Viscoplastic Problems Research Articles

An improved iterative method for large strain viscoplastic problems

In this work, some techniques are proposed to improve the usual Newton–Raphson Method (NRM) used in the numerical analysis of large strain viscoplastic problems. These techniques, based on the first-order perturbation technique, allow to define an adaptive step strategy and to improve the trial solution (guessed solution) for the first iteration at each step. To assess the efficiency of the proposed techniques, a number of numerical examples are presented: necking of a circular bar, plane strain bending, and an axisymmetric hydrostatic bulging process. Compared with the classical Newton–Raphson method, the proposed scheme requires less matrix inversions, less time steps and less CPU time. For a typical axisymmetric hydrostatic bulging process (606 degrees of freedom), the proposed method needs 5 times less matrix inversions, 1·7 times less time steps, and 4 times less CPU time. Copyright © 1999 John Wiley & Sons, Ltd.

An improved iterative method for large strain viscoplastic problems

An improved iterative method for large strain viscoplastic problems

On Rate‐Type Viscoplastic Problems with Linear Boundary Conditions

AbstractIn this paper we introduce the abstract concept of “linear boundary conditions” in the study of deformable bodies. We establish two existence and uniqueness results concerning respectively quasistatic and dynamic problems involving such type of boundary conditions. We also apply these existence results in the study of viscoplastic problems involving classical boundary conditions.

Finite element/boundary element analysis of viscoelastic and viscoplastic problems a bibliography (1994–1996)

This bibliography contains references to papers, conference proceedings and theses/dissertations dealing with finite element/boundary element analysis of viscoelastic and viscoplastic problems that were published in 1994–1996.

Analysis of a quasistatic viscoplastic problem involving tresca friction law

The quasistatic evolution of an elastic-viscoplastic body in bilateral contact with a rigid foundation is considered. The contact involves viscous friction of Tresca type. Two variational formulations of the problem are proposed, followed by existence and uniqueness results. Some properties involving the equivalence between the previous variational formulations, the continuous dependence of the solution with respect to the data as well as a convergence result with respect to the friction yield limit are also obtained.

Modelling of the thermomechanical coupling during hot forming of viscoplastic materials

In the present paper, a strategy for solving transient thermoviscoplastic problems is described. The general thermal and mechanical coupling phenomena are discussed by assuming material properties to be temperature-dependent and accounting for the mechanical terms in the non-stationary heat conduction equation. Time integration is carried out with an implicit scheme. An interesting version of the three-field approach is presented to obtain an accurate solution for the stress field. Finally, a global product coupling algorithm, in which the unknown nodal velocities and temperatures are solved with a two-step solution procedure is suggested. The proposed fractional step method partitions the initial coupled problem into an isothermal viscoplastic problem followed by a purely heat conduction problem at fixed configuration. Applications in metal forming analyses involving full thermomechanical coupling conclude the paper.

Coupling of FEM and BEM for Interface Problems in Viscoplasticity and Plasticity with Hardening

This paper is concerned with three-dimensional interface (or transmission) problems in solid mechanics which consist of the quasi-static equilibrium condition in a bounded Lipschitz domain $\Omega $, a first order evolution inclusion in $\Omega $, and the homogeneous linear elasticity problem in an unbounded exterior domain $\Omega _2 $. The evolution problem in $\Omega $ models viscoplasticity and Prandtl–Reuß plasticity with hardening. In a former paper [SIAM J. Math. Anal., 25 (1994), pp. 1468–1487] the author stated the interface problem as well as an equivalent formulation and proved existence and uniqueness of solutions. Here, the numerical approximation of the solutions is considered via a symmetric coupling of the FEM (finite element method) and BEM (boundary element method) in space and the generalized midpoint rule in time. Besides existence, uniqueness, and uniform boundedness of the solutions of the discrete problems, error estimates proving strong convergence of the numerical procedure are given. For the implicit Euler method the author also proves strong convergence for the time derivatives. Finally, the Uzawa algorithm is adopted to be used for the computation of the discrete solutions.

A Newton-Multigrid algrithm for elasto-plastic/viscoplastic problems

This work is concerned with large scaled nonlinear systems of equations resulting from discretization of problems in plasticity and viscoplasticity in the context of the finite element method. The main purpose is to show, how standard linear multigrid methods can be applied for solving the associated linear systems of equations in the frame of the Newton-algorithm. To this end, a so-called Galerkin-approach is used for construction of coarse grid matrices by transformation of fine grid matrices. It will be shown, how this transformation can be performed very efficiently element-by-element wise. Stopping criteria for the inner iteration are based on theories for so-called inexact Newton methods, where the linear systems are only solved approximately, however which preserve the rapid local convergence of Newtons method. In the numerical examples it is demonstrated, how the proposed strategy reduces the CPU-time for large scaled problems compared to solution techniques, where the associated systems of linear equations are solved directly.

