Microsoft Word - numero 1 art 3.doc V. Tvergaard, Frattura ed Integrità Strutturale, 1 (2007) 25-28 25 1 INTRODUCTION Many procedures for the analysis of crack propagation are based on using critical values of parameters character- ising the crack-tip stress and strain fields, such as the stress intensity factor, the J-integral, the crack-tip open- ing displacement, or the crack-tip opening angle. Alterna- tively, the prediction of crack growth may be directly based on the fracture mechanism operating on the micro- scale, either by incorporating the failure mechanism in the constitutive equations for the material, or by repre- senting the failure mechanism through a cohesive zone model of the fracture process zone. The present paper will give a survey of a number of investigations where the prediction of crack growth has been based on models of the actual fracture mechanism. One of the most well known material models that ac- counts for the micromechanics of damage is the modified Gurson model [1,2], which models the evolution of duc- tile fracture by the nucleation and growth of voids to coa- lescence. Some of the analyses using this model to pre- dict ductile crack growth will be discussed. Also for creep failure in metals at high temperatures material models [3] have incorporated the micromechanisms of diffusive cavity growth in grain boundaries, leading to open micro-cracks at grain boundary facets at a rate strongly affected by grain boundary sliding. Results on creep crack growth based on this failure model will be mentioned. The term continuum damage mechanics is used for constitutive relations, which are able to represent the effect of damage evolution on the macro level, by de- veloping appropriate expressions in which free material parameters can be fitted to experiments, as in the case of low cycle fatigue [4]. As an example, predictions of mi- cro-crack formation in a metal matrix composite, based on this material model, will be presented here. Cohesive zone models have been used in recent years in a number of analyses of crack growth resistance in elastic- plastic solids [5]. Some of the predictions obtained in these studies will be briefly mentioned here. 2 MATERIAL MODELS WITH DAMAGE EVOLUTION When the failure mechanism is incorporated in in the constitutive relations, the crack growth follows directly from the predicted loss of stress carrying capacity in one or more integration points in an element. Then it is natu- ral to kill the failed elements, by using the element vanish technique [6]. This procedure has been used for the pre- dictions of crack growth to be discussed in the following three subsections. Crack growth by ductile failure Much interest has been devoted to the development of elastic-plastic or viscoplastic constitutive equations that account for the effect of ductile damage development. The most well known model is that suggested by Gurson [1], which makes use of an approximate yield condition ( , , ) 0ij fMσ σΦ = for a material containing a volume fraction f of voids, where ijσ is the average macro- scopic Cauchy stress tensor and Mσ is an equivalent tensile flow stress representing the actual microscopic stress-state in the matrix material. With some modifica- tions to improve predictions of plastic flow localization Numerical modelling in non linear fracture mechanics ( da ESIS Newsletter 2005) Viggo Tvergaard Dept. of Mechanical Engineering, Solid mechanics, Technical University of Denmark, Nils Koppels Allé, Building 404, DK-2800 Kgs. Lyngby, Denmark ABSTRACT: Some numerical studies of crack propagation are based on using constitutive models that ac- count for damage evolution in the material. When a critical damage value has been reached in a material point, it is natural to assume that this point has no more carrying capacity, as is done numerically in the ele- ment vanish technique. In the present review this procedure is illustrated for micromechanically based mate- rial models, such as a ductile failure model that accounts for the nucleation and growth of voids to coales- cence, and a model for intergranular creep failure with diffusive growth of grain boundary cavities leading to micro-crack formation. The procedure is also illustrated for low cycle fatigue, based on continuum dam- age mechanics. In addition, the possibility of crack growth predictions for elastic-plastic solids using cohe- sive zone models to represent the fracture process is discussed. KEYWORDS: Damage evolution, crack growth, coesive zone V. Tvergaard., Frattura ed Integrità Strutturale, 1 (2007) 25-28 26 ( ) ( ) ( ) ( ) i local 1 ˆˆ ˆyi i i i V f y f w y y dV W y = −∫& & [7] and of final failure by void coalescence [8] this yield condition is of the form ( ) 2 2* *2 1 12 2 cosh 1 02 k e k M M q q f q f σ σ σ σ ⎛ ⎞ ⎡ ⎤Φ = + − + =⎜ ⎟ ⎢ ⎥⎣ ⎦⎝ ⎠ (1) where ( )½3 / 2ije ijs sσ = is the macroscopic effective Mises stress, and / 3ij ij ij kks Gσ σ= − is the stress devia- tor. This material model accounts for the growth of the void volume fraction f due to plastic flow of the material around voids and due to the nucleation of new voids, and final failure is directly predicted when f reaches the critical value, at