A continuum damage model for ductile fracture of weld heat affected zone

In this paper, the ductile plastic damage behaviour of weld heat affected zone (HAZ) is studied by use of continuum damage mechanics (CDM). Based on a continuum damage variable, D, the effective stress concept and the thermodynamics, a general continuum damage model for Isotropie ductile fracture is derived from a new dissipation potential chosen by the author herein. A comparison between the damage model and experimental results is presented and a good agreement is found. The model is also used to analyse the ductile plastic damage evolution in thermally simulated welding coarse-grained HAZ of a low alloy steel. The effects of stress triaxiality on plastic damage evolution and on ductile fracture of the coarse-grained HAZ are discussed.     

A continuum damage mechanics model for crack initiation in mixed mode ductile fracture

This paper presents the development of a fracture criterion based on the postulation that the threshold condition of crack initiation in mixed mode ductile fracture is satisfied when the overall damage w in an element at the prospective direction of crack path reaches its critical value w_c. The validity of the proposed criterion is checked by predicting the fracture loads of thin aluminium plates containing an isolated crack inclined at the angle of =30, 45, 60 and 75 degrees and the predicted loads are compared satisfactorily with those determined experimentally. The analysis is performed based on the anisotropic model of continuum damage mechanics theory proposed earlier by the authors, thus providing additional proof of the consistency, applicability and versatility of the model. When the fracture loads of the mixed mode plates calculated using conventional fracture mechanics are compared with those determined using the proposed damage model, a maximum close to 30 percent over-estimation of the loads from the conventional approach is observed as opposed to within 7 percent discrepancy between the computed and measured fracture loads using the damage approach. The observation reveals the importance of including damage consideration in any ductile fracture analysis.The effect of varying damage coefficients on the fracture loads is examined and it is found that the crack initiation load decreases with the increase of anisotropic damage coefficient.     

A finite element analysis of continuum damage mechanics for ductile fracture

A finite element formulation of an anisotropic theory of continuum damage mechanics for ductile fracture is presented. The formulation is based on a generalized model of anisotropic continuum damage mechanics of elasticity and plasticity proposed earlier by the authors. The validity of the proposed anisotropic damage model and finite element formulation is verified by comparing the predicted fracture load of center-cracked tension specimen made of thin aluminium alloy 2024-T3 with those determined experimentally and excellent agreement is achieved. The proposed finite element analysis can thus provide an important design tool to solve practical problems of engineering significance which may have hitherto been found difficult using the conventional fracture mechanics concept.

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Résumé On présente une formulation par éléments finis d'une théorie anisotrope de la mécanique d'endommagement d'un continuum, applicable aux ruptures ductiles. Cette formulation est basées sur la généralisation d'un modèle de mécanique d'endommagement d'un continuum anisotrope pour l'élasticité et la plasticité, proposé précédemment par les auteurs. On vérifie la validité du modèle d'endommagement anisotrope proposé et de sa formulation par éléments finis, en comparant aux valeurs expérimentales la charge de rupture prévue pour une éprouvette mince de traction d'alliage d'aluminium 2024-T3 présentant une fissure centrale. On trouve un excellent accord. L'analyse par éléments finis proposée peut ainsi constituer un outil de conception important pour résoudre des problèmes pratiques de construction que l'on aurait trouvé difficiles à traiter par les concepts de la mécanique de rupture traditionnelle.

     

A variational void coalescence model for ductile metals

We present a variational void coalescence model that includes all the essential ingredients of failure in ductile porous metals. The model is an extension of the variational void growth model by Weinberg et al. (Comput Mech 37:142–152, 2006). The extended model contains all the deformation phases in ductile porous materials, i.e. elastic deformation, plastic deformation including deviatoric and volumetric (void growth) plasticity followed by damage initiation and evolution due to void coalescence. Parametric studies have been performed to assess the model’s dependence on the different input parameters. The model is then validated against uniaxial loading experiments for different materials. We finally show the model’s ability to predict the damage mechanisms and fracture surface profile of a notched round bar under tension as observed in experiments.     

