Logarithmic Strain Measure Research Articles

Numerical simulation of the large elastic–plastic deformation behavior of hydrostatic stress-sensitive solids

The present paper deals with the numerical simulation of the large elastic–plastic deformation and localization behavior of metals which are plastically dilatant and sensitive to hydrostatic stresses. The model is based on a generalized macroscopic theory taking into account macroscopic as well as microscopic experimental data obtained from tests with iron based metals. It shows that hydrostatic components may have a significant effect on the onset of localization and the associated deformation modes, and that they generally lead to a notable decrease in ductility. The continuum formulation relies on the mixed-variant metric transformation tensor which leads to the definition of an appropriate logarithmic strain measure. Its rate is additively decomposed into elastic and plastic as well as isochoric and volumetric strain rate tensors. Particular attention is focused on the formulation of a generalized I1–J2 yield criterion to describe the effect of the hydrostatic stress on the plastic flow properties in metals. In contrast to classical theories of metal plasticity, the evolution of the plastic part of the strain rate tensor is determined by a non-associated flow rule based on a plastic potential function which is expressed in terms of stress invariants and kinematic parameters. Numerical analyses of the elastic–plastic deformation and localization behavior of hydrostatic stress-sensitive metals will demonstrate the influence of the constitutive description on critical strains as well as on localization behavior.

Large viscoplastic deformations of shells. Theory and finite element formulation

The paper is concerned with large viscoplastic deformations of shells when the constitutive model is based on the concept of unified evolution equations. Specifically the model due to Bodner and Partom is modified so as to fit in the frame of multiplicative viscoplasticity. Although the decomposition of the deformation gradient in elastic and inelastic parts is employed, no use is made of the concept of the intermediate configuration. A logarithmic elastic strain measure is used. An algorithm for the evaluation of the exponential map for nonsymmetric arguments as well as a closed form of the tangent operator are given. On the side of the shell theory itself, the shell model is chosen so as to allow for the application of a three-dimensional constitutive law. The shell theory, accordingly, allows for thickness change and is characterized by seven parameters. The constitutive law is evaluated pointwise over the shell thickness to allow for general cyclic loading. An enhanced strain finite element method is given and various examples of large shell deformations including loading-unloading cycles are presented.

Strain Rates and Material Spins

The material time rate of Lagrangean strain measures, objective corotational rates of Eulerian strain measures and their defining spin tensors are investigated from a general point of view. First, a direct and rigorous method is used to derive a simple formula for the gradient of the tensor-valued function defining a general class of strain measures. By means of this formula and the chain rule as well as Sylvester's formula for eigenprojections, explicit basis-free expressions for the material time rate of an arbitrary Lagrangean strain measure can be derived in terms of the right Cauchy–Green tensor and the material time rate of any chosen Lagrangean strain measure, e.g. Hencky's logarithmic strain measure. These results provide a new derivation of Carlson–Hoger's general gradient formula for an arbitrary generalized strain measure and supply a new, rigorous proof for Carlson–Hoger's conjecture concerning the n-dimensional case. Next, by virtue of the aforementioned gradient formula, a general fact for objective corotational rates and their defining spin tensors is disclosed: Let Ω = ϒ ( B, D, W) be any spin tensor that is continuous with respect to B, where B, D and B are the left Cauchy–Green tensor, the stretching tensor and the vorticity tensor. Then the corotational rate of an Eulerian strain measure defined by Ω is objective iff Ω = W + υ ( B, D), where Υ is isotropic. By means of this fact and certain necessary or reasonable requirements, it is further found that a single antisymmetric function of two positive real variables can be introduced to characterize a general class of spin tensors defining objective corotational rates. A general basis- free expression for all such spin tensors and accordingly a general basis-free expression for a general class of objective corotational rates of an arbitrary Eulerian strain measure are established in terms of the left Cauchy–Green tensor B and the stretching tensor B as well as the introduced antisymmetric function. By choosing several particular forms of the latter, all commonly-known spin tensors and corresponding corotational rates are shown to be incorporated into these general formulas in a natural way. In particular, with the aid of these general formulae it is proved that an objective corotational rate of the Eulerian logarithmic strain measure ln V is identical with the stretching tensor D and moreover that in all possible strain tensor measures only ln V enjoys this property.

