Elastic-viscoplastic Constitutive Law Research Articles

Multiphysics modeling of continuous casting of stainless steel

Abstract During the solidification of stainless steel, the mechanical behavior of the solidifying shell follows nonlinear elastic-viscoplastic constitutive laws depending on metallurgical phase fraction calculations (liquid, ferrite and austenite). A multiphysics model that couples thermal and mechanical behavior in a Lagrangian reference frame, including both classical time-independent plasticity and creep, with turbulent fluid flow in the liquid phase in an Eulerian frame, is applied to determine realistic temperature and stress distributions in the solidifying shell of stainless steel in a commercial continuous caster. Compositional effects are incorporated through the use of phase diagrams to define the phase fraction variations with temperature during the process. The behavior at these high temperatures can be adequately captured using specific constitutive equations for each phase and careful decisions about switching between them. Results for a 409L ferritic stainless steel show that, due to its phase fraction history, solidification stresses differ significantly from those in plain carbon steels. Specifically, they include a secondary sub-surface compression peak due to phase change expansion between γ-austenite and δ-ferrite through the thickness of the shell.

Dynamic contact problem with thermal effect

Abstract We consider a model of a dynamic frictional contact between the body and the foundation. In this model the contact is bilateral. The behaviour of the material is described by the elastic-viscoplastic constitutive law with thermal effect. The variational formulation of this model leads to a system of two evolution hemivariational inequalities. The aim of this paper is to prove that this system of inequalities has a unique solution. The proof is based on the Banach fixed point theorem and some results for hemivariational inequalities.

A dynamic contact problem for elastic–viscoplastic materials with normal damped response and damage

We consider a mathematical model which describes the frictional contact between a deformable body and a foundation. The process is assumed to be dynamic and the material behavior is described by a elastic–viscoplastic constitutive law with damage. The frictional contact is modeled with subdifferential boundary conditions. We derive the variational formulation of the problem which is a coupled system of a hemivariational inequality for the displacement and a parabolic variational inequality for the damage field. Then we prove the existence of a unique weak solution to the model. The proof is based on arguments of hyperbolic hemivariational inequality, a classical existence, and uniqueness result on parabolic inequalities and a fixed point argument.

Multiscale elastic-viscoplastic computational analysis

The objective of this work is to develop an efficient strategy for quasi-static problems with elastic-viscoplastic constitutive laws. Our approach is based on the multiscale LATIN method for domain decomposition, and particularly on the use of the Proper Generalized Decomposition (PGD) method, which allows a drastic decrease in computation costs. We present the method in its general form applicable to problems with constitutive laws expressed using internal variables; then we discuss the technical features which are necessary in order to deal with elastic-viscoplastic models. We illustrate the method in detail through a onedimensional example using a Chaboche-type elastic-viscoplastic constitutive law.

Multiscale elastic-viscoplastic computational analysis

The objective of this work is to develop an efficient strategy for quasi-static problems with elastic-viscoplastic constitutive laws. Our approach is based on the multiscale LATIN method for domain decomposition, and particularly on the use of the Proper Generalized Decomposition (PGD) method, which allows a drastic decrease in computation costs. We present the method in its general form applicable to problems with constitutive laws expressed using internal variables; then we discuss the technical features which are necessary in order to deal with elastic-viscoplastic models. We illustrate the method in detail through a onedimensional example using a Chaboche-type elastic-viscoplastic constitutive law.

An evolution problem in nonsmooth elasto-viscoplasticity

In this note we summarize some of our recent results on the mathematical modeling of contact problems. We concentrate on a dynamical model which describes the frictional contact between a deformable body and a foundation. The material behavior is described with a nonlinear elastic–viscoplastic constitutive law and the frictional contact is modeled with multivalued subdifferential boundary conditions. We derive the variational formulation of the problem which is in the form of a system involving an integral equation coupled with an evolutionary hemivariational inequality. We prove the existence and the regularity properties of a unique weak solution to the model.

Analysis of a dynamic Elastic-Viscoplastic contact problem with friction

We consider a mathematical model which describes the frictional contact between a deformable body and a foundation. The process is dynamic, the material behavior is described with an elastic-viscoplastic constitutive law and the frictional contact is modeled with subdifferential boundary conditions. We derive the variational formulation of the problem which is in the form of a system involving an integral equation coupled with an evolutionary hemivariational inequality. Then we prove the existence of a unique weak solution to the model. The proof is based on arguments of abstract second order evolutionary inclusions with monotone operators and a fixed point theorem.

