A computational model for the finite element analysis of thermoplasticity coupled with ductile damage at finite strains

An updated Lagrangian implicit FEM model for the analysis of large thermo‐mechanically coupled hyperelastic‐viscoplastic deformations of isotropic porous materials is considered. An appropriate framework for constitutive modelling is introduced that includes a stress‐free thermally expanded configuration and a plastically deformed unstressed damaged configuration. A two‐level iterative scheme is employed at each time increment to solve the field equations governing the conservation of momentum (mechanical step) and the conservation of energy (thermal step) for the coupled thermo‐mechanical problem. Exact linearizations for the calculation of the tangent stiffness are performed in each of these solution steps. A fully implicit, thermo‐mechanically coupled and incrementally objective Euler‐backward radial return based map is developed for the time integration of the constitutive equations. The present model is used to analyse a number of benchmark examples including metal forming processes wherein temperature and the accumulated damage play an important role in influencing the deformation mechanism and the nature of the deformed workpiece. Copyright © 1999 John Wiley & Sons, Ltd.     

Multiaxial creep–fatigue under anisothermal conditions

Because creep–fatigue is mainly studied in uniaxial tension, it is shown here how to proceed to perform both experiments and calculations under multiaxial loading and when the temperature varies both in time and space. The constitutive equations used are those of elasto‐visco‐plasticity coupled or not, to damage, with isotropic and kinematic hardening. It is shown that the unified damage law first proposed for ductile failure and then for fatigue may also be applied to multiaxial creep–fatigue interactions with a new expression for the damage threshold. The procedure for the identification of material parameters is described in detail. Finally, it is shown that the uncoupled calculation procedure, where damage is calculated as a post‐processing of an elasto‐visco‐plastic computation, gives satisfactory results in comparison to the fully coupled analysis; the latter being more accurate but very expensive in computer time.     

Fatigue life prediction for butt‐welded joints considering weld‐induced residual stresses and initial damage,relaxation of residual stress,and elasto‐plastic fatigue damage

Fatigue damage of butt‐welded joints is investigated by a damage mechanics method. First, the weld‐induced residual stresses are determined by using a sequentially coupled thermo‐mechanical finite element analysis. The plastic damage of material is then calculated with the use of Lemaitre's plastic damage model. Second, during the subsequent fatigue damage analysis, the residual stresses are superimposed on the fatigue loading, and the weld‐induced plastic damage is considered as the initial damage via an elasto‐plastic fatigue damage model. Finally, the fatigue damage evolution, the relaxation of residual stress, and the fatigue lives of the joints are evaluated using a numerical implementation. The predicted results agree well with the experimental data.     

Fractional step methods for thermo‐mechanical damage analyses at transient elevated temperatures

In the interest of computational efficiency this paper describes the implementation of a coupled thermo‐damage constitutive model into a coupled time‐stepping analysis using fractional step methods. To begin it is demonstrated that a thermo‐damage model can be presented in a thermodynamic framework with the evolution equations satisfying the first and second laws of thermodynamics. The equations of evolution are partitioned in two ways, thus defining two fractional step methods: an isothermal method and an isentropic method. When implemented into a time‐stepping algorithm the isentropic method maintains a precise energy balance for the entire analysis where as the isothermal method can only provide an energy balance at the end of each thermal time step. In addition, a stability analysis shows that the isentropic analysis is unconditionally stable while a isothermal analysis is at best conditionally stable. Simulations of thermal fracture in a restrained specimen under heat show stable growth of damage to failure. Copyright © 2000 John Wiley & Sons, Ltd.     

Meshfree simulation of failure modes in thin cylinders subjected to combined loads of internal pressure and localized heat

This paper focuses on the non‐linear responses in thin cylindrical structures subjected to combined mechanical and thermal loads. The coupling effects of mechanical deformation and temperature in the material are considered through the development of a thermo‐elasto‐viscoplastic constitutive model at finite strain. A meshfree Galerkin approach is used to discretize the weak forms of the energy and momentum equations. Due to the different time scales involved in thermal conduction and failure development, an explicit–implicit time integration scheme is developed to link the time scale differences between the two key mechanisms. We apply the developed approach to the analysis of the failure of cylindrical shell subjected to both heat sources and internal pressure. The numerical results show four different failure modes: dynamic fragmentation, single crack with branch, thermally induced cracks and cracks due to the combined effects of pressure and temperature. These results illustrate the important roles of thermal and mechanical loads with different time scales. Copyright © 2008 John Wiley & Sons, Ltd.     

