A finite element formulation for elastoplasticity based on a three-field variational equation

A three-field variational equation, which expresses the momentum balance equation, the plastic consistency condition, and the dilatational constitutive equation in a weak form, is proposed as a basis for finite element computations in hardening elastoplasticity. The finite element formulation includes algorithms for the integration of the elastoplastic rate constitutive equations which are similar to members of the “return mapping” family of algorithms employed in displacement formulations, except that the proposed algorithms are not required to explicitly satisfy the plastic consistency condition at the end of each time step. This condition is imposed globally by the inclusion of a variational equation that suitably constrains the solution. The plastic incompressibility constraint is also treated in an appropriate variational sense. Solution of the nonlinear finite element equations is obtained by use of Newton's method and details of the linearization of the variational equation are given. The formulation is developed for an associative von Mises plasticity model with general nonlinear isotropic and kinematic strain hardening. A number of numerical test examples are provided.     

Variational inequality in classical plasticity. Applications to Armstrong–Frederick elasto-plastic model

In this paper the quasi-static initial and boundary value problem for an elasto-plastic mixed hardening material is reformulated within the constitutive framework of small strains. The plastic factor plays the basic role in describing the rate independent evolution equations for the plastic strain and hardening variables. The plastic factor is equivalently represented as the solution of an appropriate local inequality involving the yield function. The main idea was to introduce the variational inequality at any time $t$ to be solved for the velocity field and the complementary plastic factor. There is the plastic factor in a strain-driven process. The solution procedure proposed here to solve the initial and boundary value problem is based on the solutions of the variational inequality at time $t$, coupled with an update algorithm in order to evaluate the current state of the material for an incremental deformation process. This time the return mapping algorithm is avoided as the values of the plastic factor and the velocity are known at time $t.$ As we developed a procedure to simultaneously solve the equilibrium equation coupled with the rate-independent evolution equations, no necessity to compute the algorithmic elasto-plastic tangent moduli occurs. The numerical simulations are done for the mixed hardening elasto-plastic model involving Armstrong–Frederick kinematic hardening. To validate the proposed numerical algorithms, we compare the solutions based on the variational inequality and those based on return mapping algorithm, computed for the same Prager kinematic hardening law.     

An efficient and accurate alternative to subincrementation for elastic-plastic analysis by the finite element method

In this paper a new and efficient alternative to subincrementation is developed for analysis of solid media with rate independent elastic-plastic material behavior. This alternative method is not unlike the subincrementation procedure in that it represents an Euler integration of the nonlinear constitutive equations. However, it takes advantage of the fact that the Euler integration procedure assumes proportional loading steps so that when the uniaxial stress-strain curve is idealized as a piecewise linear relation very large forward integration steps give accurate results. The new procedure, which we call the ζ method, is equally appropriate for cyclic loading with combined isotropic and kinematic hardening. However, due to the nonuniqueness of the monotonic uniaxial stress-strain relation in rate dependent media, the method is not appropriate for use in viscoplastic media. Although the algorithm deals only with the evaluation of a classical plasticity based constitutive law, numerical results are reported herein for an assortment of problems by the finite element method. It is shown via these results that the ζ method discussed herein provides not only accuracy which is superior to the subincrementation method, but the resulting algorithm also shows improved numerical efficiency.     

On plastic incompressibility within time-adaptive finite elements combined with projection techniques

This article treats the interpretation of quasi-static finite elements applied to constitutive equations of evolutionary-type as a solution scheme to solve globally differential-algebraic equations. This concept is applied to finite strain viscoplasticity based on a model with non-linear kinematic hardening under the assumption of plastic incompressibility. The model is based on multiple multiplicative decomposition both for the deformation gradient into an elastic and an inelastic part as well as for the inelastic part into a kinematic hardening (energy storage) and a dissipative part. Both intermediate configurations are described by inelastic right Cauchy-Green tensors satisfying inelastic incompressibility in the theoretical context. The attention in view of the numerical treatment within finite elements is focused on diagonally implicit Runge-Kutta methods which destroy the assumption of plastic incompressibility during the time-integration due to an additive structure of the integration step. In combination with a Multilevel-Newton algorithm these algorithms embed the classical strain-driven radial-return method. To this end, a concept of geometric numerical integration is applied, where the plastic incompressibility condition is taken into account as an additional side-condition. Since the literature states large integration errors if the side-condition is not taken into account, a particular focus lies on the application of a time-adaptive procedure. Accordingly, the article investigates (i) the algorithmic treatment of kinematic hardening within time-adaptive finite elements, (ii) the influence of the Perzyna-type viscoplasticity approach in view of an order reduction phenomenon, and (iii) the influence of taking into account the exact fulfillment of plastic incompressibility using a projection method having the advantage of simple implementation.     