On theory and numerics of large viscoplastic deformation

A multiplicative theory and corresponding computations of finite elastic-viscoplastic deformations based on unified constitutive equations are presented. Basic features of theory are: (1) The inelastic part of the deformation gradient is understood as a material stretch-type tensor, no need arises for the notion of the intermediate configuration. (2) In the isotropic case, the use of an elastic logarithmic strain tensor is shown to lead to the identification that the elastic strains are given by the quantity C −1 p C with C p being an inelastic Cauchy-Green type tensor. (3) In spite of the fact that the logarithmic strain measure is used, closed forms of the tangent operator within an implicit time integration scheme are given circumventing very involved elaborations when the spectral decomposition is used. (4) Evolution equations of the Bodner and Partom type are employed. A general formalism is developed for the application of the evolution laws of the unified type within the theoretical framework. The formalism can be used to generalize any unified evolution equations formulated for infinetismal strains to the range of finite strains. (5) For the axisymmetric case, in addition to the displacement-based 4-node and 9-node finite elements, an enhanced strain 4-node element is presented. It is shown, anyhow, that in the highly nonlinear regime the enhanced strain formulation may exhibit numerical instabilities. Different numerical examples of finite strain deformations are presented.

Hypo-Elasticity Model Based upon the Logarithmic Stress Rate

Recently these authors have proved [46, 47] that a smooth spin tensor Ωlog can be found such that the stretching tensor D can be exactly written as an objective corotational rate of the Eulerian logarithmic strain measure ln V defined by this spin tensor, and furthermore that in all strain tensor measures only ln V enjoys this favourable property. This spin tensor is called the logarithmic spin and the objective corotational rate of an Eulerian tensor defined by it is called the logarithmic tensor-rate. In this paper, we propose and investigate a hypo-elasticity model based upon the objective corotational rate of the Kirchhoff stress defined by the spin Ωlog, i.e. the logarithmic stress rate. By virtue of the proposed model, we show that the simplest relationship between hypo-elasticity and elasticity can be established, and accordingly that Bernstein's integrability theorem relating hypo-elasticity to elasticity can be substantially simplified. In particular, we show that the simplest form of the proposed model, i.e. the hypo-elasticity model of grade zero, turns out to be integrable to deliver a linear isotropic relation between the Kirchhoff stress and the Eulerian logarithmic strain ln V, and moreover that this simplest model predicts the phenomenon of the known hypo-elastic yield at simple shear deformation.

The appropriate corotational rate, exact formula for the plastic spin and constitutive model for finite elastoplasticity

The exact formulae for the plastic and the elastic spin referred to the deformed configuration are derived, where the plastic spin is a function of the plastic strain rate and the elastic spin a function of the elastic strain rate. With these exact formulae we determine the macroscopic substructure spin that allows us to define the appropriate corotational rate for finite elastoplasticity.Plastic, elastic and substructure spin are considered and simplified for various sub-classes of restricted elastic-plastic strains. It is shown that for the special cases of rigid-plasticity and hypoelasticity the proposed corotational rate is identical with the Green-Naghdi rate, while the ZarembaJaumann rate yields a good approximation for moderately large strains.To compare our exact plastic spin formula with the constitutive assumption for the plastic spin introduced by Dafalias and others, we simplify our result for small elastic-moderate plastic strains and introduce a simplest evolution law for kinematic hardening leading to the Dafalias formula and to an exact determination of its unknown coefficient. It is also shown that contrary to statements in the literature the plastic spin is not zero for vanishing kinematic hardening.For isotropic-elastic material with induced plastic flow undergoing isotropic and kinematic hardening constitutive and evolution laws are proposed. Elastic and plastic Lagrangean and Eulerian logarithmic strain measures are introduced and their material time derivatives and corotational rates, respectively, are considered. Finally, the elastic-plastic tangent operator is derived.The presented theory is implemented in a solution algorithm and numerically applied to the simple shear problem for finite elastic-finite plastic strains as well as for sub-classes of restricted strains. The results are compared with those of the literature and with those obtained by using other corotational rates.