which the yield surface has shrunk to a point. This material model has been applied in a number of nu- merical studies of crack growth, including some studies where two populations of void nucleating particles are modelled; large weak particles that nucleate voids at rela- tively small strains and small strong particles that nucle- ate voids at much larger strains. For an edge cracked specimen under dynamic loading [9] results of a plane strain analysis are shown in Fig. 1, where contours of constant void volume fraction define the predicted crack growth path in a case of a random distribution of the lar- ger inclusions ahead of the initial crack-tip. Also a full three dimensional analysis has been used to analyse this type of specimen [10]. Here the computer re- quirements were much larger, but the advantage is that more realistic spherical shapes of the larger inclusions can be accounted for, and that 3D modes of growth are accounted for, such as tunnelling and shear lip formation. Continuations of the 3D fracture study have been carried out recently in analyses that do not directly focus on crack growth, e.g. the failure of a metal matrix composite [11] or of a Charpy V-notch specimen cut through a weld [12]. Some attempts to include a damage dependent mate- rial length scale in this constitutive model have been car- ried out by Leblond et al. [13] and Tvergaard and Nee- dleman [14], using an integral condition on the rate of increase of the void volume fraction. The expressions used in [14] are (2) (3) where 0L > is the material characteristic length, i j ijz g y y= , and 8p = , 2q = . The usual local formu- lation corresponds to the limit 0L → , and it has been shown, as for other non-local continuum models, that the mesh dependence of numerical solutions in a softening regime are removed by taking 0L > . This nonlocal damage model has been applied by Needleman and Tver- gaard [15] to predict ductile crack growth in the edge cracked specimen under dynamic loading also analysed in [9,10]. Figure 1: Crack growth indicated by contours of constant void volume fraction, f , for random distribution of larger particles. (a) 1.5 ,t sμ= 0.09 mmaΔ = ; (b) 1.6 ,t sμ= 0.27 mmaΔ = . (From [9]). Creep crack growth High temperature failure leading to crack growth has been modelled in terms of continuum damage mechanics (Hayhurst et al. [16]), where damage parameters are fit- ted to material behaviour on the macro level. The micro- mechanisms of creep failure in polycrystalline metals in- volve the nucleation and growth of small voids to coales- cence; but here diffusion plays an important role, and the cavities occur primarily on grain boundary facets perpen- dicular to the maximum principal tensile stress (e.g. Ashby and Dyson [17]), where a creep constraint on the rate of cavitation is often a dominant mechanism. Cavity coalescence on a grain boundary facet leads to a micro- crack, and final intergranular failure occurs as such mi- cro-cracks link up. Grain boundary sliding is an impor- tant mechanism that further complicates the analysis of creep failure. A micromechanically based constitutive model for creep failure in a polycrystalline metal has been proposed (Tvergaard [3,18]), in which the macro- scopic creep strain rate is given by the expression ( ) ( )0 0 3 1 * 2 n nijC C e ij e e s C f σ η ε ε σ σ ⎛ ⎞ ⎡ = + +⎜ ⎟ ⎢ ⎝ ⎠ ⎣ && 2* * * *3 1 2 2 1 1 ij n n ij e e e s S Sn m n n σ σ ρ σ σ σ ⎤⎧ ⎫⎛ ⎞− −−⎪ ⎪⎥+⎨ ⎬⎜ ⎟ + + ⎥⎝ ⎠⎪ ⎪⎩ ⎭⎦ (4) Here, n is the creep power, 0C > represents sub- structure induced acceleration of creep, and expressions for other parameters are determined by axisymmetric cell ( ) ( ) ( ) ( )1 ˆˆ, 1 / q i i i i p V w y W y w y y dV z L ⎡ ⎤ = = −⎢ ⎥ +⎢ ⎥⎣ ⎦ ∫ V. Tvergaard., Frattura ed Integrità Strutturale, 1 (2007) 25-28 27 model studies for a grain with a cavitating facet and slid- ing boundaries [3]. If there is no sliding, *f is unity, *ρ is the density of cavitating facets *ijm is a direction tensor for cavitating facets, and * nS σ− is the difference between the maximum principal stress and the normal stress on a cavitating facet. The material model has been used to predict crack growth [18], by applying the ele- ment vanish technique when cavity coalescence was pre- dicted on a grain boundary. For a double edge cracked panel under tension Fig. 2 shows the predicted damage near the crack-tip at two stages of time, where the dam- age parameter a/b is the cavity radius divided by the cavity half spacing on a facet, and vanished triangular elements are painted black. Figure 2: Distributions of creep damage ahead of a crack-tip. Continuous cavity nucleation, no grain boundary sliding, and 40C = . (a) 0/ 0.064ft t = . (b) 0/ 0.686ft t = . (From [18]). Plane strain multi-grain cell models for a polycrystal- line aggregate have been used by van der Giessen and Tvergaard [19] to study the final creep fracture process, as microcracks formed at grain boundary facets link up. Such analyses are however limited by the unrealistic grain geometry and the reduced constraint on sliding. But a great advantage is that large grain arrays can be ana- lysed if a crude mesh is used within each grain, and this allows for