A solid-like shell element allowing for arbitrary delaminations

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1 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING Int. J. Numer. Meth. Engng 23; 58: (DOI: 1.12/nme.97) A solid-like shell element allowing for arbitrary delaminations Joris J. C. Remmers ;, Garth N. Wells and Rene de Borst Koiter Institute Delft; Delft University of Technology; P.O. Box 558; NL-26 GB Delft; The Netherlands SUMMARY In this contribution a new nite element is presented for the simulation of delamination growth in thin-layered composite structures. The element is based on a solid-like shell element: a volume element that can be used for very thin applications due to a higher-order displacement eld in the thickness direction. The delamination crack can occur at arbitrary locations and is incorporated in the element as a jump in the displacement eld by using the partition of unity property of nite element shape functions. The kinematics of the element as well as the nite element formulation are described. The performance of the element is demonstrated by means of two examples. Copyright? 23 John Wiley & Sons, Ltd. KEY WORDS: solid-like shell element; delamination; partition of unity 1. INTRODUCTION The application of layered composite structures in the aerospace and automotive industries has increased signicantly over the last decades. Since the material can be tailored to meet special demands, laminated composite structures can be lighter and made with superior characteristics when compared to traditional single-phase materials. However, the use of these materials introduces new failure mechanisms such as delamination, i.e. the debonding of individual layers. In general, the presence of delaminations in the material will lead to a signicant reduction of the residual strength. To arrive at a better understanding of this phenomenon, numerical simulations can be of assistance. In nite element models, a delamination crack is normally modelled using interface elements [1 3]. These elements consist of two surfaces which are connected to the continuum elements that model the adjoining layers of the laminate. A perfect bond prior to delamination growth is simulated with a high dummy stiness. Debonding is governed by a decohesion relationship, which can be derived on the basis of a damage mechanics theory [1] or can be Correspondence to: Joris J. C. Remmers, Koiter Institute Delft, Delft University of Technology, P.O. Box 558, NL-26GB Delft, The Netherlands. J.J.C.Remmers@lr.tudelft.nl Received 7 October 22 Revised 6 May 23 Copyright? 23 John Wiley & Sons, Ltd. Accepted 23 June 23

2 214 J. J. C. REMMERS, G. N. WELLS AND R. DE BORST formulated in a plasticity framework [2]. These relations have been extended to meet more practical applications such as mixed-mode delamination [4]. In an alternative approach, delamination cracks can be incorporated in continuum elements by exploiting the partition-of-unity property of the nite element shape functions [5]. Initially, this approach was used to simulate stationary cracks in a nite element model by adding the span of the near-tip solution of linear-elastic fracture mechanics to the displacement eld [6, 7]. Later, the method was cast in a cohesive framework to model progressive crack growth [8, 9]. In this fashion, a crack is modelled as a jump (a discontinuity) in the displacement eld of the continuum elements. The magnitude of the jump is determined by an extra set of degrees of freedom, which are added to existing nodes of the nite element mesh. The separation process can be specied by the same decohesion relations as used with interface elements, including those that describe mixed-mode fracture. The discontinuity, which crosses elements irrespective of the structure of the nite element mesh, can be extended into the next element by simply adding extra degrees of freedom to the nodes that support this element. The advantages of this new approach are evident. First, the total number of unknowns in the system reduces signicantly, since degrees of freedom are only added to the model when the delamination surface propagates, i.e. when the discontinuity is extended. As a consequence, the use of high dummy stinesses to simulate the perfect bond is avoided. These high dummy stinesses can lead to traction oscillations ahead of the delamination front [1]. Furthermore, since the cohesive surface is incorporated in the continuum element, a consistent extension to a nite strain framework is feasible [11]. Finally, it is possible to model a complete laminate with just one continuum element in thickness direction [12], which allows for the analysis of delamination growth on a structural level. Herein, this approach is extended to a shell model. A key feature is the choice of a proper element. Conventional volume elements show an overly sti behaviour when used in thin applications (Poisson thickness locking [13]) due to a constant strain distribution in thickness direction. An alternative is the solid-like shell element [14]. Here, an additional set of internal degrees of freedom is used to add a quadratic term to the displacement eld in thickness direction, the internal stretch of the element. Hence, the corresponding strain eld varies linearly over the thickness instead of being constant and Poisson thickness locking is avoided. It has been shown that the solid-like shell element can be used for modelling laminate structures by either stacking multiple elements, or by modelling multiple layers within one element [15]. In the latter situation, the element is divided into a number of sub-domains, each of which has dierent material parameters. When, as is proposed here, this model is extended with the possibility to initiate and propagate delaminations at arbitrary locations within the shell during the loading process, the complete mechanical behaviour of laminated structures can be simulated with just one shell element in thickness direction. This contribution is ordered as follows. In the next section, a concise description of the kinematic relations for the original solid-like shell element is given. The derivation of the enhanced solid-like shell element with the displacement jump is presented in Sections 3 5. Section 6 discusses the nite element discretisation. The incorporation of the constitutive models and some aspects regarding linearisation of the governing equations are presented in Sections 7 and 8. Key aspects of implementation are addressed in Section 9. The performance of the enhanced element is demonstrated by means of two numerical examples in Section 1, after which some conclusions are drawn. Copyright? 23 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 23; 58:213 24

