Stress Update Algorithm Research Articles

A Coupled Thermo-Viscoplastic Formulation at Finite Strains for the Numerical Simulation of Superplastic Forming

This paper is concerned with the numerical simulation of hot metal forming, especially superplastic forming. A complete thermo-viscoplastic formulation at finite strains is derived and a unified stress update algorithms for thermo-elastoplastic and thermo-elastoviscoplastic constitutive equations is obtained. The resulting unified implicit algorithm is both efficient and very inexpensive. Finally, numerical simulations of superplastic forming are exposed.

Formulation of an implicit algorithm for finite deformation viscoplasticity

Within the framework of additive plasticity, an objective stress update algorithm has been proposed. The procedure is implemented in such way that the extension from a standard small strain FE code to the finite strain range is straightforward, and objectivity can be retained for any choice of the intermediate configuration. The additional computational cost only includes some geometrical manipulations. For the Newton–Raphson iteration method, a closed-form solution of the consistent tangent is derived by direct linearization of the stress update algorithm. Numerical examples show a quadratic rate of convergence with the proposed viscoplastic model. The analysis of a tensile test first shows a shear band with a finite thickness independent of the finite element size. At large deformation, the shear band pattern transforms into a necking failure mode. As a second example, a thin sheet tensile test is analyzed. A necking failure mode leads to global softening even though, locally, the material is still hardening. Although viscoplastic regularization is used, the results still show a mesh dependence for deformations over 40%.

Aspects of ductile fracture and adaptive mesh refinement in damaged elasto-plastic materials

Numerical simulation of elasto-plastic problems involving multi-fracturing materials requires a reliable failure prediction technique and a robust solution algorithm. This work approaches ductile fracture by means of continuum damage mechanics, from which two new failure criteria based on coupled and uncoupled damage analysis are derived. A two-parameter stress update algorithm for damaged materials based on a Newton–Raphson iterative procedure is presented. A posteriori error estimators using ductile failure concepts are also discussed. Copyright © 2001 John Wiley & Sons, Ltd.

A comparative study of stress update algorithms for rate-independent and rate-dependent crystal plasticity

A comparative study of stress update algorithms for rate-independent and rate-dependent crystal plasticity

Accuracy of Two Stress Update Algorithms for Shear-Free Large Deformation Paths

The behavior of two stress update algorithms for shear-free large deformation paths is analyzed. The first algorithm has a truncation error of order 1. The second algorithm has a truncation error of order 2. As a consequence, the global performance of the second algorithm is clearly superior. However, for the particular case of shear-free deformation paths, the first algorithm correctly predicts null shear stresses, while the second one does not. This behavior was reported in a previous paper for an extension-rotation test. In this note a general shear-free deformation path is considered in full detail.

A formulation of finite elastoplasticity based on dual co- and contra-variant eigenvector triads normalized with respect to a plastic metric

The article presents a new formulation of isotropic elastoplasticity at large strains in both the Lagrangian and the Eulerian geometric setting and addresses aspects of its numerical implementation. The key ingredients on the theoretical side are the introduction of a plastic metric for the description of the local history-dependent inelastic material response, the definition of a convex elastic domain in the space of the local stress-like variable conjugate to the plastic metric, denoted as the plastic force, and a fully equivalent Lagrangian and Eulerian representation of all constitutive functions in spectral form for general non-Cartesian coordinate charts in terms of dual co- and contra-variant eigenvector triads which are normalized with respect to the plastic metric. On the numerical side, we propose a stress update algorithm for general non-associative isotropic elasto(visco)plastic response with an arbitrary number of scalar internal variables. The algorithm is based on an exponential map integrator and is recast into a general return mapping scheme, methodically organized with tensorial pre- and postprocessing and a constitutive box in the eigenvalue space. Furthermore, we propose a new perturbation stabilization technique which dramatically enhances the convergence of the general return algorithm. The theoretical and numerical developments are applied to a constitutive model problem with large elastic and large plastic strains: the von Mises-/Tresca-type associative plastic flow in Ogden-type large-strain elastic materials.