Thermo-mechanical models of steel solidification based on two elastic visco-plastic constitutive laws

Two thermo-mechanical models based on different elastic-visco-plastic constitutive laws are applied to simulate temperature and stress development of a slice through the solidifying shell of 0.27%C steel in a continuous casting mold under typical commercial operating conditions with realistic temperature dependant properties. A general form of the transient heat equation, including latent-heat from phase transformations such as solidification and other temperature-dependent properties, is solved numerically for the temperature field history. The resulting thermal stresses are solved by integrating the elastic-visco-plastic constitutive laws of Kozlowski [P.F. Kozlowski, B.G. Thomas, J.A. Azzi, H. Wang, Simple constitutive equations for steel at high temperature, Metall. Trans. 23A (1992) 903–918] for austenite in combination with the Zhu power-law [H. Zhu, Coupled thermal–mechanical finite-element model with application to initial solidification, PhD thesis, University of Illinois, 1993] for delta-ferrite with ABAQUS [ABAQUS Inc., User Manuals v6.6, 2006] using a user-defined subroutine UMAT [S. Koric, B.G. Thomas, Efficient thermo-mechanical model for solidification processes, Int. J. Num. Meth. Eng. 66 (2006) 1955–1989], and the Anand law for steel [L. Anand, Constitutive equations for the rate dependant deformation of metals at elevated temperatures, ASME J. Eng. Mater. Technol. 104 (1982) 12–17; S.B. Brown, K.H. Kim, L. Anand, An internal variable constitutive model for hot working of metals, Int. J. Plasticity 6 (1989) 95–130] using the integration scheme recently implemented in ANSYS [ANSYS Inc., User Manuals v100, 2006]. The results from these two approaches are compared and CPU times are benchmarked. A comparison of one-dimensional constitutive behavior of these laws with experimental tensile test data [P.J. Wray, Plastic deformation of delta-ferritic iron at intermediate strain rates, Metall. Trans. A 7A (1976) 1621–1627; P.J. Wray, Effect of carbon content on the plastic flow of plain carbon steel at elevated temperatures, Metall. Trans. A 13 (1982) 125–134] and previous work [A.E. Huespe, A. Cardona, N. Nigro, V. Fachinotti, Visco-plastic constitutive models of steel at high temperature, J. Mater. Process. Technol. 102 (2000) 143–152] shows reasonable agreement for both models, although the Kozlowski–Zhu approach is much more accurate for low carbon steels. The thermo-mechanical models studied here are useful for efficient and accurate analysis of steel solidification processes using convenient commercial software.

Calibration of a viscoplastic cohesive zone for crazing in PMMA

In a numerical analysis of mode I fracture in amorphous polymers, Estevez et al. [Estevez R, Tijssens MGA, Van der Giessen E. Modelling of the competition between shear yielding and crazing in glassy polymers. J Mech Phys Solids 2000;48:2585–617] have shown that the material toughness is governed by the competition between the time scales related to shear yielding and crazing. The present study aims at calibrating the parameters involved in this description, for a commercial PMMA. An elastic–viscoplastic constitutive law featuring softening upon yielding and hardening at continued deformation is used for the bulk while crazing is described with a viscoplastic cohesive zone. The three steps of crazing with initiation for a critical stress state, thickening of the craze surfaces and breakdown of the craze fibrils for a critical opening are characterized separately. In particular, it is demonstrated that the use of a viscoplastic cohesive zone is necessary to capture the variation of the toughness with loading rate. For PMMA, the related energy release rate is shown to depend primarily on the craze critical opening and the craze thickening kinetics while craze initiation is of minor importance for the quasi-static loading conditions under consideration here.

An experimental mock-up for the study of steel behaviour at high temperature during phase transformation

In most numerical simulations of welding, due to the very short times involved, viscoplastic effects are neglected. However an experimental mock-up developed at INSA has evidenced the possible importance of such effects, which must therefore be incorporated in a more refined description of the material behaviour. This paper presents such a description; an extension of Leblond’s model for transformation plasticity, accounting for the dependence of mechanical properties upon strain rate, is proposed. Finite element analyses using elastic-plastic or elastic-viscoplastic constitutive laws are compared with experiments, and it is found that incorporation of viscoplastic effects significantly improves the quality of the agreement between numerical and experimental results.

