Characterization of strain hardening for a simple pressure-sensitive plasticity model

Summary Paralleling the development of strain hardening for the pressure-independent von Mises criterion, a simple plasticity model in strain space was presented to characterize strain hardening for pressure-sensitive compressible materials. Two hardening moduli,H _T andH _C , which emerged from the constitutive equations and can becalculated from uniaxial stress-strain curves in tension and compression, were used to characterize the strainhardening responses forgeneral and special stress systems. The results indicated the implications and restrictions of the yield function on the hardening responses. It was also shown that strain softening, under general stress systems, can be a natural consequence of pressure-sensitive yielding. Consequently, a strain-space formulation is recommended for most (if not all) pressure-sensitive plasticity models. Preliminary application to the yielding of polymers under hydrostatic pressure gave reasonable results for polyethylene at moderate pressure and small strains; the results for polycarbonate were generally poor. Finally, the advantages and limitations of the present approach were discussed.With 6 Figures     

Comparison of plasticity models for tantalum and a modification of the PTW model for wide ranges of strain,strain rate,and temperature

Four well-known constitutive models for plastic deformation of materials, i.e., Johnson–Cook (JC), Zerilli–Armstrong (ZA), Voyiadjis and Abed (VA), and Preston–Tonks–Wallace (PTW), have been compared with reference to existing deformation data of tantalum in wide ranges of strain, strain rate, and temperature. All of these models reasonably describe the flow stress and the strain-hardening behavior only in the certain ranges of strain, strain rate, and temperature for which the models were developed. The PTW model with appropriate parameters most effectively describes the effects of strain rate and temperature in a wider range, except for strain hardening. The strain-hardening term of PTW was thus modified in the current work and the modified PTW demonstrated very good prediction for the constitutive behavior of tantalum in wide ranges of strain, strain rate, and temperature.     

A simple orthotropic finite elasto–plasticity model based on generalized stress–strain measures

 In this paper we present a formulation of orthotropic elasto-plasticity at finite strains based on generalized stress–strain measures, which reduces for one special case to the so-called Green–Naghdi theory. The main goal is the representation of the governing constitutive equations within the invariant theory. Introducing additional argument tensors, the so-called structural tensors, the anisotropic constitutive equations, especially the free energy function, the yield criterion, the stress-response and the flow rule, are represented by scalar-valued and tensor-valued isotropic tensor functions. The proposed model is formulated in terms of generalized stress–strain measures in order to maintain the simple additive structure of the infinitesimal elasto-plasticity theory. The tensor generators for the stresses and moduli are derived in detail and some representative numerical examples are discussed. Received: 2 April 2002 / Accepted: 11 September 2002     

Geometrical numerical algorithms for a plasticity model with Armstrong–Frederick kinematic hardening rule under strain and stress controls

In this paper, we approach the numerical integration problem of a plasticity model with the Armstrong–Frederick kinematic hardening rule on back stress through a combination of the techniques of integral representation and geometrical integrator. First, the internal symmetry group of the constitutive model is investigated. Then, we develop two geometrical integrators for strain control and stress control, respectively. These integrators are obtained by a discretization of the integral representation of the constitutive equations and an exponential approximation of the quasilinear differential equations system for the relative stress, which guarantee to retain the consistency condition exactly without the need for any iterations. Some numerical examples are used to assess the performance of the new algorithms. The measures in terms of stress relative errors and also isoerror maps confirm that our schemes are superior to the classical radial return methods. Copyright © 2005 John Wiley & Sons, Ltd.     

On hybrid stress,hybrid strain and enhanced strain finite element formulations for a geometrically exact shell theory with drilling degrees of freedom

The paper is concerned with the finite element formulation of a recently proposed geometrically exact shell theory with natural inclusion of drilling degrees of freedom. Stress hybrid finite elements are contrasted by strain hybrid elements as well as enhanced strain elements. Numerical investigations and comparison is carried out for a four-node element as well as a nine-node one. As far as the four-node element is concerned it is shown that the stress hybrid element and the enhanced strain one are equivalent. The hybrid strain formulation corresponds to the hybrid stress formulation only in shear dominated problems, that is the case of the plate. © 1998 John Wiley & Sons, Ltd.     

