This work deals with the formulation of a thermodynamically consistent, isotropic damage cohesive model for mixed-mode delamination under variable mode ratio. The proposed model is based on the introduction of an internal friction angle in the tensile case, that allows for an accurate modelling of the interaction between normal and shear openings

Confalonieri, F.

Perego, U.

F. Confalonieri

U. Perego

Archivio istituzionale della ricerca - Politecnico di Milano

6th ECCOMAS Thematic Conference on the Mechanical Response of CompositesCOMPOSITES 2017J.J.C. Remmers and A. Turon (Editors)A MIXED-MODE COHESIVE MODEL FOR DELAMINATIONWITHISOTROPIC DAMAGE AND INTERNAL FRICTIONFederica Confalonieri∗, Umberto Perego∗?Politecnico di MilanoPiazza Leonardo da Vinci, 32federica.confalonieri@polimi.it, umberto.perego@polimi.itKeywords: Mixed-mode delamination, cohesive modelSummary: This work deals with the formulation of a thermodynamically consistent, isotropicdamage cohesive model for mixed-mode delamination under variable mode ratio. The proposedmodel is based on the introduction of an internal friction angle in the tensile case, that allowsfor an accurate modelling of the interaction between normal and shear openings.1. INTRODUCTIONDelamination is a common failure mechanism in laminated composite materials, often char-acterized by mixed-mode loading conditions with variable mode ratio. The capability to accu-rately predict the progressive mixed-mode delamination in real life engineering applications isa key ingredient in the development of robust design tools. The computational strategies forthe numerical simulation of the delamination process are often based on the use of interfaceelements within the cohesive zone approach (see, for instance, [1, 2, 3]). Among the maindifficulties arising in the definition of a robust cohesive model, there are, on the one hand, thenecessity to ensure its accuracy and thermodynamic consistency for arbitrary loading paths and,on the other hand, the capability to correctly reproduce the typical growth of fracture energy inpassing from Pure Mode I to Pure Mode II. A number of experimental works on compositematerials (see e.g. [4] for a comprehensive literature survey) show that the fracture energy inPure Mode II is typically higher than in Pure Mode I, as a result of a change in the involved mi-cromechanical mechanisms, with a transition from pure mode I loading characterized by matrixcleavage and fiber pull-out, to mode II conditions dominated by the formations of cusps andhackles [5, 6].A new cohesive model, specifically conceived for mixed mode delamination with variablemode ratio, is presented in this work. The model is based on an isotropic damage formulationand is thermodynamically consistent. The coupling between the normal and the shear behaviouris achieved by projecting the cohesive stress vector onto three unit vectors defining three distinctdamage modes, accounting for mixed-mode interaction through a parameter playing the role ofan angle of internal friction. The proposed model is able to capture the non-monotonic increaseof fracture energy at increasing mode ratios without the need of introducing any empiricallaw, as demonstrated by the comparison between the outcome of the present cohesive law andthe data resulting from experimental tests performed with the Mixed Mode Bending test [7]apparatus.ISBN: 978-90-386-4366-3Federica Confalonieri, Umberto PeregoFigure 1. Damage modes2. FORMULATIONThe coupling between the normal and the shear behaviour is achieved by defining the threedamage modes, represented in Figure 1, in the plane of non-dimensional cohesive tractions (i.e.the cohesive stresses divided by the corresponding pure Mode peak values). Each damage modeis identified by a unit normal vector ni depending on an angle α playing the role of an angleof internal friction.The first damage mode is