A non-proportional loading finite element analysis of continuum damage mechanics for ductile fracture

This paper presents the development of a finite element analysis based on an anisotropic model of continuum damage mechanics theory proposed recently by the authors for ductile fracture under non-proportional loading. The condition of non-proportional loading is formulated by introducing a dynamic co-ordinate system of principal damage allowing the principal direction of damage during the loading to rotate accordingly. The finite element analysis developed under non-proportional loading is applied to predict the crack initiation load of a centre-cracked plate under uniform loading. The predicted load agrees satisfactorily with those determined experimentally with centre-cracked thin plates made of aluminium alloy 2024-T3. The analysis also reveals under non-proportional loading the hysteresis effect of the principal directions of damage and stress. In addition, the influence of varying anisotropic damage coefficients on the crack initiation load and the crack tip displacement profile is also examined. The larger the degree of the anisotropy, the higher the crack initiation load. The magnitude of the crack tip displacement profile is found to be proportional to the degree of material anisotropy.     

A normalized damage variable for ductile metals based on toughness performance

The normalized damage function based on elastic modulus is proposed to evaluate damage for ductile metals under a unified criterion. An equivalent half-sine model is employed to illustrate plastic stress-strain relationship. Toughness indicator is introduced to describe the toughness character in metals. An exponential rule is employed to calculate toughness indicator, and the parameter of toughness indicator is formulated by conventional mechanical properties of metals. Repeated loading-unloading tensile tests are conducted on stainless steel and aluminium alloy. The good correlation coefficient and dispersion are presented between the experimental values and predicted values of damage variable (within 12%). The damage function with toughness indicator can be applied to predict the effective elastic modulus.     

A damage model for ductile crack initiation and propagation

Damage-induced ductile crack initiation and propagation is modeled using a constitutive law with asymmetrical contraction of the yield surface and tip remeshing combined with a nonlocal strain technique. In practice, this means that the void fraction depends on a nonlocal strain. Finite strain plasticity is used with smoothing of the complementarity condition. The prototype constitutive laws take into account pressure sensitivity and the Lode angle effect in the fracture strain. Two plane idealizations are tested: plane stress and plane strain. Thickness variation in the former is included by imposing a null out-of-plane normal stress component. In plane strain, pressure unknowns and bubble enrichment are adopted to avoid locking and ensure stability of the equilibrium equations. This approach allows the representation of some 3D effects, such as necking. The nonlocal approach is applied to the strains so that the void fraction value evolves up to one and this is verified numerically. Three verification examples are proposed and one validation example is shown, illustrating the excellent results of the proposed method. One of the verification examples includes both crack propagation in the continuum and rigid particle decohesion based on the same model.     

A damaged evolution model for strain fatigue of ductile metals

This paper, based on Lemaitre's potential of dissipation, derives a damaged evolution model for strain fatigue of ductile metals. Then the equation of fatigue-life prediction and the criterion of cumulative fatigue damage are deduced. Experiments for steel were conducted and the experiment data agreed with the model excellently. Thus the model can predict the damaged evolution during the strain fatigue process.     

A model of continuum damage mechanics for fatigue failure

This paper describes the development of a generalized model of continuum damage mechanics for fatigue fracture. With the introduction of a new damage effect tensor, the necessary constitutive equations of elasticity and plasticity coupled with damage are for the first instance derived. This is followed by the formulations of fatigue damage dissipative potential function and a fatigue damage criterion which are required for the development of a fatigue damage evolution equation. The fatigue evolution model is based on the hypothesis that the overall fatigue damage is induced by the summation of elastic and plastic damages.The validity of the damage model proposed is verified by comparing the predicted and measured number of cycles to failure for ten tensile specimens each applied with different load ranges and excellent agreement has been achieved.University of Science and Technology of China     

An anisotropic theory of elasticity for continuum damage mechanics

This paper presents the development of an anisotropic elastic damage theory. This is achieved by deriving a modified damage effect tensor M(D) for the effective stress equations capable of including the effect of anisotropic material damage. The modified tensor removes the restriction of a priori knowledge of the directions of principal stresses imposed by a damage effect tensor developed earlier and can now be made for general practical engineering applications of failure analysis. Reduction of the proposed tensor to a scalar for isotropic damage is shown to be possible when it is expressed not only in the principal directions but also in any arbitrary coordinate system, a necessary condition to verify the validity of the proposed tensor. Uniaxial tension and pure torsion are chosen to illustrate the application of the theory as well as associated damage variables that may be experimentally determined using laboratory size specimens. The measured damage variables confirm the presence of anisotropic damage from an initially isotropic material specimen and the magnitude is more pronounced at higher stresses and strains.

