Continuum Damage Mechanics Theory Research Articles

On the Use of Low and High Cycle Fatigue Damage Models

In this paper, Continuum Damage Mechanics (CDM) theory is applied to low cycle and high cycle fatigue problems. Damage evolution laws are derived from thermodynamic principles and the fatigue number of cycles to crack initiation is expressed in terms of the range of applied stresses, triaxiality function and material constants termed as damage parameters. Low cycle fatigue damage evolution law is applied to adhesively bonded single lap joint. Damage parameters as function of stress are extracted from the fatigue tests and the damage model. High cycle fatigue damage model is applied to fretting fatigue test specimens and is integrated within a Finite Element Analysis (FEA) code in order to predict the number of cycles to crack initiation. Fretting fatigue problems involve two types of analyses; namely contact mechanics and damage/fracture mechanics. The high cycle fatigue damage evolution law takes into account the effect of different parameters such as contact geometry, axial stress, normal load and tangential load.

An Elastoplastic Damage Constitutive Model for Cementitious Materials under Wet-Dry Cyclic Sulfate Attack

The mechanical properties of cement mortars subjected to wet-dry cyclic sulfate attack were studied by the compression strength test. The results showed that the ultimate compressive strength increased with number of cycles at the initial stage. However, after a certain time, it started to decrease with further increases in the number of cycles. Moreover, the concentration of the sodium sulfate solution proved to be an important factor affecting the ultimate compressive strength. Based on continuum damage mechanics theory, an elastoplastic damage constitutive model is presented to describe the mechanical behavior of cementitious materials under compressive stress. The results obtained agree well with the experimentally observed elastic, plastic, and damage characteristics of cement mortars under compressive stress.

The effect of reheat treatments on recovering the creep behavior of a corrosion-resistant nickel-based superalloy

The present work is an attempt to possibly obtain the most suitable reheat treatment, which could provide the optimal microstructural characteristics for creep-exposed superalloys. After creep at 850°C/350MPa for 96h and 140h a cast nickel-based superalloy was reheat-treated with different programs, and then the specimens were subjected to creep again at the same condition. Creep tests indicate that during reheat treatment applying aging at lower temperature, 850°C, for proper time (4h) before aging at 1050°C rather than a repeated standard heat treatment is beneficial to restore creep life. Creep-induced changes in the microstructure, such as excessive grain-boundary secondary M23C6 and/or M6C carbides formation, and primary MC carbides decomposition, are noticeably more advanced in the reheat-treated alloys. A quantitative study on the dissolution of σ phases and the formation of cavitations is presented. After the interrupted creep, the γ′ particles at dendrite cores remained cubic-shaped. However, the rafting of cuboidal γ′ precipitates was observed in the same creep condition by the reheat treatment programs. Design of a reheat treatment for recovering the creep properties should emphasize the promotion of ductility rather than strength of the alloy. An equation based on the continuum damage-mechanics theory, was suggested to evaluate the remaining creep life.

Compressive Strength Reduction of Concrete Subjected to Hydrostatic Compression

An experimental study of the damage behavior of two kinds of concrete with different strength grades has been performed using 100mm cubes subjected to increasing hydrostatic loading history, namely, the isotropic compression at high pressure. The compressive strength and ultrasonic velocity are measured before and after loading history, respectively. The damage degree of these cubes is defined as the reduction of compressive strength based on the continuum damage mechanics theory. Linear and exponential curve fit of experimental data is performed respectively to describe the evolution of damage as well as the descent of ultrasonic velocity with respect to the loading history. It can be seen that, the influence of hydrostatic loading history upon strength and ultrasonic velocity could really reflect that upon the degree of damage development. In general, ultrasonic inspection is convenient and applicable to estimation of damage of concrete due to loading history in engineering practice.

Assessment of Low Cycle Fatigue Life of Steam Turbine Rotor Based on a Thermodynamic Approach

A new nonlinear model is proposed to assess the low cycle fatigue life of a 300 MW steam turbine rotor. Manson-Coffin equation and cyclic stress-strain relationship are employed to eliminate the unmeasured parameters, so all the parameters in model are measurable. Through comparison with that from the linear accumulation theory and continuum damage mechanics theory the results show this new nonlinear model describes the damage accumulation well and precisely in accordance with the practical test data. This approach supplies a new way to assess the damage of steam turbine rotor with satisfactory precision in engineering.