A new strain hardening model for rate-dependent crystal plasticity

This paper presents a crystal plasticity based finite element analysis employing the new microstructure-based strain hardening model recently presented by Saimoto and Van Houtte (2011) [7] to simulate formability and texture evolution in the commercial aluminum alloy 5754. Simulations are performed to compare the predictive capability of the new hardening model against the common work hardening models using a rate-dependent plasticity formulation. The parameters in the numerical models are calibrated using the X-ray data published by Iadicola et al. (2008) [9] for the aluminum sheet alloy 5754. The predictions of the model for balanced biaxial tension and in-plane plane-strain tests are compared against experimental observations presented in Iadicola et al. (2008) [9]. It is concluded that the new model provides the best predictions of the large strain behavior of Aluminum sheet alloy 5754 subjected to various strain paths.     

A finite strain model of combined viscoplasticity and rate-independent plasticity without a yield surface

A new internal variable formulation dealing with mechanisms with different characteristic times in solid materials is proposed within a finite deformation framework. The framework relies crucially on the consistent combination of a general viscoplastic theory and a rate-independent theory (generalized plasticity) which does not involve the yield surface concept as a basic ingredient. The formulation is developed initially in a material setting and then is extended to a covariant one by applying some basic elements and results from the tensor analysis on manifolds. The covariant balance of energy is systematically employed for the derivation of the mechanical state equations. It is shown that unlike the case of finite elasticity, for the proposed formulation the covariant balance of energy does not yield the Doyle–Ericksen formula, unless a further assumption is made. As an application, by considering the material (intrinsic) metric as a primary internal variable accounting for both elastic and viscoplastic (dissipative) phenomena within the body, a constitutive model is proposed. The ability of the model in simulating several patterns of the complex response of metals under quasi-static and dynamic loadings is assessed by representative numerical examples.     

Assessing and ensuring parameter identifiability for a physically-based strain hardening model for twinning-induced plasticity

Physically-based strain hardening models have become important ingredients in metal forming simulations over the last years, since they allow for the modeling of multi-stage forming processes based on the evolution of physically meaningful internal variables. Although these models are physically-based, there are still many fitting parameters involved which have to be identified from experiments. As a matter of fact, for each physical effect that is included in the model, a separate equation with new fitting parameters is introduced, such that physically-based models tend to contain a large number of fitting parameters. Parameter estimation is often based on the macroscopic response of a specimen which is tested in compression, tension or shear at various strain rates and temperatures. It is not guaranteed that this macroscopic information suffices to estimate parameters in model equations that describe (sub-) microscopic phenomena, since the effect of one parameter on the course of strain hardening can be compensated by other parameters. Since such parameter correlations are hard to detect from the model equations alone, the parameter estimation process may be ill-conditioned, i.e. numerous parameter sets can be found for such models that deliver almost the same minimum value of the error function in the parameter identification process. Given that parameter estimation involves a series of costly experiments, methods are needed that allow for analyzing the identifiability of the model parameters before costly experiments are performed. In this paper, an approach is presented that analyzes model parameter dependencies and quantifies the identifiability of the model parameters. The model considered in this study calculates the flow stress based on the evolution of three dislocation densities and the evolution of deformation twins. The analysis shows that correlations between the model parameters exist and that it is not possible to determine all model parameters based on an experimental set of flow curves in a single curve fitting procedure. An adapted fitting strategy is presented in which fitting is performed step-wise so that in each fitting step, only identifiable parameters are estimated, allowing for successful parameter identification.     

A large strain plasticity model for implicit finite element analyses

The theoretical basis and numerical implementation of a plasticity model suitable for finite strains and rotations are described. The constitutive equations governing J ₂ flow theory are formulated using strains-stresses and their rates defined on the unrotated frame of reference. Unlike models based on the classical Jaumann (or corotational) stress rate, the present model predicts physically acceptable responses for homogeneous deformations of exceedingly large magnitude. The associated numerical algorithms accommodate the large strain increments that arise in finite-element formulations employing an implicit solution of the global equilibrium equations. The resulting computational framework divorces the finite rotation effects on strain-stress rates from integration of the rates to update the material response over a load (time) step. Consequently, all of the numerical refinements developed previously for small-strain plasticity (radial return with subincrementation, plane stress modifications, kinematic hardening, consistent tangent operators) are utilized without modification. Details of the numerical algorithms are provided including the necessary transformation matrices and additional techniques required for finite deformations in plane stress. Several numerical examples are presented to illustrate the realistic responses predicted by the model and the robustness of the numerical procedures.     

