The Governing Parameter Method for implicit integration of viscoplastic constitutive relations for isotropic and orthotropic metals

A general algorithm of implicit stress integration in viscoplasticity, based on the governing parameter method (GPM) is briefly presented. It is assumed that the associative viscoplastic constitutive relations are governed by the Perzyna formulation with a generalization suggested by Simo and Hughes. The algorithm is first applied to isotropic metals obeying the von Mises yield condition with mixed hardening and then, to orthotropic metals with a generalized Hill's yield condition including a mixed hardening assumption. Derivation of consistent tangent moduli is presented for both viscoplastic material models. The proposed computational procedures are efficient, since they reduce the problem of stress integration to the solution of one nonlinear equation, can use large time steps and are applicable to 2-D, 3-D, shell and beam structures. The tangent elastic viscoplastic matrix provides high convergence rate in the overall equilibrium iterations. Numerical examples illustrate the main characteristics of the developed computational procedures.     

Numerical integration and FEM-implementation of a viscoplastic Chaboche-model with static recovery

This paper is concerned with the implementation of a viscoplastic material model of the Chaboche type in the framework of the finite element method (FEM). The equations of the used constitutive law, that incorporates isotropic hardening, back stress evolution with static recovery terms and drag stress evolution, are introduced. A representation of their numerical integration using the implicit backward Euler method under the assumption of small deformations and an isothermal formulation follows. The use of the backward Euler method leads to a nonlinear algebraic system of three equations, which is solved by a combination of the Pegasus method and a fixed-point iteration. After considering the accuracy of the presented integration algorithm in form of iso-error maps, the derivation of the consistent viscoplastic tangent operator is shown. The integration scheme and the calculation of the consistent viscoplastic tangent operator are implemented in the commercial finite element code ABAQUS, using the possibility of the user-defined material subroutine (UMAT). Finally a numerical example in form of a notched bar under tension is presented.     

A dynamic elasto/viscoplastic solution for a thick-walled spherical container subjected to impulsive loading

A general method for the solution of dynamic elasto/viscoplastic solid problems is presented. This method is an extension of ray theory to cater for dynamic elasto/viscoplastic deformations. The method reduces the elasto/viscoplastic problem to a sequence of elastic problems with initial strains. The solution of this problem is determined by using four displacement functions. Using the foregoing method, the solution is derived for the dynamic elasto/viscoplastic behaviour of a thick-walled spherical shell subjected to internal impact load. The numerical results show how the dynamic stresses in a sphere with viscoplastic properties vary with time.     

A new stress-updating algorithm for viscoplasticity

Material behavior beyond the elastic limit can be rate-dependent, and this rate sensitivity can be captured by the viscoplastic material models. To describe the viscoplastic material behavior in structural analysis, an efficient numerical framework is necessary. In this paper an algorithm is proposed for metals for which von Mises yield surface along with Peri?’s viscoplastic model is employed. The efficiency and accuracy of the technique is examined by comparison with different numerical studies. The convergence rate of the proposed algorithm is investigated. Characteristics of the viscoplastic behavior such as relaxation are illustrated in the selected case studies. Finally, application of the algorithm in practice is demonstrated by a boundary value problem.

     

A model for predicting grain boundary cracking in polycrystalline viscoplastic materials including scale effects

A model is developed herein for predicting the mechanical response of inelastic crystalline solids. Particular emphasis is given to the development of microstructural damage along grain boundaries, and the interaction of this damage with intragranular inelasticity caused by dislocation dissipation mechanisms. The model is developed within the concepts of continuum mechanics, with special emphasis on the development of internal boundaries in the continuum by utilizing a cohesive zone model based on fracture mechanics. In addition, the crystalline grains are assumed to be characterized by nonlinear viscoplastic mechanical material behavior in order to account for dislocation generation and migration. Due to the nonlinearities introduced by the crack growth and viscoplastic constitution, a numerical algorithm is utilized to solve representative problems. Implementation of the model to a finite element computational algorithm is therefore briefly described. Finally, sample calculations are presented for a polycrystalline titanium alloy with particular focus on effects of scale on the predicted response. This revised version was published online in July 2006 with corrections to the Cover Date.     