A thermodynamic framework for constitutive modeling of time- and rate-dependent materials. Part I: Theory

A general thermodynamic-based framework for deriving coupled temperature-dependent viscoelasticity, viscoplasticity, viscodamage, and micro-damage healing constitutive models for constitutive modeling of time- and rate-dependent materials is presented. Principle of virtual power, Clausius–Duhem inequality, and the principle of maximum rate of dissipation are used to construct this general thermodynamic framework. A micro-damage healing natural configuration is introduced to enhance the continuum damage mechanics theories in modeling the healing phenomenon. This healing configuration can be considered as the extension of the well-known Kachanov’s effective (undamaged) configuration (Kachanov, 1958). The viscoplasticity loading condition is defined from the microforce balance derived directly from the principle of virtual power. Moreover, for the first time, viscoelasticity, viscodamage, and micro-damage healing microforce balances are derived directly from the principle of virtual power. It is also shown that the generalized non-associative plasticity/viscoplasticity theories can be a direct consequence of postulating the principle of virtual power. The emphasis in this paper is placed on the decomposition of thermodynamic conjugate forces into energetic and dissipative components. It is shown that this decomposition is necessary for accurate estimation of the rate of energy dissipation. The energetic components are related to the Helmholtz free energy, whereas the dissipative components are related to the rate of energy dissipation. This thermodynamic framework is used for deriving more comprehensive viscoelastic, viscoplastic, and viscodamage, and micro-damage healing constitutive models.

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.

Creep–fatigue interaction damage model and its application in modified 9Cr–1Mo steel

Creep–fatigue interaction damage evolution of the nuclear engineering materials modified 9Cr–1Mo steel is studied with Continuum Damage Mechanics (CDM) theory. Based on the Norton creep damage and fatigue dissipate potential theory, an effective stress controlled creep–fatigue interaction damage model has been developed in this paper, in which the creep and fatigue damage function are both considered as nonlinear variables. The damage evolution function consists of the stress amplitude and the range of mean strain. The damage parameters in the model have clear physical meaning and can be determined easily. A stress controlled creep–fatigue interaction experiment has been performed with the P91 steel to obtain the damage model. The experimental results indicated that the damage model is applicable to describe the damage evolution for creep–fatigue interaction.

Bi-variable damage model for fatigue life prediction of metal components

Based on the theory of continuum damage mechanics, a bi-variable damage mechanics model is developed, which, according to thermodynamics, is accessible to derivation of damage driving force, damage evolution equation and damage evolution criteria. Furthermore, damage evolution equations of time rate are established by the generalized Drucker’s postulate. The damage evolution equation of cycle rate is obtained by integrating the time damage evolution equations, and the fatigue life prediction method for smooth specimens under repeated loading with constant strain amplitude is constructed. Likewise, for notched specimens under the repeated loading with constant strain amplitude, the fatigue life prediction method is obtained on the ground of the theory of conservative integral in damage mechanics. Thus, the material parameters in the damage evolution equation can be obtained by reference to the fatigue test results of standard specimens with stress concentration factor equal to 1, 2 and 3.

A Damage-coupled Multi-axial Time-dependent Low Cycle Fatigue Failure Model for SS304 Stainless Steel at High Temperature

Based on the time-dependent strain cyclic characteristics and fatigue behaviors of SS304 stainless steel under multi-axial cyclic loading at 700 ◦C, and in the frame of unified visoco-plastic cyclic constitutive model and continuum damage mechanics theory, the damage-coupled multi-axial time-dependent constitutive model and fatigue failure model were proposed. In the model, the evolution equation of damage was introduced in and the time-dependent effects, e.g. holding time, loading rate, were taken into account. The model was applied to the simulation of whole-life cyclic deformation behaviors and prediction of LCF life for SS304 stainless steel in multiaxial time-dependent low cycle fatigue tests. It is shown that the simulated results agree well with experimental ones.