3 SOLID-LIKE SHELL ELEMENT ALLOWING FOR ARBITRARY DELAMINATIONS KINEMATICS OF THE SOLID-LIKE SHELL ELEMENT We consider the thick shell shown in Figure 1. The position of a material point in the shell in the undeformed conguration can be written as a function of the three curvilinear co-ordinates [; ; ]: X(; ; )=X (; )+D(; ) (1) where X (; ) is the projection of the point on the mid-surface of the shell and D(; ) is the thickness director in this point: X (; )= 1 2 [X t(; )+X b (; )] (2) D(; )= 1 2 [X t(; ) X b (; )] (3) The subscripts ( ) t and ( ) b denote the projections of the variable onto the top and bottom surface, respectively. The position of the material point in the deformed conguration x(; ; ) is related to X(; ; ) via the displacement eld M(; ; ) according to where x(; ; )=X(; ; )+M(; ; ) (4) M(; ; )=u (; )+u 1 (; )+(1 2 )u 2 (; ) (5) In this relation, u and u 1 are the displacements of X on the shell mid-surface, and the thickness director D, respectively: u (; )= 1 2 [u t(; )+u b (; )] (6) u 1 (; )= 1 2 [u t(; ) u b (; )] (7) and u 2 (; ) denotes the internal stretching of the element, which is colinear with the thickness director in the deformed conguration and is a function of an additional stretch parameter w: u 2 (; )=w(; )[D + u 1 (; )] (8) Figure 1. Kinematic relations of the regular solid-like shell element in undeformed and deformed position. The dash dotted line denotes the mid-surface of the shell. Copyright? 23 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 23; 58:213 24

4 216 J. J. C. REMMERS, G. N. WELLS AND R. DE BORST In the remainder, we will consider the displacement eld M as a function of two kinds of variables; the ordinary displacement eld u, which will be split in a displacement of the top and bottom surfaces u t and u b, respectively, and the internal stretch parameter w: M = M(u t ; u b ;w) (9) 3. ENHANCED KINEMATIC RELATIONS The thick shell with constant thickness of Figure 2 is crossed by a discontinuity surface d; which divides the domain into two parts, + and. The discontinuity surface is assumed to be parallel to the mid-surface of the thick shell. The displacement eld M(; ; ) can now be regarded as a continuous regular eld ˆM with an additional continuous eld M that determines the magnitude of the displacement jump [8]. The position of a material point in the deformed conguration can then be written as: x = X + ˆM + H d; M (1) where H d; represents the Heaviside step function, dened as H d; (X)= { if X 1 if X + Since the displacement eld M is a function of the variables u t, u b decompose these three terms as (11) and w, we need to u t = û t + H d; ũ t u b = û b + H d; ũ b w =ŵ + H d; w (12) Figure 2. Thick shell crossed by a discontinuity d; (heavy line). The vectors n u; and u d; are perpendicular to the shell surface and the discontinuity surface, respectively. Copyright? 23 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 23; 58:213 24