Viscoplastic behaviour of steels during welding

In most numerical simulations of welding, viscoplastic phenomena are neglected, because of shortness of the process. The aim of this paper is to accurately assess the effect of such phenomena upon predicted residual stresses and distortions, through comparison of simulations using both elastic–plastic and elastic–viscoplastic constitutive laws, and experiments. These experiments are performed on a mock-up developed at INSA Lyon and especially designed for the study of the material behaviour in the heat affected zone of a welded component; this mock-up is sufficiently simple to allow for cheap 2D axisymmetric simulations, but nevertheless complex enough to represent a real welded structure, albeit in a schematic way. It is found that incorporation of viscous effects into the description of the material behaviour has a marginal influence upon predicted residual stresses but a significant one upon predicted residual distortions, which agree much better with measured ones when viscous effects are accounted for.

Failure Prediction of Aluminum Laser-Welded Blanks

The formability of aluminum-3% magnesium laser-welded blanks manufactured by 3kW YAG laser welding process is investigated in this paper. Previous experimental results showed that shear failure occurs in the welds of these blanks during tensile testing with the weld line perpendicular to the tensile axis. In order to further understand the failure in these blanks under uniaxial and biaxial straining conditions, finite element computations under generalized plane strain conditions are conducted to understand the effects of weld geometry and strength on the shear failure. The results indicate that the average plastic stress-strain behavior of the weld should be softer than that in the base metal in the direction perpendicular to the weld line. The results also show that the weld geometry has significant effects on strain localization in the weld under the tensile loading perpendicular to the weld line. The stress-strain histories of the material elements of the weld metal in tensile specimens obtained from the finite element computation are used for a failure/localization analysis. The imperfection approach of Marciniak and Kuczynski is employed to examine the failure/localization in the weld of the tensile specimens. The failure/localization analysis is based on an elastic-viscoplastic constitutive law that accounts for the potential surface curvature, material plastic anisotropy, material rate sensitivity and softening due to nucleation, growth and coalescence of microvoids. The material imperfection parameters for void nucleation and growth are obtained by fitting the results of the failure/localization analysis to those of the uniaxial tensile tests. The material imperfection parameters are then used to predict the failure strains of the blanks under biaxial straining conditions based on the stress-strain histories of the material elements obtained from finite element computations under generalized plane strain conditions. The results indicate that the forming limits of the blanks under plane strain tension and equal biaxial tension are reduced significantly when compared with that under uniaxial tension. Finally, a finite element simulation of the welded sheet under uniaxial tension is carried out with consideration of void nucleation and growth for all the material elements in the weld. Our computational results indicate that without consideration of the failure/localization analysis presented in this paper, an unrealistically high volume fraction for void nucleation must be assumed in our finite element simulation of the tensile tests in order to match the failure strains observed in the uniaxial tensile tests.

3D Inverse Analysis Model Using Semi-Analytical Differentiation for Mechanical Parameter Estimation

An inverse method is developed in order to estimate constitutive parameters of a material from compression tests. The direct model used to simulate mechanical tests is FORGE3®. It solves a transient thermo-mechanical problem using a finite element method. From velocity, pressure and temperature fields, any output of a mechanical test can be computed and compared with experimental data. A Gauss-Newton algorithm is implemented to solve the least-square problem associated with the inverse problem. The optimisation module is coupled with a semi-analytical sensitivity analysis method. This method is fast and stable when using a remeshing algorithm. A confidence interval estimator is proposed. The stability of the optimisation module and the confidence interval estimation are tested for numerical test cases. Finally, constitutive parameters of a steel grade are estimated for two elastic-viscoplastic constitutive laws.