A computationally efficient algorithm to simulate the butt curl deformation in casting processes

The aim of this work is to present a computationally efficient algorithm to simulate the butt curl deformation. In our previous articles [1, 2] the nonlinearities due to the viscoplastic law and the contact condition with the bottom block were solved by means of duality methods involving two multipliers. In [1] these multipliers were computed with a fixed point algorithm and in [2] with a generalized Newton’s method. In this work we improve the viscoplastic algorithm by means of a generalized duality method with variable parameters. We will present numerical results showing the applicability of the resultant algorithm to casting processes.     

Sensitivity analysis of viscoplastic deformation process with application to metal preform design optimization

The sensitivity analysis of rigid viscoplastic deformation processes with application to metal preform design optimization is investigated. For viscoplastic constitutive models, the deformation process is path-dependent in nature and thus the sensitivity analysis of the deformation history is formulated in an incremental procedure. To this end, an algorithm is derived on the basis of the time integration scheme used in the primary finite element analysis, where the contact conditions are treated with the penalty method. The discretized equilibrium equations, as well as the time integration equations, are directly differentiated with respect to the design variables. The discrete form of the sensitivity equations is then solved with procedures similar to those used in the direct analysis, where the secant matrix decomposed in the direct analysis can also be utilized at each time instant. Thus the sensitivity of the deformation history is evaluated in a step-wise procedure. The present algorithm can be employed for the optimization of metal forming processes. The accuracy of the proposed sensitivity analysis as well as its applicability are demonstrated by numerical examples with reference to preform design optimization problems, where the aggregate function method is employed for converting the non-smooth Min–max type objective function into a numerically tractable one.     

Computational analysis of PTFE shaft seals

An endochronic viscoplastic approach, derived from the theory of finite viscoplasticity based on material isomorphisms, is presented, in order to describe the nonlinear material behaviour of filled polytetrafluoroethylene (PTFE) in a computational analysis of PTFE shaft seals. The model allows to characterize viscoplastic material behaviour with an equilibrium hysteresis using a rate-independent elastoplastic model (with an endochronic flow rule and a logarithmic elastic law) in parallel connection with a nonlinear Maxwell model. Due to the endochronic flow rule, an elastic range limited by a yield stress is not needed in the present approach. The volumetric stress contribution is assumed to be purely elastic. The proposed model is applied to simulate the mounting process of PTFE shaft seals in an axially symmetric finite element analysis. The numerical results (radial force, pressure in the contact zone) are in fair agreement with the experimental data.     

Limit analysis of viscoplastic flows using an extended general algorithm sequentially: convergence analysis and validation

The paper presents the extension of a general algorithm to limit analysis of viscoplastic flow problems. Based on the concept of sequential limit analysis, the paper treats viscoplastic flow problems as a sequence of limit analysis problems. In the formulation, a general plane-strain problem involving incompressible rigid-viscoplastic materials was stated in the upper bound formulation. The von Mises yield criterion was employed to model the yield behavior. In each step of a deformation sequence, limit load was computed by using a combined smoothing and successive approximation (CSSA) algorithm. Especially, the extended CSSA algorithm was shown to be unconditionally convergent by utilizing the Hölder inequality. Finally, numerical and analytical studies of a thick-walled cylinder under internal pressure were performed to demonstrate the applicability and accuracy of the computational procedure presented here. It is found that the computed limit loads are rigorous upper bounds and agree very well with the analytical solutions.     

Steady-state isothermal flow of a viscoplastic dispersed system in an elbow pipe of circular cross section

The equation of flow of a viscoplastic system in an elbow pipe of noncircular cross section is presented in natural form. Using the equation so derived, an approximate solution is obtained for the motion of a viscoplastic system in an elbow pipe of circular cross section.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 18, No. 6, pp. 1077–1084, June, 1970.     