5 SOLID-LIKE SHELL ELEMENT ALLOWING FOR ARBITRARY DELAMINATIONS 217 Inserting Equation (12) into Equations (6) (8) gives where u = û + H d; ũ u 1 = û 1 + H d; ũ 1 u 2 = û 2 + H d; ũ 2 (13) û = 1 2 [û t + û b ]; ũ = 1 2 [ũ t + ũ b ] û 1 = 1 2 [û t û b ]; ũ 1 = 1 2 [ũ t ũ b ] (14) û 2 =ŵ[d + û 1 ]; ũ 2 = w[d + û 1 + ũ 1 ]+ŵũ 1 Note that the enhanced part of the internal stretch parameter u 2, i.e. ũ 2, contains regular variables and additional variables. The base vectors at the material point in undeformed conguration G i can be found by dierentiating the position vector X with respect to the isoparametric coordinates i =[; ; ]: = E + D ; ; =1; 2 (15) G 3 = D (16) where ( ) ; denotes the partial derivative with respect to. E is the covariant surface vector, which is the projection of the base-vector G on the mid-surface and is dened as E The base vectors of the shell in the deformed conguration g i are found in a similar fashion: = E + û ; + D ; + û 1; + H d; [ũ ; + ũ 1; ]+h:o:t: d (18) g 3 = D + û 1 2û 2 + H d; [ũ 1 2ũ 2 ]+h:o:t: d (19) Owing to the displacement jump, the base vectors in the deformed conguration g i are not continuous at the discontinuity d;. The higher order terms (h.o.t.) in Equations (18) and (19) contain terms up to the fourth order in the thickness co-ordinate and the derivatives of the stretch parameter u 2 with respect to and. In the remainder, these terms will be neglected without a signicant loss of accuracy of the kinematic model [14]. The metric tensors G and g can be determined by using the base vectors G i and g i in Equations (15) (19): G ij = G i G j ; g ij = g i g j (2) Copyright? 23 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 23; 58:213 24

6 218 J. J. C. REMMERS, G. N. WELLS AND R. DE BORST Elaboration of the expressions yields the following components of the metric tensor in the undeformed conguration: G = E E + [E D ; + E D ; ] G 3 = E D + D ; D G 33 = D D (21a) (21b) (21c) and in the deformed conguration (again neglecting the terms which are quadratic in thickness direction): g =(E + û ; + H d; ũ ; ) (E + û ; + H d; ũ ; ) + [(E + û ; + H d; ũ ; ) (D ; + û 1; + H d; ũ 1; ) +(E + û ; + H d; ũ ; ) (D ; + û 1; + H d; ũ 1; )] (22a) g 3 =(E + û ; + H d; ũ ; ) (D + û 1 + H d; ũ 1 ) + [(D + û 1 + H d; ũ 1 ) (D ; + û 1; + H d; ũ 1; ) 2(E + û ; + H d; ũ ; ) (û 2 + H d; ũ 2 )] (22b) g 33 =(D + û 1 + H d; ũ 1 ) (D + û 1 + H d; ũ 1 ) 4(D + û 1 + H d; ũ 1 ) (û 2 + H d; ũ 2 ) (22c) We can rewrite these relations as G ij = G ij + G 1 ij g ij = g ij + g 1 ij (23) where G ij and g ij correspond to the constant terms in Equations (21) and (22), whereas G 1 ij and g 1 ij correspond to the terms that vary linearly with respect to the thickness. Apart from the covariant base vectors in the undeformed state, we will need the contravariant counterparts to derive the strain eld later on. They can be derived as follows: G j =(G ij ) 1 G i (24) The contravariant base vector G j is related to the contravariant surface vector E k via the so-called shell tensor j k [16]: G j = j k Ek ; j k =( j k G j k ) (25) In Equation (25), G j k denotes the mixed variant metric tensor which is calculated with the contravariant and the covariant tensor components as follows, see Equation (23) [14]: G j k = G jm G 1 mk (26) Copyright? 23 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 23; 58:213 24

