A model of continuum damage mechanics for high cycle fatigue of metallic materials

A non-linear continuum damage model was presented based on the irreversible thermodynamics framework developed by LEMAITRE and CHABOCHE. The proposed model was formulated by taking into account the influence of loading frequency on fatigue life. The parameters H and c are constants for frequency-independent materials, but functions of cyclic frequency for frequency-dependent materials. In addition, the expression of the model was discussed in detail at different stress ratios （R）. Fatigue test data of AlZnMgCu1.5 aluminium alloy and AMg6N alloy were used to verify the proposed model. The results showed that the model possesses a good ability of predicting fatigue life at different loading frequencies and stress ratios.

Aero-engine Blade Fatigue Analysis Based on Nonlinear Continuum Damage Model Using Neural Networks

Fatigue life and reliability of aero-engine blade are always of important significance to flight safety.The establishment of damage model is one of the key factors in blade fatigue research.Conventional linear Miner＇s sum method is not suitable for aero-engine because of its low accuracy.A back propagation neutral network（BPNN） based on the combination of Levenberg-Marquardt（LM） and finite element method（FEM） is used to describe process of nonlinear damage accumulation behavior in material and predict fatigue life of the blade.Fatigue tests of standard specimen made from TC4 are carried out to obtain material fatigue parameters and S-N curve.A nonlinear continuum damage model（CDM）,based on the BPNN with one hidden layer and ten neurons,is built to investigate the nonlinear damage accumulation behavior,in which the results from the tests are used as training set.Comparing with linear models and previous nonlinear models,BPNN has the lowest calculation error in full load range.It has significant accuracy when the load is below 500 MPa.Especially,when the load is 350 MPa,the calculation error of the BPNN is only 0.4%.The accurate model of the blade is built by using 3D coordinate measurement technology.The loading cycle in fatigue analysis is defined from takeoff to cruise in 10 min,and the load history is obtained from finite element analysis（FEA）.Then the fatigue life of the compressor blade is predicted by using the BPNN model.The final fatigue life of the aero-engine blade is 6.55 104 cycles（10 916 h） based on the BPNN model,which is effective for the virtual design of aero-engine blade.

New approach based on continuum damage mechanics with simple parameter identification to fretting fatigue life prediction

A new continuum damage mechanics model for fretting fatigue life prediction is established. In this model, the damage evolution rate is described by two kinds of quantities. One is associated with the cyclic stress characteristics obtained by the finite element （FE） analysis, and the other is associated with the material fatigue property identified from the fatigue test data of standard specimens. The wear is modeled by the energy wear law to simulate the contact geometry evolution. A two-dimensional （2D） plane strain FE implementation of the damage mechanics model and the energy wear model is presented in the platform of ABAQUS to simulate the evolutions of the fatigue damage and the wear scar. The effect of the specimen thickness is also investigated. The predicted results of the crack initiation site and the fretting fatigue life agree well with available experimental data. Comparisons are made with the critical plane Smith- Watson-Topper （SWT） method.

A continuum damage model for piezoelectric materials

In this paper, a constitutive model is proposed for piezoelectric material solids containing distributed cracks. The model is formulated in a framework of continuum damage mechanics using second rank tensors as internal variables. The Helrnhotlz free energy of piezoelectric mate- rials with damage is then expressed as a polynomial including the transformed strains, the electric field vector and the tensorial damage variables by using the integrity bases restricted by the initial orthotropic symmetry of the material. By using the Talreja＇s tensor valued internal state damage variables as well as the Helrnhotlz free energy of the piezoelectric material, the constitutive relations of piezoelectric materials with damage are derived. The model is applied to a special case of piezoelectric plate with transverse matrix cracks. With the Kirchhoff hypothesis of plate, the free vibration equations of the piezoelectric rectangular plate considering damage is established. By using Galerkin method, the equations are solved. Numerical results show the effect of the damage on the free vibration of the piezoelectric plate under the close-circuit condition, and the present results are compared with those of the three-dimensional theory.

