A 3D delamination modelling technique based on plate and interface theories for laminated structures

A 3D delamination modelling technique based on plate and interface theories for laminated structures
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  European Journal of Mechanics A/Solids 24 (2005) 127–149 A 3D delamination modelling technique based on plateand interface theories for laminated structures Domenico Bruno ∗ , Fabrizio Greco, Paolo Lonetti  Department of Structural Engineering, University of Calabria, Cosenza, Italy Received 11 May 2004; accepted 30 November 2004Available online 6 January 2005 Abstract In this work a technique for analyzing mixed mode delamination problems in laminated composite plates under generalloading conditions is developed. The technique adopts the first-order shear deformable laminated plate theory and the interfacemethodology, which in turn is based on fracture and contact mechanics. In the thickness direction, an assembly of laminatedplate and interfaces models the laminated structure. Delamination may occur along these interfaces, which otherwise perfectlyconnect the plate models by considering the limit case of interface stiffness coefficients approaching infinity. Lagrange andpenalty methods are adopted in order to simulate adhesion and contact effects. Analytical expressions for total energy releaserate and its mode components along the delamination front are given in terms of interface variables and of plate stress resultantdiscontinuities. By means of these expressions the influence of transverse shear on interface fracture analysis is investigatedand comparisons with other plate-based delamination models adopted in the literature are established. Numerical results for theenergy release rate distributions are given for typical mixed mode delamination problems by implementing the method in a 2Dfinite element analysis, which utilizes shear deformable plate elements and interface elements. Comparisons with full 3D finiteelement models show the accuracy and the computational efficiency of the proposed procedure. Some applications are proposedto point out the influence of delamination faces interaction on delamination analysis and the convergence of the mode partitionprocedure as the delamination front element size decreases, also when oscillatory singularities exist. On the basis of the latterresult it transpires that the proposed interface model may represent the physical situation of a very thin adhesive layer. © 2004 Elsevier SAS. All rights reserved. Keywords:  Laminated structure; Mixed mode delamination; Interface elements; Plate elements; Contact; Convergence 1. Introduction Due to the relatively low resistance in the transverse direction, fibre-reinforced composite laminates exhibit damage in theform of interlaminar delamination, when subjected to transverse loading conditions. Delamination may compromise seriouslythe performance of a laminate structure and, consequently, must be accurately modelled.In the context of a fracture mechanics approach, the propagation of an existing delamination is analyzed by comparingthe amount of energy release rate with interface toughness. When mixed mode conditions are involved, such as in actualdelamination problems for composite structures, the decomposition of the total energy release rate  G  into mode I  (G I ) , mode II * Corresponding author. Tel.: +39984496914; fax: +39984494045.  E-mail address: (D. Bruno).0997-7538/$ – see front matter  © 2004 Elsevier SAS. All rights reserved.doi:10.1016/j.euromechsol.2004.11.005  128  D. Bruno et al. / European Journal of Mechanics A/Solids 24 (2005) 127–149 ( G II )  and mode III ( G III )  components, becomes a necessary task due to the mode-mix dependency of interface toughness (see,for instance, Reeder, 1993). The interface crack problem in layered elastic structures has been widely studied in the literature inorder tocompute G and itsmode decomposition intheframework of theelasticitytheory (see,for example, Suoand Hutchinson,1990; Schapery and Davidson, 1990; Davidson et al., 1995; Suo, 1990).Due to the high gradient of stress and strain states in the neighborhood of the delamination front, a very accurate solutionis indispensable, which can be obtained by adopting solid finite elements and an appropriate mesh at the delamination front.Illustrations of this methodology can be found both for two-dimensional delamination problems (see, for instance, Raju etal., 1988; Beuth, 1996; Sun and Qian, 1997) and three-dimensional delamination problems (Whitcomb, 1989; Davidson etal., 1996a; Hitchings et al., 1996; Riccio et al., 2001). However, a consistent computation of the energy release rate modecomponents requiresmany layersof solidelementsthrough thelaminatethickness and avery finemesh inthein-plane directionsin the neighborhood of the delamination front, especially for complex layered structure. As a matter of fact, previous studieshave shown that delamination tip elements should be at least one-quarter to one-half the ply thickness of a composite laminate(see Raju et al., 1988; Davidson et al., 2000). Therefore, the computational cost of the analysis may increase notably and theapproach may become not advantageous from the engineering point of view. Moreover, additional complications