A new computational model for Tresca plasticity at finite strains with an optimal parametrization in the principal space

Abstract A new computational model for the rate-independent elasto-plastic solids characterized by yield surfaces containing singularities and general nonlinear isotropic hardening is presented. Within the context of fully implicit return mapping algorithms, a numerical scheme for integration of the constitutive equations is formulated in the space of principal stresses. As a direct consequence of the principal stress approach, the representation of a yield surface is cast in terms of ‘optimal’ parameterization, which for the Tresca yield criterion takes a simple linear form. The associated return mapping equations then reduce to a remarkably simple format. In addition, due to assumed isotropy of the models, the associated algorithmic (incremental) constitutive functionals can be identified as particular members of a class of isotropic tensor functions of one tensor in which the function eigenvalues are expressed in terms of the eigenvalues of the argument. This observation leads to a simple closed form derivation of the consistent tangent moduli associated with the described integration algorithms. The extension of the present model to finite strains is carried out following standard multiplicative plasticity described in terms of logarithmic stretches and exponential approximation to the flow rule. The efficiency and robustness of the computational model are illustrated on a range of numerical examples.

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

[2]  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 .

[3]  C. Miehe,et al.  Aspects of the formulation and finite element implementation of large strain isotropic elasticity , 1994 .

[4]  D. Owen,et al.  A model for finite strain elasto-plasticity based on logarithmic strains: computational issues , 1992 .

[5]  D. Owen,et al.  Design of simple low order finite elements for large strain analysis of nearly incompressible solids , 1996 .

[6]  Scott W. Sloan,et al.  Removal of singularities in tresca and mohr–coulomb yield functions , 1986 .

[7]  M. A. Crisfield,et al.  Plasticity computations using the Mohr—Coulomb yield criterion , 1987 .

[8]  Donald E. Carlson,et al.  The derivative of a tensor-valued function of a tensor , 1986 .

[9]  Christian Miehe,et al.  Computation of isotropic tensor functions , 1993 .

[10]  J. C. Simo,et al.  Non‐smooth multisurface plasticity and viscoplasticity. Loading/unloading conditions and numerical algorithms , 1988 .

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

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

[13]  R. Hill.  The mathematical theory of plasticity , 1950 .

[14]  Ioannis Vardoulakis,et al.  Constitutive laws for engineering materials: theory and application , 1987 .

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

[16]  E. A. de Souza Neto,et al.  A computational framework for a class of fully coupled models for elastoplastic damage at finite strains with reference to the linearization aspects , 1996 .

[17]  E. P. Popov,et al.  Accuracy and stability of integration algorithms for elastoplastic constitutive relations , 1985 .

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

[19]  D. Owen,et al.  Finite elements in plasticity : theory and practice , 1980 .

[20]  J. Moreau.  Evolution problem associated with a moving convex set in a Hilbert space , 1977 .

[21]  Roger Temam,et al.  Mathematical Problems in Plasticity , 1985 .

[22]  J. C. Simo,et al.  Geometrically non‐linear enhanced strain mixed methods and the method of incompatible modes , 1992 .

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

[24]  O. C. Zienkiewicz,et al.  Elasto‐plastic stress analysis. A generalization for various contitutive relations including strain softening , 1972 .

[25]  Weimin Han,et al.  Computational plasticity : the variational basis and numerical analysis , 1995 .

[26]  R. De Borst,et al.  Integration of plasticity equations for singular yield functions , 1987 .

[27]  A. P. Green.  THE PLASTIC YIELDING OF NOTCHED BARS DUE TO BENDING , 1953 .

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

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

[30]  W. Drugan,et al.  Theory of plasticity , 1987 .

[31]  F. Dimaggio,et al.  MATERIAL MODEL FOR GRANULAR SOILS , 1971 .

[32]  E. Pramono,et al.  Implicit integration of composite yield surfaces with corners , 1989 .