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Résumé On présente un développement d'une théorie sur l'endommagement élastique anisotrope en déduisant un tenseur modifié décrivant l'effet de l'endommagement pour un système d'équations de contraintes effectives susceptible d'inclure l'effet d'un endommagement dans un matériau anisotrope. Le tenseur modifié supprime la restriction de la connaissance a priori des directions des contraintes principales imposées par un tenseur d'effet d'endommagement développé précédemment; il peut à présent entrer dans les applications pratiques en construction de l'analyse des ruptures.On montre qu'il est possible de réduire le tenseur proposé à une valeur scalaire dans le cas d'un dommage isotrope, dès lors qu'il est exprimé non seulement suivant les directions principales, mais dans un système de coordonnées arbitraires, ce qui est une condition nécessaire pour en vérifier la validité.On choisit une traction multiaxiale et une torsion pure pour illustrer l'application de la théorie ainsi que des variables d'endommagement associées, susceptibles d'être déterminées expérimentalement à l'aide d'éprouvettes de laboratoire.Les variables d'endommagement mesurées confirment la présence d'un dommage anisotrope dans le cas d'une éprouvette d'un matériau initialement isotrope; son amplitude est plus prononcée à des contraintes ou des déformations plus importantes.

     

Constitutive modeling for ductile metals behavior incorporating strain rate, temperature and damage mechanics

One the most serious limitation to an extensive use of computational techniques in simulating and predicting structures and components behavior under dynamic loading is given by the inadequacy of constitutive models to fairly represent failure process. In this paper a new constitutive model for ductile metals has been developed using the innovative solid state equation proposed by Milella (1998) and the non-linear damage model proposed by Bonora (1997). These two models are physically based and require a very limited number of constants that can be experimentally identified according to the procedures given by the authors. The implementation of the model in commercial finite element code is simple and cost effective with respect to similar nucleation and growth (NAG) models with the additional feature that hydrostatic pressure effect on ductile damage is correctly taken into account.     

Development of a viscoelastic continuum damage model for cyclic loading

A previously developed spectrum model for linear viscoelastic behavior of solids is used to describe the rate-dependent damage growth of a time dependent material under cyclic loading. Through the use of the iterative solution of a special Volterra integral equation, the cyclic strain history is described. The spectrum-based model is generalized for any strain rate and any uniaxial load history to formulate the damage function. Damage evolution in the body is described through the use of a rate-type evolution law which uses a pseudo strain to express the viscoelastic constitutive equation with damage. The resulting damage function is used to formulate a residual strength model. The methodology presented is demonstrated by comparing the peak values of the computed cyclic strain history as well as the residual strength model predictions to the experimental data of a polymer matrix composite.     

On continuum models of ductile fracture

Several fracture criteria are reviewed with respect to ductile fracture. It is suggested that both critical crack-tip displacement, 2V _c ^*, and critical fracture strain,∃ ^*, criteria may describe the fracture of a ductile second phase rod in a ductile matrix. As a first approximation, this is experimentally verified by observations of ductile stainless steel fibres fracturing in an age-hardened aluminium matrix. For 0.05, 0.10 and 0.20 volume fraction composites, the average fracture strains are calculated to be 1.15 as compared to a measured average of 0.93 while the average critical crack-tip displacement is calculated to be 0.50 mm as compared to an “observed” average of 0.40 mm. The statistical variation in the fracture strain was not sufficiently small to allow any choice between these proposed criteria. In fact, both the experimental and theoretical evidence point to the equivalency of these criteria as given by 2V _c ^*=π/^*∃^* where /^* is the microstructural unit in front of the crack over which the strain is greater than or equal to∃ ^*.     

