Two stress update algorithms for large strains: accuracy analysis and numerical implementation

Two algorithms for the stress update (i.e., time integration of the constitutive equation) in large-strain solid mechanics are compared from an analytical point of view. The order of the truncation error associated to the numerical integration is deduced for each algorithm a priori, using standard numerical analysis. This accuracy analysis has been performed by means of a convected frame formalism, which also allows a unied derivation of both algorithms in spite of their inherent dierences. Then the two algorithms are adapted from convected frames to a xed Cartesian frame and implemented in a small-strain nite element code. The implementation is validated by means of a set of simple deformation paths (simple shear, extension, extension and compression, extension and rotation) and two benchmark tests in non-linear mechanics (the necking of a circular bar and a shell under ring loads). In these numerical tests, the observed order of convergence is in very good agreement with the theoretical order of convergence, thus corroborating the accuracy analysis. ? 1997 John Wiley & Sons, Ltd.

[1]  Clifford Ambrose Truesdell,et al.  The Simplest Rate Theory of Pure Elasticity , 1955 .

[2]  J. Altenbach.  Zienkiewicz, O. C., The Finite Element Method. 3. Edition. London. McGraw‐Hill Book Company (UK) Limited. 1977. XV, 787 S. , 1980 .

[3]  H. Saunders,et al.  Finite element procedures in engineering analysis , 1982 .

[4]  L. Brillouin.  Les tenseurs en mécanique et en élasticité , 1987 .

[5]  L. E. Malvern.  Introduction to the mechanics of a continuous medium , 1969 .

[6]  Cv Clemens Verhoosel,et al.  Non-Linear Finite Element Analysis of Solids and Structures , 1991 .

[7]  K. Bathe,et al.  FINITE ELEMENT FORMULATIONS FOR LARGE DEFORMATION DYNAMIC ANALYSIS , 1975 .

[8]  Antonio Huerta,et al.  A note on a numerical benchmark test: an axisymmetric shell under ring loads , 1997 .

[9]  T. Hughes.  Numerical Implementation of Constitutive Models: Rate-Independent Deviatoric Plasticity , 1984 .

[10]  C. Truesdell,et al.  Corrections and Additions to 'The Mechanical Foundations of Elasticity and Fluid Dynamics' , 1953 .

[11]  T. Hughes,et al.  Finite rotation effects in numerical integration of rate constitutive equations arising in large‐deformation analysis , 1980 .

[12]  J. C. Simo,et al.  A framework for finite strain elastoplasticity based on maximum plastic dissipation and the multiplicative decomposition. part II: computational aspects , 1988 .

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

[14]  P. Pegon,et al.  Finite strain plasticity in convected frames , 1986 .

[15]  R. D. Krieg,et al.  On the numerical implementation of inelastic time dependent and time independent, finite strain constitutive equations in structural mechanics☆ , 1982 .

[16]  P. Pinsky,et al.  Numerical integration of rate constitutive equations in finite deformation analysis , 1983 .

[17]  O. C. Zienkiewicz,et al.  The finite element method, fourth edition; volume 2: solid and fluid mechanics, dynamics and non-linearity , 1991 .

[18]  J. Oldroyd.  On the formulation of rheological equations of state , 1950, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.