7 SOLID-LIKE SHELL ELEMENT ALLOWING FOR ARBITRARY DELAMINATIONS GREEN LAGRANGE STRAIN FIELD The Green Lagrange strain tensor S is dened conventionally in terms of the deformation gradient F: S = 1 2 (FT F I) (27) The deformation gradient F can be written as a function of the covariant basevector in the deformed conguration g i and the contravariant basevector in the undeformed reference con- guration G i : F = g i G i (28) Inserting this relation in Equation (27) we can write the Green Lagrange strain tensor in terms of the contravariant basis G j : S = ij G i G j where ij = 1 2 (g ij G ij ) (29) Substituting Equation (25) in this relation yields 2S =(g ij G ij )( i k G i k)( j l G j l )Ek E l (3) After some manipulations, the strain tensor can be written in terms of the membrane midsurface strain ij and the bending strain ij according to 2S =( ij + ij )E i E j (31) where 2 = E (û ; + H d; ũ ; )+E (û ; + H d; ũ ; ) +(û ; + H d; ũ ; ) (û ; + H d; ũ ; ) (32a) 2 3 = E (û 1 + H d; ũ 1 )+D (û ; + H d; ũ ; ) +(û ; + H d; ũ ; ) (û 1 + H d; ũ 1 ) 2 33 =2D (û 1 + H d; ũ 1 )+(û 1 + H d; ũ 1 ) (û 1 + H d; ũ 1 ) (32b) (32c) 2 = E (û 1; + H d; ũ 1; )+E (û 1; + H d; ũ 1; ) + D ; (û ; + H d; ũ ; )+D ; (û ; + H d; ũ ; ) +(û 1; + H d; ũ 1; ) (û ; + H d; ũ ; ) +(û 1; + H d; ũ 1; ) (û ; + H d; ũ ; ) G [E (û ; + H d; ũ ; )+E (û ; + H d; ũ ; ) +(û ; + H d; ũ ; ) (û ; + H d; ũ 1; )] G [E (û ; + H d; ũ ; )+E (û ; + H d; ũ ; ) +(û ; + H d; ũ ; ) (û ; + H d; ũ 1; )] (32d) Copyright? 23 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 23; 58:213 24

8 22 J. J. C. REMMERS, G. N. WELLS AND R. DE BORST 2 3 = D ; (û 1 + H d; ũ 1 )+D (û 1; + H d; ũ 1; ) +(û 1 + H d; ũ 1 ) (û 1; + H d; ũ 1; ) 2 33 = 4(D + û 1 + H d; ũ 1 ) (û 2 + H d; ũ 2 ) (32e) (32f) Note that the strain elds on either side of the discontinuity d; are not necessarily equal. This implies that it is possible to capture phenomena which are restricted to just one layer of the laminate such as delamination buckling [11, 12]. The strain eld is still dened in the isoparametric frame of axes E i. In order to obtain the strains in the element local frame of reference l j, they must be transformed using: kl =( ij + ij )T i kt j l ; Ti k = E i l k (33) The magnitude of the displacement jump v at the internal discontinuity d; is equal to the magnitude of the additional displacement eld at the discontinuity d. In the spirit of previous assumptions, we neglect the terms that vary quadratically in the thickness direction, so that v = ũ + d ũ 1 (34) The displacement jump does not have to be transformed, since it is already dened in the element local frame of reference. 5. WEAK FORM OF THE EQUILIBRIUM EQUATIONS The static equilibrium equations and boundary conditions for the body without body forces with respect to the undeformed conguration can be written as: P = in (35a) n u; P = t on u; (35b) n d; P = t on d; (35c) where P is the nominal stress tensor, t the applied external load in the reference conguration and n u; is the outward unit normal vector to the body; t is the traction at discontinuity d; with n d; the inward unit normal to +, see also Figure 2. Equation (35c) represents the tractions at the discontinuity d;, which can be conceived as an internal boundary. The strong governing equations can be written as the weak equations of equilibrium by multiplying Equation (35a) by an admissible displacement variation M and integrating the result over the domain : M ( P)d = (36) which must hold for all admissible variation of the displacement eld M. Following a standard Bubnov Galerkin approach, the space of admissible displacement variations is taken the same Copyright? 23 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 23; 58:213 24

