Poromechanics of saturated media based on the logarithmic finite strain

In this paper, we introduce the mathematical formulation and numerical implementation of a coupled thermo-hydro-mechanical model for saturated poromaterials undergoing logarithmic finite deformation and corotational rates. The model combines (i) the thermodynamics of standard materials, (ii) the frame indifferent hyperelastoplasticty, (iii) the orthogonality condition of maximum dissipation, as well as (iv) the principles of conservations of mass, energy and momenta. This formulation involves new developments based on the logarithmic strain measures and corotational rates which overcome the aberrant oscillations classically encountered in large simple shear. It also takes into accounts recent findings on the thermodynamics of dissipative materials which consist of deriving the yielding conditions and flow rules from suitable free energy and dissipation functions. This framework resulted in the implementation of a new finite element algorithm based on Galerkin’s method. The numerical procedures used in this paper involve the spectral decomposition of the logarithmic strain measures, the gradient split techniques as well as the return mapping method. The formulation is validated using the classical problems of Terzaghi and strip loading consolidation.

[1]  M. Biot.  MECHANICS OF DEFORMATION AND ACOUSTIC PROPAGATION IN POROUS MEDIA , 1962 .

[2]  M. Kleiber,et al.  Modelling and numerical analysis of stresses and strains in the human lung including tissue-gas interaction , 1994 .

[3]  H. Xiao,et al.  Strain Rates and Material Spins , 1998 .

[4]  Jacob Lubliner,et al.  A maximum-dissipation principle in generalized plasticity , 1984 .

[5]  J. C. Simo,et al.  On the computational significance of the intermediate configuration and hyperelastic stress relations in finite deformation elastoplasticity , 1985 .

[6]  A. Nádai.  Plastic Behavior of Metals in the Strain‐Hardening Range. Part I , 1937 .

[7]  P. M. Naghdi,et al.  SOME REMARKS ON ELASTIC-PLASTIC DEFORMATION AT FINITE STRAIN , 1971 .

[8]  Alexander M. Puzrin,et al.  Principles of Hyperplasticity: An Approach to Plasticity Theory Based on Thermodynamic Principles , 2010 .

[9]  Ronaldo I. Borja,et al.  Stabilized low-order finite elements for coupled solid-deformation/fluid-diffusion and their application to fault zone transients , 2008 .

[10]  Nicola Castelletto,et al.  A fully coupled 3-D mixed finite element model of Biot consolidation , 2010, J. Comput. Phys..

[11]  Ali Karrech,et al.  Frame indifferent elastoplasticity of frictional materials at finite strain , 2011 .

[12]  P. M. Naghdi,et al.  A general theory of an elastic-plastic continuum , 1965 .

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

[14]  Alexander M. Puzrin,et al.  A thermomechanical framework for constitutive models for rate-independent dissipative materials , 2000 .

[15]  Lev Davidovich Landau,et al.  Mécanique des fluides , 1989 .

[16]  Ali Karrech,et al.  Time-dependent, irreversible entropy production and geodynamics , 2010, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[17]  A. Karrech,et al.  A unified multi-scale thermodynamical framework for coupling geomechanical and chemical simulations , 2010 .

[18]  J. Dienes.  On the analysis of rotation and stress rate in deforming bodies , 1979 .

[19]  B. Schrefler,et al.  The Finite Element Method in the Static and Dynamic Deformation and Consolidation of Porous Media , 1998 .

[20]  Ali Karrech,et al.  Continuum damage mechanics for the lithosphere , 2011 .

[21]  Boumediene Nedjar,et al.  Frameworks for finite strain viscoelastic-plasticity based on multiplicative decompositions. Part II: Computational aspects , 2002 .

[22]  Boumediene Nedjar,et al.  Frameworks for finite strain viscoelastic-plasticity based on multiplicative decompositions. Part I: Continuum formulations , 2002 .

[23]  W. Brocks,et al.  On internal dissipation inequalities and finite strain inelastic constitutive laws: Theoretical and numerical comparisons , 2006 .

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

[25]  R. Tolman,et al.  On the irreversible production of entropy , 1948 .

[26]  Klaus Regenauer-Lieb,et al.  Towards a Self Consistent Plate Mantle Model that Includes Elasticity: Simple Benchmarks and Application to Basic Modes of Convection , 2005 .

[27]  Walter Noll,et al.  The thermodynamics of elastic materials with heat conduction and viscosity , 1963 .

[28]  Andrew M. Stuart,et al.  A First Course in Continuum Mechanics: Bibliography , 2008 .

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

[30]  J. C. Simo,et al.  Consistent tangent operators for rate-independent elastoplasticity☆ , 1985 .

[31]  Francisco Armero,et al.  Formulation and finite element implementation of a multiplicative model of coupled poro-plasticity at finite strains under fully saturated conditions , 1999 .

[32]  Abimael F. D. Loula,et al.  Improved accuracy in finite element analysis of Biot's consolidation problem , 1992 .

[33]  O. Bruhns,et al.  On objective corotational rates and their defining spin tensors , 1998 .

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

[35]  Olaf Kolditz,et al.  Finite element analysis of poro-elastic consolidation in porous media: Standard and mixed approaches , 2006 .

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

[37]  Ronaldo I. Borja,et al.  Finite element formulation for transient pore pressure dissipation: A variational approach , 1986 .

[38]  A. Karrech,et al.  A damaged visco-plasticity model for pressure and temperature sensitive geomaterials , 2011 .

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

[40]  Frédéric Valentin,et al.  Enriched finite element methods for unsteady reaction-diffusion problems , 2005 .

[41]  S. Nemat-Nasser.  On finite deformation elasto-plasticity , 1982 .

[42]  J. C. Simo,et al.  Remarks on rate constitutive equations for finite deformation problems: computational implications , 1984 .

[43]  Heng Xiao,et al.  Elastoplasticity beyond small deformations , 2006 .

[44]  Patrick de Buhan,et al.  The constitutive equations of finite strain poroelasticity in the light of a micro-macro approach , 1998 .

[45]  Giang D. Nguyen,et al.  A coupled damage–plasticity model for concrete based on thermodynamic principles: Part I: model formulation and parameter identification , 2008 .

[46]  O. Bruhns.  Objective Rates in Finite Elastoplasticity , 2003 .

[47]  J. Mandel,et al.  Equations constitutives et directeurs dans les milieux plastiques et viscoplastiques , 1973 .