Igor M. Dunaev

Vladislav I. Dunaev

Comptes Rendus Mécanique (CRM)

t.defect. Inolidconddy-odèle, onième mo-n dente.C. R. Mecanique 332 (2004) 789–794Analysis of the thermodynamic conditions for brittle fractureIgor M. Dunaeva,∗, Vladislav I. Dunaevba MSD, Kuban State Technology University, Moskovskaya str. 2, Krasnodar, 350072, Russiab AMD, Kuban State University, Stavropolskaya str. 149, Krasnodar, 350040, RussiaReceived 15 April 2003; accepted after revision 1 June 2004Available online 10 August 2004Presented by Évariste Sanchez-PalenciaAbstractThe analysis of the thermodynamic conditions for brittle fracture is given for the two known models of an isolated defecIn the first model the stresses on the external surface of the solid remain the same as before and after formation of athe second model the displacements on the external surface ofth solid at a defect formation remain the same as in the swithout a defect. It is shown that the first model in the isothermal case of deformation leads to Griffith condition and the semodel leads to the other proposed energy condition of fracture which, unlike the Griffith condition, containsan increment of theentropy component of the internal energy which is not zero in a general case.To cite this article: I.M. Dunaev, V.I. Dunaev,C. R. Mecanique 332 (2004). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméAnalyse des conditions thermodynamique pour la rupture fragile. On présente une analyse des conditions thermonamiques pour la rupture fragile, dans le cadre des deux modèles du défaut isolé bien connus. Dans le premier mmaintient les mêmes contraintes à la surface extérieure du solide avant et après l’apparition du défaut. Dans le deuxdèle, ce sont les déplacements que l’on garde constants. Nous démontrons que le premier modèle conduit à la conditioGriffith dans le cas d’une déformation isothermique, tandis que le deuxième modèle produit une condition bien différeSelon cette condition, les variations de l’entropie ne sont plus nulles, contrairement à la condition de Griffith.Pour citer cetarticle : I.M. Dunaev, V.I. Dunaev, C. R. Mecanique 332 (2004). 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.Keywords: Rupture; Energy condition; Internal energy; EntropyMots-clés : Rupture ; Condition sur l’énergie ; Energie interne ; Entropie* Corresponding author.E-mail address: idunaev@hotmail.com (I.M. Dunaev).1631-0721/$ – see front matter 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crme.2004.06.008790 I.M. Dunaev, V.I. Dunaev / C. R. Mecanique 332 (2004) 789–794idera-onfssmentssion ofe criticalteds on the,s to beher-ed.icaly-entropymics forid withwhich,eGriffithntropylye1. IntroductionGriffith [1,2] was the first to approach the theory of brittle fracture from the point of view of energy constions and he proposed an energy condition of brittle fracturedUp − γp dΣ = dA (1)Here,Up = U(0)p − U(1)p is the change of a potential energy caused by the defect formation;U(0)p andU(1)p arepotential energies of a solid without and with defect, respectively;γp is the energy necessary for the formatiof a unit area of defect surfaceΣ , which is equal to the specific surface energy;A is the change of the work oexternal forces during a defect formation. It is well-known that condition (1) is inapplicable for strength asseat compression. In particular, for problems of material fracture under uniaxial and biaxial tension/comprea plate with a defect, the condition (1) leads to the temperature independent and identical in absolute valustresses, contradicting experimental data for practicallyall known materials. This short-coming was not eliminausing assumption [2,3] that defects are closing at compression, consequently the sliding friction appeardefect surface. However, to explain the compressive streng h exceeding (in absolute value) the tensile strengthin the calculations using this assumption [2,3] the value of sliding friction coefficient on defect surface haaccepted improbably large in comparison with experimentaldata. In numerous investigations [2,4,5] based on tmodynamics, a generalized energy approach to the fracture was