An implicit time integration scheme for inelastic constitutive equations with internal state variables

In a broad class of inelastic constitutive models for the deformation of metals the inelastic strain rates are functions of the current state of stress and internal state variables only. All known models are in some regions of application mathematically stiff and therefore difficult to integrate. The unconditionally stable implicit Euler rule is used for integration. It leads to a system of highly nonlinear algebraic equations which have to be solved by an iterative process. The general Newton-Raphson method, which converges under very broad conditions, requires repeated solution of the finite element system and is infeasible for large inelastic problems. But for the inelastic strains and internal state variables the Jacobian can be computed analytically and therefore the NRI can be used. For the stresses the Jacobian cannot be computed analytically and therefore the accelerated Jacobi iteration is used. A new method for computing the relaxation parameter is introduced which increases the rate of convergence significantly. The new algorithm is applied on Hart's model. A comparison with prior computations using an approximation is made.     

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.     

An integration scheme for Prandtl-Reuss elastoplastic constitutive equations

We investigate the Generalized Midpoint Rule for the time integration of elastoplastic constitutive equations for pressure-independent yield criteria. The incremental equations are divided into one scalar hydrostatic pressure/dilation rate equation, and a stress deviator/strain rate deviator tensorial equation, the solution of which reduces to one single scalar equation in the plastic multiplier. The existence and uniqueness of an incremental solution is discussed. The pressure/deviator decomposition is the basis for reduced integration of the pressure term in the Principle of Virtual Work, in order to avoid locking and spurious pressure oscillations. It is also shown that an optimal choice of the parameter of the Midpoint Rule can be computed by reference to the analytical solution of the equations assuming no work hardening. A benchmark test shows that this choice allows increased time steps. This formulation is applied to two classical problems: bulging of a tube under internal pressure and tension test on a notched specimen, and a comparison with the analytical solution is performed. Finally, the hypothesis which sustains these formulations of elastoplasticity (constant strain rate during an increment) is discussed with reference to elastic unloading and residual stress computation.     

An experimental method to validate viscoplastic constitutive equations in the dynamic response of plates

In the present paper measured and simulated vibrations of viscoplastic plates under impulsive loadings are compared to each other. The aim is to determine how accurately the measured deformations can be calculated by the chosen constitutive and structural theories. A first-order shear deformation shell theory assuming small strains and moderate rotations as well as viscoplastic laws are used. In the experimental part of this study short time measurement techniques are applied to shock tubes in order to record fast loading processes and plate deformations.     

An implicit unitless error and step‐size control method in integrating unified viscoplastic/creep ODE‐type constitutive equations

A series of numerical analyses are carried out to investigate the difficulties in numerical integration of unified viscoplastic/creep constitutive equations, which are normally represented as a system of ordinary differential equations (ODEs). The problems of numerically integrating the constitutive equations are identified and analysed. To overcome the stiffness problems, implicit methods are used for the numerical integration and a generic technique is introduced to calculate the Jacobian matrix. A normalization technique is introduced in the paper to convert the integration errors for each time increment to unitless errors. Thus, a single tolerance can be used to control the accuracy and step size in integrating a set of unified viscoplastic/creep constitutive equations. In addition, an implicit step‐size control method is proposed and used in the integrations. This method reduces the possibility of rejection of an integration increment due to poor accuracy. Copyright © 2007 John Wiley & Sons, Ltd.     

Nonlinear viscoelastic and viscoplastic constitutive equations with growing damage

Nonequilibrium thermodynamics, rate-process theory, viscoelastic fracture mechanics and various experimentally-motivated simplifications are used to develop constitutive equations that account for effects of viscoelasticity, viscoplasticity, growing damage and aging. Their form is more general than previously developed by the author, and allows for relatively general tensorial effects of damage. Some important special cases are then covered, with emphasis on viscoelasticity. Evolution equations for the damage expressed in terms of internal state variables (ISVs) are discussed, comparing formulations using scalar ISVs and tensor ISVs. Finally, some experimental support for the theory is described. An Appendix illustrates the theory for an aging, linear viscoelastic material with growing cracks. This revised version was published online in July 2006 with corrections to the Cover Date.     