Crack velocity dependent toughness in rate dependent materials

Mode I, quasi-static, steady-state crack growth is analyzed for rate dependent materials under plane strain conditions in small scale yielding. The solid is characterized by an elastic–viscoplastic constitutive law and the plane ahead of the crack tip is embedded with a rate dependent fracture process zone. The macroscopic work of fracture of the material is computed as a function of the crack velocity and the parameters characterizing the fracture process zone and the solid. With increasing crack velocity a competition exists between the strain rate hardening of the solid, which causes elevated tractions ahead of the crack tip that tend to drive crack propagation, and the rate strengthening of the fracture process zone which tends to resist fracture. Results for material parameters characteristic of polymers show that the toughness of the material can either increase or decrease with increasing crack velocity. To motivate the model, the cohesive zone parameters are discussed in terms of failure mechanisms such as crazing and void growth ahead of the crack tip. The toughness of rubber modified epoxies is explained by employing the fracture model along with micromechanical void cell calculations.

Seismic Wave Propagation in Elastic-Viscoplastic Shear Layers

A computational method is introduced for simulating seismic wave propagation in elastic-viscoplastic shear layers. The fundamental dynamics are expressed by two partial differential equations for shear stress and velocity, the balance of momentum, and the elastic-viscoplastic constitutive law. They are converted to those for the upward and downward components of horizontally-polarized motion and then a finite difference method is applied. Seismic wave propagation is computed, considering the reflection and refraction rule at the contact boundary of two layers, as well as the boundary conditions at the bottom and top. Simulation analyses for the observed records show that the proposed method is effective and practical, although it cannot simulate liquefaction phenomena.

ON THE DYNAMIC SHEARING PROBLEM FOR RATE- AND TEMPERATURE-DEPENDENT MEDIA

An elastic-viscoplastic constitutive law with thermal softening and without hardening is used. The local existence of the solution and the lower semicontinuous dependence of the blow-up time upon the initial data are obtained. The homogeneous solution is introduced and the finite time evolution of the perturbations of this solution is studied

Deformation and plastic waves in dynamic tension tests

The deformation history of a round tensile specimen at high testing speeds is investigated via a one-dimensional approach. Numerical results illustrating the effects of inertia are generated employing a one-dimensional large-strain finite element program. The rate-dependent characteristics of the material are modelled using an elastic-viscoplastic constitutive law with strain and strain-rate hardening properties. Some features of one-dimensional plastic wave propagation are discussed in relation to previously reported experimental results.

Plastic buckling of metal matrix laminated plates

A method is proposed for the determination of plastic bifurcation buckling load of metal matrix composite plates. The metallic matrix behavior is described by an elastic-viscoplastic constitutive law, while the fibers are assumed to be either elastic or elastic-viscoplastic material. The approach is based on the load level and history dependent instantaneous effective properties of the inelastic plate, which are established by a micromechanical analysis. An incremental procedure is developed in which a buckling condition has to be established and its fulfilment must be checked at each increment. The method is applied for the prediction of the plastic buckling of boron/aluminum composite plates in various situations, by employing the classical and higher order shear deformation plate theories.

Inertial effects on necking in tension

A one-dimensional finite element program has been developed to study the effects of inertia on necking instability in a round bar under tension. An elastic-viscoplastic constitutive law is employed to model the material behaviour of solids with strain and strain-rate hardening characteristics. Necking is initiated by prescribing a geometric imperfection in the form of an area defect in the initial undeformed state. The entire load-deformation-time response from this state is computed. Key features of necking under dynamic conditions are contrasted with static behaviour and with certain aspects of recent similar studies on dynamic effects. Mesh sensitivity resulting from the one-dimensionalization of the formulation is also discussed.

Analysis of forming rate sensitive materials

To study how the rate of deformation effects forming rate sensitive materials, the Bodner-Partom elastic-viscoplastic constitutive law is incorporated into a finite element program. This law postulates both elastic and plastic components of deformation at any stress level. The stress is a Hookean function of the elastic strain, while the plastic deformation rate is a function of the deviatoric stress and an internal state variable defining the load history. The finite element derivation adopts the small strain assumption with updated coordinates. The equilibrium rate equation is formulated using total velocities with the nonlinearities incorporated into an equivalent plastic load vector depending upon the current stress. The resulting equation explicitly includes time and is a true rate equation. The program calculates the current stress field using the incremental equilibrium equation. An iterative technique is used to ensure that the assumed current load rate used to calculate the current stress field is correct. Convergence of the iteration procedure needs only be monitored at the velocity specified nodes. To demonstrate the applicability of this method, two plane strain problems, a tensile bar and strip rolling, with rate sensitive materials are investigated.