A theoretical and computational model for isotropic elastoplastic stress analysis in shells at large strains

The article presents new aspects of large-strain isotropic elastoplastic analysis of shells. On the shell theory side, we consider several possible continuum based parametrizations of the shell deformation and investigate their applicability with respect to the implementation of general three-dimensional local constitutive models. As an aspect of computational shell analysis we then discuss in detail the numerical implementation of a shell based on the parametrization of the top and bottom surface in terms of a new 8-node brick-type mixed finite shell element on the basis of assumed strain and enhanced strain variational approaches. On the side of plasticity theory we adopt a recently proposed intermediate-configuration-free formulation of large-strain plasticity based on the notion of a plastic metric and consider its representation with respect to the parameter manifold of the shell. This frame is formulated in terms of principal strains and principal stresses with respect to dual co- and contra-variant eigenvector triads associated with a mixed-variant elastic strain tensor. The new aspect of computational plasticity is the development of a stress update algorithm for the plasticity model mentioned above which includes a general return mapping in the eigenvalue space. The algorithm can be applied to general coordinate charts and is particularly convenient when formulated with respect to the space of the curvilinear coordinates of the shell parametrization. We demonstrate the performance of the finite element formulation and constitutive algorithms by means of several numerical examples.

Two stress update algorithms for large strains: accuracy analysis and numerical implementation

Two algorithms for the stress update (i.e., time integration of the constitutive equation) in large-strain solid mechanics are compared from an analytical point of view. The order of the truncation error associated to the numerical integration is deduced for each algorithm a priori, using standard numerical analysis. This accuracy analysis has been performed by means of a convected frame formalism, which also allows a unified derivation of both algorithms in spite of their inherent differences. Then the two algorithms are adapted from convected frames to a fixed Cartesian frame and implemented in a small-strain finite element code. The implementation is validated by means of a set of simple deformation paths (simple shear, extension, extension and compression, extension and rotation) and two benchmark tests in non-linear mechanics (the necking of a circular bar and a shell under ring loads). In these numerical tests, the observed order of convergence is in very good agreement with the theoretical order of convergence, thus corroborating the accuracy analysis. © 1997 John Wiley & Sons, Ltd.

EXPONENTIAL MAP ALGORITHM FOR STRESS UPDATES IN ANISOTROPIC MULTIPLICATIVE ELASTOPLASTICITY FOR SINGLE CRYSTALS

This paper presents a new stress update algorithm for large-strain rate-independent single-crystal plasticity. The theoretical frame is the well-established continuum slip theory based on the multiplicative decomposition of the deformation gradient into elastic and plastic parts. A distinct feature of the present formulation is the introduction and computational exploitation of a particularly simple hyperelastic stress response function based on a further multiplicative decomposition of the elastic deformation gradient into spherical and unimodular parts, resulting in a very convenient representation of the Schmid resolved shear stresses on the crystallographic slip systems in terms of a simple inner product of Eulerian vectors. The key contribution of this paper is an algorithmic formulation of the exponential map exp: sl(3) SL(3) for updating the special linear group SL(3) of unimodular plastic deformation maps. This update preserves exactly the plastic incompressibility condition of the anisotropic plasticity model under consideration. The resulting fully implicit stress update algorithm treats the possibly redundant constraints of single-crystal plasticity by means of an active set search. It exploits intrinsically the simple representation of the Schmid stresses by formulating the return algorithm and the associated consistent elastoplastic moduli in terms of Eulerian vectors updates. The performance of the proposed algorithm is demonstrated by means of a representative numerical example.

Multisurface thermoplasticity for single crystals at large strains in terms of eulerian vector updates

This paper presents a formulation of large-strain rate-independent multisurface thermoplasticity for single crystals and addresses aspects of its numerical implementation. The theoretical frame is the well-established continuum slip theory based on the multiplicative decomposition of the deformation gradient into elastic and plastic parts. A key feature of the present paper is the introduction and computational exploitation of a particularly simple hyperelastic stress response function based on a further multiplicative decomposition of the elastic deformation gradient into spherical and unimodular parts, resulting in a very convenient representation of the Schmid resolved shear stresses on the crystallographic slip systems in terms of a simple inner product of Eulerian vectors. This observation is intrinsically exploited on the numerical side by formulating a new fully implicit stress update algorithm and the associated consistent elastoplastic moduli in terms of these Eulerian vectors. The proposed return mapping algorithm treats the possibly redundant constraints of large-strain multisurface plasticity for single crystals by means of an active set search. Furthermore, it satisfies in an algorithmically exact way the plastic incompressibility condition in situations of multislip. The performance of the proposed formulation is demonstrated for two representative numerical simulations of thermoplastic deformation processes in single crystals with isotropic Taylor-type hardening.