associated to an opening dominated decohesionprocess, while the second and third damage modes are the result of the interaction between shearand normal relative displacements. The unit vector n1 is associated to the opening-dominatedmode, while n2 and n3 define the two shear-dominated, symmetrical modes.n1 = [1 0]n2 = [sinα cosα] (1)n3 = [sinα − cosα]The modelling of the delamination under mixed-mode conditions with variable mode ratiosand arbitrary loading paths is obtained formulating an isotropic damage cohesive model in athermodynamically consistent framework. The free energy per unit surface Ψ is defined as:Ψ =12K (〈δn〉−)2 + 12(1− d)K (〈δn〉+)2 + 12(1− d)K (δs)2 (2)being δn and δs the normal and tangential relative displacements, K the elastic stiffness of theinterface and d the isotropic damage variable. The unilateral effect is accounted for by theintroduction of the Macauley brackets 〈 〉 that allow to distinguish between the tensile and thecompressive cases. Within a classical thermodynamic framework, the cohesive tractions tn andts and the strain energy per unit of damage growth Y can be defined by means of the followingstate equations:Federica Confalonieri, Umberto Peregotn =∂Ψ∂δn= K〈δn〉− + (1− d)K〈δn〉+ (3)ts =∂Ψ∂δs= (1− d)Kδs = (1− d) ts0δs0δs (4)Y = −∂Ψ∂d=12K (〈δn〉+)2 + 12K (δs)2 (5)with tn0 = Kδn0 and ts0 = Kδs0, being tn0 and ts0 the strengths in pure Modes I and II and δn0 andδs0 the relative displacements at the onset of delamination, i.e. corresponding to tn0 and ts0. Forthe sake of simplicity, only the tensile case, i.e. δn ≥ 0, will be considered in the following.Let us introduce the vector t, collecting the non-dimensional cohesive tractions, i.e:t =[tnts ]T=[tntn0tsts0]T(6)A vector of effective cohesive stresses s is introduced to account for the interaction betweennormal and tangential behaviour: each component si, i = 1, 2, 3 is computed by projecting tonto the direction defined by one of the three normals in eqn. 1, identifying a distinct damagemode:s1 = tTn1 = tns2 = tTn2 = tnsinα + tscosα (7)s3 = tTn3 = tnsinα− ts cosαIn matrix form:s = Nt (8)where matrix N = [n1 n2 n3]T gathers the unit vectors ni. The same approach can be followedalso for the relative displacements. Let δ define the vector collecting the non-dimensionalopening displacements in the normal and in the tangential directions as:δ =[δnδs ]T=[δnδn0δsδs0]T(9)A vector w of effective relative displacements, conjugated to the effective cohesive stresses sin the expression of the free energy density Ψ, is defined by introducing three structural vectorsmi, defined as:m1 = a [1 0]Tm2 = b [sin θ cos θ]T (10)m3 = b [sin θ − cos θ]TFederica Confalonieri, Umberto PeregoAs depicted in figure 1, m1 is aligned to n1 for symmetry considerations, while the orientationofm2 andm3 is defined as a function of the angle θ, different from the angle of internal frictionα. The effective relative displacements wi, i = 1, 2, 3 are obtained by the the projection of thenon-dimensional relative displacements vector δ onto a structural vector mi as:w1 = δTm1w2 = δTm2 (11)w3 = δTm3In matrix form:w = Mδ (12)being M the matrix gathering the components of the three structural vectors mi.The strain energy Ψ can be expressed as a function of the effective quantities defined ineqns. 7 and 11 to underline the contribution of each single damage mode.