Viscoplastic plane‐strain analysis of infinite solids using elastic supports

A highly efficient novel Finite Element Boundary Element Method (FEBEM) is proposed for the elasto‐viscoplastic plane‐strain analysis of displacements and stresses in infinite solids. The proposed method takes advantage of both the Finite Element Method (FEM) and the Boundary Element Method (BEM) to achieve higher efficiency and accuracy by using the concept of elastic supports to simulate the effects of unbounded solid mass surrounding the region of interest. The BEM is used to compute the stiffnesses of elastic supports and to estimate the location of the truncation boundary for the finite element model. As compared to the conventional coupled FEBEM, the proposed method has three main computational advantages. Firstly, the symmetrical and highly banded form of the standard finite element stiffness matrix is not disturbed. Secondly, the proposed technique may be implemented simply by using standard codes for elasto‐viscoplastic finite element analysis and elastic boundary element analysis. Thirdly, the yielded zone is approximately located in advance by using the BEM and hence, an unnecessarily large extent of the domain does not have to be discretized for the finite element modelling. The efficiency and accuracy of the proposed method are demonstrated by computing elastic and elasto‐plastic displacements and stresses around ‘deep’ underground openings in rock mass subject to hydrostatic and non‐hydrostatic in situ stresses. Results obtained by the proposed method are compared with ‘exact’ solutions and with those obtained by using a BEM and a coupled FEBEM. Copyright © 1999 John Wiley & Sons, Ltd.     

Energy–entropy–momentum integration schemes for general discrete non‐smooth dissipative problems in thermomechanics

We present the theory of novel time‐stepping algorithms for general nonlinear, non‐smooth, coupled, and thermomechanical problems. The proposed methods are thermodynamically consistent in the sense that their solutions rigorously comply with the two laws of thermodynamics: for isolated systems, they preserve the total energy and the entropy never decreases. Extending previous works on the subject, the newly proposed integrators are applicable to coupled mechanical systems with non‐smooth kinetics and can be formulated in arbitrary variables. The ideas are illustrated with a simple non‐smooth problem: a rheological model for a thermo‐elasto‐plastic material with hardening. Numerical simulations verify the qualitative features of the proposed methods and illustrate their excellent numerical stability, which stems precisely from their ability to preserve the structure of the evolution equations they discretize. Copyright © 2017 John Wiley & Sons, Ltd.     

Low cycle thermo‐mechanical fatigue: damage operator approach

The strain‐life approach is standardized and widely accepted for determining fatigue damage under strain‐controlled low cycle fatigue (LCF) loading. It was first extended to non‐isothermal cases by introducing an equivalent temperature approach (ETA). The paper presents its extension that is the damage operator approach (DOA) enabling online continuous damage calculation for isothermal and non‐isothermal loading with mean stress correction. The cycle closure point, cycle equivalent temperature, threshold temperature and separate rainflow counting obligatory for the ETA are not necessary for the DOA any more. Both approaches are equivalent for the second and subsequent runs of block loading if temperature is constant. However, for non‐isothermal cases, the DOA is within the worst and the best case scenarios of the ETA. The approaches are compared to the simple stress histories and several thermo‐mechanical fatigue (TMF) cycle types.     

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.     

Implicit integration and consistent tangent modulus of a time‐dependent non‐unified constitutive model

This paper describes the implicit integration and consistent tangent modulus of an inelastic constitutive model with transient and steady strain rates, both of which are time‐ and temperature‐dependent; the transient rate is influenced by the evolution of back stress decomposed into parts, while the steady rate depends only on applied stress and temperature. Such a non‐unified model is useful for high‐temperature structural analysis and is practical owing to the ease in determining material constants. The implicit integration is shown to result in two scalar‐valued coupled equations, and the consistent tangent modulus is derived in a quite versatile form by introducing a set of fourth‐rank constitutive parameters into the discretized evolution rule of back stress. The constitutive model is, then, implemented in a finite element program and applied to a lead‐free solder joint analysis. It is demonstrated that the implicit integration is very accurate if the multilinear kinematic hardening model of Ohno and Wang is employed, and that the consistent tangent modulus certainly affords quadratic convergence to the Newton–Raphson iteration in solving nodal force equilibrium equations. Copyright © 2003 John Wiley & Sons, Ltd.     

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.     

Implicit algorithm for finite deformation hypoelastic–viscoplasticity in fcc metals

An implicit objective stress update algorithm is proposed for a hypoelastic–viscoplastic model. A thermal/dynamic yield function, which is derived based on the thermal activation analysis and dislocation interaction mechanisms, is used, along with the Consistency approach and the framework of additive viscoplasticity, in deriving the proposed model for fcc metals. The corotational formulation approach is utilized in developing the proposed model in the finite deformation field. For the case of the Newton–Raphson iteration method, a new expression for the consistent (algorithmic) tangent stiffness matrix of rate‐dependent metals is derived by direct linearization of the stress update algorithm. Finite element simulations are performed by implementing the proposed viscoplasticity constitutive models in the commercial finite element program ABAQUS. Numerical implementation for a simple tensile problem is used for validating the material parameters of the OFHC Copper under low and high strain rates and temperatures. The numerical results of the adiabatic true stress–true strain curves compare very well with the experimental data. The effectiveness of the present approach is tested by studying strain localization in a simple plane strain problem. Results indicate excellent performance of the present framework in describing the strain localization problem and in obtaining mesh‐independent results. Copyright © 2006 John Wiley & Sons, Ltd.     

