On Some New General and Complementary Energy Theorems for the Rate Problems in Finite Strain. Classical Elastoplasticity

ABSTRACT General variational theorems for the rate problem of classical elastoplasticity at finite strains, in both Updated Lagrangian (UL) and Total Lagrangian (TL) rate forms, and in terms of alternate measures of stress-rate and conjugate strain-rates, are critically studied from the point of view of their application. Attention. is primarily focused on the derivation of consistent complementary energy rate principles which could form the basis of consistent and rational assumed stress-type finite element methods, and two such principles, in both UL and TL forms, are newly stated. Systematic procedures to exploit these new principles in the context of a finite element method are discussed. Also discussed are certain general modified variational theorems which permit an accurate numerical treatment of near incompressible behavior at large plastic strains

[1]  Ernst Hellinger,et al.  Die allgemeinen Ansätze der Mechanik der Kontinua , 1913 .

[2]  S. Atluri,et al.  Finite Elasticity Solutions Using Hybrid Finite Elements Based on a Complementary Energy Principle , 1978 .

[3]  B. D. Veubeke,et al.  A new variational principle for finite elastic displacements , 1972 .

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

[5]  K. Washizu.  Variational Methods in Elasticity and Plasticity , 1982 .

[6]  Incremental hybrid finite element methods for finite deformation problems (with special emphasis on complementary energy principle) , 1978 .

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

[8]  A. Needleman,et al.  Necking of biaxially stretched elastic-plastic circular plates , 1977 .

[9]  Samuel W. Key,et al.  A variational principle for incompressible and nearly-incompressible anisotropic elasticity , 1969 .

[10]  P. Tong.  An assumed stress hybrid finite element method for an incompressible and near-incompressible material☆ , 1969 .

[11]  C. Truesdell,et al.  The Nonlinear Field Theories in Mechanics , 1968 .

[12]  H. Hibbitt,et al.  A finite element formulation for problems of large strain and large displacement , 1970 .

[13]  A general variational theorem for the rate problem in elasto-plasticity , 1972 .

[14]  P. M. Naghdi,et al.  On the time derivative of tensors in mechanics of continua , 1961 .

[15]  J. Rice,et al.  Finite-element formulations for problems of large elastic-plastic deformation , 1975 .

[16]  S. Nemat-Nasser.  V – General Variational Principles in Nonlinear and Linear Elasticity with Applications , 1974 .

[17]  Alf Samuelsson,et al.  Finite Elements in Nonlinear Mechanics , 1978 .

[18]  Satyanadham Atluri,et al.  On the hybrid stress finite element model for incremental analysis of large deflection problems , 1973 .

[19]  Prager.  AN ELEMENTARY DISCUSSION OF DEFINITIONS OF STRESS RATE. Technical Report No. 53 , 1960 .

[20]  R. Hill.  Eigenmodal deformations in elastic/plastic continua , 1967 .

[21]  Alan Needleman,et al.  A numerical study of necking in circular cylindrical bar , 1972 .

[22]  E. F. Masur.  On the definition of stress rate , 1961 .

[23]  T. H. H. Pian,et al.  Notes on finite elements for nearly incompressible materials , 1976 .

[24]  C. Popelar,et al.  On the use of the complementary energy in the solution of buckling problems , 1976 .

[25]  Karl S. Pister,et al.  On a variational theorem for incompressible and nearly-incompressible orthotropic elasticity , 1968 .

[26]  Recent developments in the theory of finite element approximations of boundary value problems in nonlinear elasticity , 1979 .

[27]  J. G. Oldroyd,et al.  Non-Newtonian effects in steady motion of some idealized elastico-viscous liquids , 1958, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[28]  J. M. Alexander,et al.  Progress in Applied Mechanics , 1964 .

[29]  M. Taya,et al.  Model studies of ductile fracture-I. Formulation , 1976 .

[30]  E. Reissner.  On a Variational Theorem in Elasticity , 1950 .

[31]  Rodney Hill,et al.  Some basic principles in the mechanics of solids without a natural time , 1959 .