A CONTINUUM DAMAGE MODEL OF AGING CONCRETE

There is up to now no constitutive model in the current theoriesof CDM that could give a description for the degradation of agingconcrete. The two internal state variable β and ω are intro- ducedin this paper β is called cohesion variable as an additionalkinematic parameter, reflecting the cohesion state among materialparticles. ω is called damage factor for micro-defects such s voids.Then a damage model and a series of constitutive equations aredeveloped on Continuum Mechanics. The model proposed could give avalid description for the whole-course-degradation of aging concretedue to chemical and mechanical actions. Finally, the validity of themodel is evaluated by an example and experimental results.

Ductile Fracture Characterization for Medium Carbon Steel Using Continuum Damage Mechanics

This paper presents the ductility characterization for a medium carbon steel, for two microstructural conditions, that has been evaluated using the continuum damage mechanics theory, as proposed by Kachanov and developed by Lemaitre. Tensile tests were carried out using loading-unloading cycles in order to capture the gradual deterioration of the elastic modulus, which may be linked to the ductile damage increase with increasing plastic strain. The mechanical parameters for the isotropic damage evolution equation were obtained and then used as inputs for a plasticity-damage coupled nu- merical algorithm, validated through numerical simulations of the experimental tensile tests. A comparison between the SAE 1050 steels studied and two carbon steel alloys (obtained from the literature), provided some basic understanding of the influence of the carbon level on the evolution of the damage parameters. An empiric relationship for this set of parameters, which can provide useful data for preliminary studies envisaging prediction of ductile failure in carbon steels, is also presented.

THE ZONAL DISINTEGRATION MECHANISM OF SURROUNDING ROCK AROUND DEEP SPHERICAL TUNNELS UNDER HYDROSTATIC PRESSURE CONDITION: A NON-EUCLIDEAN CONTINUUM DAMAGE MODEL

A new non-Euclidean continuum damage model is proposed to investigate the zonal disintegration phenomenon of the surrounding rocks around deep spherical tunnels under hydrostatic pressure condition as well as the total elastic stress field distributions.The elastic stress fields of the surrounding rocks around deep spherical tunnels under hydrostatic pressure condition axe obtained.If the elastic stresses of the surrounding rocks satisfy the strength criterion of the deep rock masses,the number,size and location of fractured and nonfractured zones are determined.The effect of physico-mechanical parameters of the surrounding rocks on the zonal disintegration phenomenon is studied and numerical computation is carried out.It is found from numerical results that the number,size and location of fractured and non-fractured zones are sensitive to the physico-mechanical parameters of the surrounding rocks.

Fatigue-creep interaction based on continuum damage mechanics for AISI H13 hot work tool steel at elevated temperatures

AISI H13 （4Cr5MoSiV1） is one of the commonly used materials for extrusion tool, and it suffers from fatigue-creep damage during the hot extrusion process. Stress-controlled fatigue and creep-fatigue interaction tests were carried out at 500℃ to investigate its damage evolution. The accumulated plastic strain was selected to define the damage variable due to its clear physical meaning. A new fatigue-creep interaction damage model was proposed on the basis of continuum damage mechanics. A new equivalent impulse density for fatigue-creep tests was proposed to incorporate the holding time effect by transforming creep impulse density into fatigue impulse density. The experimental results indicated that the damage model is able to describe the damage evolution under these working conditions.