9 SOLID-LIKE SHELL ELEMENT ALLOWING FOR ARBITRARY DELAMINATIONS 221 as the eld of actual displacements. Referring to Equation (1), we therefore write M = ˆM + H d; M (37) Substituting this relation into Equation (36) gives ˆM ( P)d + H d; M ( P)d = (38) which must hold for all admissible variations ˆM and M. The equation can be expanded using Gauss theorem. The Heaviside function can be eliminated by changing the integration domain into + [8]: (P ˆM)d ˆM : P d + (P M)d M : P d = (39) + + In this equation, ˆM and M are the deformation gradients of the regular and the additional parts of the admissible displacement variations [11]. Applying the conditions at the boundary of the domain, Equation (35b), and the condition at the discontinuity, Equation (35c), the above equation can be written as ˆM : P d + + M : P d + ṽ t d = d; ˆM t d + u; H d; M t d u; The terms ˆM : P and M : P can be replaced by the work-conjugate terms Ŝ : and S : [17], where Ŝ and S are the regular and additional parts of the variation of the Green Lagrange strain eld and is the second Piola Kirchho stress tensor: Ŝ : d + + (4) S : d + ṽ t d = d; ˆM t d + u; H d; M t d u; (41) For completeness, the variation of the strain eld S is now given as a function of the displacement components: 2 = e (û ; + H d; ũ ; )+e (û ; + H d; ũ ; ) (42a) 2 3 = d (û ; + H d; ũ ; )+e (û 1 + H d; ũ 1 ) 2 33 =2d (û 1 + H d; ũ 1 ) (42b) (42c) 2 = e (û 1; + H d; ũ 1; )+e (û 1; + H d; ũ 1; ) + d ; (û ; + H d; ũ ; )+d ; (û ; + H d; ũ ; ) G [e (û ; + H d; ũ ; )+e (û ; + H d; ũ ; )] G [e (û ; + H d; ũ ; )+e (û ; + H d; ũ ; )] (42d) Copyright? 23 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 23; 58:213 24

10 222 J. J. C. REMMERS, G. N. WELLS AND R. DE BORST 2 3 = d (û 1; + H d; ũ 1; )+d ; (û 1 + H d; ũ 1 ) 2 33 = 8u 2 (û 1 + H d; ũ 1 ) 4d d(ŵ + H d; w) (42e) (42f) where e = E + û ; + H d; ũ ; ; d = D + û 1 + H d; ũ 1 (43) d ; = D ; + û 1; + H d; û 1; ; u 2 = û 2 + H d; ũ 2 In an incremental iterative solution scheme, the strain increment S is normally needed. It can be derived as 2 = e u ; + e u ; +u ; u ; 2 3 = e u 1 + d u ; +u ; u =2d u 1 +2u 1 u 1 (44a) (44b) (44c) 2 = e u 1; + d ; u ; +u 1; u ; + e u 1; + d ; u ; +u 1; u ; G [e u ; + e u ; +u ; u ; ] G [e u ; + e u ; +u ; u ; ] 2 3 = d ; u 1 + d u 1; +u 1; u 1 (44d) (44e) 2 33 = 4d u 1 2wu 1 u 1 2wd d 4wu 1 d 2wu 1 u 1 (44f) where u 1,u 1;,u 1;,d ; and w are dened in the spirit of Equation (43): u 1 =û 1 + H d; ũ 1 ; u ; =û ; + H d; ũ ; u 1; =û 1; + H d; ũ 1; ; d ; =ˆd ; + H d; d ; (45) w =ŵ + H d; w 6. FINITE ELEMENT IMPLEMENTATION The solid-like shell element can be modelled as an eight-node or as a sixteen-node element, see Figure 3. In both cases, a linear distribution of the internal stretch is assumed, so that only four internal degrees of freedom (situated at the four corners of the mid-surface of the element mid-surface) are necessary [14]. Each geometrical node i contains three degrees of freedom a i =[a x ;a y ;a z ] i that set up the regular displacement eld û and a set of three additional degrees of freedom b i =[b x ;b y ;b z ] i that describes the enhanced displacement Copyright? 23 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 23; 58:213 24

