Elastic-plastic Solids Research Articles

Development of a three-dimensional finite element program for large strain elastic-plastic solids

In recent years the progress in computational mechanics has made it possible to adopt more sophisticated constitutive models for solving complicated plasticity problems. For the formulation of elastic—plastic theory at finite strain, E.H. Lee decomposed the deformation gradient into a product of the elastic and plastic parts instead of assuming that the strain rate is the combination of the elastic and plastic strain rates. A finite element model has been formulated based on E.H. Lee's theory. Accordingly, a three-dimensional general purpose finite element software package has been developed. The simple tension problem and the simple shear problem are solved as examples so that the numerical solutions and the exact analytical solutions can be compared. Finally, a tensile test of a thin-walled aluminum alloy tube is solved and compared with the experimental data. The good agreement between the finite element solution and the experimental data demonstrates the validity and applicability of the proposed numerical model.

Acceleration instability in elastic-plastic solids. II. Analytical techniques

Acceleration instability occurs when a body is accelerated by surface tractions. This situation resembles classic Rayleigh–Taylor instability, but differs due to the temporal and spatial variation of the stress field in the accelerated body caused by wave propagation and the time dependence of accelerating forces. We wish to investigate the extent to which the acceleration instability response of elastic-plastic solids can be approximately modeled using a modal or one-degree-of-freedom technique assuming incompressible flow. We find that neither the response equation derived assuming a linear elastic shear response nor the so-called ‘‘minimum amplitude’’ instability criterion derived from rigid plastic analyses is sufficient to describe overall stability characteristics at very large driving pressures. Several versions of the modal approximation technique for an elastic-plastic material response are derived and compared. The features which must be included in such an analysis in order to obtain qualitative agreement with instability results seen in numerical experiments are indicated. However, quantitative agreement with numerical results cannot be obtained in all cases. The sources of the disagreement and the regions of applicability of the analysis are discussed.

Acceleration instability in elastic-plastic solids. I. Numerical simulations of plate acceleration

Acceleration instability occurs when a body is accelerated by surface tractions. This situation resembles classic Rayleigh–Taylor instability, but differs due to the temporal and spatial variation of the stress field in the accelerated body caused by wave propagation and the time dependence of the accelerating forces. These factors produce phenomena in acceleration instability which are without precedent in classical Rayleigh–Taylor analyses. An extensive numerical study of acceleration instability using a Lagrangian finite-difference wavecode has determined the influence of various parameters including amplitude and wavelength of initial surface perturbations, material yield strength, and time dependence of the driving force. The nature of the criteria determining stability or instability is established, and the fundamental physical quantity controlling perturbation growth at an interface is shown to be the local stress gradient.

On constitutive inequalities and bifurcation in elastic-plastic solids with a yield-surface vertex

A constitutive inequality for time-independent elastic-plastic models of polycrystalline metals at finite strain is derived from the known constitutive framework for single metal crystals obeying the normality and symmetry postulates. Consequences of the inequality are examined for a general class of nonlinear constitutive rate equations at a yield-surface vertex. It is demonstrated that existence of a strain-rate potential and the normality flow rule can be inferred from the inequality alone. With the help of the constitutive inequality, it is proved that bifurcation in velocities is ruled out by positive definiteness of the quadratic functional based on the tangent moduli. Micromechanically-based justification for linearization of the bifurcation problem along a regular deformation path is obtained in that way, without the need of specifying the full constitutive law. As another example of application of the constitutive inequality, explicit restrictions on phenomenological models of material behaviour at a yield-surface vertex are derived.

Discontinuities in elastic-plastic solids

Stress and strain discontinuities propagating in an elastic-plastic solid are examined within the small-deformation three-dimensional framework. Since in elastoplasticity, field quantities, in general, are history dependent, their possible jumps across a discontinuity surface depend on their variation within the thin transition zone which is often modeled as a discontinuity surface. The discontinuity relations associated with dynamic elasto-plastic problems are formulated in such a manner that all discontinuous field data are described in terms of only two stress components. Various special cases, such as incompressible solids, adiabatic deformations, plane strain problems, and quasistatic advance of discontinuities, are examined; for the quasistatic case, in particular, it is shown that all stress components are continuous and that, if the solid has positive work-hardening, then all strain components are also continuous.

An energy approach to the boundary element method. Part II: Elastic-plastic solids

On the basis of the energy formulation presented in a twin paper [1], the use of the BEM to solve the most typical problems of elastoplasticity is studied. To this aim a two-fold discretization is operated by boundary elements, as well as by cell elements, the latter having the only purpose of suitably modelling the plastic behaviour of the continuum. Global plastic yielding laws for the individual cell elements are derived by means of the maximum plastic work theorem. In such way, the BE/cell discretized model is shown to be governed by discrete-form equations fully equivalent to those of FE models. In particular, provided that the cell plastic yielding laws are piecewise linear, all the mathematical programming methods used to solve plasticity problems within the FEM can be entirely utilized also within the BEM. However, the numerical implementation of this approach and its competitiveness with respect to the FEM are questions open to further study.