Continuum damage mechanics based modeling progressive failure of woven-fabric composite laminate under low velocity impact

A continuum damage mechanics (CDM) meso-model was derived for both intraply and interply progressive failure behaviors of a 2D woven-fabric composite laminate under a transversely low velocity impact.An in-plane anisotropic damage constitutive model of a 2D woven composite ply was derived based on CDM within a thermodynamic framework,an elastic constitutive model with damage for the fibre directions and an elastic-plastic constitutive model with damage for the shear direction.The progressive failure behavior of a 2D woven composite ply is determined by the damage internal variables in different directions with appropriate damage evolution equations.The interface between two adjacent 2D woven composite plies with different ply orientations was modeled by a traction-separation law based interface element.An isotropic damage constitutive law with CDM properties was used for the interface element,and a damage surface which combines stress and fracture mechanics failure criteria was employed to derive the damage initiation and evolution for the mixed-mode delamination of the interface elements.Numerical analysis and experiments were both carried out on a 2D woven glass fibre/epoxy laminate.The simulation results are in agreement with the experimental counterparts,verifying the progressive failure model of a woven composite laminate.The proposed model will enhance the understanding of dynamic deformation and progressive failure behavior of composite laminate structures in the low velocity impact process.

Application of viscoelastic continuum damage approach to predict fatigue performance of Binzhou perpetual pavements

For this study, the Binzhou perpetual pavement test sections constructed in Shandong Province, China, were simulated for long-term fatigue performance using the layered viscoelastic pavement analysis for critical distresses （LVECD） finite element software package. In this framework, asphalt concrete was treated in the context of linear visco- elastic continuum damage theory. A recently developed unified fatigue failure criterion that defined the boundaries of the applicable region of the theory was also incorporated. The mechanistic modeling of the fatigue mechanisms was able to accommodate the complex temperature variations and loading conditions of the field pavements in a rigorous manner. All of the material models were conveniently characterized by dynamic modulus tests and direct tension cyclic fatigue tests in the laboratory using cylindrical specimens. By comparing the obtained damage characteristic curves and failure criteria, it is found that mixtures with small aggregate particle sizes, a dense gradation, and modified asphalt binder tended to exhibit the best fatigue resistance at the material level. The 15 year finite element structural simulation results for all the test sections indicate that fa- tigue performance has a strong dependence on the thickness of the asphalt pavements. Based on the predicted location and severity of the fatigue damage, it is recommended that Sections 1 and 3 of the Binzhou test sections be emoloved for perpetual pavement design.

Continuum damage mechanics for sintered powder metals

Sintered metals are characterized by the high porosity(8%)and voids/micro-cracks in microns.Inelastic behavior of the materials is coupled with micro-crack propagation and coalescence of open voids.In the present work the damage evolution of the sintered iron under multi-axial monotonic loading conditions was investigated experimentally and computationally.The tests indicated that damage of the sintered iron initiated already at a stress level much lower than the macroscopic yield stress.The damage process can be divided into the stress-dominated elastic damage and the plastic damage described by the plastic strain.Based on the uniaxial tensile tests an elastic-plastic continuum damage model was developed which predicts both elastic damage and plastic damage in the sintered iron under general multi-axial monotonic loading conditions.Computational predictions agree with experiments with different multi-axial loading paths.A phenomenological continuum damage model for the sintered metal is developed based on the experimental observations to predict the inelastic behavior and damage process to failure under multi-axial loading conditions.The proposed damage model is experimentally verified under different loading conditions.

Low-cycle fatigue lifetime estimation of Ti–6Al–4V welded joints by a continuum damage mechanics model

The low-cycle fatigue （LCF） behavior of directionally solidified nickel-based superalloy Ti-6A1-4V was studied under bare and electron beam welding condi- tions at room temperature. Results show that： （1） under the same test conditions, all the joints exhibit lower LCF lifetime than Ti-6A1-4V; （2） the failure of welded structures is mainly ascribed to the welding defect. A novel lifetime prediction methodology based on continuum damage mechanics is proposed to predict the lifetime of Ti-6A1-4V and its welded joints.