arise due to theoscillatory behavior of stress and displacement fields near the delamination front, predicted by the three-dimensional elasticitytheory (see Raju et al., 1988).In order to reduce the cost of computation the “global/local analysis” concept has been proposed and illustrated (Suo andHutchinson, 1990; Schapery and Davidson, 1990). In this method the classical plate theory is adopted to predict  G , whereasmode decomposition into individual energy release rates is completed by means of a separate local continuum problem withreference to a small element containing the delamination front. The above approach has shown its effectiveness for both 2D(Sundararaman and Davidson, 1997) and 3D (Davidson et al., 1996b; 2000) delamination problems and is known as “Crack Tip Element (CTE) method” (Davidson et al., 1995). Since the classical plate theory is generally used, contributions from sheardeformations of delaminated and undelaminated members to energy release rates are usually neglected. On the other hand, ithas been shown that for laminates with a relatively low transverse shear modulus or moderate thickness, the shear deformationmust be incorporated otherwise both total and individual energy release rates may be notably underestimated (Bruno and Greco,2001a, 2001b). The classical solution by Suo and Hutchinson (1990), Schapery and Davidson (1990) has been then revisited byWang andQiao(2004) inorder tocapturetheeffect of shear deformation inthe2D interfacecrackproblem. For3D delaminationsituations in order to avoid this restriction, the CTE formulation was revised by Davidson et al. (2000) by introducing sheardeformation kinematics to determinate the local loading of the crack tip element, and the accuracy of the analysis was thusimproved, as shown by a good agreement with prediction by 3D FE analysis for in-plane forces and bending moment loadingconditions. However when transverse shear forces must be considered, this method cannot be applied since only contributionsfrom in-plane shear deformations are incorporated, whereas transverse shear deformation is not completely accounted for inthe analysis. Moreover, the Crack Tip Element formulation cannot be adopted to analyze configurations where contact betweendelamination faces (as, for example, multiple delaminations) occurs. In such cases one must resort to the full three-dimensionalFE procedure, which may involve a large computational effort.A more efficient alternative for delamination analysis is to use plate theory. Classical delamination models (for instance,the models used in Cochelin and Potier-Ferry, 1991; Kim, 1997; Bruno and Greco, 2000), which consider the laminate as anassemblage of two plates in the cracked zone and of a unique plate in the undelaminated one, may give an accurate estimate of the total energy release rate  G . Unfortunately, they are unable to accurately predict the individual energy release rates due to aninaccurate description of the local crack tip strain state and to take into account for shear deformations. As a consequence, theclassical plate-based delamination models have been improved according to different methods.For example, delamination wasmodeled by adopting sublaminates governed by transverse shear deformable laminatetheory,thus obtaining a reasonable approximation to the mode separation solution (Zou et al., 2001).On the other hand, interface models were introduced in conjunction with plate theories, thus obtaining an intermediateapproach between classic delamination models and that based on a continuum analysis. The interface technique is also ableto take into account for non-linear effects due to bridging mechanisms or to damage and to easily incorporate the unilateralcontact condition (see, for instances, Allix et al., 1995; Allix and Csrcliano, 1996; Bui et al., 2000; Greco et al., 2002). Inparticular, for 2D delamination problems the undelaminated region of the laminate has been modeled as two first-order sheardeformable plate elements, in place of a single plate element as in classical models, and a linear elastic interface model has beenintroduced to reconstruct interlaminar stresses and thus to compute energy release rate mode components (Bruno and Greco,2001a; 2001b; Greco et al., 2002). In this way shear deformations was incorporated in the solution and a refined energy releaserate computation was obtained. Moreover, it was pointed out that in the context of a plate-based model the accuracy of modepartition is influenced by the laminate kinematics assumptions and it was concluded that a more reliable mode partition couldbe obtained by using improved plate kinematical models. Consequently the two-layer plate kinematical model was refined bydeveloping a coupled interface-multilayer approach (Bruno et al., 2003), showing that mode partition may be performed to thedesired accuracy by introducing an appropriate number of plate models in each sublaminate.   