proposed, and in particular condition (1) obtainHowever, in these investigations the authors did not state explicitly the fundamental physical and mathematassumptions of their models of fracture leading to condition (1). In particular, they did not precise the physical hpotheses and mathematical substantiations in their models of fracture on which basis the increment of thecomponent of internal energy was assumed equal to zero.In this Note the necessary energy conditions of fracture are formulated on the basis of thermodynathermoelastic deformation of solids (Section 2). Analyzing the conditions for the two known models of a sola defect (Section 3), it is shown that one of the models leads to a Griffith type energy condition of fracture,unlike condition (1), contains an increment of an entropy component of the internal energy which is nonzero in thgeneral case. This condition allows the elimination of the above mentioned short-comings of the originaltheory. Another model of a solid with a defect leads to a thermodynamically incomplete (increment of the ecomponent of the internal energy is zero) condition (1) and, therefore, to physically groundless and experimentalunsupported results.2. Necessary energy conditions of fractureLet us consider a linear thermoelastic solid where a defect (a crack) is forming and propagating. Then, wintroduce notations for values with indexes(0) and (1) for the solid without and with a defect respectively:u(0),η(0), ε(0)ij , σ(0)ij , u(0)i , T(0), V0 is a volume,S0 is surface area,u(1), η(1), ε(1)ij , σ(1)ij , u(1)i , T(1), V1, S1 = S0 + Σ ,whereΣ is the surface area of the defect. Here, indexes(0) and(1) are used for:u the specific internal energy,η thespecific entropy,εij the components of the strain tensor,σij the components of the stress tensor,ui the componentsof the displacement vector,T the absolute temperature. For the thermoelastic solid we have [6]:u = 12σij εij + 32α0K0Θ(T − T0) + 3α0K0T0Θ + cε(T − T0) (2)η = 3α0K0Θ + cε ln TT0≈ 3α0K0Θ + cε T − T0T0, atT − T0T0 1 (3)εij = 12(∂ui∂xj+ ∂uj∂xi), σij = 2µεij + λΘδij − 3α0K0(T − T0)δij , i, j = 1,2,3 (4)I.M. Dunaev, V.I. Dunaev / C. R. Mecanique 332 (2004) 789–794 791n is per-,ervationefecteat andtynd (6)dng andsd by onlyn) letwherexi are Cartesian coordinates of the solid points. In the expression (2) and beyond, the summatioformed over repeating indicesi, j , Θ = εij δij is the first invariant of the strain tensor,δij is the Kronecker deltaα0 is the linear coefficient of thermal expansion,cε is the specific heat at constant strain,µ andλ are Lame’scoefficients, 2µ = E/(1 + ν), λ = νE/[(1 + ν)(1 − 2ν)], 3K0 = E/(1 − 2ν), E is the Young’s modulus,ν isthe Poisson’s ratio. Using the integral form of the first and second laws of thermodynamics and the consmechanical energy theorem [3,6–8], let us write the energy conditions for formation and propagation of the d(crack)dK + dU − dU∗ = dA + dQ (5)dS − dS∗ =∫V0dQ(0)T (0)dV −∫V1dQ(1)T (1)dV (6)In expressions (5) and (6)K = K(0) − K(1), U = U(0) − U(1), A = A(0) − A(1), Q = Q(0) − Q(1), S = S(0) − S(1) (7)U(q) =∫Vqu(q) dV, S(q) =∫Vqη(q) dV , q = 0,1are the respective changes of the kinetic energy, internal energy, work of the external forces, influx of hentropy, caused by defect propagation during timet ,U∗ =∮Σu∗ ds = γΣ, S∗ =∫Ση∗ ds (8)are the internal energy of defect formation of a surfaceΣ and the entropy of defect formation, ds is an elemenof surface area of defect,u∗ > 0 is a specific internal energy,γ is an average specificinternal energy,η∗ is thespecific entropy. The condition (5) is the thermodynamic requirement to define crackpropagation. The entropof defect formation (8) could be obtained from Eq. (6). To solve applied problems, the conditions (5) amust be complemented by geometrical, physical and mathematical model of the defect, loading conditions analso by the condition which determines the relative position of the defect and it trajectory. In static loadiT (0) = T (1) = T0 = const, dK = 0, dQ = 0 from conditions (5) and (6) considering (8) we find:dU − γ dΣ = dA (9)dS − dS∗ = 0 (10)With these conditions, unlike condition (1), the