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 finite element approach to anisotropic damage of ductile materials in large deformations Part I: an anisotropic ductile elastic-plastic-damage model

During past decades, many material models using the continuum damage mechanics (CDM) approach have been proposed successfully in the small deformation regime to describe inelastic behaviors and fracturing phenomena of a material. For ductile materials, large deformation takes place at the level of damage appearance. Damage is anisotropic in nature. In this paper, the ductile damage at finite deformations is modeled as an anisotropic tensor quantity. Then, a fourth-order symmetric stress correction tensor is proposed for computationally efficient and easy implementation in the finite element formulations. Consequently, an explicit form of the fourth-order constitutive equations of the proposed elastic-plastic-damage model is derived. Both isotropic and kinematic hardening effects are included in the formulation. The new constitutive model can predict not only the elastic-plastic behaviors, but also the sequential variations of ductile materials. An evaluation of the constitutive and damage evolution equations is presented. This revised version was published online in August 2006 with corrections to the Cover Date.     

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 finite element approach to anisotropic damage of ductile materials in large deformations Part II: finite element formulation and applications

A finite element analysis model for material and geometrical non-linearities due to large plastic deformations of ductile materials is presented using the continuum damage mechanics approach. To overcome limitations of the conventional plastic analysis, a fourth-order tensor damage, defined in Part I of this paper to represent the stiffness degradation in the finite strain regime, is incorporated. General forms of an updated Lagrangian (U.L.) finite element procedure are formulated to solve the governing equations of the coupled elastic–plastic-damage analysis, and a computer program is developed for two-dimensional plane stress/strain problems. A numerical algorithm to treat the anisotropic damage is proposed in addition to the non-linear incremental solution algorithm of the U.L. formulation. Selected examples, compared with published results, show the validity of the presented finite element approach. Finally, the necking phenomenon of a plate with a hole is studied to explore plastic damage in large strain deformations. This revised version was published online in August 2006 with corrections to the Cover Date.     

Comparison of different finite deformation inelastic damage models within multiplicative elastoplasticity for ductile materials

The new contribution of this study is to formulate two wellknown isotropic elastoplastic damage concepts for ductile materials in the framework of geometrically exact finite multiplicative elastoplasticity. For the model originally proposed by Lemaitre the damage evolution follows from a dissipation potential and the hypothesis of general associativity. In contrast, the Gurson model takes into account the balance of mass separately to formulate damage evolution. In this contribution both formulations are based on logarithmic Hencky strains leading to a simple application of the so called exponential map stress integrator which is the algorithmic counterpart of the multiplicative elastoplastic formulation adopted. Special emphasis is directed towards the numerical implementation of these models within the framework of finite element analysis of inelastic boundary value problems. To compare the results of numerical computations several standard examples within finite elastoplasticity are analysed with both damage models and the results are contrasted to the outcome of an analysis with the classical v. Mises model. thereby, the dramatic influence of damage on the behaviour within necking and localization computations is highlighted. The different behaviour of the two models considered within compression dominated problems is appreciated.     

Damage evolution in ductile materials: from micro- to macro-damage

This research presents a new simulation concept of damage evolution for metallic materials under large displacements and deformations. The complete damage range is subdivided into both the micro-damage and the macro-damage range. The micro-damage phase is described by the Cocks/Ashby void-growth model for isotropic, ductile materials under isothermal conditions. After having reached a critical void-volume fraction, a macro-crack is introduced into the model. With such a concept the damage evolution from nucleation and growth of first micro-voids to initiation of macro-cracks and complete failure of the material can be simulated. Applying the Finite Element Method for the numerical formulation, at every incremental macro-crack step the Finite Element mesh is adapted such that the crack path remains independent of the initial mesh.     

A finite element treatment of ductile crack extension

Abstract

To provide a realistic analysis for fracture, this paper is devoted to finite element approaches which attempt to describe ductile crack extension. A rigorous hybrid displacement finite‐element modeling, involving the translation of entire “singular” near‐tip elements and consideration of global energy balance, is successfully developed. To investigate the variation of global energy release rate G* and J‐integral for single‐step extensions at various levels of applied loads (extension ^δa = constant), the Kfouri et al.’s center‐cracked problem together with A533B compact tension specimen are solved. In addition, an available experimental J resistance curve for the compact tension specimen (made of Ni‐Cr‐Mo‐V rotor steel) is simulated by the present finite element analysis procedure to study the actual process of ductile fracture. The effect of unloading process occurred in the plastic region behind the advancing crack‐tip is also discussed.     

Three dimensional modelling of non-local ductile damage: element technology

This paper presents an engineering scale numerical analysis framework of ductile damage and focusses on the three dimensional element technology required for it. A three dimensional low-order tetrahedral element is developed which is free from locking and efficient from the calculation time point of view. The element is tested using a well known benchmark problem and the results show that the new element performs better compared to the standard element.     

An exponential ductile continuum damage model for metals

In the frame of continuum damage mechanics an isotropic ductile plastic damage model is derived. The model is based on void damage variable, defined using effective stress concept and thermodynamics. The damage evolution from this model is exponential with equivalent plastic strain as experienced in some low carbon steel like AISI 1015. The damage model is sensitive to stress triaxiality and emulates the damage evolution as recorded in experiments conducted on such metals. The model is validated by comparison with the Rice-Tracey model and other experimental results published in the literature. This model can be used to study the growth and coalescence of micro voids, influence of stress triaxiality on strain to rupture and crack initiation phenomena.     

A novel damage variable to characterize evolution of microstructure with plastic deformation for ductile metal materials under tensile loading

Damage and fracture of ductile metal materials are greatly associated with evolution of their microstructure under loading, which meso-dimensionally corresponds to grain deformation as well as nucleation, growth, and coalescence of microvoids in later local deformation. The evolution behavior of microstructure is important to realize ductile damage and fracture mechanism of materials. In the present work, a novel damage variable of shape factor of grains was put forward to quantitatively describe the character of microstructure; its evolution with the plastic deformation was built-up by microanalytical and mechanical experiments for Armco iron and mild steel tensile bars. The evolution rule based on the damage variable of shape factor is possible to be extended into all of ductile metal materials.     

Effect of viscosity on cavity growth in ductile damage

In this paper the Budiansky approach is used in order to reproduce the void growth measured experimentally by 3D tomography in metals. It permits to avoid the non physical fitting of the alpha coefficient in Huang’s law. This model considers the viscoplasticity of the material to account for the interaction between the voids. It is shown that strain rate sensitivity of the material is playing a key role on growth of cavity and thus on damage of materials. Moreover finite elements calculations are carried out to check the influence of geometrical parameters (fraction of porosity, distance between voids) on the damage behaviour. The results are in agreement with the Budiansky model.     

A note on calibration of ductile failure damage indicators

After a review of the literature on the prognosis of ductile fracture, a damage indicator is developed which is based on micromechanical results and adapted to experiments. A calibration of this damage indicator is possible by inspecting in detail a load displacement curve for a long and smooth specimen. Relation to currently published damage indicators is discussed. The damage indicators can be used to indicate the onset of a local crack in a ductile structureDedicated to Prof. Franz Kollmann, Darmstadt, on the occasion of his 60th birthday.     

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.