Sharp V‐notches in viscoplastic solids: Strain energy rate density rule and fracture toughness

Different from Neuber's rule or Glinka's energy method which are always adopted to characterize the notch tip field under elastoplastic condition, in this paper, the strain energy rate density (SERD) rule is used for viscoplastic materials. In particular, based on the definition of generalized notch stress intensity factor (G‐NSIF) for sharp V‐notch in viscoplastic solids, the concept of SERD for sharp V‐notch in viscoplastic solids is presented. Subsequently, by taking as a starting point the SERD, the averaged strain energy density (SED) for sharp V‐notch in viscoplastic solids is derived with integration of time. The fracture toughness relation between sharp V‐notch specimens and crack specimen in viscoplastic materials is given based on the transformation of SERD. A numerical approach is presented to compute the SERD and SED based on finite element method. Some crucial comments on the G‐NSIF have been discussed. Some typical solutions for SERD and SED for sharp V‐notched specimens are investigated.     

Introducing realistic tire–pavement contact stresses into Pavement Analysis using Nonlinear Damage Approach (PANDA)

Realistic tire–pavement interface contact areas and stresses were incorporated into the Pavement Analysis using Nonlinear Damage Approach (PANDA) user interface (PUI). PANDA is a software library developed to simulate the complex thermo-viscoelastic–viscoplastic–viscodamage responses of the pavement to mechanical and environmental loads. The PUI is an interface generating a finite element representation of the pavement within PANDA. The application of realistic tire loading is necessary to calculate accurate pavement responses. The PUI incorporates a database of tire contact areas and stresses obtained from tire finite element simulations. The database includes tire interface characteristics with pavements for various applied loads, tire inflation pressures, vehicle speeds and scenarios of different rolling simulations. A parametric study was conducted to investigate the effect of simulations of tire contact stresses that match field measurements on viscoelastic and viscoplastic pavement responses. Pavement responses are greatly affected using realistic tire loading contact stresses and contact geometry as compared to simplified contact models. The impact on rutting and damage predictions cannot be ignored if reliable projections of pavement performance are to be made. This study confirms the importance of considering realistic three-dimensional contact stresses to design and analyse pavements.     

Measurement of viscoplastic properties of a liquid in experiments with a torsion viscosimeter

The results of a numerical solution of the problem of vibrations of a torsion viscosimeter filled with an incompressible viscoplastic liquid are presented. It has been shown that in the vicinity of the rotation axis there appears a dead zone, whose boundary changes in the process of vibrations. The influence of plastic properties of the liquid on the characteristics of viscosimeter vibrations has been determined. A method for identifying viscoplastic properties by the observed parameters of vibrations is proposed. __________ Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 80, No. 1, pp. 124–127, January–February, 2007.     

Improvement of a frictional contact algorithm for strongly curved contact problems

One of the challenges in contact problems is the prediction of the actual contact surface and the kind of contact that is established in each region. In numerical simulation of deep drawing problems the contact conditions change continuously during the forming process, increasing the importance of a correct evaluation of these parameters at each load step. In this work a new contact search algorithm devoted to contact between a deformable and a rigid body is presented. The rigid body is modelled by parametric Bézier surfaces, whereas the deformable body is discretized with finite elements. The numerical schemes followed rely on a frictional contact algorithm that operates directly on the parametric Bézier surfaces. The algorithm is implemented in the deep drawing implicit finite element code DD3IMP. This code uses a mechanical model that takes into account the large elastoplastic strains and rotations. The Coulomb classical law models the frictional contact problem, which is treated with an augmented Lagrangian approach. A fully implicit algorithm of Newton–Raphson type is used to solve within a single iterative loop the non‐linearities related with the frictional contact problem and the elastoplastic behaviour of the deformable body. The numerical simulations presented demonstrate the performance of the contact search algorithm in an example with complex tools geometry. Copyright © 2003 John Wiley & Sons, Ltd.     