On the thermodynamic consistency of the equivalence principle in continuum damage mechanics

We consider theories of continuum damage mechanics involving damage effect variables of different tensorial ranks. It turns out that orthotropic damage together with the use of Lemaitre's equivalence principle for the elastic part does not allow thermodynamic potentials such as the free enthalpy to exist. As the existence of these potentials is, however, a strict thermodynamic requirement, a theory employing orthotropic damage in this way is inconsistent. We show that the use of a rank-4 damage effect variable allows a consistent use of the equivalence principle.

Study on hydromechanical damage coupled model of Boom clay during tunnel excavation and construction

Based on the theory of continuum damage mechanics, a modified Mohr–Coulomb failure criterion according to plastic damage evolution and the hydromechanical coupling is applied to analyse the excavation damage of Boom clay influenced by the evolution of pore pressure and plastic strain. Considering the actual case of construction of the connecting gallery of radioactive waste disposal in the deep Boom clay formation in Belgium, a nonlinear finite element damage model for simulating shield tunnelling is proposed to study the evolution of damage, pore pressure and permeability of the surrounding rock. The variations of damage and permeability around the tunnel with time are analysed in detail. The proposed model is able to effectively depict the main features of hydromechanical behaviours of Boom clay. Therefore, the present model can provide references for the distribution, degree and evolution of damage and fractures self‐sealing process analysis of surrounding rock.

Flexural Fatigue Behavior of PAN Fiber Reinforced Concrete under Cyclic Loading

The flexural fatigue performance of polyacrylonitrile (PAN) fiber reinforced concrete (PANFRC）was investigated by third-point loading tests. Based on the previous research work, optimum mixture proportions of PANFRC for highway overlays and bridge decks that satisfied both the minimum compressive and bending strengths, and showed excellent mechanical properties, were selected for fatigue testing. The experimental program included a total of 69 flexural specimens, 15 of which were plain concrete specimens, and the remaining 54 specimens were PANFRC specimens. Three mixes containing 0.0%, 0.1 %, and 0.15% of PAN fiber volume fractions were selected. For each mix, 4 different target load ranges were applied: 10–75%, 10–80%, 10–85%, and 10–90% of the ultimate flexural capacity, as obtained from the corresponding control static test. The bending fatigue life of PANFRC specimens under various stress ratios are proved to follow two-parameter Weibull distribution. Both a semi-logarithm and a double-logarithm P-S-N equations with various failure probabilities are derived from the experimental measurements. The denifition of the fatigue damage variable and damage evolution equation for PANFRC are furtherly proposed based on theory of continuum damage mechanics.

Hybrid forming processes for production of lightweight high strength automotive panel parts

AbstractThe research concentrates on a heat treatable AA 6082 aluminium alloy. A set of unified constitutive equations has been developed and determined from experimental data. In addition to modelling viscoplastic flow of the material at different temperatures, the equations contain other two specific features. One is to predict the failure of the material under various deformation conditions based on continuum damage mechanics theories. The other is to model the precipitation formation and growth under straining and aging conditions; thus, the strength distribution of formed parts can be predicted via process modelling. The determined unified constitutive equations are then implemented into the commercial finite element code ABAQUS/Explicit via the user defined subroutine, VUMAT. A finite element process simulation model and numerical procedures are established for the modelling of a hot stamping and cold die quenching processes for a spherical part with a central hole. To validate the simulation result...

Soil–structure interaction effects on seismic inelastic analysis of 3-D tunnels

This paper investigates the importance of seismic soil–structure interaction in three-dimensional lined tunnels, assuming inelastic material behaviour for both the concrete liner and the soft rock type of soil. The seismic response of the soil–structure system is determined by the finite element method (FEM) in the time domain. Viscous absorbing boundaries are used in conjunction with the discretization of the rock medium. Both the rock medium and the concrete liner are assumed to behave inelastically on the basis of the continuum damage mechanics theory. The seismic waves are assumed to have any arbitrary time variation and direction of propagation. The system is analysed with and without soil–structure interaction in order to assess its importance on the response of the system. Through parametric studies, the influence of the most critical parameters affecting the structural response is determined and critically discussed.