11 SOLID-LIKE SHELL ELEMENT ALLOWING FOR ARBITRARY DELAMINATIONS 223 η 15 ζ ξ 12 IV I II 2 geometrical node 5 internal node 1 Figure 3. Geometry and location of the nodes for the sixteen-node solid-like shell element. Each geometrical node contains a regular set of three degrees of freedom [a x;a y;a z] i and an additional set of three degrees of freedom [b x;b y;b z] i. The internal nodes contain one regular and one additional degree-of-freedom (p j and q j, respectively). eld ũ. Likewise, there are two sets of internal degrees of freedom: p j that constructs the regular stretch term ŵ and q j that constructs the additional stretch term w. It was shown by Babuska et al. [5] that the basis of the nite element shape functions can be enriched with enhanced bases by virtue of the fact that the nite element shape functions i form a partition of unity. Hence, a eld u can be interpolated in terms of nodal values according to: ( ) u(x;t)= n i(x) a i (t)+ m k (X)b ik (t) (46) i=1 k=1 In this specic case, the function k can be replaced by the Heaviside step function H d;. The projected displacements at the bottom and top surfaces of the element u b and u t can be constructed using the N bottom and N top nodes of the element respectively (consequently, the number of nodes in one element equals 2N). For example, the sixteen-node element has eight bottom and eight top nodes. Hence, the continuous value u b can be constructed from the eight bottom nodes, using the eight quadratic shape functions for a quadrilateral element i. In the eight-node element, this value can be constructed using the four bottom nodes and the four linear isoparametric shape functions for a quadrilateral element. Since the internal stretch w is constructed with four internal degrees of freedom in both elements, we use the four linear shape function j. The projection of the displacement vector can be constructed by interpolating the regular and enhanced degrees of freedom at the bottom nodes of the element (i =1;N): u b = û b + H d; ũ b = N i=1 ia i + H d; N i=1 ib i (47) where represents the conventional matrix which contains the element shape functions for node i in three directions x; y; z: i = i I 3 (48) Copyright? 23 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 23; 58:213 24

12 224 J. J. C. REMMERS, G. N. WELLS AND R. DE BORST where I 3 is the identity matrix. The projected displacements on the top surface u t can be interpolated using the degrees of freedom at the top nodes of the element (numbered i = N + 1;N +2;:::;2N ) and shape functions i : u t = û t + H d; ũ t = N N ia i+n + H d; ib i+n (49) i=1 Similarly, the internal degrees of freedom are calculated using the regular and additional internal degrees of freedom p j and q j, respectively: w =ŵ + H d; w = 4 4 j p j + H d; j q j (5) j=1 The discretized displacement of the shell mid-surface and the displacement of the thickness director, u and u 1 respectively, can be found using relations (14), (47) and (49): where N and N 1 are dened as i=1 j=1 û = N a; ũ = N b (51) û 1 = N 1 a; ũ 1 = N 1 b (52) N = 1 2 [ 1; 2 ;:::; N ; 1 ; 2 ;:::; N ] T (53) N 1 = 1 2 [ 1; 2 ;:::; N ; 1 ; 2 ;:::; N ] T (54) From Equation (5) we dene the interpolation matrix for the internal degrees of freedom: N w =[ 1 ; 2 ; 3 ; 4 ] T (55) Trivially, the derivatives of the displacement vectors u and u 1 with respect to the isoparametric coordinates and can be written as û ; = N ; a; ũ ; = N ; b (56) û 1; = N 1; a; ũ 1; = N 1; b (57) where the matrices N ; and N 1; contain the derivatives of the shape functions with: N ; = 1 2 [ 1;; 2; ;:::; N; ; 1; ; 2; ;:::; N; ] T (58) N 1; = 1 2 [ 1;; 2; ;:::; N; ; 1; ; 2; ;:::; N; ] T (59) i; I 3 (6) There are three kinds of admissible variations that need to be discretised, namely those of the Green Lagrange strain tensor Ŝ and S, that of the displacement jump, ṽ, and that of the Copyright? 23 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 23; 58:213 24

