Viscoplastic Formulation Research Articles

Prediction of the initiation of fatigue cracks in notched test pieces (La prévision de l'amorçage des fissures de fatigue à partir d'entailles): Lieurade, H.P., Flavenot, J.F. and Truchon, M. Rev. Fr. Mec. 1992 (1), 25–39 (in Frenche)

In this work, we investigate the mechanical behavior of superplastic materials under various loading conditions and examine the nature of anisotropy. This and other phenomena are modeled within a continuum theory of viscoplasticity which includes an overstress function, ‘yield surface’ and internal variables. Anisotropic hardening is represented by an internal stress tensor whose evolution (including plastic spin) consists of hardening, dynamic recovery, and static recovery terms. The ‘dynamic’ yield surface (within a viscoplasticity formulation) is assumed to be isotropic and depends on both J2 and J3. Various experiments on a PbSn superplastic alloy including tension, jump test, stress relaxation, fixed-end torsion, and combined tension-torsion tests were performed and the results are used to calibrate and verify the theoretical model. It is shown that the isotropic dynamic surface can successfully model the uniaxial behavior of superplastic deformation, including stress relaxation, and produces axial stresses in fixed-end torsion.

On a class of constitutive equations in viscoplasticity: Formulation and computational issues

AbstractThe viscoplastic constitutive model is formulated based on the existence of the dissipation potential which embodies the notion of the gauge (Minkowski) function of the convex set. A perturbation method is used for a solution of stiff differential equations characterizing the associated problem of evolution. It relies on a discrete formulation of viscoplasticity which results from the regularized version of the principle of maximum plastic dissipation. The operator split methodology and the Newton‐Raphson method are used to obtain the numerical solution of the discretized equations of evolution. The consistent tangent modulus is expressed in a closed form as a result of the exact linearization of the discretized evolution equations. For several variants of the flow potential function, including some representative stiff functional forms, numerical tests of the integration algorithm based on iso‐error maps are provided. Finally, a numerical example is presented to illustrate the robustness and the effectiveness of the proposed approach.

Thermal stress analysis due to welding processes by the finite element method

Abstract A finite element model has been developed to predict the residual stresses generated in weldments during fabrication. The thermal history of the weld piece is computed using a three-dimensional heat flow model which serves as input for computation of stresses. A micro-structural model, based on the Avrami equation, and the grain growth law have also been employed to predict the grain growth due to welding. A coupled thermal elastic visco-plastic formulation including the micro-structural changes has been developed to predict the overall deformations and residual stresses caused by a welding thermal cycle. The model is applied to an austenitic type of steel, namely AISI 308, and the predictions are in good agreement with experimental results reported in the literature. The longitudinal stresses are found to be as high as the yield stress, and the transverse stresses are found to be almost half of the longitudinal stresses.

Theoretical Aspects of the Elastoplastic-Viscoplastic Bounding Surface Model for Cohesive Soils

A generalized three-dimensional constitutive model for isotropic cohesive soils, based on the concept of the bounding surface in stress space, is developed within the framework of coupled elastoplasticity-viscoplasticity and critical state soil mechanics. The prominent feature of the bounding surface concept is the fact that inelastic deformations can occur for stress points within the bounding surface. The consideration of viscoplastic characteristics sets the current model apart from previous bounding surface formulations for soils, and introduces rate and time effects. The coupling between plasticity and viscoplasticity for stress states within the bounding surface differentiates the present work from the classical formulation of pure viscoplasticity (no coupling), or from formulations involving plasticity and viscoplasticity with a yield surface (coupling only for states on the yield surface). General incremental constitutive relations applicable to cohesive soils are developed and suitably specialized for isotropic cohesive soils. A minimum number of parameters are introduced and a procedure for their determination is outlined. The predictive capabilities of the model are treated in a companion paper.

Variational Formulation, Discrete Conservation Laws, and Path-Domain Independent Integrals for Elasto-Viscoplasticity

A discrete variational formulation of plasticity and viscoplasticity is developed based on the principle of maximum plastic dissipation. It is shown that the Euler-Lagrange equations (spatial conservation laws) emanating from the proposed discrete Lagrangian yield the equilibrium equation, the strain-displacement relations, the stress-strain relations, the discrete flow rule and hardening law in the form of closest-point-projection algorithm, and the loading/unloading conditions in Kuhn-Tucker form. Lack of invariance of the discrete Lagrangian relative to the group of material translations precludes the classical Eshelby law from being a conservation law. However, a discrete inhomogeneous form of Eshelby’s conservation law is derived which leads to a path-domain independent integral that generalizes the classical J-integral to elasto-viscoplasticity. It is shown that this path-domain independent integral admits a physical interpretation analogous to Budiansky and Rice interpretation of the classical J-integral.

