Formulation and finite element implementation of a multiplicative model of coupled poro-plasticity at finite strains under fully saturated conditions

In this paper we present the formulation and numerical analysis of a finite deformation model of coupled poro-plasticity. Fully saturated conditions are assumed. The model is based on a multiplicative decomposition of the deformation gradient in elastic and plastic parts, together with an additive elastoplastic decomposition of the fluid content. The final relations describing the evolution of the different fields are then obtained in the thermodynamic framework furnished by the principle of maximum plastic dissipation. The particular case of an associated Drucker-Prager model in effective stresses is considered as a model example. A complete characterization of the dissipative structure of the problem of evolution is described through the derivation of an a priori stability estimate. We describe in detail the finite element implementation of the model in the context of staggered algorithms for the solution of the physically coupled problem. Finally, we present representative numerical simulations to illustrate the features of the proposed models. In particular, the case of strain localization in globally undrained conditions is studied in the proposed framework.

[1]  Francisco Armero,et al.  A new unconditionally stable fractional step method for non‐linear coupled thermomechanical problems , 1992 .

[2]  J. C. Simo,et al.  Algorithms for static and dynamic multiplicative plasticity that preserve the classical return mapping schemes of the infinitesimal theory , 1992 .

[3]  M. Biot.  General Theory of Three‐Dimensional Consolidation , 1941 .

[4]  Bernard Schrefler,et al.  On convergence conditions of partitioned solution procedures for consolidation problems , 1993 .

[5]  O. C. Zienkiewicz,et al.  Unconditionally stable staggered solution procedure for soil-pore fluid interaction problems , 1988 .

[6]  J. Z. Zhu,et al.  The finite element method , 1977 .

[7]  Klaus-Jürgen Bathe,et al.  A hyperelastic‐based large strain elasto‐plastic constitutive formulation with combined isotropic‐kinematic hardening using the logarithmic stress and strain measures , 1990 .

[8]  Jacob Lubliner,et al.  Normality rules in large-deformation plasticity , 1986 .

[9]  Morton E. Gurtin,et al.  Thermodynamics and stability , 1975 .

[10]  J. Moreau.  Application of convex analysis to the treatment of elastoplastic systems , 1976 .

[11]  J. C. Small,et al.  Elasto-plastic consolidation of soil , 1976 .

[12]  J. Bear.  Dynamics of Fluids in Porous Media , 1975 .

[13]  M. Biot.  THEORY OF ELASTICITY AND CONSOLIDATION FOR A POROUS ANISOTROPIC SOLID , 1955 .

[14]  Bernhard A. Schrefler,et al.  The Finite Element Method in the Deformation and Consolidation of Porous Media , 1987 .

[15]  E. H. Lee,et al.  Finite‐Strain Elastic—Plastic Theory with Application to Plane‐Wave Analysis , 1967 .

[16]  M. Biot.  THEORY OF DEFORMATION OF A POROUS VISCOELASTIC ANISOTROPIC SOLID , 1956 .

[17]  I. Vardoulakis.  Deformation of water-saturated sand: I. uniform undrained deformation and shear banding , 1996 .

[18]  M. Biot.  Theory of finite deformations of porous solids , 1972 .

[19]  J. C. Small,et al.  The analysis of finite elasto-plastic consolidation , 1979 .

[20]  N. N. Yanenko,et al.  The Method of Fractional Steps , 1971 .

[21]  Sanjay Govindjee,et al.  Coupled stress-diffusion: Case II , 1993 .

[22]  I. Vardoulakis.  Deformation of water-saturated sand: II. effect of pore water flow and shear banding , 1996 .

[23]  Bernhard A. Schrefler,et al.  Shear band localization in saturated porous media , 1995 .

[24]  Paul Germain,et al.  Applications of methods of functional analysis to problems in mechanics : Joint symposium IUTAM/IMU held in Marseille, September 1-6, 1975 , 1976 .

[25]  En-Jui Lee.  Elastic-Plastic Deformation at Finite Strains , 1969 .

[26]  R. Borja,et al.  A mathematical framework for finite strain elastoplastic consolidation Part 1: Balance laws, variational formulation, and linearization , 1995 .

[27]  W. T. Koiter.  Stress-strain relations, uniqueness and variational theorems for elastic-plastic materials with a singular yield surface , 1953 .

[28]  M. Biot.  Thermoelasticity and Irreversible Thermodynamics , 1956 .

[29]  Michael Ortiz,et al.  A unified approach to finite deformation elastoplastic analysis based on the use of hyperelastic constitutive equations , 1985 .

[30]  T. R. Hughes,et al.  Mathematical foundations of elasticity , 1982 .

[31]  Jean-Herve Prevost,et al.  Dynamic Strain Localization in Fluid‐Saturated Porous Media , 1991 .

[32]  Chandrakant S. Desai,et al.  Numerical Methods in Geotechnical Engineering , 1979 .

[33]  W. L. Wood.  Practical Time-Stepping Schemes , 1990 .

[34]  Constantine M. Dafermos,et al.  Contraction semigroups and trend to equilibrium in continuum mechanics , 1976 .

[35]  Craig J. Hickey,et al.  Mechanics of porous media , 1994 .

[36]  J. C. Small,et al.  An investigation of the stability of numerical solutions of Biot's equations of consolidation , 1975 .

[37]  L. Anand,et al.  Finite deformation constitutive equations and a time integrated procedure for isotropic hyperelastic—viscoplastic solids , 1990 .

[38]  Jean H. Prevost.  Consolidation of Anelastic Porous Media , 1981 .

[39]  J. C. Simo,et al.  A priori stability estimates and unconditionally stable product formula algorithms for nonlinear coupled thermoplasticity , 1993 .

[40]  E. Wilson,et al.  FINITE-ELEMENT ANALYSIS OF SEEPAGE IN ELASTIC MEDIA , 1969 .

[41]  William Alan Day,et al.  Continuum thermodynamics based on a notion of rate of loss of information , 1975 .

[42]  Bernard D. Coleman,et al.  On thermodynamics and the stability of motions of materials with memory , 1973 .

[43]  Maurice A. Biot,et al.  Theory of Stress‐Strain Relations in Anisotropic Viscoelasticity and Relaxation Phenomena , 1954 .

[44]  K. C. Park,et al.  Stabilization of partitioned solution procedure for pore fluid-soil interaction analysis , 1983 .

[45]  Pierre Maurice Marie Duhem,et al.  Traité D'Énergétique ou de Thermodynamique Générale , 1913 .

[46]  J. Mandel,et al.  Plasticité classique et viscoplasticité , 1972 .

[47]  Francisco Armero,et al.  On the formulation of enhanced strain finite elements in finite deformations , 1997 .

[48]  John M. Ball,et al.  Lyapunov functions for thermomechanics with spatially varying boundary temperatures , 1986 .

[49]  J. C. Simo,et al.  Associated coupled thermoplasticity at finite strains: formulation, numerical analysis and implementation , 1992 .

[50]  M. Ortiz,et al.  A material‐independent method for extending stress update algorithms from small-strain plasticity to finite plasticity with multiplicative kinematics , 1992 .