Fatigue Reliability Analysis of Fiber-Reinforced Laminated Composites by Continuum Damage Mechanics

This paper investigates the reliability of composite laminates with various lay-ups under fatigue loading.The prediction of failure probability of composite laminates subjected to different loads involves many uncertainties associated with mechanical properties,loading,and boundary conditions.Failure in the composite material is truly hard to trace because there are individual faults in each ply,and we face a stochastic process due to the scatter in the mechanical properties.The continuum damage mechanics(CDM),as a powerful approach,is applied to model the damage of fiber,matrix,and fiber/matrix debonding.This method defines criteria for damage detection and determines safe zones.The material constitutive equations are executed using a subroutine inAbaqus.The first-order reliability method and second-order reliability method have been applied to examine the reliability of laminated composites.The results are compared with those of the Monte Carlo simulation.Different composite laminates under different stress levels are considered for the failure probability investigation.The limit state functions and random variables have been determined based on the CDM model.Finally,the effects of the number of cycles,applied stress,and stacking sequence of the laminate on the reliability and fatigue life in fiber-reinforced laminated composites are assessed.

Creep life assessment of aero-engine recuperator based on continuum damage mechanics approach

The creep life of an aeroengine recuperator is investigated in terms of continuum damage mechanics by using finite element simulations.The effects of the manifold wall thickness and creep properties of brazing filler metal on the operating life of the recuperator are analyzed.Results show that the crack initiates from the brazing filler metal located on the outer surface of the manifold with the wall thickness of 2 mm and propagates throughout the whole region of the brazing filler metal when the creep time reaches 34900 h.The creep life of the recuperator meets the requirement of 40000 h continuous operation when the wall thickness increases to 3.5 mm,but its total weight increases by 15%.Decreasing the minimum creep strain rate with the enhancement of the creep strength of the brazing filler metal presents an obvious effect on the creep life of the recuperator.At the same stress level,the creep rupture time of the recuperator is enhanced by 13 times if the mismatch between the minimum creep rate of the filler and base metal is reduced by 20%.

Damage evolution model of strain hardening cementitious composites under the uniaxial stress state

The deformation and damage behaviors of strain hardening cementitious composites （SHCC） under the uniaxial stress state were investigated in this paper. Two ductile failure-based constitutive models were introduced to describe the uniaxial tension and compression properties of SHCC only using a few parameters. The computation method of model parameters was developed to ease the simulation procedures. Damage evolution of the SHCC was simulated by the formulation of continuum damage mechanics subsequently. The results show that the proposed models fit the stress-strain curves reasonably well, and the damage variables show different growth rules under uniaxial tension and compression. It is concluded that the proposed method can not only simply simulate the constitutive behavior of SHCC with the reasonable accuracy but also capture the characteristic of material degradation.

Revised damage evolution equation for high cycle fatigue life prediction of aluminum alloy LC4 under uniaxial loading

The fatigue life prediction for components is a difficult task since many factors can affect the final fatigue life. Based on the damage evolution equation of Lemaitre and Desmorat, a revised two-scale damage evolution equation for high cycle fatigue is presented according to the experimental data, in which factors such as the stress amplitude and mean stress are taken into account. Then, a method is proposed to obtain the material parameters of the revised equation from the present fatigue experimental data. Finally, with the utilization of the ANSYS parametric design language （APDL） on the ANSYS platform, the coupling effect between the fatigue damage of materials and the stress distribution in structures is taken into account, and the fatigue life of specimens is predicted. The outcome shows that the numerical prediction is in accord with the experimental results, indicating that the revised two-scale damage evolution model can be well applied for the high cycle fatigue life prediction under uniaxial loading.

A fatigue damage model for asphalt mixtures under controlled-stress and controlled-strain modes

A fatigue damage model based on thermodynamics was deduced for asphalt mixtures under controlled-stress and controlled-strain modes. By employing modulus of resilience as the damage hardening variable, a damage variable related with dynamic modulus was extracted as the evaluation index. Then, the damage evolution law under two control modes was proposed, and it has a similar form to the Chaboche fatigue model with a nonnegative material parameter m related to its loading level. Experimental data of four loading levels were employed to calibrate the model and identify the parameter in both control modes. It is found that the parameter m shows an exponential relationship with its loading level. Besides, the difference of damage evolution under two control modes was explained by the law. The damage evolves from fast to slow under a controlled-strain mode. However, under a controlled-stress mode, the evolution rate is just the opposite. By using the damage equivalence principle to calculate the equivalent cycle numbers, the deduced model also interprets the difference of damage evolution under two control modes on the condition of multilevel loading. Under a controlled-strain mode, a loading sequence from a low level to a high level accelerates damage evolution. An inverse order under the controlled-stress mode can prolong fatigue life.