A return mapping algorithm for cyclic viscoplastic constitutive models

This paper presents a return mapping algorithm for cyclic viscoplastic constitutive models that include material memory effects. The constitutive model is based on multi-component forms of kinematic and isotropic hardening variables in conjunction with von Mises yield criterion. Armstrong–Frederick (A–F) type rules are used to describe the nonlinear evolution of each of the multi-component kinematic hardening variables. A saturation type (exponential) rule is used to describe the nonlinear evolution of each of the isotropic hardening variables. The concept of memory surface is used to describe the strain range dependent material memory effects that are induced by the prior strain histories. In this paper, the above class of cyclic viscoplastic constitutive models is formulated within a consistent thermodynamic framework that encompasses the standard generalized materials framework. Furthermore, a complete algorithmic treatment of the above rate-dependent constitutive model is also presented for any desired stress or strain constrained configuration subspace. A generalized midpoint algorithm is used to integrate the rate constitutive equations. The consistent tangent operator is obtained by linearizing the return mapping algorithm, and is found to be unsymmetric due to the presence of nonlinear evolution rules for the kinematic hardening variables. Several numerical examples representing the cyclic hardening and softening behavior, transient and stabilized hysteresis behavior, and the non-fading memory effects of the material are presented. These examples demonstrate the accuracy and robustness of the present algorithmic framework for modeling the cyclic viscoplastic behavior of the material.     

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.     

Operator split methods for the numerical solution of the elastoplastic dynamic problem

The elastoplastic dynamic problem is first formulated in a form that facilitates the application of product formula techniques. The additive decomposition of the dynamic equations into elastic and plastic parts is taken as a basis for the definition of product algorithms that exploit such decomposition. In the context of a finite element discretization, these product algorithms entail, for every time step, the solution of an elastic problem followed by the application of plastic algorithms that operate on the stresses and internal variables at the integration points and bring in the plastic constitutive relations. Suitable plastic algorithms are discussed for the cases of perfect and hardening plasticity and viscoplasticity. The proposed formalism does not depend on any notion of smoothness of the yield surface and is applicable to arbitrary convex elastic regions, with or without corners. The stability properties of the product algorithm are identical to those of the elastic algorithm used whereas the computational expense is practically equal to that of an elastic problem.     

Numerical techniques for plasticity computations in finite element analysis

Common numerical techniques for plasticity computations in finite element analysis are examined. The plasticity theory considered is the simple rate-independent von Mises criterion for small strains. Work hardening is represented by a general isotropic model or by a linear, isotropic-kinematic mixed model. Algorithms to integrate the rate equations, strategies for stress updating over a time (load) step in implicit codes, and tangent operators consistent with the integration algorithm are discussed. The elastic predictor-radial return algorithm and a consistent tangent operator satisfy the requirements for a stable, accurate and efficient numerical procedure. An extension of this model for plane stress with mixed hardening is described. Two numerical examples are given to demonstrate the accuracy and efficiency for plane stress analyses.     

Consistent tangent operators for rate-independent elastoplasticity

It is shown that for problems involving rate constitutive equations, such as rate-independent elastoplasticity, the notion of consistency between the tangent (stiffness) operator and the integration algorithm employed in the solution of the incremental problem, plays a crucial role in preserving the quadratic rate of asymptotic convergence of iterative solution schemes based upon Newton's method. Within the framework of closest-point-projection algorithms, a methodology is presented whereby tangent operators consistent with this class of algorithms may be systematically developed. To wit, associative J₂ flow rules with general nonlinear kinematic and isotropic hardening rules, as well as a class of non-associative flow rules are considered. The resulting iterative solution scheme preserves the asymptotic quadratic convergence characteristic of Newton's method, whereas use of the socalled elastoplastic tangent in conjunction with a radial return integration algorithm, a procedure often employed, results in Newton type of algorithms with suboptimal rate of convergence. Application is made to a set of numerical examples which include saturation hardening laws of exponential type.     

