Lie symmetries of finite strain elastic–perfectly plastic models and exactly consistent schemes for numerical integrations

Liu, Chein‐Shan

doi:10.1016/j.ijsolstr.2003.11.008

Cited by 16 publications

(6 citation statements)

References 41 publications

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“…The foregoing algorithms stem from the structure of J 2 ‐flow elastoplasticity models with the high symmetry property of von Mises yield function and initial material isotropy. In this respect, rich internal symmetry and Lie‐type structure have been uncovered recently by Hong and Liu and Liu for small‐strain and finite‐strain perfect J 2 ‐flow elastoplasticity models. In the just‐mentioned works, a new integration algorithm of second order accuracy is established for perfect J 2 ‐flow elastoplasticity, which preserves the internal symmetry and automatically fulfils the plastic consistency condition.…”

Section: Introductionmentioning

confidence: 89%

Automatic consistency integrations for the stress update of J₂ elastoplastic rate constitutive equations with combined hardening

Yin

Xiao

2014

Numerical Meth Engineering

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SUMMARYA novel method is introduced to study numerical integrations of J2 elastoplastic rate constitutive equations with general combined hardening. The basic idea is to transform the usual time rate constitutive equations into those with reference to the equivalent plastic strain. By virtue of tensorial matrix operations, we show that these transformed equations may be converted to a linear differential system governing the shifted stress and the plastic multiplier. From this system, we derive explicit integrations for the shifted stress and then for the back stress and the Cauchy stress. We demonstrate that these results are accurate up to within a third order term of the equivalent plastic strain increment. In particular, for pure kinematic hardening, we show that the integrations obtained can achieve automatic enforcement of both the plastic consistency condition and the loading condition, thus bypassing the numerical treatment of the latter two. Furthermore, we explain that, with the new algorithm for the stress update, the continuum tangent moduli may be used to ensure a quadratic rate of convergency in Newton's iteration scheme for the balance equation. Numerical examples suggest that the new algorithm may be more accurate and efficient than the widely used return algorithm. Copyright © 2014 John Wiley & Sons, Ltd.

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Section: Introductionmentioning

confidence: 89%

Automatic consistency integrations for the stress update of J₂ elastoplastic rate constitutive equations with combined hardening

Yin

Xiao

2014

Numerical Meth Engineering

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“…Liu and Hong [12] have developed an effective algebraic procedure to construct the spinor map from SL(2, R) onto SO o (2, 1), SL(2, C) onto SO o (3,1) and also SL(2, H) onto SO o (5,1), where H is a quaternion field. The last result as advocated by Liu [10] is very useful in the computational plasticity of large deformation models.…”

Section: Lorentz Group So O (2 1)mentioning

confidence: 96%

“…(27) and (28), andẋ(t) andẏ(t) by Eqs. (9) and (10). The phase portraits of (x,ẋ) and (y,ẏ) for the above two cases are plotted in Figs.…”

Section: (B) and 3(c)mentioning

confidence: 99%

A study of type I intermittency of a circular differential equation under a discontinuous right-hand side

Liu

2007

Journal of Mathematical Analysis and Applications

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In this paper we study a circular differential equation under a discontinuous periodic input, developing a quadratic differential equations system on S 1 and a linear differential equations system in the Minkowski space M 3 . The symmetry groups of these two systems are, respectively, PSO o (2, 1) and SO o (2, 1). The Poincaré circle map is constructed exactly, and a critical value α c of the parameter is identified. Depending on α of the input amplitude the equation may exhibit periodic, subharmonic or quasiperiodic motions. When α varies from α > α c to α < α c , there undergoes an inverse tangent bifurcation; consequently, the resultant Poincaré circle map offers one route to the quasiperiodicity via a type I intermittency. In the parameter range of α < α c the orbit generated by the Poincaré circle map is either m-periodic or quasiperiodic when n/m is a rational or an irrational number.

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“…The simple shear problem is a classical one within the context of large deformation plasticity (see, e.g., Nagtegaal and De Jong [1982], Dafalias [1983], Atluri [1984], Bruhns et al [1999], Liu and Hong [2001], Eshraghi et al [2013] and also the theoretical developments given in Liu [2004], Liu and Hong [2001], Cheviakov et al [2013]) and is defined as…”

Section: Simple Shearmentioning

confidence: 99%

“…(6) These ideas led to the concept of the self-consistent (elastic-plastic) model, which may be defined as one which in the absence of plastic deformation, results in hyperelastic (conservative) response [Bruhns et al, 1999]. Using these derivations as a starting point further research has been conducted dealing with the elastic-plastic torsion problem , the application of the Sturm's comparison theorem in the finite shear problem Liu and Hong [2001] and the Lie symmetries of the governing equations [Liu, 2004]. Very recent developments are those by Zhu et al [2014]; Xiao et al [2014] dealing with the constitutive modeling of metals under cyclic loadings and Brepols et al [2014] dealing with a material model which can be used in industrial metal forming processes.…”

Section: Introductionmentioning

confidence: 99%

Logarithmic Spin, Logarithmic Rate and Material Frame-Indifferent Generalized Plasticity

Soldatos

Triantafyllou

2016

Int. J. Appl. Mechanics

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In this work we present a new rate type formulation of large deformation generalized plasticity which is based on the consistent use of the logarithmic rate concept. For this purpose, the basic constitutive equations are initially established in a local rotationally neutralized configuration which is defined by the logarithmic spin. These are then rephrased in their spatial form, by employing some standard concepts from the tensor analysis on manifolds. Such an approach, besides being compatible with the notion of (hyper)elasticity, offers three basic advantages, namely:(i) The principle of material frame-indifference is trivially satisfied.(ii) The structure of the infinitesimal theory remains essentially unaltered.(iii) The formulation does not preclude anisotropic response.A general integration scheme for the computational implementation of generalized plasticity models which are based on the logarithmic rate is also discussed. The performance of the scheme is tested by two representative numerical examples.

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Lie symmetries of finite strain elastic–perfectly plastic models and exactly consistent schemes for numerical integrations

Cited by 16 publications

References 41 publications

Automatic consistency integrations for the stress update of J₂ elastoplastic rate constitutive equations with combined hardening

Automatic consistency integrations for the stress update of J₂ elastoplastic rate constitutive equations with combined hardening

A study of type I intermittency of a circular differential equation under a discontinuous right-hand side

Logarithmic Spin, Logarithmic Rate and Material Frame-Indifferent Generalized Plasticity

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Lie symmetries of finite strain elastic–perfectly plastic models and exactly consistent schemes for numerical integrations

Cited by 16 publications

References 41 publications

Automatic consistency integrations for the stress update of J2 elastoplastic rate constitutive equations with combined hardening

Automatic consistency integrations for the stress update of J2 elastoplastic rate constitutive equations with combined hardening

A study of type I intermittency of a circular differential equation under a discontinuous right-hand side

Logarithmic Spin, Logarithmic Rate and Material Frame-Indifferent Generalized Plasticity

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Automatic consistency integrations for the stress update of J₂ elastoplastic rate constitutive equations with combined hardening

Automatic consistency integrations for the stress update of J₂ elastoplastic rate constitutive equations with combined hardening