Effect of low cycle fatigue pre-damage on very high cycle fatigue

Carbon-manganese steel is often applied in components of pipes in nuclear plant. Ultrasonic fatigue tests following low cycle fatigue （LCF） cycles damaged are used to study the strength of very high cycle fatigure （VHCF）. The comparison of test results of simple VHCF and cumulative fatigue （LCF plus VHCF） shows that LCF load influences the following VHCF strength. Continuum damage mechanics model is extended to VHCF region.

Physico-mechanical properties of granite after cyclic thermal shock

Understanding rock mechanical behaviors after thermal shock is critically important for practical engineering application.In this context,physico-mechanical properties of Beishan granite,Gansu Province,China after cyclic thermal shock were studied using digital image correlation(DIC),acoustic emission(AE)monitoring,and microscopic observation.The results show that the peak strength and elastic modulus decreased gradually with increase in thermal shock cycle.However,the above two parameters showed no further changes after 10 thermal shock cycles.The loading stress ratio(i.e.the ratio of the current loading stress level to the peak stress in this state)corresponding to the occurrence of the uneven principal strain field and the local strain concentration zone on the surface of the granite specimen decreased with increase in thermal shock cycle.Three transformation forms of the standard deviation curves of the surface principal strain were found.For granite with fewer thermal shock cycles(e.g.no more than 2 cycles),the standard deviation curves exhibited approximately exponential growth in exponential form.With increase in thermal shock cycle,the S-shaped curve was dominant.After 10 thermal shock cycles,an approximate ladder-shaped curve was observed.It is displayed that AE activity was mainly concentrated around the peak strength zone of the granite specimen when the rock samples underwent fewer thermal shock cycles.With increase in thermal shock cycle,AE activity could occur at low loading stress levels.Microscopic observation further confirmed these scenarios,which showed that more microcracks were induced with increase in thermal shock cycle.The number of induced microcracks at the edge location of the granite specimen was significantly larger than that at the interior location.Finally,a continuum damage model was proposed to describe the damage evolution of the granite specimen after cyclic thermal shock during loading.

Investigation of FE Model Size Definition for Surface Coating Application

An efficient prediction mechanical performance of coating structures has been a constant concern since the dawn of surface engineering. However, predictive models presented by initial research are normally based on traditional solid mechanics, and thus cannot predict coating performance accurately. Also, the high computational costs that originate from the exclusive structure of surface coating systems （a big difference in the order of coating and substrate） are not well addressed by these models. To fill the needs for accurate prediction and low computational costs, a multi-axial continuum damage mechanics （CDM）-based constitutive model is introduced for the investigation of the load bearing capacity and fracture properties of coatings. Material parameters within the proposed constitutive model are determined for a typical coating （TIN） and substrate （Cu） system. An efficient numerical subroutine is developed to implement the determined constitutive model into the commercial FE solver, ABAQUS, through the user-defined subroutine, VUMAT. By changing the geometrical sizes of FE models, a series of computations are carried out to investigate （1） loading features, （2） stress distributions, and （3） failure features of the coating system. The results show that there is a critical displacement corresponding to each FE model size, and only if the applied normal loading displacement is smaller than the critical displacement, a reasonable prediction can be achieved. Finally, a 3D map of the critical displacement is generated to provide guidance for users to determine an FE model with suitable geometrical size for surface coating simulations. This paper presents an effective modelling approach for the prediction of mechanical performance of surface coatings.