A micromechanical constitutive model for dynamic damage and fracture of ductile materials

This paper proposes a detailed theoretical analysis of the development of dynamic damage in plate impact experiments for the case of high-purity tantalum. Our micro-mechanical model of damage is based on physical mechanisms (void nucleation and growth). The model is aimed to be general enough to be applied to a variety of ductile materials subjected to high tensile pressure loading. In this respect, the work of Czarnota et al. (J Mech Phys Solids 56:1624–1650, 2008) has been extended by introducing the concept of nucleation law and by entering a nonlinear formulation of the elastic response based on the Mie-Grüneisen equation of state. This later aspect allows us to consider high impact velocities. All model parameters are directly assessed by experimental measurements to the exception of the nucleation law which is characterized by the way of an inverse identification method using three free-surface velocity profiles (at low, intermediate and high impact velocities). It is shown that the nucleation law can be consistently determined in the range of operating pressures. The nucleation law being identified, the development of internal damage happens to be a natural outcome of the modelling. The model is applied to predict damage development and free-surface velocity profiles for various test conditions. The variety and the quality of results support the physical basis (in particular micro-inertia effects) upon which the proposed model of dynamic damage is based.     

An analysis of dynamic fracture in concrete with a continuum visco-elastic visco-plastic damage model

We present a rate-dependent continuum model for the dynamic modelling of concrete. Rate dependency is included by means of visco-elastic and visco-plastic constitutive relationships. The inclusion of the Stefan effect, described by a visco-elastic model, results in an increased tensile strength with increasing loading rate. An additional rate effect comes from the visco-plastic contribution which, from a mechanical standpoint, can be related to the micro-inertia of the material surrounding the crack tip. Static and dynamic examples show the capabilities of the model with respect to localization, hysteresis and dissipation of energy.     

Prediction of creep deformation in ductile two-phase alloys by a continuum mechanics model

The creep deformation of the ductile two-phase alloys was analysed on the basis of the continuum mechanics model which incorporated the projection concept proposed by Evans and Wilshire. The calculated creep curves were compared with the experimental ones in ferrjte-pearlite steels. It was found that the continuum mechanics model was able to predict the whole creep deformation process of the ductile two-phase alloys from the onset of creep loading to the final rupture, if the creep-deformation and creep-rupture data of the individual phases which constituted the two-phase alloys were known. A steady-state creep in the ductile two-phase alloys was predicted by the continuum mechanics model to occur, even if the constituent phases did not have the inherent steady-state creep. This was caused by the internal stresses arising from the creep-strength difference between second-phase and matrix in the two-phase alloys. This steady-state creep was observed in ferrite-pearlite steels during creep at 873 K. The predicted rupture life on the basis of the continuum mechanics model was correlated well with the experimental results in ferrite-pearlite steels, although the former was somewhat shorter than the latter under higher creep stresses. The continuum mechanics model was able to apply to the life prediction and the creep-strength design of the ductile two-phase alloys.     

An updated continuum damage model to investigate fracture process of structures in DBTT region

In the Ductile–Brittle Transition Temperature (DBTT) region, it is not realistic to take unique fracture stress or fracture strain as the fracture criterion to investigate the fracture properties. In this paper, an updated continuum damage model was proposed, in which the fracture energy density, a function of the stress triaxiality, temperature and strain rate in the transition region was taken as the critical damage factor. Uniaxial tension tests were carried out to get the basic material properties at different temperatures, to calibrate the fracture model constants and verify the validity of the damage model. The fracture behaviour of pipes with penetrating cracks under the internal pressure was experimentally investigated with the load–deflection curves and the crack propagation length captured from tests. The J–R curves were obtained from the testing results for different temperatures. Based on the Finite Element Analyses (FEA) with the proposed fracture criterion of the updated continuum damage model, the loading level of pipes with penetrating cracks were estimated and compared with the experimental results. Meanwhile the fracture processes of the pipeline structures in the transition region were reproduced. The experimental and numerical results agreed very well in present calculations. It has been shown that the fracture process in the transition region strongly depends on both the stress and strain states, and could be effectively predicted using the continuum damage model.