direct modelling of intergranular crack growth in a plane strain multi-grain aggregate (Onck and van der Giessen [20]). Fatigue cracking Among the many applications of continuum damage me- chanics [4], studies of failure by low cycle fatigue are an important example, where a material model directly based on the micro mechanics of failure has not been de- veloped. As the development of fatigue fracture depends strongly on the plastic strain range in each cycle, an accu- rate cyclic plasticity model is needed (e.g. Ohno and Wang [21]), with damage mechanics incorporated. The scalar damage parameter D is taken to be zero initially, but when the accumulated plastic strain p reaches a threshold value dp , it is assumed that damage starts to develop according to the evolution law 1 , if ( ) , 0 , if d d p pY D p p p pS α α ≥⎧ = = ⎨ <⎩ & & (5) Here, S is a material parameter describing the energy strength of damage, the strain energy release rate is given by ( )( )22 / 2 1e VY R E Dσ= − , and the expression for VR depends on the mean stress / 3 ,kkσ so that fatigue devel- ops more rapidly under tensile stresses. When the damage parameter reaches a critical value cD , this is taken to represent such a high density of microcracks that coales- cence into a macrocrack occurs. In a finite element analy- sis this failure event is represented in terms of the ele- ment vanish technique, such that the model can be used to predict the growth of a macroscopic crack. This type of numerical study has been carried out in [22] for a metal matrix composite, where the fatigue crack growth occurs in the metal matrix around short brittle fibres. 3 MODELLING BY COESIVE ZONE As an alternative to the continuum models discussed above, a number of crack growth analyses describe the fracture process separately in terms of a traction separa- tion law for the crack surface, while the inelastic defor- mations around the crack are accounted for by standard plasticity without damage. This gives an attractive possi- bility for separating effects of fracture process parameters from effects of the material parameters determining ine- lastic deformations, e.g. in relation to determining crack growth resistance curves. Thus, analyses of this type de- termine directly the ratio between the remote fracture toughness and the local fracture toughness determined by the assumed cohesive model. In [5] a rather general case of crack growth along the in- terface between an elastic-plastic solid and a rigid solid was studied. Here, a cohesive zone model was needed that accounts for both normal and tangential separation, or mixtures of these, not only in order to study effects of remote mixed mode loading, but also because of the os- cillating elastic singularity resulting from the elastic mismatch across the interface, which gives varying mix- tures of normal stress and shear stress along the interface. This work has been continued in a number of different studies of interface debonding, for different types of ma- terial systems. Thus, in [23] resistance curves have been determined numerically for crack growth along an inter- face joining two elastic-plastic solids, or an elastic-plastic solid to an elastic substrate. The steady-state value ss K of the remote fracture toughness is found when the resis- tance curves reach their maximum, which depends on the local mode mixity 0ψ near the crack-tip. As an example Fig. 3 shows such steady-state values for a case with an elastic substrate, where the elastic modulus 2E in the substrate is twice that in the elastic-plastic solid. The an- gular measure 0ψ is near o0 for mode I loading and would be near o90 or o90− for mode II loading. The steady-state toughnesses are normalised by the value 0K corresponding to a purely elastic solid, for the separation V. Tvergaard., Frattura ed Integrità Strutturale, 1 (2007) 25-28 28 energy assumed in the traction separation law. The dif- ferent curves correspond to different values of the peak stress σ̂ for the traction separation law, normalised by the initial yield stress. The curves show two typical fea- tures of such results, that the fracture toughness level is very sensitive to small increases of the peak stress, and that the curves have minima for near mode I conditions at the crack-tip. Figure 3: Steady-state interface toughness as a function of the local mixity measure 0ψ , for 1 1/ 0.003Y Eσ = and 2Yσ ∞ , considering different values of 1/ ,Yσ σ 2 1/ 2E E = . (From [23]). 4 REFERENCES [1] A.L. Gurson, Engng. Mater. Technol., 99 (1977) 2. [2] V. Tvergaard, Advances in Applied Mechanics, Academic Press, Inc., 27 (1990) 83. [3] V. Tvergaard, Acta Metallurgica, 32 (1984) 1977. [4] J. Lemaitre, A Course on Damage Mechanics. Springer-Verlag, (1992). [5] V. Tvergaard, J.W. Hutchinson, J. Mech. Phys. Sol- ids, 41 (1993) 1119. [6] V. Tvergaard, J. Mech. Phys. Solids, 30 (1982) 399. [7] V. Tvergaard, Int. J. 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Onck, E. van der Giessen, Mech. Mater, 26 (1997) 109. [21] N. Ohno, J.-D. Wang, Int. J. of Plasticity, 9 (1993) 375. [22] V. Tvergaard, T.Ø. Pedersen, Arch. Mech., 52 (2000) 799. [23] V. Tvergaard, J. Mech. Phys. Solids, 49 (2001) 2689.