A Newton-Multigrid algorithm for elasto-plastic viscoplastic problems

A Newton-Multigrid algorithm for elasto-plastic viscoplastic problems

Interface Problems in Viscoplasticity and Plasticity

This paper is concerned with three-dimensional interface (or transmission) problems in solid mechanics which consist of the (quasi-static) equilibrium condition and a first order evolution inclusion in a bounded Lipschitz domain $\Omega $ and the homogeneous linear elasticity problem in an unbounded exterior domain $\Omega _2 $. The evolution problem in $\Omega $ models viscoplasticity and Prandtl–Reus plasticity with hardening as well as perfect plasticity. The exterior part of the interface problem is rewritten in terms of boundary integral operators using the Poincare–Steklov operator. This symmetric coupling approach takes the total system of the Calderon projector into account. Then, existence and uniqueness results are obtained in a mixed variational formulation.

Regularized Method in Limit Analysis

The regularized problem proposed by Friaâ is discussed in this paper. The Norton‐Hoff viscoplastic law is first generalized and can be proved to be associated with the yield function of a plastic material. It can be demonstrated that, for any value of the coefficient of viscosity p, the viscoplastic‐strain rates satisfy also the plastic‐normality conditions. The creep‐strain rates tend to plastic‐strain rates when the coefficient p tends to one. Two dual variational formulations are then established for viscoplastic problems. These formulations have been proved well posed mathematically in view of the existence of solutions, and lead to efficient numerical computations. They are very useful for limit analysis, which is characterized by the minimization of a nonlinear functional in the space of kinematically admissible velocity fields. The optimized kinematically admissible velocity field is then used to justify the stability of structures under external loads. Different criterion functions of limit loads are discussed and the best one is proposed. An analytical example is presented and some numerical problems are presented to illustrate the performance of the regularized method.

Micromechanical analysis for thermoviscoplastic behavior of unidirectional fibrous composites: Robertson, D.D. and Mall, S. Compos. Sci. Technol. (1994) 50(4), 483–496

Elastic–visco-plastic deformations modeled by the Bodner–Partom constitutive equations [S.R. Bodner, Y. Partom, Constitutive equations elastic viscoplastic strain-hardening materials, J. Appl. Mech. 42 (1975) 385–389] have been evaluated by the adaptive finite element method [J.M. Bass, J.T. Oden, Adaptive finite element methods for a class of evolution problems in viscoplasticity, Int. J. Engrg. Sci. 25 (623–653) (1987); L. Demkowicz, J.T. Oden, W. Rachowicz, O. Hardy, Toward a universal hp-adaptive finite element strategy. Part 1: Constrained approximation and data structure, Comput. Methods Appl. Mech. Engrg. 77 (1989) 79–112]. We use the explicit residual error estimate [I. Babuška, W.C. Rheinboldt, Error estimates for adaptive finite element computations, Int. J. Numer. Methods Engrg. 12 (1978) 1597–1615] supplemented with a special term accounting for the fact that the right hand side of the equations we use is only approximated since it depends on the inelastic strains which are computed only at the Gauss integration points. We also address the problems of adaptation strategy and the method of solution transfer to a new mesh. Results of recent tests confirm efficiency and robustness of the approach.

Hierarchical simulation of hot composite structures: Chamis, C.C., Murthy, P.L.N. and Singhal, S.N. NASA Technical Memorandum TM-106200 (1993) 16 pp

Elastic–visco-plastic deformations modeled by the Bodner–Partom constitutive equations [S.R. Bodner, Y. Partom, Constitutive equations elastic viscoplastic strain-hardening materials, J. Appl. Mech. 42 (1975) 385–389] have been evaluated by the adaptive finite element method [J.M. Bass, J.T. Oden, Adaptive finite element methods for a class of evolution problems in viscoplasticity, Int. J. Engrg. Sci. 25 (623–653) (1987); L. Demkowicz, J.T. Oden, W. Rachowicz, O. Hardy, Toward a universal hp-adaptive finite element strategy. Part 1: Constrained approximation and data structure, Comput. Methods Appl. Mech. Engrg. 77 (1989) 79–112]. We use the explicit residual error estimate [I. Babuška, W.C. Rheinboldt, Error estimates for adaptive finite element computations, Int. J. Numer. Methods Engrg. 12 (1978) 1597–1615] supplemented with a special term accounting for the fact that the right hand side of the equations we use is only approximated since it depends on the inelastic strains which are computed only at the Gauss integration points. We also address the problems of adaptation strategy and the method of solution transfer to a new mesh. Results of recent tests confirm efficiency and robustness of the approach.