An approximated computational method for fast stress reconstruction in large strain plasticity

This article describes a numerical method to reconstruct the stress field starting from strain data in elastoplasticity. Usually, this reconstruction is performed using the radial return algorithm, commonly implemented also in finite element codes. However, that method requires iterations to converge and can bring to errors if applied to experimental strain data affected by noise. A different solution is proposed here, where an approximated numerical method is used to derive the stress from the strain data with no iterations. The method is general and can be applied to any plasticity model with a convex surface of the yield locus in nonproportional loading. The theoretical basis of the method is described and then it is implemented on two constitutive models of anisotropic plasticity, namely, Hill48 and Yld2000-2D. The accuracy of the proposed method and the advantage in terms of computational time with respect to the classical radial-return algorithm are discussed. The possibility of using such method to reconstruct the stress field in case of few temporal data and noisy strain fields is also investigated.     

A plane stress softening plasticity model for orthotropic materials

A plane stress model has been developed for quasi-brittle orthotropic materials. The theory of plasticity, which is adopted to describe the inelastic behaviour, utilizes modern algorithmic concepts, including an implicit Euler backward return mapping scheme, a local Newton–Raphson method and a consistent tangential stiffness matrix. The model is capable of predicting independent responses along the material axes. It features a tensile fracture energy and a compressive fracture energy, which are different for each material axis. A comparison between calculated and experimental results in masonry shear walls shows that a successful implementation has been achieved. © 1997 John Wiley & Sons, Ltd.     

Hybrid stress finite element model for non-linear shell problems

The present paper describes a hybrid stress finite element formulation for geometrically non-linear analysis of thin shell structures. The element properties are derived from an incremental form of Hellinger-Reissner's variational principle in which all quantities are referred to the current configuration of the shell. From this multi-field variational principle, a hybrid stress finite element model is derived using standard matrix notation. Very simple flat triangular and quadrilateral elements are employed in the present study. The resulting non-linear equations are solved by applying the load in finite increments and restoring equilibrium by Newton-Raphson iteratioin. Numerical examples presented in the paper include complete snap-through buckling of cylindrical and spherical shells. It turns out that the present procedure is computationally efficient and accurate for non-linear shell problems of high complexity.     

The state of stress and strain of an elastic cylindrical shell under acoustic action

The article investigates the state of stress and strain of the surface of an elastic cylindrical shell whose outer part is subjected to the effect of an incident acoustic wave. A qualitative and quantitative analysis of the state of strain of the surface is carried out and recommendations are given for optimizing the functional purpose of the shell and minimizing the arising stresses.Translated from Problemy Prochnosti, No. 4, pp. 43–47, April, 1991.     

Study of the effects of stress and strain on martensite transformation: Kinetics and transformation plasticity*

In this paper, the effects of stress and strain on the kinetics and plasticity during martensitic transformation are studied. The mathematical models of transformation kinetics and plasticity under stress are developed. According to experimental results, the transformation plasticity parameter k is concluded not to be a constant, but it varies with the stresses.     

A complete plane strain fictitious stress boundary element method for poroelastic media

A fully coupled generalized plane strain boundary element model for determining the distribution of stress and pore pressure around underground openings in poroelastic media is developed. It is based on the indirect boundary element method of fictitious stress extended to complete plane strain analysis. New fundamental solutions for longitudinal forces were derived for the development of the model which uses elements with a constant variation of fictitious forces and sources in both space and time. As an example, the problem of a borehole drilled in an arbitrary direction in a triaxial stress field is considered. The results indicate that the fictitious stress method is an accurate and suitable means for complete plane strain poroelastic analyses of underground openings.     

Families of 4-node and 9-node finite elements for a finite deformation shell theory. An assesment of hybrid stress, hybrid strain and enhanced strain elements

In this paper we discuss and compare three types of 4-node and 9-node finite elements for a recently formulated finite deformation shell theory with seven degrees of freedom. The shell theory takes thickness change into account and circumvents the use of a rotation tensor. It allows for the applicability of three-dimensional constitutive laws and equipes the configuration space with the structure of a vector space. The finite elements themselves are based either on a hybrid stress functional, on a hybrid strain functional, or on a nonlinear version of the enhanced strain concept. As independent variables either the normal and shear resultants, the strain tensor related to the deformation of the midsurface, or the incompatible enhanced strain field are taken as independent variables. The fields of equivalence of these different formulations, their limitations as well as possible improvements are discussed using different numerical examples. Received 10 December 1998