Ψ =12tTδ =12s1w1︸ ︷︷ ︸Ψ1+12s2w2︸ ︷︷ ︸Ψ2+12s3w3︸ ︷︷ ︸Ψ3(13)The two unknown constants a and b are, then, determined by imposing that Ψ remains the samein passing from the direct to the effective variables, i.e.12(tnδn + tsδs) =12(s1w1 + s2w2 + s3w3)(14)thus, obtaining:a = (tn0δn0 − ts0δs0 tanα tan θ) b =ts0δs02 cosα cos θ(15)By considering the strain energy Ψi associated to each damage mode, the following stateequations define three effective strain energies release rates Y i:Y 1 = −∂Ψ1∂d=12(tn0δn0 − ts0δs0 tanα tan θ)(δn)2Y 2 = −∂Ψ2∂d=14ts0δs0[tanα tan θ(δn)2+(δs)2+ (tanα + tan θ) δnδs](16)Y 3 = −∂Ψ3∂d=14ts0δs0[tanα tan θ(δn)2+(δs)2− (tanα + tan θ) δnδs]that play the role of driving forces acting on the corresponding damage modes, with:Y 1 + Y 2 + Y 3 =12tn0δn0(δn)2+12ts0δs0(δs)2= Y ≥ 0 (17)Federica Confalonieri, Umberto PeregoFigure 2. First activation domain: a) for increasing values of α and k = 4; b) for increasingvalues of k and α = 30◦In the following, it is assumed that:θ = arctan(tn0δn0ts0δs0tanα)(18)so that in the tensile case Y 1 ≥ 0 for α < 45◦. On the contrary, either Y 2 or Y 3 can be negative,but their sum Y 2 + Y 3, representing the fraction of the strain energy release rate associated tothe shear-dominated damage modes, is always non-negative.The damage activation criterion is written as a function of the strain energy release rates Y ias:ϕ =(Y 1χ10 + χ1)k+H(Y 2)(Y 2χ20 + χ2)k+H(Y 3)(Y 3χ30 + χ3)k− 1 ≤ 0 (19)being the exponent k an input parameter of the model, H() the Heaviside function, introducedto avoid negative contributions of Y 2 and Y 3 to the damage activation function, and χi + χi0the current threshold for the i-th damage mode, evolving during the decohesion process as afunction of the damage variable d. χi0 represents the initial threshold, while χi is the internalvariable, governing the threshold evolution for increasing damage and determines the shape ofthe softening branch. Figure 2 shows the damage activation surface at the onset of decohesion,for increasing values of the internal friction angle α and k = 4 (a) and for increasing values ofthe exponent k, while maintaining a constant value of the angle α = 30◦ (b).In the framework of a classical thermodynamic formulation, the cohesive model is com-pleted by the introduction of the damage evolution law and of the Kuhn-Tucker conditions. Thedamage rate is obtained by imposing that ϕ = 0 and ϕ˙ = 0, i.e.:d˙ = −∂φ∂δnδ˙n + ∂φ∂δsδ˙s∂φ∂d=∑3i=1(∂φ∂Y i∂Y i∂δn)δ˙n +∑3i=1(∂φ∂Y i∂Y i∂δs)δ˙s∑3i=1(∂φ∂χi∂χi∂d) (20)Federica Confalonieri, Umberto PeregoFigure 3. Bilinear traction-separation laws in pure Modes I and IIwhile the loading-unloading conditions read as:φ ≤ 0 d˙ ≥ 0 φd˙ = 0 (21)The thermodynamic consistency of the cohesive model can be proven considering the Clausius-Duhem inequality for isothermal processes and showing that the mechanical dissipation is al-ways non negative.D = Y 1d˙+ Y 2d˙+ Y 3d˙ =(Y 1 + Y 2 + Y 3)d˙ = Y d˙ ≥ 0 (22)The expressions of χi0 and χi can be found by prescribing a fixed functional form for thetwo traction-separation laws in pure Modes. In this work, the case of a bilinear shape, shownin Figure 3, is envisaged. Under this hypothesis, the two branches of the pure Mode traction-separation law depicted in Figure 4 can be described as:t =t0δ0δ for δ ≤ δ0 (23)t = (1− d) t0δ0δ = t0δcr − δδcr − δ0 for δ ≥ δ0 (24)From eqn. 24 it is possible to derive the relation between the damage variable d and the relativedisplacement δ for δ ≥ δ0, i.e.δ =δcrδ0δcr − (δcr − δ0)d (25)Pure Mode I and II conditions can be retrieved by imposing that:δn 6= 0 δs = 0 for pure Mode I (26)δn = 0 δs 6= 0 for pure Mode II (27)Federica Confalonieri, Umberto PeregoThe effective strain energies per unit of damage growth under pure Mode I and II loadingconditions can be found by substituting eqns. 27 into eqn. 16:Y 1mI =12(tn0δn0 − ts0δs0 tanα tan θ)(δn)2Y 2mI =14ts0δs0(tanα tan θδn)s(28)Y 3mI =14ts0δs0(tanα tan θδn)sY 1mII = 0Y 2mII =14ts0δs0(δs)s(29)Y 3mII =14ts0δs0(δs)swith the subscriptsmI andmII denoting pure Mode I and Mode II loading conditions. It couldbe noticed that the behaviour in pure Mode II is uncoupled since Y 1mII = 0, while in pure ModeI one has Y 2mI 6= 0 and Y 3mI 6= 0. For that reason, it is necessary to deal first with the pure ModeII case and, then, treat the Pure Mode I case on the basis of the results obtained for Mode II.Because of the symmetry of the two shear dominated damage modes with respect to theaxis ts = 0, it turns out that Y 2mII = Y3mII , χ20 = χ30 and χ2 = χ3. The initial thresholds χ20and χ30 can be determined by imposing that the activation criterion is fulfilled at the onset ofdelamination, i.e. ϕ = 0 for d = 0 and δs= 1.