GEOMETRICALLY NON-LINEAR AND ELASTOPLASTIC THREE-DIMENSIONAL SHEAR FLEXIBLE BEAM ELEMENT OF VON-MISES-TYPE HARDENING MATERIAL

A three-dimensional elastoplastic beam element being capable of incorporating large displacement and large rotation is developed and examined. Elastoplastic constitutive equations are applied to the beam element based upon the assumption of small deformational strain leading to a material formulation which is completely objective for the application of stress update procedures. The continuum-type equations of plastic model of J2 mixed hardening are transformed into the beam equations by satisfying beam hypotheses. An effective stress update algorithm is proposed to integrate elastoplastic rate equations by means of the so-called multistep method which is a method of successive control of residuals on yield surfaces. It avoids severe divergence when the displacement increments become large which is usual for the continuation methods. Material tangent stiffness matrix is derived by using consistent elastoplastic modulus resulting from the integration algorithm and is combined with geometric tangent stiffness matrix. Different from other elements, the present element is shear flexible and can satisfy the plasticity condition in a pointwise fashion. A great number of numerical examples are analysed and compared with the literature. The proposed beam element is verified to be not only quite accurate but also very effective for the analyses of pre-buckling and large deflection collapse of spatial framed structures.

Coupled creep-elastoplastic-damage analysis for isotropic and anisotropic nonlinear materials

This paper deals with the computational aspects of a coupled creep-elastoplastic-damage analysis for anisotropic, and as a special case isotropic nonlinear materials. A three phase backward Euler integration algorithm for stress update is proposed. For anisotropic nonlinear materials a general direct stress return mapping algorithm, utilising Newton-Raphson iteration, is derived. The stress vector and scalar variables quantifying the incremental creep, plasticity and damage are updated simultaneously. For isotropic materials the elasto-plastic stress update algorithm for plane stress by Jetteur (1986, Engng Camp.3, 251–253) is extended to include creep and damage. In addition, a simple stress algorithm for the general three-dimensional isotropic case is also presented. The resulting algorithms are suitable for inclusion in general structural analysis codes. The consistent tangent matrix is also formulated for use in a global Newton iterative procedure, in which structural displacements are sought as the problem unknowns. Examples are given using the general purpose code LUSAS in which the algorithms have been implemented.

A consistent formulation for the integration of combined plasticity and creep

AbstractWithin the context of the consistent tangent update, this paper outlines a stress update algorithm for combined creep and plasticity. The algorithm is implicit, providing unconditional stability, and utilizes local Newton iteration to solve SCALAR forms of the coupled constitutive equations for the creep and plastic strain increments. The tangent for the local iteration is obtained accurately providing quadratic convergence at the Gauss point level. Quadratic convergence of the global iteration procedure is also maintained using an explicitly derived consistent tangent for combined plasticity and creep. Further, combination with an automatic time‐stepping scheme provides an efficient, stable, accurate and robust computational algorithm. The algorithm has been implemented in the general purpose FE package LUSAS1.

Universal anisotropic yield criterion based on superquadric functional representation: Part 1. Algorithmic issues and accuracy analysis

A robust and efficient stress update algorithm for the anisotropic yield criterion represented in the form of superquadric function introduced by Barlat et al. is presented. It relies on a discrete variational formulation of the Lagrangian functional which results from the principle of maximum plastic dissipation. Numerical solution of the discretized equations of evolution is based on the operator split methodology and the Newton-Raphson method. A line search algorithm is implemented to overcome numerical instabilities and to expand the convergence region of the basic Newton-Raphson solution procedure. The consistent tangent modulus is expressed in a closed form as a result of the exact linearization of the discrete Kuhn-Tucker condition. Numerical testing of the integration algorithm based on iso-error maps is provided for several variants of the yield function.

Symmetry‐preserving return mapping algorithms and incrementally extremal paths: A unification of concepts

AbstractIn this work we seek to characterize the conditions under which an elastic–plastic stress update algorithm preserves the symmetries inherent to the material response. From a numerical standpoint, the aim is to determine under what conditions a stress update algorithm produces symmetric consistent tangents when applied to materials obeying normality. For the ideally plastic solid we show that only the fully implicit or closest point return mapping algorithm is symmetry preserving. For hardening plasticity, symmetry cannot be preserved in general unless suitable restrictions are imposed on the constitutive equations. We show that these restrictions amount to the existence of a pseudo‐internal energy function acting as a joint potential for both the direction of plastic flow and the hardening moduli. In view of the fact that holonomic methods based on incrementally extremal paths also result in update rules possessing a potential structure and, hence, in symmetric tangents, we address the question of whether any connections exist between the two approaches. We show that holonomic methods and the fully implicit algorithm may indeed be brought into correspondence.

Viscoplastic behaviour of stainless steels AISI 316L and 316H

The theory of viscoplasticity based on total strain and overstress is used in order to simulate the sensitivity to the rate of loading of two commonly used stainless steels, namely AISI 316L and 316H. The consitutive model has been implemented within a transient finite element computer code using a stress update algorithm based on the elastic predictor-return mapping concept. Both monotonic and cyclic loading conditions are considered in one or more space dimensions.