A penny-shaped cohesive crack model for material damage

More documents

Recommendations

Info

304 G. Wang, S.F. Li / Theoretical and Applied Fracture Mechanics 42 (2004) 303–316In principle, a more realistic micromechanicsmodel and a feasible homogenization schemeshould lead to a better constitutive law at macrolevel.For Gurson model, failure mechanism dueto void growth is supported by many experimentalobservations on failures of ductile materials (e.g.[8,10,20,24,29]). On the other hand, in most brittle,quasi-brittle, and even some ductile materials(such as concrete, rocks, ceramics and some metals),materialÕs failure mechanism may be attributedto propagation, nucleation and coalescenceof micro-cracks as well. Although several microcrackbased damage models have been proposedto describe elastic damage processes (e.g.[4,9,15,16,19] and others), few micro-crack damagemodels are available for inelastic damageprocesses.The motif of contemporary micromechanics isaimed at discovering unknown but importanteffective constitutive information by homogenizingmassive numbers of micro-objects with simplestructures. In this paper, new cohesive damagemodels are derived based on homogenization ofrandomly distributed penny-shaped cohesive cracks(Barenblatt–Dugdale type [1,2,7]) in an elasticRVE. At micro-level, the cohesive damage modelmimics realistic interactions among atomistic bondforces at crack tips, hence it may capture the overalldamage effects due to crack opening andpropagation.In general, damage means material degradationcaused by defects or deformation. Damage may bedefined as surface separations, permanent latticedistortion, various irreversible effects due to endochronicdissipation etc. In the context of thispaper, the term damage is strictly referred to thematerial degradation due to specific defect—permanentcrack opening, or volume fraction of permanentmicro-crack opening, which may beviewed as a second phase in a composite material.This definition of damage has been extensivelyused in engineering literature. The definition ofdamage used in Gurson model (e.g. void growth)belongs to this category as well.To distinguish the damage caused by deviatoricstress, such as dislocation, disclination, and surfacesliding, with the damage caused by hydrostaticstress, such as permanent crack opening,may simplify constitutive modeling. Of course inreality, material damage may be susceptible andrelated to both hydrostatic and deviatoric stressstates and it is sensitive to their combinations.However, it is a reasonable approximation to assumethat the damage due to permanent crackopening is only related to hydrostatic stress state,which renders a tractable homogenization solution.Throughout this paper, vectors and tensors aredenoted as bold-face letters, while their componentsare written as italic. The symbol ÔÆÕ denotesinner product of two vectors a Æ b = a i b i or contractionof adjacent indices of a vector and a tensor(e.g. n Æ r = n i r ij ). The symbol Ô:Õ denotes aninner product of two second-order tensors (e.g.c:d = c ij d ij ) or a double contraction of adjacentindices of tensors of rank two and higher (eg.C:e = C ijkl e kl ).2. Average theorem for an RVE with cohesivecracksOne of the fundamental concept in classicalmicromechanics is so called representative volumeelement (RVE). An RVE for a material point usuallycontains a very large number of microstructuresand it is statistical representative of thelocal continuum properties. Despite heterogeneityof microstructures, averaged stress or strain propertiesof an RVE can be derived under specificboundary conditions. For an RVE with randomlydistributed cohesive cracks inside, traditionalmicromechanics averaging theory for traction-freedefects [23] cannot be applied. In this section, averagingtheorem for RVE containing cohesive defectswith constant cohesive traction would bederived for our purpose.Define the volume average operator h Æ i, anddefine macro stress tensor R as the volume averageof micro stress tensor in an RVE,R ¼hri ¼ 1 ZrdVð1ÞV VFirst, consider the average stress in a three-dimensionalelastic RVE with a single penny-shapedBarenblatt–Dugdale crack at the center. Neglect-
G. Wang, S.F. Li / Theoretical and Applied Fracture Mechanics 42 (2004) 303–316 305ing body force, the equilibrium equation inside anRVE takes the formrr ¼ 0 8x 2 V ð2ÞAssume that the prescribed tractions on the remoteboundary of the RVE (oV 1 ) are generatedby a constant stress tensor r 1 . Let oV ec denotetraction-free part of a cohesive crack surface,and let oV pz denote the cohesive part of the cracksurface where constant traction force t is applied.Using divergence theorem and Eq. (2), it isstraightforward to show thathri ¼ 1 ZrdV ¼ 1 Zfr ðr xÞg T dVV V V V¼ 1 ZZr 1 dV fn ðr xÞg T dSV VoVZecfn ðr xÞg T dSoV pzZ¼ r 1 1fn ðr xÞg T dSV oV pzZ¼ r 1 1x tdSð3ÞV oV pzwhere t is the constant cohesive traction.Note that oV pz = oV pz+ [ oV pz and the surfaceareas joV pzþ j¼joV pz j¼ 1 joV 2 pzj, where subscriptÔ+Õ and Ô Õ are used to distinguish upperand lower part of the crack surfaces. So the lastterm in Eq. (3) becomesZ1x tdS ¼ 1 V oV pzVZZx t þ dS þoV pzþoV pz¼ 12V x ðtþ þ t ÞjoV pz j¼ 0x t dS!ð4Þwhere t + = t are the cohesive tractions actingon oV pz+ and oV pz respectively.Therefore, the average stress inside the RVEwill equal to remote stressR ¼hri ¼r 1ð5ÞBy superposition, it is straightforward to generatethis result to an RVE with N cohesive cracks randomlydistributed inside (see Fig. 1),hri ¼r 1 1VX Na¼1ZoV ðaÞpzx t ðaÞ dS ¼ r 1ð6ÞFig. 1. Randomly distributed micro-cracks within an RVE.Hence, the averaging theorem follows:Theorem. Suppose1. an elastic representative volume element containsN Barenblatt–Dugdale penny-shaped cracks withcohesive tractions in the cohesive zones;2. the tractions on the remote boundary of the RVEis generated by a constant stress tensor, i.e.,t 1 = n Æ r 1 and r 1 is constant.Then, macro stress of the RVE equals to the remoteconstant stress, i.e. R = hri = r 1 .3. Penny-shaped crack under uniform triaxialtensionBefore homogenization, the analytical solutionof three-dimensional (3D) penny-shaped crack inan RVE that is under uniform triaxial tension isoutlined in this section (see Fig. 2).Penny-shaped Dugdale crack problem has beenstudied by several authors. The early contributionwas made by Keer and Mura, who used the Trescayield criterion to link the cohesive strength tomicro yield stress [17]. In their study, only uniaxialtension loading was considered. More recently,
Page 4 and 5: 306 G. Wang, S.F. Li / Theoretical
Page 10: 312 G. Wang, S.F. Li / Theoretical
Page 13 and 14: G. Wang, S.F. Li / Theoretical and

A penny-shaped cohesive crack model for material damage

Create successful ePaper yourself

Delete template?

Save as template?