ϕ =(Y 2mII |1χ20)k+(Y 3mII |1χ30)k− 1 = 2(Y 2mII |1χ20)k− 1 = 0 → χ20 = χ30 = 21k14ts0δs0 (30)Similarly, the expressions of χ2 and χ3 can be found by imposing that the activation criterion ismet for a generic point belonging to the softening branch, i.e that the activation function is zerofor d > 0 and δs> 1:φ =(Y 2mIIχ20 + χ2)k+(Y 3mIIχ30 + χ2)k− 1 = 0 → χ2 = χ3 = 2 1k 14ts0δs0[(δs)2− 1](31)By substituting eqn. 25 into eqn. 31, one obtains:χ2 = χ3 = 21k14ts0δs0[δscrδscr − (δscr − δs0) d]2− χ20 (32)Let us now focus on pure Mode I. The expressions of χ10 and χ1 can be determined with ananalogous procedure, considering the delamination onset and the softening branch, respectively.Federica Confalonieri, Umberto PeregoFigure 4. Pure mode bilinear lawFigure 5. Mixed-mode traction-separation lawFor δn= 1 and d = 0, it holds that:ϕ =(Y 1mI |1χ10)2+(Y 2mI |1χ20)2+(Y 3mI |1χ30)2− 1 = 0 → χ10 =12(tn0δn0 − ts0δS0 tanα tan θ)[1− (tanα tan θ)2] 12(33)while for a generic non-dimensional opening displacement δn> 1 one has:ϕ =(Y 1mIχ10 + χ1)k (Y 2mIχ20 + χ2)k+(Y 3mIχ30 + χ2)k− 1 = 0 (34)andχ1 =(tn0δn0 − ts0δS0 tanα tan θ){1−[(δncrδscrδscr−(δscr−δs0)dδncr−(δncr−δn0 )d)2tanα tan θ]k} 1k 12(δncrδncr − (δncr − δn0 )d)2− χ10 (35)The resulting mixed-mode response is depicted in Figure 5. Even though the pure Modeslaws are assumed to be bilinear, the softening branch in the mixed-mode law is, in general,Federica Confalonieri, Umberto Peregocurvilinear, except that for the particular case of identical pure Modes, i.e. tn0 = ts0 = t0,δn0 = δs0 = δ0 and δncr = δscr = δcr. In this particular case, the resulting mixed-mode traction-separation law can be determined analytically in the case of a radial path, expressing the non-dimensional normal and tangential relative displacements as:δn= (1− β) δ δs = βδ (36)By substituting eqns. 36 into enqs. 16 and 19, the activation function turns out to be:ϕ =(1− β)2 [1− (tanα)2k] 1k δ2[δcrδcr−(δcr−δ0)d]2k+ [(1− β) tanα + β]2 δ221/k[δcrδcr−(δcr−δ0)d]2k++ [(1− β) tanα− β]2 δ221/k[δcrδcr−(δcr−δ0)d]2k− 1 = 0 (37)being α = θ according to eqn. 18. Solving eqn. 37 for d, the following relationship betweenthe damage variable d and δ holds:d =δcrδcr − δ0(1Cβαδ)for ϕ = 0 withCβα ={(1− β)2k [1− tan2k α]+ 0.5 [(1− β) tanα + β]2k + 0.5 [(1− β) tanα− β]2k} 12k(38)Thus, from eqns. 3 and 4, for ϕ = 0 one obtains:tn = (1− β) t0[−Cβαδ0δ + δcr] (39)ts = βt0[−Cβαδ0δ + δcr] (40)3. NUMERICAL EXAMPLESOne of the strengths of the proposed cohesive model is that the overall fracture energy is anoutcome of the interaction between damage modes without the need of introducing any empiri-cal law and without making any assumption on the loading path. In these numerical examples,the effectiveness of the model in capturing the variation of the fracture energy with the mode-mixity ratio is assessed considering the experimental data of three different fiber reinforcedcomposite materials, namely HMF/5322 I[8], IM7/8552 [9] and AS4/PEEK [10], resulting fromMixed Mode Bending tests [7]. The input parameters required for the definition of the proposedcohesive model, i.e. the fracture energies GIc, GIIc and the peak tractions tn0 , ts0 in pure ModesI and II, the internal friction angle α and the exponent k appearing in the activation function φ,Federica Confalonieri, Umberto PeregoMaterial GIc GIIc t0n t0s K α kmJmm2mJmm2MPa MPa Nmm2degHMF/5322 0.3043 0.8039 10 18 10,000 32 10IM7/8552 0.2085 0.7713 60 90 20,000 36 10AS4/PEEK 0.969 1.719 75 80 100,000 42.8 1.56Table 1. Adopted parameters.Figure 6. HMF/5322. Experimental vs numerical mixed-mode fracture energies. Dots: experi-mental data [8]. Solid line: result of the present model.are listed in Table 1. Figures 6, 7 and 8 show the comparison between the experimental data andthe curve obtained with the proposed cohesive law: in all the three cases a good agreement canbe observed. In addition, for comparison purposes, the empirical Benzeggagh-Kenane (B-K)law [11] Gc = GIc + (GIIc −GIc)(GIIGI+GII)ηis depicted in Figures 7 and 8, considering theexponent η determined in [9] and [10] by a fitting of the experimental data.4. CONCLUSIONSA new isotropic damage cohesive model for the simulation of mixed-mode delaminationhas been presented in this work. The proposed model is based on the introduction of a dissipa-tion mechanism described by a parameter qualitatively similar to an angle of internal friction,leading in a natural way to a coupling between normal and shear behaviours. The model isthermodynamically consistent and is able to accurately reproduce the fracture energy undermixed-mode loading conditions, as shown in the numerical examples.5. AcknowledgementsThe financial support of Tetra Pak Packaging Solutions is gratefully acknowledged.Federica Confalonieri, Umberto PeregoFigure 7. IM7/8552. Experimental vs numerical mixed-mode fracture energies. Dots: experi-mental data [9]. Dashed line: B-K law [11]. Solid line: result of the present model.Figure 8. AS4/PEEK. Experimental vs numerical mixed-mode fracture energies. Dots: experi-mental data [10]. Dashed line: B-K law [11]. Solid line: result of the present model.Federica Confalonieri, Umberto PeregoReferences[1] P.P. Camanho, C.G. Davila, M.F. de Moura, Numerical Simulation of Mixed-mode Pro-gressive Delamination in Composite Materials. Journal of Composite Materials, 37, 1415-1438, 2003.[2] M.J. Van den Bosch, P.J.G. Schreurs, M.G.D. Geers, An improved description of the ex-ponential Xu and Needleman cohesive zone law for mixed-mode decohesion. EngineeringFracture Mechanics, 73, 1220-1234, 2006.[3] K. Park, G.H. Paulino, J.R. Roesler, A unified potential-based cohesive model of mixed-mode fracture. Journal of the Mechanics and Physics of Solids, 57, 891-908, 2009.[4] B.D. Davidson, W. Zhao, An accurate mixed-mode delamination failure criterion for lam-inated fibrous composites requiring limited experimental input. Journal of Composite Ma-terials, 41, 679-702, 2007.[5] E.S. Greenalgh, C. Rogers, P. Robinson, Fractographic observations on delaminationgrowth and the subsequent migration through the laminate. Composites Science and Tech-nology, 69, 2345-2351, 2009.[6] R. Marat-Mendes, M. de Freitas, Fractographic analysis of delamination in glass/fibreepoxy composites. Journal of Composite Materials, 47, 1437-1448, 2013.[7] J.R. Reeder, J.H. Crews, Mixed-mode bending method for delamination testing. AIAAJournal, 28, 1270-1276, 1990.[8] N.B. Adeyemi, K.N. Shivakumar, V. S. Avva, Delamination fracture toughness of woven-fabric composites under mixed-mode loading. AIAA journal, 37, 517-520, 1999.[9] R. Krueger, Development and application of benchmark examples for mixed-mode I/IIquasi-static delamination propagation predictions. NASA report, 2012.[10] P. Naghipour, M. Bartsch, L. Chernova, J. Hausmann, H. Voggenreiter, Effect of fiberangle orientation and stacking sequence on mixed mode fracture toughness of carbon fiberreinforced plastics: Numerical and experimental investigations. Materials Science andEngineering: A, 527, 509-517, 2010.[11] M.L. Benzeggagh, M. Kenane, Measurement of mixed-mode delamination fracture tough-ness of unidirectional glass/epoxy composites with mixed-mode bending apparatus. Com-posites Science and Technology, 56, 439–449, 1996.

A mixed-mode cohesive model for delamination with isotropic damage and internal friction

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A mixed-mode cohesive model for delamination with isotropic damage and internal friction

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