Multiaxial fatigue life prediction of rubber-like materials using the continuum damage mechanics approach

The purpose of this study is to propose a fatigue criterion based on the continuum damage mechanics (CDM) theory to predict the fatigue life of rubber-like materials. The fatigue life of a styrene-butadiene rubber was investigated under tension and twist loadings. Using a generalized Ogden strain energy density function, a CDM model was derived in order to express the fatigue life as a function of an equivalent stress. Since the obtained model was unable to fit all the experimental results, a modified version, which shows a fair agreement with our data, is finally proposed.

Recent developments on damage modeling and finite element analysis for composite laminates: A review

Under complex environments such as continuous or cyclic loads, the stiffness degradation for the laminated composites such as the carbon fiber reinforced polymer matrix composites is an important physical and mechanical response to the damage and failure evolution. It is essential to simulate the initial and subsequent evolution process of this kind of damage phenomenon accurately in order to explore the mechanical properties of composite laminates. This paper gives a comprehensive review on the general methodologies on the damage constitutive modeling by continuum damage mechanics (CDM), the various failure criteria, the damage evolution law simulating the stiffness degradation, and the finite element implementation of progressive failure analysis in terms of the mechanical response for the variable-stiffness composite laminates arising from the continuous failure. The damage constitutive modeling is discussed by describing the evolvement of damage tensors and conjugate forces in the CDM theory. The failure criteria which interpret the failure modes and their interaction are compared and some advanced methods such as the cohesive theory which are used to predict the damage evolution properties of composites are also discussed. In addition, the solution algorithm using finite element analysis which implements progressive failure analysis is summarized and several applicable methods which deal with the numerical convergence problem due to singular finite element stiffness matrices are also compared in order to explore the whole failure process and ultimate load-bearing ability of composite laminates. Finally, the multiscale progressive failure analysis as a popular topic which associates the macroscopic with microscopic damage and failure mechanisms is discussed and the extended finite element method as a new finite element technique is expected to accelerate its practical application to the progressive failure analysis of composite laminates.

A Study on Al-6061-T6 Tube Drawing Limit Based on Critical Damage Value

he Al-6061-T6 tube’s drawing limit and the drawing process’s elasto-plastic behavior were investigated based on the foundational theories of larger deformation and continuum damage mechanics. A mathematical computation equation about the maximum Cockcroft-Latham damage value was converted to an appropriate discrete expression which is easy for FE code to programme, thus the corresponding finite element numerical algorithm for damage computation was developed. By an approach that physical experiments and numerical simulation provide mutual support for the critical damage value, the crack criterion of Al-6061-T6 was evaluated as 1.34. The crack criterion obtained was introduced as important design considerations of tube drawing process. An 3D graph which reflects maximum damage variation according with the diameters at different tube thickness was achieved, according to which the drawing process’s safe and unsafe areas of Al-6061-T6 tube with diameter 10mm at temperature 20°C and drawing velocity 100mm/s was ploted.

Continuum damage model of low-cycle fatigue and computation of ultimate strength of locally fatigue damaged plates under uniaxial compression

The aim of this study is to present a simple method to assess the effects of locally low-cycle fatigue (LCF) damages on the reduced ultimate strength of unstiffened ship plate structures. A LCF damage model is formulated and then it is applied to the finite element analysis (FEA) in order to identify the continuum damage mechanics (CDM) theory to be introduced in the ship structure designs. A series of non-linear FEA of simply supported plates with various locations and sizes of damages have been carried out. The large deformation theory is utilized during plastic material conditions. Based on the FEA results, an analytical formula is derived for estimating the reduced ultimate strength of ship plates with localized damages under uniaxial compression. In the assessment method, the reduced elastic modulus ˜ E is defined

A straightforward numerical technique for finite element implementation of non-local gradient-dependent continuum damage mechanics theories

This paper presents a direct algorithm to implement non-local gradient-enhanced damage mechanics theories in the existing finite element codes with minor modifications and without the need to formulate a higher-order element. This method extends the algorithm of Abu Al-Rub and Voyiadjis (2005) to gradient-dependent damage theories and to three-dimensional (3D) problems. The presented algorithm is implemented in the well-known finite element code Abaqus via the user material subroutine UMAT. The potential of the proposed numerical algorithm for non-local gradient-enhanced damage theories in eliminating mesh-dependent simulations is validated by conducting various numerical tests of the localised damage.