D. Bruno et al. / European Journal of Mechanics A/Solids 24 (2005) 127–149  129 In the present study a simple but appropriate model for the analysis of 3D delamination problems (mode I, mode II andmode III) is presented for laminated composite structures, based on the multi-layer shear deformable plate modeling and in-terface technique. The study can be regarded as an extension of the previous authors’ work (Bruno et al., 2003) in the 2Ddelamination problem to the one in the more complex 3D situation. The aim is to take advantage of the computational efficiencyof plate-based models and to provide a reasonable accuracy in the determination of both the total and the individual energyrelease rates in comparison with 3D models.In the present formulation the laminated structure is schematized by a through-the thickness sequence of first-order sheardeformable plates connected through interfaces, one of which contains an arbitrary delamination. Interfacial constitutive lawsare developed by using fracture mechanics and contact mechanics concepts. Displacement continuity and contact are simulatedby the use of a penalty method in the delaminated interface, whereas Lagrange’s method is adopted to impose interfacialcontinuity in the undelaminated interfaces. Non-linearities are restricted to contact modeling, the formulation of the modelbeing otherwiselinear. Simpleanalytical expressions for energy releaseratein termsof plate strainsand stressresultantsare alsoobtained by using fracture mechanics concepts, gaining a better insight into the reliabilityof plate-based models in delaminationanalysis. Applications, developed by implementing the model by means of a FE formulation, demonstrate the accuracy of theproposed method in comparison with results obtained by using 3D solid finite elements available in the literature. 2. Mechanics of the laminated composite structure Consider the delamination problem of Fig. 1, where a laminate structure, composed of unidirectional fiber reinforced plies,contains an in-plane delamination crack of area  Ω D  and arbitrary front  Γ  D . The delaminated structure whose thickness isdenoted as  h 0 , is divided by the delamination plane  Ω  into two sub-laminates of respective thickness  h 1 ,  h 2 .Each sublaminate is schematized by an assemblage of first order shear deformable plate bonded by zero-thickness interfacesin the transverse direction: the upper one is subdivided into  n u  plates whereas the lower one into  n l . In the subsequent resultsand figures the adoption of a laminate assembly comprising  n u  plate elements for the upper sublaminate and  n l  plate elementsfor the lower one will be referred to as ( n u  −  n l ) . The first plate model is the lowest one and the thickness of the  i -th plateelement is denoted by  t  i . The kinematics of the  i -th plate element, according to a global reference system with the  xy -planetaken to be the midplane of the laminate and the  z -axis taken positive downward from the midplane, assumes the form:  U  i (x,y,z)  = u i (x,y) + (z − z i ) · ψ xi (x,y),V  i (x,y,z)  = v i (x,y) + (z − z i ) · ψ yi (x,y),W  i (x,y,z)  = w i (x,y), (1)where  U  i ,  V  i  refer to the in-plane displacements and  W  i  to the transverse displacements,  u i ,  v i  and  w i  are the mid-surfacein-plane and transverse displacements, respectively, and  ψ xi ,  ψ yi  denote rotations of transverse normals about  y  and  x , respec-tively. In addition,  z i  denotes the coordinate along the  z -direction of the  i -th mid-plane.The membrane strain vector  ε i  at the reference surfaces, the curvature  κ i  and transverse shear  γ  i  strains, respectively, aredefined as: Fig. 1. Mechanics of the delaminated composite structure.  130  D. Bruno et al. / European Journal of Mechanics A/Solids 24 (2005) 127–149  ε xxi ε yyi γ  xyi  =  ∂u i ∂x∂v i ∂y∂u i ∂y  +  ∂v i ∂x  ,  κ xxi κ yyi κ xyi  =  ∂ψ xi ∂x∂ψ yi ∂y∂ψ xi ∂y  +  ∂ψ yi ∂x  ,  γ  yzi γ  xzi  =  ψ yi  +  ∂w i ∂y ψ xi  +  ∂w i ∂x  . By considering each plate as composed by one or several physical fiber-reinforced plies with their material axes arbitrarilyoriented, the constitutive relations between stress resultants and corresponding strains are:  N  i M  i  =  A i  B i B i  D i  ε i κ i  , (2) T  i  = H  i γ  i , where  N  i  = { N  xxi  N  yyi  N  xyi  } T is the membrane force resultant vector,  M  i  = { M  xxi  M  yyi  M  xyi  } T the moment re-sultant vector and T  i  = { T  yzi  T  xzi  } T the transverse shear force resultant vector. In addition, A i , D i , B i  denote the classicalextensional stiffness matrix, bending stiffness matrix and bending-extensional coupling stiffness matrix, respectively, whereas H  i  is the shear stiffness matrix (see, for instance, Barbero, 1999). 2.1. Interfacial constitutive relationships As shown in Fig. 1, in the undelaminated region  Ω – Ω D  each sublaminate is assumed to be composed of perfectly bondedplate elements, whereas in the delaminated domain  Ω D  contact between sub-laminates may occur. Accordingly, a linear inter-face model is introduced along theundelaminated portion of the delaminaton plane  Ω – Ω D , whose constitutive law involves twostiffness parameters,  k z ,  k xy , imposing displacement continuity in the  z , and  x – y  directions, respectively, by considering themas penalty parameters. Denoting the interlaminar normal and shear stresses, by  σ  zz  and  σ  zy ,  σ  zx , respectively, the constitutiverelation assumes the following form: σ  zz  = k z w, σ  zx  = k xy u, σ  zy  = k xy v,  (3)where  w ,  u  and  v  are the corresponding interface relative displacements, evaluated as the difference between