Some computational aspects of large deformation,rate-dependent plasticity problems

The governing equations for a finite element formulation of boundary value problems for large deformation metal forming processes are derived using a principle of virtual work formulated in a Lagrangian reference system. The updated Lagrangian method is used to simplify the equations. The resulting nonlinear equilibrium equations are solved using Newton's method.A constitutive model for large deformation, rate-dependent plasticity is developed. The model incorporates a one parameter, implicit integration operator for stability and accuracy. The stressrate/strain-rate relation is written in terms of the Jaumann rate of stress.Numerous example problems are solved to demonstrate the effectiveness of the numerical algorithms.     

A plastic nonlocal damage model

In the present paper, a plastic nonlocal damage model is proposed for studying the mechanical response of structural elements made of cementitious materials. A new isotropic damage model, which is able to describe the behavior of a wide class of cementitious materials, is presented. A regularization technique, based on the introduction of the damage Laplacian in the damage limit function, is adopted to overcome the analytical and computational problems induced by the softening constitutive law. A Drucker–Prager type of plastic limit function is proposed considering isotropic hardening. A numerical procedure, based on an implicit `backward-Euler' technique for the time integration of the plastic and damage evolution equations, is presented. To solve each nonlinear step, a predictor–corrector iterative method is developed within the splitting method. In particular, the damage evolution is determined solving a constrained minimization problem of a convex functional. The proposed algorithm is implemented in a finite element code and it is used to study the structural behavior of elements made of masonry materials.     

Simulation of strength difference for adhesive materials in finite deformation elasto-plasticity

The experimental investigation of certain adhesive materials reveals elastic strains, plastic strains and hardening, respectively. Additionally a pronounced strength difference between tension, torsion or combined loading is observed. The purpose of this work is the simulation of these phenomena in the framework of large strain elasto-plasticity. To this end a yield function dependent on the first and second basic invariants of the Cauchy stress tensor is introduced. Furthermore, a plastic potential with the same mathematical structure is used to formulate the evolution equations under the assumption of small elastic strains. Upon considering thermodynamic consistency of the model equations some restrictions on the material parameters are derived. Furthermore numerical aspects are addressed concerning the integration of the constitutive relations and the finite element equilibrium iteration. In the numerical examples, firstly, we compare simulated and experimental results exhibiting the yield strength difference between tension and torsion for the adhesive material Betamate 1496. A second example investigates the deformation evolution of a compact tension specimen with an adhesive zone.     

A finite element formulation for transient analysis of viscoplastic solids with application to stress wave propagation problems

A brief note on a finite element formulation for the transient analysis of viscoplastic solids is presented. Attention is confined to small strains in the present discussion and the emphasis is on stress wave propagation problems. The algorithm is based on straightforward explicit integration of both the equations of motion and the plastic rate equations. The explicit central difference method, which appears suitable for wave propagation problems, is used for integration of the equation of motion. The stress update is accomplished by means of a forward gradient scheme based on an estimate of the plastic flow over a time increment (Peirce, Shih and Needleman, Comput. Struct.18, 875–887, 1984). A number of simple numerical examples are presented to illustrate the method.     

An investigation of ultrasonic nanocrystal surface modification machining process by numerical simulation

As a method for surface severe plastic deformation (S²PD), ultrasonic nanocrystal surface modification (UNSM) enhances metal surface properties through striker peening, a metal dimpling process driven by ultrasonic vibration energy. UNSM treatment introduces residual stress, surface hardening, and nano-crystalline structures into metal surfaces which are beneficial for reducing wear, fatigue, and corrosion properties. In this paper, the process of UNSM is described and a simplified physical model created using the equivalent static loading method is presented. Along with the simplified physical model, a finite elements simulation model was developed. Effective plastic strain was considered as a parameter for evaluating the level of work hardening produced in the simulation. The dynamic processes and energy dissipation were also examined, and it was found that different kinds of energy dissipation occur during UNSM treatment. Comparisons between the processing parameters (processing velocity, static load, and feed rate) were performed using a simulated example of UNSM linear processing. The results show that the linear processing produces a uniform region containing identical distributions of residual stress and effective plastic strain. The effects of the parameters on the processing results (residual stress, plastic deformation and work hardening) were likewise studied using UNSM linear processing. Compared to processing velocity, a high static load produced more work hardening and higher compressive residual stress. Surface deformation and residual stress results were also more sensitive to static load than processing velocity. Feed rate was found to be an important parameter as well, greatly influencing both surface deformation and work hardening.     

