Non-linear creep-fatigue interaction based on a viscoplastic law coupled with damage

The purpose of this work is to develop an approach in crack initiation assessment in components operating at high temperature. Components of fossil power plants, such as rotors, casings or headers, have to face the combination of highly damaging processes due to the operating temperatures and the high numbers of start-ups and shut-downs. The processes involved are indeed creep, thermal fatigue and creep-fatigue interaction. Our approach is based on viscoplastic constitutive equations coupled with a three dimensional law of continuum damage evolution. These laws use a formalism based on the thermodynamic of irreversible processes. Our goal is to represent the crack initiation of structures loaded with low cycles thermal loads more accurately. To identify the different constitutive coefficients, we have chosen to study a P22 (2,25CrMo) steel from a retired header of one of our coal fired plants. The methodology of identification is given and the influence of each variable is discussed. We focus our interest on the creep-fatigue interaction which is a major parameter in the estimation of residual life of low used fossil power plant. A numerical simulation is given, which shows the necessity to use viscoplastic constitutive equations including a kinematic hardening, to take into account the non-linear aspect of the creep-fatigue interaction. Then, we detail an example of a finite elements calculation. The hypothesis and the particularities of this calculation are explained. The results are analysed and linked to experimental datas. We expect to apply soon those laws to various industrial materials and structures such as turbines (casings and rotors) or header to improve the residual life estimation of our plants.     

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.     

高温合金材料时间相关热机械疲劳寿命预测技术

本文在变温非线性运动强化规律所描述的高温合金材料热机械疲劳应力－应变循环特性的基础上,重点讨论了应变控制的时间相关热机械疲劳寿命预测技术。对于温度循环的影响,采用由应变能密度表示的损伤参数,并且引入了温度损伤系数。对于循环时间的影响,引入了蠕变─疲劳相互作用的损伤机制,采用韧性耗散损伤模型。在确定模型的一些参数时,采用等温力学试验和疲劳试验的数据,把等温疲劳研究成果推广到变温疲劳分析领域。     

Analysis on the fatigue damage evolution of notched specimens with consideration of cyclic plasticity

This paper presents a damage mechanics method applied successfully to assess fatigue life of notched specimens with plastic deformation at the notch tip. A damage‐coupled elasto‐plastic constitutive model is employed in which nonlinear kinematic hardening is considered. The accumulated damage is described by a stress‐based damage model and a plastic strain‐based damage model, which depend on the cyclic stress and accumulated plastic strain, respectively. A three‐dimensional finite element implementation of these models is developed to predict the crack initiation life of notched specimens. Two cases, a notched plate under tension‐compression loadings and an SAE notched shaft under bending‐torsion loadings including non‐proportional loadings, are studied and the predicted results are compared with experimental data.     

Evaluating a strain energy fatigue method using cyclic plasticity models

For the development of constitutive equations that describe the behaviour of materials under cyclic plastic strains, different kinds of formulations can be adopted. Recently, an energy‐based fatigue damage parameter has been developed to present energy‐fatigue life curves using a calculation of the total strain energy. In this study, the damage criterion is examined by calculation of the plastic strain energy from stress–strain hysteresis loops in the cyclic plasticity models under condition of multi‐axial fatigue. These cyclic plasticity models are the Garud multi‐surface model and the Chaboche nonlinear kinematic hardening model. The models are briefly explained and the general features of their computational procedure are presented. Then, the hysteresis loops of these models will be obtained and the fatigue lives are predicted and compared to experimental data by the ratio of predicted life to experimental life. Consequently, a weighting factor on shear plastic work is presented to decrease the life factors.     

Efficient iteration schemes for anisotropic hyperelasto‐plasticity

Numerical issues arising when integrating hyperelasto‐plastic constitutive equations with elastic anisotropy, stemming from anisotropic damage, and plastic anisotropy as represented by kinematic hardening are discussed. In particular, solution algorithms for the corresponding non‐linear system of equations due to implicit integration algorithms are addressed. It is shown that algorithms like staggered iteration and quasi‐Newton techniques are superior to a pure Newton technique when the cpu time is compared. However, the drawback of the staggered and the quasi‐Newton technique is that rather small time steps must be taken to ensure convergence, which can be of importance when applying complex constitutive models in a finite element programme. Copyright © 2005 John Wiley & Sons, Ltd.     