High Strain Rate Behavior of Metals

Experimental results on the high strain rate response of polycrystalline metals are reviewed, with emphasis on the behavior of pure metals. A strong increase in flow stress with increasing strain rate is reported for strain rates of approximately 105s−1 and higher. This increase is observed in pressure-shear plate impact experiments at nominally constant strain rates from 105s−1 to 106s−1. To improve understanding of the increased rate sensitivity at high strain rates, pressure-shear, strain-rate-change experiments have been conducted on OFHC copper specimens. These experiments have been analyzed using a conventional viscoplasticity formulation and an internal variable formulation in which the hardening rate depends on the rate of deformation. Only the latter formulation is successful in describing the observed response to the change in strain rate. This observation is discussed in terms of its implications for interpreting other dynamic plasticity experiments and for improved understanding of the underlying dislocation mechanisms. The enhanced rate sensitivity at high strain rates is concluded to be related primarily to the rate sensitivity of strain hardening, not the rate sensitivity of the flow stress at constant structure.

On the microscopic origin of the plastic spin

Considering the configuration of a single slip and employing a scale invariance argument, it is possible to deduce a set of “microscopic” plastic flow relations having a direct counterpart in the “macroscopic” formulation of plasticity and viscoplasticity. In particular, a microscopic form of the plastic spin and its macroscopic counterpart for the case of anisotropy induced by kinematic hardening are obtained in terms of elementary physical arguments. Moreover the evolution equation for the back-stress is rigorously derived. Parameters which were assumed to be constant and/or independent from each other in a macroscopic development, are now found to be interrelated and dependent on the accumulated plastic strain. These findings are used for the analysis of the simple shear problem, especially in evaluating the development of the axial stress normal to the shear plane. A preliminary qualitative comparison with available data from fixed-end torsion experiments is discussed.

A viscoplastic formulation with elasticity for transient metal forming

Abstract A formulation for elastic-viscoplastic flow is presented which retains the basic structure of viscoplastic formulations. The constitutive theory is based on a multiplicative decomposition of the deformation gradient into elastic parts. The decomposition is cast in a rate form and the reference configuration for the decomposition is updated at each solution time to coincide with the intermediate configuration. Small elastic strains due to applied stresses are assumed. Volumetric strains due to thermal expansion are allowed to be finite. The inelastic behavior is described by a rate-dependent internal variable constitutive model. In the finite element formulation the elasticity is included in an integral sense over a time step by modifying the flow properties of the material to account for the change in the elastic strain over the step. The geometric update is made by a semi-implicit Euler integration of the velocity field so that the current geometry and state depend on the current solution for the velocity. Application to analytical and experimental benchmarks in metal forming indicate the accuracy and stability of the method.

A unified approach to finite deformation elastoplastic analysis based on the use of hyperelastic constitutive equations

By assuming from the outset hyperelastic constitutive behavior, an alternative approach to finite deformation plasticity and viscoplasticity is proposed whereby the need for integration of spatial rate constitutive equations is entirely bypassed. To enhance the applicability of the method, reference is made to a general formulation of plasticity and viscoplasticity which embodies both the multiplicative and additive theories. A new return mapping algorithm capable of accommodating general yield conditions, arbitrary flow and hardening rules and non-constant tangent elasticities is proposed. Finally, a numerical example is presented which illustrates the excellent performance of the method for very large time steps.

Analytical Formulation of a Rate and Temperature Dependent Stress-Strain Relation

The equation for plastic strain rate in the Bodner-Partom viscoplastic formulation is integrated under conditions of uniaxial stress, constant plastic strain rate, and isotropic hardening to give an analytical expression for the stress as a function of plastic strain and strain rate. Temperature dependence is introduced which leads to a general relationship between stress, strain, strain rate, and temperature. The resulting equation indicates an asymptotic saturation stress whose dependence on strain rate and temperature appears to agree with experimental results. Strain hardening given by the analytical equation also seems to be consistent with experiments. A possible new definition of yield stress is a consequence of the rate dependent stress-strain relation.

Plastic and viscoplastic analysis of axisymmetric shells

Abstract A general formulation of large deformation analysis of plastic and viscoplastic problems is presented first. The equilibrium equations are derived from an incremental variational formulation using the Lagrangian mode of description of motion. The symmetric Piola-Kirchhoff stress and Lagrangian strain are used in all the constitutive relations. Using degenerate isoparametric elements, permitting relaxation of the Kirchhoff-Love hypothesis, the procedure is specialized for the finite element analysis of shells of revolution subjected to axisymmetric loading. A modified incremental method, which applies an equilibrium correction at each step, is used for the solution of the linearized incremental equilibrium equations. Two approaches are presented for adapting the viscoplasticity formulation to provide inviscid plasticity solutions- one involving the extrapolation of results as the viscosity coefficient tends to infinity, and the other in which plasticity solutions are obtained by using time as an artifice in the viscoplastic analysis until equilibrium states are achieved at each succeeding load level. A detailed study of the nonlinear behavior of a torispherical pressure vessel is presented to illustrate the effectiveness of the numerical techniques.