On time integration of viscoplastic constitutive models suitable for creep

Creep of critical components such as electrical solder connections may occur over long periods of time. Efficient numerical simulations of such problems generally require the use of quasi‐static formulations with conjugate‐gradient techniques for solving the large number of algebraic equations. Implicit in the approach is the need to solve the constitutive equation several times for large time steps and for loading directions that may have no resemblance to the actual solution. Therefore, an unconditionally stable and efficient algorithm for solving the constitutive equation is essential for the overall efficiency of the solution procedure. Unfortunately, constitutive equations suitable for simulating the materials of interest are notoriously difficult to solve numerically and most existing algorithms have a stability limit on the time step which may be several orders of magnitude smaller than the desired time step. Here an algorithm is proposed which is a combination of the use of a trapezoidal rule and an iterative Newton–Raphson method for solving implicitly the non‐linear equations. The key to the success of the proposed approach is to always use an initial guess based on the steady‐state solution to the constitutive equation. A representative viscoplastic constitutive equation is used as a model for illustrating the approach. The algorithm is developed and typical numerical results are provided to substantiate the claim that stability has been achieved. Copyright © 2001 John Wiley & Sons, Ltd.     

Effective constitutive equations for fiber-reinforced viscoplastic composites exhibiting anisotropic hardening

Effective elastic-viscoplastic stress-strain relations are derived for fiber-reinforced composites whose constituents are elastic-viscoplastic materials displaying anisotropic hardening. The derivation is based on a recently developed high-order continuum theory with microstructure for the modeling of viscoplastic composites, and is generalized here to incorporate anisotropic hardening effects. A specific reduction of the theory gives the effective rate-dependent elastic-plastic behavior of the composite which exhibits plastic anisotropy. In the special case of perfectly elastic constituents, the approximate overall moduli of the fiber-reinforced composite are obtained. Rate-dependent average stress-strain curves are given for numerous modes of cyclic loading of the composite. The effective behavior of periodically bilaminated viscoplastic composites is determined as a special case.     

An implicit multi-time step integration method for structural dynamics problems

A multi-time step integration algorithm is developed based on the trapezoidal rule time integration method for finite element equations of motion. This algorithm uses nodal groups to partition the mesh into subdomains that are updated with different time steps. A␣stability analysis of the method shows that the scheme retains the unconditionally stable behavior of the trapezoidal rule and conserves the same pseudo energy as the parent algorithm. Several numerical examples are used to verify the stability of the method and to investigate the accuracy of the scheme.     

An efficient method for solving implicit and explicit stiff differential equations

When the s‐stage fully implicit Runge–Kutta (RK) method is used to solve a system of n ordinary differential equations (ODE) the resulting algebraic system has a dimension ns. Its solution by Gauss elimination is expensive and requires 2s³n³/3 operations. In this paper we present an efficient algorithm, which differs from the traditional RK method. The formal procedure for uncoupling the algebraic system into a block‐diagonal matrix with s blocks of size n is derived for any s. Its solution is s²/2 times faster than the original, nondiagonalized system, for s even, and s³/(s−1) for s odd in terms of number of multiplications, as well as s² times in terms of number of additions/multiplications. In particular, for s=3 the method has the same precision and stability properties as the well‐known RK‐based RadauIIA quadrature of Ehle, implemented by Hairer and Wanner in RADAU5 algorithm. Unlike RADAU5, however, the method is applicable with any s and not only to the explicit ODEs My′=f(x, y), where M=const., but also to the general implicit ODEs of the form f(x, y, y′)=0. The block‐diagonal form of the algebraic system allows parallel processing. The algorithm formally differs from the implicit RK methods in that the solution for y is assumed to have a form of the algebraic polynomial whose coefficients are found by enforcing y to satisfy the differential equation at the collocation points. Locations of those points are found from the derived stability function such as to guarantee either A‐ or L‐stability properties as well as a superior precision of the algorithm. If constructed such as to be L‐stable the method is a good candidate for solving differential‐algebraic equations (DAEs). Although not limited to any specific field, the application of the method is illustrated by its implementation in the multibody dynamics described by both ODEs and DAEs. Copyright © 2000 John Wiley & Sons, Ltd.     

Mixed finite element method for generalized convection-diffusion equations based on an implicit characteristic-based algorithm

Summary A mixed finite element method for generalized convection-diffusion equations is proposed. The primitive variable with its spatial gradient and the diffusion flux are interpolated as independent variables. The variational (weak) form of the governing equations is given on the basis of the extended Hu-Washizu three-field variational principle. The mixed element is formulated with stabilized one point quadrature scheme and particularly implicit characteristic-based algorithm for eliminating spurious numerical oscillations. The numerical results illustrate good performances in accuracy and efficiency of the proposed mixed element in comparison with standard finite element.     