Fatigue and ratcheting assessment of AISI H11 at 500°C using constitutive theory coupled with damage rule

Hot extrusion is one of the most commonly used manufacturing methods for metal plastic deformation, and the consumption of extrusion tooling is considerably high due to its fatigue damage under cyclic serving condition. Hot‐work tool steel AISI H11 is one of these typical materials employed in extrusion tooling. This work is dedicated to calculating the stress/strain state of AISI H11 and predicting its lifetime at high temperature 500°C by building a unified constitutive model coupled with Lemaitre's damage law. Tensile tests and strain/stress reversed cycling tests have been conducted at 500°C to investigate mechanical properties and damage evolution. A unified constitutive model with Armstrong‐Fredrick/Ohno‐Wang kinematic hardening rule and a new proposed isotropic hardening rule is built; Lemaitre's damage law is employed as well. Parameters are determined based on tests and are temperature dependent. Finite element simulation of the deformation behaviour and fatigue lifetime is implemented into commercial software ABAQUS Standard v6.14‐2 with user material subroutine to validate the proposed method. The comparison shows good agreement with experimental results, and this part of work is essential and crucial to subsequent structure analysis.     

Implementation of cyclic plasticity models based on a general form of kinematic hardening

This paper deals with implementation of cyclic plastic constitutive models in which a general form of strain hardening and dynamic recovery is employed to represent the multilinear, as well as non‐linear, evolution of back stress. First, in order to incorporate such a general form of kinematic hardening in finite element methods, successive substitution and its convergence are discussed for implicitly integrating stress; moreover, a new expression of consistent tangent modulus is derived by introducing a set of fourth‐rank constitutive parameters into discretized kinematic hardening. Then, the constitutive parameters introduced are specified in three cases of the general form of kinematic hardening; the three cases have distinct capabilities of simulating ratcheting and cyclic stress relaxation. Numerical examples are given to verify the convergence in successive substitution and the new expression of consistent tangent stiffness. Error maps for implicitly integrating stress under non‐proportional as well as proportional loading are also given to show that the multilinear case of the general form provides high accuracy even if strain increment is very large. Copyright © 2002 John Wiley & Sons, Ltd.     

Localized fracture phenomena in thermo-visco-plastic flow processes under cyclic dynamic loadings

Summary The main objective of the paper is the investigation of localized fatigue fracture phenomena in thermo-viscoplastic flow processes under cyclic dynamic loadings. Recent experimental observations for cycle fatigue damage mechanics at high temperature and dynamic loadings of metals suggest that the intrinsic microdamage process does very much depend on the strain rate and the wave shape effects and is mostly developed in the regions where the plastic deformation is localized. The microdamage kinetics interacts with thermal and load changes to make failure of solids a highly rate, temperature and history dependent, nonlinear process.A general constitutive model of elasto-viscoplastic damaged polycrystalline solids developed within the thermodynamic framework of the rate type covariance structure with a finite set of the internal state variables is used (cf. Dornowski and Perzyna [16], [17], [18]). A set of the internal state variables is assumed and interpreted such that the theory developed takes account of the effects as follows: (i) plastic nonnormality; (ii) plastic strain induced anisotropy (kinematic hardening); (iii) softening generated by microdamage mechanisms (nucleation, growth and coalescence of microcracks); (iv) thermomechanical coupling (thermal plastic softening and thermal expansion); (v) rate sensitivity; (vi) plastic spin.To describe suitably the time and temperature dependent effects observed experimentally and the accumulation of the plastic deformation and damage during a dynamic cyclic loading process the kinetics of microdamage and the kinematic hardening law have been modified. The relaxation time is used as a regularization parameter. By assuming that the relaxation time tends to zero, the rate independent elasticplastic response can be obtained. The viscoplastic regularization procedure assures the stable integration algorithm by using the finite difference method. Particular attention is focussed on the well-posedness of the evolution problem (the initial-boundary value problem) as well as on its numerical solutions. The Lax-Richtmyer equivalence theorem is formulated, and conditions under which this theory is valid are examined. Utilizing the finite difference method for a regularized elasto-viscoplastic model, the numerical investigation of the three-dimensional dynamic adiabatic deformation in a particular body under cyclic loading condition is presented.Particular examples have been considered, namely a dynamic adiabatic cyclic loading process for a thin plate with sharp notch. To the upper edge of the plate is applied a cyclic constraint realized by rigid rotation of the edge of the plate while the lower edge is supported rigidly. A small localized region, distributed asymmetrically near the tip of the notch, which undergoes significant deformation and temperature rise, has been determined. Its evolution until occurrence of fatigue fracture has been simulated.The propagation of the macroscopic fatigue damage crack within the material of the plate is investigated. It has been found that the length of the macroscopic fatigue damage crack distinctly depends on the wave shape of the assumed loading cycle.     