A new algorithm for numerical solution of dynamic elastic-plastic hardening and softening problems

The objective of this paper is to develop a new algorithm for numerical solution of dynamic elastic-plastic strain hardening/softening problems, particularly for the implementation of the gradient dependent model used in solving strain softening problems. The new algorithm for the solution of dynamic elastic-plastic problems is derived based on the parametric variational principle. The gradient dependent model is employed in the numerical model to overcome the mesh-sensitivity difficulty in dynamic strain softening or strain localization analysis. The precise integration method, which has been used for the solution of linear problems, is adopted and improved for the solution of dynamic non-linear equations. The new algorithm is proposed by taking the advantages of the parametric quadratic programming method and the precise integration method. Results of numerical examples demonstrate the validity and the advantages of the proposed algorithm.     

Sheet metal forming and springback simulation by means of a new reduced integration solid-shell finite element technology

The paper deals with the validation of a recently proposed hexahedral solid-shell finite element in the field of sheet metal forming. Working with one integration point in the shell plane and an arbitrary number of integration points in thickness direction, highly non-linear stress states over the sheet thickness can be incorporated in an efficient way. In order to avoid volumetric locking and Poisson thickness locking at the level of integration points the enhanced assumed strain (EAS) concept with only one EAS degree-of-freedom is implemented. A key point of the formulation is the construction of the hourglass stabilization by means of different Taylor expansions. This leads to the advantage that the sensitivity with respect to mesh distortion is noticeably reduced. The hourglass stabilization includes the assumed natural strain (ANS) concept and a kind of B-Bar method. So transverse shear locking and volumetric locking are eliminated.The finite element formulation incorporates a finite strain material model for plastic anisotropy as well as non-linear (Armstrong–Frederick type) kinematic and isotropic hardening. In this context the plastic anisotropy can be modeled by representing the yield surface and the plastic flow rule as functions of so-called structural tensors. The integration of the evolution equations is performed by means of an exponential map exploiting the spectral decomposition. The element formulation and material model have been implemented into the commercial code ABAQUS/Standard by means of the UEL interface for user-defined elements. Using an implicit time integration scheme numerical results for classical deep drawing simulations as well as springback predictions are presented in comparison to experimental measurements.     

A nonsmooth Newton method for elastoplastic problems

In this work we reformulate the incremental, small strain, J₂-plasticity problem with linear kinematic and nonlinear isotropic hardening as a set of unconstrained, nonsmooth equations. The reformulation is done using the minimum function. The system of equations obtained is piecewise smooth which enables Pang's Newton method for B-differentiable equations to be used. The method proposed in this work is compared with the familiar radial return method. It is shown, for linear kinematic and isotropic hardening, that this method represents a piecewise smooth mapping as well. Thus, nonsmooth Newton methods with proven global convergence properties are applicable. In addition, local quadratic convergence (even to nondifferentiable points) of the standard implementation of the radial return method is established. Numerical tests indicate that our method is as efficient as the radial return method, albeit more sensitive to parameter changes.     

Explicit momentum conserving algorithms for rigid body dynamics

Two new explicit time integration algorithms are presented for solving the equations of motion of rigid body dynamics that identically preserve angular momentum in the absence of applied torques. This is achieved by expressing the equations of motion in conservation form. Both algorithms also eliminate the need for computing the angular acceleration. The first algorithm employs a one-pass predictor-corrector scheme while the second algorithm is based upon the staggered time integration approach of Park. Numerical results are presented comparing the new algorithms to the algorithms of Simo and Wong and Park et al. The predictor-corrector algorithm is shown to suffer weak instabilities while the staggered conserving algorithm exhibits improved performance compared to the staggered algorithm of Park et al.     

An implicit formulation of the Bodner-Partom constitutive equations

The framework for an implicit implementation of the Bodner-Partom material model is presented. All equations needed for using a Newton-Raphson algorithm to solve the stress and hardening equations at the integration points are derived. In addition, the algorithmic tangential stiffness tensor, ensuring quadratic convergence in the global loop is presented.