On shakedown of elastic plastic solids

Making reference to elastic perfectly plastic solids subjected to cyclic loads, the problem of the shakedown load factor is considered and the relevant Euler-Lagrange equations are discussed. It is proved that the solution to these equations describes the gradient, with respect to the load multiplier, of the steady-state response of the solid body to the cyclic loads at the shakedown limit, and that it thus enables one to predict the nature of the impending collapse. These results are then extended to the more general case of loads varying within a given load domain.

Elastic-plastic fields near crack tip growing step-by-step in a perfectly plastic solid

In this paper, based on the three-dimensional flow theory of plasticity, the fundamental equations for the plane strain problem of elastic-perfectly plastic solids are presented. By using these equations the elastic-plastic fields near the crack tip growing step-by-step in an elastic incompressible-perfectly plastic solids are analysed. The first order asymptotic solutions for the stress field and velocity fields near the crack tip are obtained. The solutions show the evolution process of the elastic unloading domain and the development process of the central fan domain, and reveal the possibility of the presence of the secondary plastic domain. The second order asymptotic solution for the stress field is also presented.

Plasticity of damaged solids and shear band localization

The main objective of the paper is the investigation of shear band localization conditions for finite elastic-plastic rate independent deformations of damaged solids. The first part of the paper is devoted to the formulation of the constitutive relations for elastic-plastic solids when isotropic and kinematic hardening effects and the micro-damage process are taken into consideration. The isotropic work-hardening effect is incorporated in the theory directly by defining the work-hardening-softening material function while the kinematic hardening effect and the softening effect generated by the micro-damage process are described by means of the internal state variable method. The second part of the paper aims at the investigation of the localization of plastic deformations. Different effects on the localization phenomenon are investigated. Particular attention is focused on kinematic hardening and micro-damage effects. It has been found that the influence of these two cooperative phenomena on the onset of localization within shear bands has synergetic nature. The results obtained are in good agreement with recent experimental observations.

Plasticity of damaged solids and shear band localization

The main objective of the paper is the investigation of shear band localization conditions for finite elastic-plastic rate independent deformations of damaged solids. The first part of the paper is devoted to the formulation of the constitutive relations for elastic-plastic solids when isotropic and kinematic hardening effects and the micro-damage process are taken into consideration. The isotropic work-hardening effect is incorporated in the theory directly by defining the work-hardening-softening material function while the kinematic hardening effect and the softening effect generated by the micro-damage process are described by means of the internal state variable method. The second part of the paper aims at the investigation of the localization of plastic deformations. Different effects on the localization phenomenon are investigated. Particular attention is focused on kinematic hardening and micro-damage effects. It has been found that the influence of these two cooperative phenomena on the onset of localization within shear bands has synergetic nature. The results obtained are in good agreement with recent experimental observations.

Reexamination of Jumps Across Quasi-Statically Propagating Surfaces Under Generalized Plane Stress in Anisotropically Hardening Elastic-Plastic Solids

Strong discontinuities across quasi-statically propagating surfaces in anisotropic elastic-plastic solids under generalized plane stress are reexamined allowing for some generality in constitutive response and taking into account the phenomenon of necking. Jumps in stresses are ruled out on the basis of material stability postulates and a previous approach (by Pan, 1982) is discussed. It is noted that for elastic-perfectly plastic solids, sliding velocity discontinuities occur under restrictive and exceptional conditions (when both the surface and its normal are stress characteristics) for generalized plane stress as compared to plane strain. Necks may form along (stress) characteristic directions with the relative velocity vector orthogonal to the other family of characteristics.

The determination of fracture toughness for a porous elastic-plastic solid

The near tip stress and porosity fields in a plane strain mode I crack under small scale yielding (s.s.y.) conditions have been studied using a constitutive model for porous elastic-plastic solids. These fields have been used in conjunction with criteria for fracture of tensile specimens to predict values of KIc for a mild steel over a range of temperatures. The regimes of dominance of fracture by slip-induced cleavage and void coalescence mechanisms have been determined. The predictions are shown to compare well with experimental data.

Restrictions on dynamically propagating surfaces of strong discontinuity in elastic-plastic solids

F or dynamic three-dimensional deformations of elastic-plastic materials, we elicit conditions necessary for the existence of propagating surfaces of strong discontinuity (across which components of stress, strain or material velocity jump). This is accomplished within a small-displacement-gradient formulation of standard weak continuum-mechanical assumptions of momentum conservation and geometrical compatibility, and skeletal constitutive assumptions which permit very general elastic and plastic anisotropy including yield surface vertices and anisotropic hardening. In addition to deriving very explicit restrictions on propagating strong discontinuities in general deformations, we prove that for anti-plane strain and incompressible plane strain deformations, such strong discontinuities can exist only at elastic wave speeds in generally anisotropic elastic-ideally plastic materials unless a material's yield locus in stress space contains a linear segment. The results derived seem essential for correct and complete construction of solutions to dynamic elastic-plastic boundary-value problems.