Smooth finite strain plasticity with non‐local pressure support

The aim of this work is to introduce an alternative framework to solve problems of finite strain elastoplasticity including anisotropy and kinematic hardening coupled with any isotropic hyperelastic law. After deriving the constitutive equations and inequalities without any of the customary simplifications, we arrive at a new general elasto‐plastic system. We integrate the elasto‐plastic algebraico‐differential system and replace the loading–unloading condition by a Chen–Mangasarian smooth function to obtain a non‐linear system solved by a trust region method. Despite being non‐standard, this approach is advantageous, since quadratic convergence is always obtained by the non‐linear solver and very large steps can be used with negligible effect in the results. Discretized equilibrium is, in contrast with traditional approaches, smooth and well behaved. In addition, since no return mapping algorithm is used, there is no need to use a predictor. The work follows our previous studies of element technology and highly non‐linear visco‐elasticity. From a general framework, with exact linearization, systematic particularization is made to prototype constitutive models shown as examples. Our element with non‐local pressure support is used. Examples illustrating the generality of the method are presented with excellent results. Copyright © 2009 John Wiley & Sons, Ltd.     

Service-type creep-fatigue experiments with cruciform specimens and modelling of deformation

Advanced material models for the application to component life prediction require multiaxial experiments. A biaxial testing system for cruciform test pieces has been established in order to provide data for creep, creep-fatigue and thermomechanical fatigue (TMF) experiments. For this purpose a cruciform specimen was developed with the aid of Finite element calculation and the specimen design was optimised for tension and compression load. The testing system is suitable for strain (displacement) and load control mode. A key feature deals with the opportunity to perform thermomechanical experiments. Further, a constitutive material model is introduced which is implemented as a user subroutine for Finite element applications. The constitutive material model of type Chaboche considers both isotropic as well as kinematic hardening and isotropic damage. Identification of material parameters is achieved by a combination of Neural networks and subsequent Nelder–Mead Method.     

A nonlinear CDM model for ductile failure analysis of steel bridge columns under cyclic loading

A nonlinear cyclic plasticity damage model for ductile metals, which is able to take large deformation effects into consideration, has been developed using a new damage dissipation potential formulation in order to predict the cyclic inelastic behavior of steel bridge piers. The cyclic constitutive equations that employ the combined isotropic–kinematic hardening rule for plastic deformation is incorporated into the damage mechanics in conjunction with the large strain formulation. The damage growth law is based on the experimental observations that the evolution of microvoids results in nonlinear damage accumulation with plastic deformation. The damage model parameters and the procedure for their identification are presented. The proposed model has been validated and successfully applied to thin-walled steel bridge tubular columns subjected to alternating lateral displacements to evaluate the seismic performance.     

Predicting damage and failure under low cycle fatigue in a 9Cr steel

Experimental data have been generated and finite element models developed to examine the low cycle fatigue (LCF) life of a 9Cr (FB2) steel. A novel approach, employing a local ductile damage initiation and failure model, using the hysteresis total stress–strain energy concept combined with element removal, has been employed to predict the failure in the experimental tests. The 9Cr steel was found to exhibit both cyclic softening and nonlinear kinematic hardening behaviour. The finite element analysis of the material's cyclic loading was based on a nonlinear kinematic hardening criterion using the Chaboche constitutive equations. The models’ parameters were calibrated using the experimental test data available. The cyclic softening model in conjunction with the progressive damage evolution model successfully predicted the deformation behaviour and failure times of the experimental tests for the 9Cr steels performed.     

Classification of metallic alloys for fatigue damage accumulation: A conservative model under strain control for 304 stainless steels