A viscoplastic constitutive model for dynamic fracture

The use of a viscoplastic approach to dynamic fracture prediction is proposed with the aim of producing a model which is applicable to ductile situations. A numerical (finite-element) approach is adopted with specially developed joint elements being used to simulate behaviour within the fracture-process zone. Results are presented for an expanding edge-crack problem. Experimental results are used to calibrate the numerical model by determining the material-specimen parameters under dynamic-fracturing conditions. Detailed results are presented for both LEFM and nonlinear fracture-mechanics approaches.     

An improved unified viscoplastic constitutive model for strain-rate sensitivity in high temperature fatigue

An improved unified cyclic viscoplastic material model for high temperature fatigue of P91 steel is presented. The primary enhancement over existing models is in relation to strain-rate independence of parameters, for accurate interpolation and extrapolation across a range of strain-rates and stress regimes, as relevant to flexible operation of high temperature power generation plant. The model combines a hyperbolic sine constitutive equation with anisothermal cyclic evolution of isotropic and kinematic hardening variables. The material model is developed from a thermodynamic framework and is implemented in multi-axial form within a user material subroutine. The user material subroutine is calibrated and validated for P91 steel across a range of cyclic (isothermal fatigue and thermo-mechanical fatigue) and non-cyclic high temperature loading conditions. A novel method for the identification of the cyclic viscoplastic material parameters is also presented.     

On the implicit integration of rate‐dependent inelastic constitutive models

Although there are many different algorithms for the integration of inelastic constitutive models, the fully implicit backward Euler method has become the most popular one. In this study further investigations on the accuracy of the backward Euler method have been carried out. Also the performance of the discontinuous Galerkin family and some implicit Runge–Kutta time integrators is evaluated. By using a simple scalar model problem accuracy of some integrators is studied when a single finite time step is applied. Conclusions drawn from this scalar model problem has been verified to apply also to a full six‐dimensional strain space formulation by numerical means. Special emphasis is placed on rate‐dependent inelastic creep models. Copyright © 2005 John Wiley & Sons, Ltd.     

A family of integration algorithms for constitutive equations in finite deformation elasto-viscoplasticity

A two parameter family of incrementally objective integration schemes is proposed for the analysis of a broad range of unified rate-dependent viscoplastic constitutive models in large deformation problems. A similar scheme can be applied to rate-independent solids as well. These algorithms are a generalization of the mid-point integration rule. Full linearization of the principle of virtual work is performed in an updated Lagrangian framework together with a calculation of the consistent linearized moduli. Some details of the finite element implementation are given for plane strain and axisymmetric problems. The method is compared with other objective integration schemes and is tested with several examples where large strains and rotations occur.     

Questioning numerical integration methods for microsphere (and microplane) constitutive equations

In the last few years, more and more complex microsphere models have been proposed to predict the mechanical response of various polymers. Similarly than for microplane models, they consist in deriving a one-dimensional force vs. stretch equation and to integrate it over the unit sphere to obtain a three-dimensional constitutive equation. In this context, the focus of authors is laid on the physics of the one-dimensional relationship, but in most of the case the influence of the integration method on the prediction is not investigated.Here we compare three numerical integration schemes: a classical Gaussian scheme, a method based on a regular geometric meshing of the sphere, and an approach based on spherical harmonics. Depending on the method, the number of integration points may vary from 4 to 983,040! Considering simple quantities, i.e. principal (large) strain invariants, it is shown that the integration method must be carefully chosen. Depending on the quantities retained to described the one-dimensional equation and the required error, the performances of the three methods are discussed. Consequences on stress–strain prediction are illustrated with a directional version of the classical Mooney–Rivlin hyperelastic model. Finally, the paper closes with some advices for the development of new microsphere constitutive equations.     

An efficient algorithm for calculating the cutter location point based on projection method

Aiming at improving the efficiency of calculating the cutter location point with toroid cutter based on the projection method in NC machining for surface, a new algorithm is proposed to calculate the cutter location point directly by torus surface approximating the surface to be machined. According to the geometric information of the points on the surface, the geometrical conditions of the two tangential tori are figured out, and then the contact point is obtained by solving multivariate non-linear equations. Parameters of the tangent point on the surface to be machined are calculated in the next step. Finally, the cutter location point is calculated by a small adjustment. The proposed algorithm is applied to calculate the cutter location point with toroid cutter in surface machining and compared with the existing algorithm. The results show that the computing time of the proposed algorithm in this paper saved about 63–78%.