Thermodynamics of flows of a polymerizing,nonlinearly viscoplastic fluid having a free surface

The finite elements method is used as a basis for proposing a numerical algorithm for the solution of the problem of the flow of a polymerizing viscoplastic fluid with a free surface. The effect of the main parameters of the problem on the character of the hydrodynamic process is explained.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 59, No. 5, pp. 764–771, November, 1990.     

Wave propagation and localization problems in saturated viscoplastic geomaterials

This paper presents an improved algorithm to deal with wave propagation and localization problems in saturated viscoplastic geomaterials. It consists of a mixed formulation in terms of effective stress, velocity and pore pressure that uses a fractional step algorithm allowing the use of equal order of interpolation for the three variables and the simplest element such as the linear triangle. The viscoplastic model used is of modified cam‐clay type. Viscoplasticity results in a strong source term that deteriorates the accuracy of the two‐step Taylor–Galerkin algorithm. Therefore a Runge–Kutta splitting scheme has been used to deal with the source terms, resulting in a better accuracy of the method. Copyright © 2006 John Wiley & Sons, Ltd.     

The dynamic indentation of an elastic-viscoplastic work-hardening slab by a rigid punch

A unified theory for elastic-viscoplastic work-hardening materials, which requires neither a yield criterion nor loading or unloading conditions, is implemented to solve two-dimensional dynamic problems. Specifically, the theory is applied to the dynamic indentation by a rigid indenter of a slab made of an elastic-viscoplastic material. The contact area between the indenter and the slab at any time is not known in advance but should be determined from the process of the solution. An iterative numerical procedure is proposed by which the complete solution is determined from the dynamic elastic-viscoplastic equations, the moving mixed boundary conditions, the requirement that the contact normal stresses are compressive and that no interpenetration occurs outside the contact area. The method is applied to the indentation problem of a viscoplastic slab by a long rigid circular cylinder, and by a wedge-shaped die. Comparisons with the corresponding perfectly elastic problems are given.     

On a class of constitutive equations in viscoplasticity: Formulation and computational issues

The viscoplastic constitutive model is formulated based on the existence of the dissipation potential which embodies the notion of the gauge (Minkowski) function of the convex set. A perturbation method is used for a solution of stiff differential equations characterizing the associated problem of evolution. It relies on a discrete formulation of viscoplasticity which results from the regularized version of the principle of maximum plastic dissipation. The operator split methodology and the Newton-Raphson method are used to obtain the numerical solution of the discretized equations of evolution. The consistent tangent modulus is expressed in a closed form as a result of the exact linearization of the discretized evolution equations. For several variants of the flow potential function, including some representative stiff functional forms, numerical tests of the integration algorithm based on iso-error maps are provided. Finally, a numerical example is presented to illustrate the robustness and the effectiveness of the proposed approach.     

EFFICIENT AND ACCURATE EXPLICIT INTEGRATION ALGORITHMS WITH APPLICATION TO VISCOPLASTIC MODELS

Several explicit integration algorithms with self-adaptive time integration strategies are developed and investigated for efficiency and accuracy. These algorithms involve the Runge–Kutta second order, the lower Runge–Kutta method of orders one and two, and the exponential integration method. The algorithms are applied to viscoplastic models put forth by Freed and Verrilli and Bodner and Partom for thermal/mechanical loadings (including tensile, relaxation, and cyclic loadings). The large amount of computations performed showed that, for comparable accuracy, the efficiency of an integration algorithm depends significantly on the type of application (loading). However, in general, for the aforementioned loadings and viscoplastic models, the exponential integration algorithm with the proposed self-adaptive time integration strategy worked more (or comparably) efficiently and accurately than the other integration algorithms. Using this strategy for integrating viscoplastic models may lead to considerable saving in computer time (better efficiency) without adversely affecting the accuracy of the results. This conclusion should encourage the utilization of viscoplastic models in the stress analysis and design of structural components.