A finite element formulation for finite strain elasto-plastic analysis based on mixed interpolation of tensorial components

A quadrilateral finite element formulation for modelling finite strain elasto-plastic deformation processes is presented. The developed formulation is based on Lee's multiplicative decomposition of the deformation gradient and on the hyperelastic expression of the associated Von Mises flow rule. The Hencky or logarithmic strain measure is used in the strain energy function. The quadrilateral element is developed using displacements interpolation and interpolation of the Hencky strain tensor covariant components. A consistent linearization is developed, obtaining a symmetric stiffness matrix.

Theory and analysis of shells undergoing finite elastic-plastic strains and rotations

In this paper theory and analysis of shells undergoing finite elastic and finite plastic strains and rotations are presented. The shell kinematics are based on a relaxed normality hypothesis allowing transverse normal material fibers to be stretched and bended, whereas shear deformations are neglected. Lagrangean logarithmic membrane and logarithmic bending strain measures are introduced, and it is shown that they can be additively decomposed into purely elastic and purely plastic parts for superposed moderately large strains and unrestricted rotations. The logarithmic strain measures are used to formulate thermodynamic-based constitutive equations for isotropic elastic and plastic material behavior with isotropic and kinematic hardening induced by continuous plastic flow. To analyse path-dependent elastic-plastic shell deformations by iterative procedures the application of logarithmic strain measures allows to realize load steps with corresponding moderate strains and unrestricted rotations. The moderate strain restriction for superposed deformations can be assured by an appropriate update procedure. Formulae are given to determine exactly the rotational change of the reference configuration during the update. Finally, the principle of virtual work with corresponding elastic-plastic material tensor is formulated and it is shown that the weak form of the virtual work leads to the Lagrangean equilibrium equations and boundary conditions well-known from the nonlinear theory of elastic shells.

A hyperelastic‐based large strain elasto‐plastic constitutive formulation with combined isotropic‐kinematic hardening using the logarithmic stress and strain measures

AbstractThis paper addresses the formulation of a set of constitutive equations for finite deformation metal plasticity. The combined isotropic‐kinematic hardening model of the infinitesimal theory of plasticity is extended to the large strain range on the basis of three main assumptions: (i) the formulation is hyperelastic based, (ii) the stress‐strain law preserves the elastic constants of the infinitesimal theory but is written in terms of the Hencky strain tensor and its elastic work conjugate stress tensor, and (iii) the multiplicative decomposition of the deformation gradient is adopted. Since no stress rates are present, the formulation is, of course, numerically objective in the time integration. It is shown that the model gives adequate physical behaviour, and comparison is made with an equivalent constitutive model based on the additive decomposition of the strain tensor.

A tensorial Hencky measure of strain and strain rate for finite deformations

A proper tensorial logarithmic strain measure and its associated time rate of change is developed using an equivalence between the expressions for an isotropic tensor function and the Sylvester-Lagrange formalism. It is shown that the specific Cauchy stress T/ρ is derivable from a potential with respect to this strain measure. A modified stress TF is shown to be derivable from a potential with respect to the left polar decomposition of the deformation gradient. Advantages and disadvantages of this formulation are discussed.

Effect of finite deformation on the elastic moduli — Part II

It is assumed that during purely elastic response, an elastic-plastic material admits a work potential. The work potential is written in terms of the principal stretches. The constitutive relations are then derived, through which the elements of the elastic moduli are calculated for a particular solid whose uniaxial stress-strain behaviour for logarithmic strain measure is known.