts, theconsistent algorithm of the pressure-dependent elastoplastic model is derived in theframework of Cosserat continuum theory, i.e. the return mapping algorithm for the integrationof the rate constitutive equation and the closed form of the consistent elastoplastic tangentmodulus matrix. The matrix inverse operation usually required in the calculation ofelastoplastic tangent constitutive modulus matrix is avoided, that ensures the second orderconvergence rate and the computational efficiency of the model in numerical solutionprocedure. The strain localization phenomena due to strain softening are numericallysimulated using the developed model with corresponding finite element method. Numerical results of the plane strain examples illustrate the capability and performance of the presentmodel in keeping the well-posedness of the boundary value problems with strain softeningbehavior incorporated and in reproducing the characteristics of strain localization problems, i.e. intense plastic straining development localized into the narrow band and a significantreduction of the load-carrying capacity of the structure in consideration.Next, the effects of constitutive parameters, such as Cosserat shear module, the softeningmodule and the internal length scale for modelling the softening behaviour and adhering tothe Cosserat continuum, on the numerical results of the simulation of the strain localization, are studied. It is pointed out that the value of Cosserat shear module chosen within certainrange has no effect on the results; the greater the absolute value of the soften modules is, thesteeper the post-failure curve of load-displacement is and the narrower the width of shearband is; the greater the value of the internal scale is, the flatter the post-failure curve ofload-displacement is and the wider the width of shear band is. The numerical study in theperformance of two types of Cosserat continuum finite elements, i.e. u8ω8(8-noded elementinterpolation approximations for both displacements u and microrotationω) and u8ω4(8-noded and 4-noded element interpolation approximations for u andωrespectively)elements is carried out. It is indicated that as compared with the u8ω8 element mesh, theu8ω4 element mesh possesses better performance in simulation of post-failure process. Bothtwo possess better performance in simulation of strain localization.Then, based on the pressure-dependent elastoplastic Cosserat continuum model, progressive failure phenomena, which occured in the typical geotechnical engineeringproblems, such as the slope, excavation, retaining wall and soil foundation, characterized bystrain localization due to the material dilatancy, i.e. non-associated plasticity, or strainsoftening, are numerically simulated. Numerical results illustrate that as compared with theperformance of the finite element procedure for the classical continuum model, the finiteelement procedure based on Cosserat continuum model is capable of preserving thewell-posedness of the localization problems widely existing in geoteclanical engineering andsimulating the entire progressive failure phenomena characterized by strain localization due tostrain softening or the non-associated yield criterion adopted.Besides, the Biot-Cosserat continuum model for coupled hydro-dynamic processes insaturated porous media is proposed by means of the combination of both Biot theory andCosserat continuum theory to simulate the strain localization phenomena due to the strainsoftening. In the present contribution, the Biot formulation of the skeleton material isextended to include the microrotation and correspondingly the coupled stresses defined in theCosserat model. The finite element formulations governing the coupled hydro-dynamicbehavior with the primary variables of the displacements and the microrotaion for the solid phase and the pressure for the fluid phase are derived on the basis of the Galerkin-weightedresidual method. The strain localization phenomena in saturated porous media due to thestrain softening are numerically simulated by using the developed model with correspondingfinite element method and the non-associated Drucker-Prager yield criterion particularlyconsidered for the pressure-dependent elasto-plastic Cosserat skeleton. Numerical results ofthe plane strain examples illustrate that the capability of the developed model in keeping thewell-posedness of the boundary value problems with strain softening behavior incorporated, and the availability of modeling the strain localization phenomena due to the strain softeningin saturated media.In addition, an elastoplastic Cosserat continuum model for CAP constitutive model withnon-smooth multiplicative yield surfaces is presented. The consistent algorithm of the CAPelastoplastic model is derived in the framework of Cosserat continuum theory, i.e. the returnmapping algorithm for the integration of the rate constitutive equation and the closed form ofthe consistent elastoplastic tangent modulus matrix. The matrix inverse operation usuallyrequired in the calculation of elastoplastic tangent constitutive modulus matrix is avoided.The strain localization phenomena of the slope due to strain softening and the failure of thetunnel due to the excavation are numerically simulated using the developed model withcorresponding finite element method. Numerical results of the plane strain examples illustratethe capability and performance of the present model in keeping the well-posedness of theboundary value problems and ensuring the second order convergence rate and thecomputational efficiency of the model in numerical solution procedure. It also illustrates thepossibility of application of Cosserat continuum theory to various geotechnical constitutivemodels.Finally, the two engineering examples, i.e. the failure of the two geotechnical structurescaused in the filling and the excavation processes respectively, are performed by using theproposed models and the developed algorithms. The non-associated perfect elastoplasticbehavior is particularly considered for the filling material and the strain softening behavior forthe excavation material. Numerical results indicate the capability and performance of Cosseratcontinuum model in keeping the well-posedness of the boundary value problems with strainsoftening behavior or non-associated perfect elastoplastic behavior incorporated and incompleting simulation of the whole failure progress.