internal energy (2) includes the entropy component (3) andγ =const has another, above mentioned,physical meaning. Further it will be shown with what additional assumptionthe conditions (9) and (10) lead to the condition (1). In the case when the defect surface may be determineone parametera (e.g. a half-length of a crack or large half-axis of an ellipse form defect in plain deformatious rewrite condition (9) as:dWdaȧ = 0, W = U − γΣ − A (11)If ȧ = 0 then there is no crack propagation. Thus the criteria of probable propagation of the crack is:dWda= 0, ȧ  0 (12)but we do not know whether it will actually happen or not (so thatȧ = 0 cannot be excluded) [7].792 I.M. Dunaev, V.I. Dunaev / C. R. Mecanique 332 (2004) 789–794modelsest. Inhsurface ofon3. Energy condition for fracture using two models for an isolated defectLet us consider in detail the energy condition (12) at plane stressed (strained) state for the two knownof an isolated defect. In model(A) the same stresses are prescribed onthe external surface of the solidS0 beforeand after formation of the defect, and stresses on the surface of the defectΣ are equal to zero. External forcin this model produce work on the external surfaceS0 on displacement, caused by the formation of the defecmodel(B), on the external surface of the solidS0 before and after formation of the defect, displacements whiccorrespond to the applied load are fixed, but before the defect was formed. Stresses are also zero on thethe defectΣ in model(B). The work of external forces on the external surface of the solidS0 uring formation ofthe defect is dA = 0 as displacements are fixed. The integrals of the int rnal energy (7) taking into considerati(2)–(4) atT (0) = T (1) = T0 may be writtenU = (U(0)p − U(1)p ) + T0(S(0) − S(1))= 12(∫V0σ(0)ij ε(0)ij dV −∫V1σ(1)ij ε(1)ij dV)+ α0T0k1(∫V0ε(0)ij δij dV −∫V1ε(1)ij δij dV)(13)wherek1 = E/(1− ν) is for the plane stressed state,k1 = E/(1− 2ν) is for the plane deformation,i, j = 1,2. Letus compute the increment of the total energyW (11). For model(A), using relations (13) and (4) atT = T0, Betti’sreciprocal theorem [6], the expression for the work of external forces, and Airy stress functionF ,σ(0)ij ε(1)ij = σ (1)ij ε(0)ij , A =∮S0σ(0)ij(u(0)i − u(1)i)nj ds, σ(q)11 =∂2F (q)∂x22, σ(q)22 =∂2F (q)∂x21(14)respectively, we obtainW = 12∫V1(σ(0)ij + σ (1)ij)(ε(0)ij − ε(1)ij)dV + 12∫Vσ(0)ij ε(0)ij dV −∮S0σ(0)ij(u(0)i − u(1)i)nj ds+ α0T0χ[∫V0(∂2F (0)∂x21+ ∂2F (0)∂x22)dV −∫V1(∂2F (1)∂x21+ ∂2F (1)∂x22)dV]− γΣ (15)HereV = V0 − V1, χ = 1 andχ = 1+ ν for the plane stressed state and theplane deformation, respectively,njare direction cosines of the external normaln to the solid boundary. Using [6] on the boundariesS0 andS0 + ΣexpressionsF(q)x1 (s) =∂F (q)∂x1= −s∫0σ(q)nx2(s)ds + Aq, F (q)x2 (s) =∂F (q)∂x2=s∫0σ(q)nx1(s)ds + Bqσ(q)nx1 = σ (q)11 n1 + σ (q)12 n2, σ (q)nx2 = σ (q)12 n1 + σ (q)22 n2, q = 0,1 (16)Green’s formula and equations of equilibrium∂σ(0)ij∂xi= 0, ∂σ(1)ij∂xi= 0, i, j = 1,2 (17)the expression (15), considering the formula (4), may be writtenW = 12∮S0+Σ(σ(0)ij + σ (1)ij)(u(0)i − u(1)i)nj ds + 12∮Σσ(0)ij u(0)i nj ds −∮S0σ(0)ij(u(0)i − u(1)i)nj dsI.M. Dunaev, V.I. Dunaev / C. R. Mecanique 332 (2004) 789–794 793{ [ ] [ ]e internalula andf the(23)lem+ α0T0χ∮S0s∫0(σ (1)nx1 − σ (0)nx1)ds dx1 +s∫0(σ (1)nx2 − σ (0)nx2)ds dx2+∮Σ( s∫0σ (1)nx1 ds)dx1 +( s∫0σ (1)nx2 ds)dx2}− γΣ,∮ΣAq dx1 + Bq dx2 = 0 (18)So, since far from the boundary conditions (16) ina solid with and without the defect in model(A) we haveσ (0)nx1 = σ (1)nx1, σ (0)nx2 = σ (1)nx2 at (x1, x2) ∈ S0; σ (1)nx1 = σ (1)nx2 = 0 at(x1, x2) ∈ Σ (19)then the last two integrals in the expression (18) are equal to zero. Therefore, the entropic component of thenergy (13) also is equal to zero. Then, on account ofthe condition (19) and the energy condition (12) we finallyobtain the increment of the total energyW (11) and Griffith criterion (1) in the formW = 12∮Σσ(0)ij u(1)i nj ds − γΣ,dWda= dda(12∮Σσ(0)ij u(1)i nj ds − γΣ)= 0 (20)Let us compute the increment of the total energy (11) for model(B), using the conditions on the boundaryS0 of asolid with and without a defect and