The role of interface and reinforcement properties on the fracture and fatigue resistance of ductile-phase toughened γ-TiAl composites: Rao, K.T.V., Ritchie, R.O. and Odette, G.R. Proc. Conf. Structural Intermetallics, Champion, PA, USA, 26–30 Sept. 1993, pp. 829–835

Elastic–visco-plastic deformations modeled by the Bodner–Partom constitutive equations [S.R. Bodner, Y. Partom, Constitutive equations elastic viscoplastic strain-hardening materials, J. Appl. Mech. 42 (1975) 385–389] have been evaluated by the adaptive finite element method [J.M. Bass, J.T. Oden, Adaptive finite element methods for a class of evolution problems in viscoplasticity, Int. J. Engrg. Sci. 25 (623–653) (1987); L. Demkowicz, J.T. Oden, W. Rachowicz, O. Hardy, Toward a universal hp-adaptive finite element strategy. Part 1: Constrained approximation and data structure, Comput. Methods Appl. Mech. Engrg. 77 (1989) 79–112]. We use the explicit residual error estimate [I. Babuška, W.C. Rheinboldt, Error estimates for adaptive finite element computations, Int. J. Numer. Methods Engrg. 12 (1978) 1597–1615] supplemented with a special term accounting for the fact that the right hand side of the equations we use is only approximated since it depends on the inelastic strains which are computed only at the Gauss integration points. We also address the problems of adaptation strategy and the method of solution transfer to a new mesh. Results of recent tests confirm efficiency and robustness of the approach.

Adaptive remeshing and error control for forming processes

ABSTRACT In order to take into account the large deformations of the material, the numerical simulation of forming problems invoices a great number of meshings. For effective treatment of meshing and remeshing, error estimators have been considered as a guiding tool. The present work especially concerns methods for the construction of adapted meshes when an optimal size criterion has been calculated from error estimators. The remeshing procedure is based on the Delaunay type mesh generator and on the uncoupling between geometric and adaptivity constraints of the remeshing problem. The procedure is tested for various elasticity and viscoplasticity problems. It is shown to be robust and reliable, then it is used to compare the efficiency of two error indicators which stem from the Z2 (Zienkiewicz and Zhu) error estimator and the Δ error estimators (a correction of the Z2 estimate in order to take into account the unbalancing of the smoothed stress tensor) which have been suggested in a previous paper. Singu...

A spectral overlay method for quasi-static viscoplasticity: Applications to anti-plane crack and comparison with closed-form solutions

The modified spectral overlay finite element method is described for a two-dimensional antiplane viscoplastic problem under quasi-static, isothermal conditions. The essential feature of the spectral overlay method is that high resolution of localized steep gradients can be achieved by overlaying a spectral interpolant on a standard finite element mesh. In order to evaluate this method and compare it with the conventional finite element method, a closed-form solution of a shear localization problem in one dimension has been devised. The results show that this method is very powerful in capturing the shape of the gradient field and its peak value in the localization zone. Solutions for an antiplane crack problem are then given to illustrate the effectiveness of this method.

A large time increment approach for cyclic viscoplasticity

The principles of a new approach for cyclic viscoplastic problems are described in this paper. A computational method of dealing with nonlinear mechanical behaviours, described by internal variables, is presented. This approach contrasts with the classical step-by-step method since it is an iterative procedure that accounts for the whole loading process in a single time increment. Special tools have been developed to simulate, within the large time increment method, several hundred cycles in a single increment. The feasibility of this strategy is shown through simple examples, with a large number of cycles in the case of viscoplastic behaviour.

The determining relationships in viscoplasticity problems

The determining relationships in viscoplasticity problems

A numerical method for a viscoplastic problem. An application to wire drawing

A stationary boundary value problem with friction for the Bingham fluid is considered. The given iterative finite element method provides an approximation of the solution. The numerical method is applied to wire drawing processes and the results are presented.

Adaptive finite element methods for a class of evolution problems in viscoplasticity

Adaptive finite element procedures are presented for the analysis of broad classes of two-dimensional problems in viscoplasticity which involve internal state variables. Several a posteriori error estimates are developed and used as a basis for refinements of meshes of triangular and quadrilateral elements. Algorithms are presented which reduce the computational complexity of the adaptive process and provide for the use of arbitrary local estimates of the approximation error. The results of numerical experiments are given which illustrate the effectiveness of the adaptive algorithms and various error indicators.