The theory of elastic-plastic deformation at finite strain with induced anisotropy modeled as combined isotropic-kinematic hardening

A T heory of elastic-plastic deformation with strain induced anisotropy based on finite-deformation-valid continuum mechanics is presented. On the foundation of nonlinear kinematics which provides strict uncoupling of elastic and plastic deformation rate terms according to their physical origins, it introduces a basis for the modified plastic rate of deformation D̂ p suggested by G.J. Creus, A.G. Groehs and E.T. Onat in a report entitled “Constitutive Equations for Finite Deformations of Elastic-Plastic Solids,” 1984, in which this variable was suggested in order to give an elegant mathematical structure to the theory. D̂ p is shown to express the resultant rate of deformation in the current configuration of the elastically-plastically deforming material which is envisaged to be generated by the pure plastic flow and the anisotropy-caused spin, both considered to be occurring in the unstressed state. From this basis an elastic-plastic theory is developed in the case where the strain-induced anisotropy takes the form of combined isotropic-kinematic hardening, although the concepts involved also apply to more general anisotropic characteristics. A general evolution equation is adopted for the back stress α, the kinematic-hardening shift of the yield surface, its rate of growth being expressed as a general form-invariant function of α and D̂ p , including a general term expressing the influence of the spin of α because it is embedded in the deforming material. By providing an expression for the total strain rate as the sum of the strain rate D̂ p and an elastic term, linear in the Jaumann derivative of Kirchhoff stress, it is shown that ( D̂ p dt) is the residual strain increment following a loading/unloading cycle imposed by a stress increment. By considering materials which obey the normality rule it is also shown that the instantaneous elastic-plastic moduli have the symmetries necessary for generating a rate potential function and hence can be incorporated into Hill's variational principle valid for solving problems involving finite deformation and convenient for finite-element exploitation.

A Simplified Analysis of Rolling Elliptical Contacts in the Elastic–Plastic Range

The paper presents an analysis of the deformations and stresses in the rolling contact of solids of revolution. The simplified model is based upon traction-free rolling and materials which are ideal elastic-plastic solids, but produces upper and lower bound solutions for groove formation and residual stress distributions which are realistic. Data for the three-dimensional problem are presented showing the process of cumulative deformation during a number of rolling cycles up to a steady state.

A More Direct and General Analysis of Moving Strong Discontinuity Surfaces in Quasi-Statically Deforming Elastic-Plastic Solids

A More Direct and General Analysis of Moving Strong Discontinuity Surfaces in Quasi-Statically Deforming Elastic-Plastic Solids

Viscosity and steady shocks

The dependence of the strain rate on the Hugoniot stress and the effective viscosity is derived for steady wave propagation, by considering the balance between viscous and driving forces in the steady wave. The strain rate in elastic-plastic solids is shown to vary as the fourth power of the Hugoniot stress, which is in agreement with experimental data.

The plastic potential in the theory of anisotropic elastic-plastic solids

In the paper the yield condition is proposed for the most general anisotropic material. It is one of the possible generalizations of the Huber-Mises-Hencky yield condition for the case of anisotropy. The body considered is anisotropic elastically as well as plastically. It is assumed that the plastic anisotropy tensor is a definite function of the elastic anisotropy tensor. The corresponding flow function is a part of the strain energy and its value remains unchanged when all normal components of stress are increased by the same value. The theory of the eigen states for fourth order tensors is used. The plastic anisotropy tensor proposed has the same deviatoric eigen states as the elastic anisotropy tensor. The proposed yield condition reduces to that of Huber-Mises-Hencky when the anisotropy is vanishingly small. The method presented in this paper can be also applied to describe other types of plastic anisotropy tensor.

On finite element formulations for large elastic-plastic deformations

Some issues arising in the formulation of finite strain plasticity problems are discussed. One issue concerns the formulation of finite deformation elastic-plastic constitutive relations for use in this class of problems. Here, a formulation is outlined with particular attention on the role of elasticity in setting the form of the stress rate-strain rate relation. The dependence of solutions to problems involving large plastic deformations on the plastic flow rule is illustrated by reviewing results on finite pure bending of a plane strain strip. The response of elastic-plastic solids is contrasted with that of a nonlinear elastic solid.

Deformation bounds for elastic-plastic solids within and out of the creep range

A bounding principle for elastic-perfectly plastic creeping and noncreeping structures subjected to mechanical and/or thermal loads varying below or above the shakedown limit is presented. This principle contains some free “perturbation functions” which, suitably chosen, enable it to specialize, so generating bounds on a variety of deformation measures (such as inelastic work dissipated within any portion of the body, inelastic strains and displacements), some of which are new results, others recover or generalize known results. The resulting bounding technique possesses a quite unified character which is useful for computational purposes. The concept of “pseudo-plastic” strain is shown to be crucial for the derivation of bounds applicable above the shakedown limit.