on the boundary of the defectΣu(0)i = u(1)i at (x1, x2) ∈ S0, σ (1)nx1 = σ (1)nx2 = 0 at(x1, x2) ∈ Σ (21)In this case the work of external forces is equal to zero. Using the relations (13) and (14), Green’s formEqs. (17), (4) for expression (11) we obtainW = U − γΣ = 12∮S0+Σ(σ(0)ij + σ (1)ij)(u(0)i − u(1)i)nj ds + 12∮Σσ(0)ij u(0)i nj ds+ α0T0k1[ ∮S0+Σ(u(0)i − u(1)i)δij nj ds +∮Σu(0)i δij nj ds]− γΣ (22)Then, on account of conditions (21) we reduce the expressions (22) and (12) todWda= dda(12∮Σσ(0)ij u(1)i nj ds + α0T0k1∮Σu(1)i δij nj ds − γΣ)= 0 (23)Therefore, the model of the defect(B) leads to condition (23) in which increment of the entropy component ointernal energy in a general case isnot zero. This is the main essential difference of the proposed conditionfrom the Griffith criterion (20).4. Test problemLet us consider the problem of the determination of critical stresses in a circular plate, radiusb with a roundshaped defect radiusa under a symmetrical loadP in the case of a plane deformation. The solution of the probof elasticity theory is:u(1)1 = c1r +c2r, u(1)2 = 0, σ (1)12 = σ (1)rϑ = 0σ(1)11 = σ (1)r =E1+ ν(c11− 2ν −c2r2), σ(1)22 = σ (1)ϑ =E1+ ν(c11− 2ν +c2r2)(24)u(0)1 =P(1− 2ν)r1+ ν , u(0)2 = 0, σ (0)11 = σ (0)r = σ (0)ϑ = P, σ (0)12 = σ (0)rϑ = 0794 I.M. Dunaev, V.I. Dunaev / C. R. Mecanique 332 (2004) 789–794ineecurvealityofwherer, ϑ are the polar coordinates of the solid points,c1, c2 are the integration constants which we determfrom the boundary conditions in models(A) and (B). For model(A), using the boundary conditions atr = a,σ(1)11 = 0, r = b, σ (1)11 = P and the solution (24), from condition (20) atn1 = 1, n2 = 0, Σ = 2πa, after integrationwe obtain equal (in absolute value) critical stresses at tensionP+ and compressionP−P± = ±√γE(1− a2/b2)2a(1− ν2)For model(B) using the boundary conditions atr = a, σ (1)11 = 0, atr = b, u(1)1 = u(0)1 , the solution (24) and thcondition (23) we obtain similarlyP± = −α0T0k1 ±√(α0T0k1)2 + γE2a(1− ν2)[1+ a2b21(1− 2ν)]2(25)The critical stresses (25) at tension and compression are different and obviously depend onT0 andα0. In [9], thesolution similar to (25) was obtained for a defect in the form of a crack. In [10], using condition (23), aof fracture in the form of an ellipse in the space of principle stressesP1, P2 has been obtained with additionlimitations imposed upon the physical and mathematical models of the defect, conditions of isotropy and convexP 21 + P 22 − 2ν∗P1P2 + 2α0T0k1(1− ν∗)(P1 + P2) = 32µ(1− ν∗)γ /[(æ+ 1)πa∗](26)wherea∗ is a half-length of a crack critical dimension,ν∗ ∈ (−1,1), æ= 3 − 4ν, æ= (3 − ν)/(1 + ν) for plaindeformation and plane stressed state, respectively.5. ConclusionThe work proposes an energy condition (23) for the brittle fracture of solids on the basis of thermodynamicsa thermoelastic deformation, considering an increment of the entropy component of the internal energy.References[1] A.A. Griffith, The phenomena of rupture and flow in solids, Philos. T. Roy. Soc. A 221 (1921) 163–198.[2] H. Liebovitz (Ed.), Fracture, an Advanced Treatise,vol. 2, Math. Fundamentals, Academic Press, NY and London, 1968.[3] F.A. McClintok, J.B. Walsh, in: Int. Proc. of the 4th U.S. National Congress of Appl. Mechanics, 1962, pp. 1015–1031.[4] G. Cherepanov (Ed.), Fracture, a Topical Encyclopedia of Current Knowledge, Krieger, Malabar, FL, USA, 1998.[5] H.D. Bui, Mécanique de la Rupture Fragile, Masson, Paris, 1977.[6] W.K. Nowacki, Thermoelasticity, Pergamon Press, Oxford, 1986.[7] G.A. Maugin, The Thermomechanics of Plasticity and Fracture, Cambridge University Press, 1992.[8] A.A. Il’yshin, Continuum Mechanics, Mosk. Gos. University, Moscow, 1990 (in Russian).[9] I.M. Dunaev, V.I. Dunaev, On the energy condition for fracture of solids, J. Dokl. Phys. 45 (5) (2000) 213–215.[10] I.M. Dunaev, V.I. Dunaev, in: ICF Int. Conf. Fracture at Multiple Dimensions, Moscow, 2003, p. 19.

Analysis of the thermodynamic conditions for brittle fracture

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Analysis of the thermodynamic conditions for brittle fracture

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