Fully reversed uniaxial tests performed under total strain and stress control on 304 stainless steels specimens show that, under strain control the fatigue damage for High–Low (H–L) cycling is more significant than that using Miner’s rule, but under stress control opposite results are obtained. This has been attributed to opposite effects of pre-hardening under strain and stress control. Classical non linear damage accumulation models are not able to take into account this difference in sequence effect. Smith–Watson–Topper (SWT) and Fatemi–Socie (FS) criterion combined to linear damage accumulation can take into account this difference in sequence effect through the presence of maximum stress. However these models require an elastic–plastic constitutive law which is difficult to propose due to the presence of high cycle secondary hardening observed on 304 stainless steel. A conservative model for damage accumulation under variable amplitude strain control loading is thus proposed, which does not require a constitutive law. Linear damage accumulation is used, while sequence effect is taken into account using the elastic–plastic memory effect through cyclic strain–stress curves (CSSC) with pre-hardening. This modeling classifies metallic alloys in two groups for damage accumulation, with a stable (independent to pre-hardening) CSSC as for aluminum alloys and with an unstable (dependent to pre-hardening) one as for austenitic stainless steels. For the former case the modeling is identical to Miner’s rule. The modeling is approved based on a large number of tests on 304 stainless steel and is compared with SWT and FS models. In presence of mean stress the modeling permits in a qualitative way to explain the fact that tensile mean stresses in constant amplitude strain control tests are more detrimental than for constant amplitude stress control tests. Moreover it is shown that the SWT model is not always able to predict accurately the fatigue life in presence of a mean stress. Finally, it is concluded that for a 304 stainless steel, in order to take into account the mean stress in fatigue life, the mean stress effect has to be decomposed into two parts: maximum and “intrinsic” mean stress effects.     

Low-cycle fatigue model of damage accumulation - The strain approach

The paper presents a model of damage accumulation designed to analyse fatigue life of structural elements exploited in multiaxial, non-proportional, low-cycle loading conditions. The discussed approach consists of two calculation blocs. In the first bloc the components of stress and strain tensor are determined. This module, in which Mroz’s multisurface model was used, contains constitutive relations and the law of kinematic hardening. The second bloc contains the dependencies which determine the growth of anisotropic measure of damage accumulation (associated with the physical plane) and crack initiation criterion. The growth of the damage accumulation measure was associated with the loading damage accumulation function and the increment of non-dilatational plastic strain on the physical plane. It was assumed that crack initiation occurs when stress or a measure of damage accumulation on any physical plane reaches critical values.     

A high-cycle fatigue life model for variable amplitude multiaxial loading

A high‐cycle fatigue life model for structures subjected to variable amplitude multiaxial loading is presented in this paper. It treats any kind of repeated blocks of variable amplitude multiaxial loading without using a cycle counting method. This model based on a mesoscopic approach is characterized by the following features: (i) the choice of a damage factor related to the accumulated mesoscopic plastic strain per stabilised cycle; (ii) the use of a mesoscopic mechanical behaviour taking into account the fatigue mechanisms such as plasticity and void growth. This behaviour is a von Mises elastoplastic model with linear kinematic hardening and hydrostatic stress dependent yield stress. The fatigue life model has six parameters identified with one SN curve and two fatigue limits. In‐phase and out‐of‐phase experimental tests from the literature are simulated. The predicted fatigue lives are compared to experimental ones.     

A unified theory of elasticity and plasticity

It is shown that a certain set of rate theory type constitutive equations can give rise to both elastic behavior and plastic yield in infinitesmal strain both in loading and unloading. Elastic behavior corresponds to uniqueness of solutions of initial value problems of differential equations and holds when the equations satisfy a Lipschitz condition. This Lipschitz condition fails when a von Mises yield condition holds, and the corresponding nonuniqueness gives rise to plastic yield during loading. During plastic yield, the basic constitutive relations automatically turn into the Prandtl-Reuss equations. Upon unloading from yield, the stable solution of the equations dictates reentry into an elastic regime. The transition to and from plastic yield is exact and sudden, not asymptotic, but the transition is smooth. It is shown that linear behavior in the elastic regimes can be approximated arbitrarily closely. The theory can be extended to include strain hardening.     

Automatic consistency integrations for the stress update of J2 elastoplastic rate constitutive equations with combined hardening

A novel method is introduced to study numerical integrations of J₂ elastoplastic rate constitutive equations with general combined hardening. The basic idea is to transform the usual time rate constitutive equations into those with reference to the equivalent plastic strain. By virtue of tensorial matrix operations, we show that these transformed equations may be converted to a linear differential system governing the shifted stress and the plastic multiplier. From this system, we derive explicit integrations for the shifted stress and then for the back stress and the Cauchy stress. We demonstrate that these results are accurate up to within a third order term of the equivalent plastic strain increment. In particular, for pure kinematic hardening, we show that the integrations obtained can achieve automatic enforcement of both the plastic consistency condition and the loading condition, thus bypassing the numerical treatment of the latter two. Furthermore, we explain that, with the new algorithm for the stress update, the continuum tangent moduli may be used to ensure a quadratic rate of convergency in Newton's iteration scheme for the balance equation. Numerical examples suggest that the new algorithm may be more accurate and efficient than the widely used return algorithm. Copyright © 2014 John Wiley & Sons, Ltd.