Topology optimization of finite strain viscoplastic systems under transient loads

A transient finite strain viscoplastic model is implemented in a gradient‐based topology optimization framework to design impact mitigating structures. The model's kinematics relies on the multiplicative split of the deformation gradient, and the constitutive response is based on isotropic hardening viscoplasticity. To solve the mechanical balance laws, the implicit Newmark‐beta method is used together with a total Lagrangian finite element formulation. The optimization problem is regularized using a partial differential equation filter and solved using the method of moving asymptotes. Sensitivities required to solve the optimization problem are derived using the adjoint method. To demonstrate the capability of the algorithm, several protective systems are designed, in which the absorbed viscoplastic energy is maximized. The numerical examples demonstrate that transient finite strain viscoplastic effects can successfully be combined with topology optimization.     

A thermo‐hydro‐mechanical model for multiphase geomaterials in dynamics with application to strain localization simulation

In this paper, the non‐isothermal elasto‐plastic behaviour of multiphase geomaterials in dynamics is investigated with a thermo‐hydro‐mechanical model of porous media. The supporting mathematical model is based on averaging procedures within the hybrid mixture theory. A computationally efficient reduced formulation of the macroscopic balance equations that neglects the relative acceleration of the fluids, and the convective terms is adopted. The modified effective stress state is limited by the Drucker–Prager yield surface. Small strains and dynamic loading conditions are assumed. The standard Galerkin procedure of the finite element method is applied to discretize the governing equations in space, while the generalized Newmark scheme is used for the time discretization. The final non‐linear set of equations is solved by the Newton method with a monolithic approach. Coupled dynamic analyses of strain localization in globally undrained samples of dense and medium dense sands are presented as examples. Vapour pressure below the saturation water pressure (cavitation) develops at localization in case of dense sands, as experimentally observed. A numerical study of the regularization properties of the finite element model is shown and discussed. A non‐isothermal case of incipient strain localization induced by temperature increase where evaporation takes place is also analysed. Copyright © 2016 John Wiley & Sons, Ltd.     

Damage Operator‐Based Lifetime Calculation Under Thermomechanical Fatigue and Creep for Application on Uginox F12T EN 1.4512 Exhaust Downpipes

Abstract:  The article focuses on the application of a recently developed damage operator‐based lifetime calculation to a thermomechanically loaded exhaust downpipe. The damage operator approach enabling online continuous damage calculations for isothermal and non‐isothermal loading with mean stress corrections is reviewed. The article also highlights an extension of the strain‐life approach to take into account viscoplastic effects and creep. The transient results from thermal and structural analyses using finite element analyses have been applied to the exhaust downpipe in LMS Virtual.Lab and the damage predicted. Tested exhaust downpipes were then subjected to the same loading conditions as in the calculation, and load cycles were repeated up to the point of failure. Simulated and test results are comparable.     

Least‐squares mixed finite elements for small strain elasto‐viscoplasticity

The main aim of this contribution is to provide a mixed finite element for small strain elasto‐viscoplastic material behavior based on the least‐squares method. The L₂‐norm minimization of the residuals of the given first‐order system of differential equations leads to a two‐field functional with displacements and stresses as process variables. For the continuous approximation of the stresses, lowest‐order Raviart–Thomas elements are used, whereas for the displacements, standard conforming elements are employed. It is shown that the non‐linear least‐squares functional provides an a posteriori error estimator, which establishes ellipticity of the proposed variational approach. Further on, details about the implementation of the least‐squares mixed finite elements are given and some numerical examples are presented. Copyright © 2008 John Wiley & Sons, Ltd.     

A new rigid‐viscoplastic model for simulating thermal strain effects in metal‐forming processes

A new rigid‐viscoplastic model that includes the effect of thermal strains when modelling steady‐state metal‐forming processes was developed. A symmetric approximation to the resulting non‐symmetric stiffness matrix was derived. The thermo‐mechanical flow formulation was implemented using the pseudo‐concentrations technique. The new formulation was numerically tested showing that it provides reliable results. Copyright © 2003 John Wiley & Sons, Ltd.     

Strain-life approach in thermo-mechanical fatigue evaluation of complex structures

This paper is a contribution to strain‐life approach evaluation of thermo‐mechanically loaded structures. It takes into consideration the uncoupling of stress and damage evaluation and has the option of importing non‐linear or linear stress results from finite element analysis (FEA). The multiaxiality is considered with the signed von Mises method. In the developed Damage Calculation Program (DCP) local temperature‐stress‐strain behaviour is modelled with an operator of the Prandtl type and damage is estimated by use of the strain‐life approach and Skelton's energy criterion. Material data were obtained from standard isothermal strain‐controlled low cycle fatigue (LCF) tests, with linear parameter interpolation or piecewise cubic Hermite interpolation being used to estimate values at unmeasured temperature points. The model is shown with examples of constant temperature loading and random force‐temperature history. Additional research was done regarding the temperature dependency of the K_p used in the Neuber approximate formula for stress‐strain estimation from linear FEA results. The proposed model enables computationally fast thermo‐mechanical fatigue (TMF) damage estimations for random load and temperature histories.