displace-ments at the interface location of the lower and upper sub-laminate.Interpenetration between delaminated sub-laminates isavoided by adopting aunilateral frictionlesscontact interface, charac-terized by azero stiffness for opening relativedisplacements ( w  0 ) and a positive stiffnessfor closing relativedisplacements( w < 0 ) : σ  zz  =  12  1 − sign (w)  k z w,  (4)where  σ  zz  is the contact stress,  k z  is the penalty number imposing contact constraint and sign is the signum function. A verylarge value for  k z  restricts sub-laminate overlapping and simulates contact condition.If a damage variable  d   is introduced, taking the value 1 for no adhesion and the value 0 for perfect adhesion, Eqs. (3) and (4)can be incorporated into the following unique constitutive law valid both for undelaminated and delaminated delamination planeportions: σ  zz  =  1 − 12 d   1 + sign (w)  k z w, σ  zx  = ( 1 − d)k xy u, σ  zy  = ( 1 − d)k xy v,  (5)where the stiffness parameters may assume, for numerical convenience, different values for the adhesive interface ( d   =  0 )  andthe contact interface ( d   =  1 ) . Interface constitutive laws are illustrated in Fig. 1.On the other hand, inside each sublaminate the displacement continuity requirements between any two adjacent plates,  i  and i  + 1 with ( i  = n l ) : u i  = v i  = w i  =  0 , (6) u i  = u i  − t  i 2 ψ xi  − u i + 1  − t  i + 1 2  ψ xi + 1 , v i  = v i  − t  i 2  ψ yi  − v i + 1  − t  i + 1 2  ψ yi + 1 , w i  = w i  − w i + 1 , are ensured by the Lagrange’s method. In this case interlaminar stresses are represented by the Lagrange multipliers.   D. Bruno et al. / European Journal of Mechanics A/Solids 24 (2005) 127–149  131 3. Theoretical delamination analysis The computation of the total energy release rate and of its individual mode components can be solved both locally by usinginterface variables (interlaminar stresses and relative displacements), and globally in terms of plate strains and stresses. Asit will be shown by resorting to fracture mechanics concepts, these two approaches are equivalent. Both formulations will bedeveloped not only for interest in mathematical aspects but also to give a better understanding of the mechanics of delamination.As a matter of fact, the local approach corresponds to the limit physical situation resulting when the thickness of a thinadhesive layer tends to zero, which can be referred to as “strong” interface model (since the interface stiffnesses approach in-finity). In the global one, the interface may be considered as a so-called “collapsed” interface model: the interface is introducedonly to impose a purely geometrical constraint ensuring perfect adhesion. As a consequence, the information about interlaminarstresses and displacements is lost and energy release rates must be computed from jumps in plate strains and stresses. Theequivalence between the two approaches is established by considering that the “strong” interface expresses a penalty-type con-straint whereas the “collapsed” interface represents a Lagrangian constraint. Moreover, it will be shown that individual energyrelease rates arising from the present formulation are well defined despite the oscillatory behavior, which may be involved inthe context of the three-dimensional fracture mechanics theory. On the other hand, numerical results obtained in the followingsections will show that individual energy release rates obtained by the present formulation are a good approximation of thosecalculated by continuum FE analyses in the case when no oscillatory singularities are involved. This shows the effectivenessand the consistency of the present approach for predicting delamination in composite laminates. 3.1. Local approach Using the interface constitutive specification (3) to compute interlaminar stresses (see also Bruno and Greco, 2001b), leadsto the following total energy release rate expression: G(s) = 12 lim k z ,k xy →∞  k z w 2 (s) + k xy u 2 (s) + k xy v 2 (s)  , w(s)  0 .  (7)In Eq. (7)  G(s)  is the local energy release rate function defined along the delamination contour  Γ  D , defined by the followingformula: ˙ Π  r  Ω D (t),λ  = −   Γ  D G(s)  ˙ Ω D (s) d s,  (8)where  Π  r is the total potential energy of the system at equilibrium, a dot denotes total differentiation with respect to a time-like parameter  t   governing a virtual monotonic delamination growth  Ω D (t) , and  ˙ Ω D (s)  (restricted to a positive increase ˙ Ω D (s)  0  ∀ s ) is the rate of normal extension of the delamination front  Γ  D  (see Fig. 1) which describes the rate of varia-tion of   Ω D (t)  (Nguyen, 2000). Note that in the limit as interface stiffness parameters approach infinity, interlaminar stresses(3) must approach infinity at the delamination front whereas interlaminar displacements must tend to zero. This ensures a finitevalue for Eq. (7).In order to determine the individual energy release rates the relative interface displacements must be expressed in the localco-ordinate system attached to the delamination front shown in Fig. 2. Denoting the unit vectors in the normal and tangential Fig. 2. Delamination front element of the  i -th plate showing stress resultant discontinuities and Lagrange point multipliers.
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