13 SOLID-LIKE SHELL ELEMENT ALLOWING FOR ARBITRARY DELAMINATIONS 225 displacement eld itself, ˆM and M. The variations of the Green Lagrange strain eld can be written as Ŝ = B u a + B w p (61) S = B u b + B w q (62) with B u and B w as dened in Appendix A. From Equation (34) one obtains the discrete variation displacement jump ṽ = N b + d N 1 b (63) Since the load can only be applied to the geometrical nodes [14], the admissible displacement terms in the external load parts of the equilibrium equations can be replaced by a modied admissible displacement eld M which is dened as or in discrete form M = ˆM + M = û + û 1 + H d; (ũ + ũ 1 ) (64) ˆM =(N + N 1 )a; M =(N + N 1 )b (65) Substituting Equations (61), (63) and (65) into the equilibrium equation (41) gives (B u a + B w p) T d + (B u b + B w q) T d + + ((N + d N 1 )b) T t d d; = ((N + N 1 )a) T t d + u; H d; ((N + N 1 )b) T t d u; (66) By taking the variation of one admissible variation a, b, p or q at the time and setting the others to zero, the following four systems of equations can be derived: B T u d = (N + N 1 ) T t d u; (67a) B T u d + (N + d N 1 ) T t d = d; H d; (N + N 1 ) T t d u; (67b) + B T w d = B T w d = + (67c) (67d) The left-hand sides of these equations constitute the internal force vectors: fa int = B T u d (68a) Copyright? 23 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 23; 58:213 24

14 226 J. J. C. REMMERS, G. N. WELLS AND R. DE BORST fb int = B T u d + (N + d N 1 ) T t d + d; fp int = B T w d fq int = B T w d while the right-hand sides represent the external force vectors: fa ext = (N + N 1 ) T t d u + fb ext = H d; (N + N 1 ) T t d u (68b) (68c) (68d) (69a) (69b) f ext p = (69c) f ext q = (69d) 7. CONSTITUTIVE RELATIONS For the bulk material and for the interface standard constitutive relations can be used. At this stage, the treatment is restricted to small strains, but still allows for arbitrarily large displacement gradients. Consequently, a linear relation can be postulated between the increments of the second Piola Kirchho stress, and the Green Lagrange strain, S, see Equation (44): where D is the tangent stiness matrix of the bulk material. For the discontinuity, the constitutive relation reads: = DS (7) t dis = C dis v dis (71) where v dis is the incremental displacement jump at the discontinuity as dened in the current orientation of the discontinuity, as denoted by the subscript dis. Transformation into the element local frame of reference in the undeformed conguration gives: t = Q T t dis = Q T Cv dis = Q T CQv (72) so that in the undeformed conguration, the tangent stiness matrix is equal to C = Q T C dis Q (73) The orthogonal transformation matrix Q performs the transformation of the orientation of the discontinuity in the current conguration into the orientation in undeformed conguration and can be composed using the base vectors in the deformed conguration, Equations (18) Copyright? 23 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 23; 58:213 24

15 SOLID-LIKE SHELL ELEMENT ALLOWING FOR ARBITRARY DELAMINATIONS 227 ĝ 3 + g 3 Ω + Ω ĝ 3 ĝ 1 + g 1 g 3,dis g 1,dis ĝ 1 Γ + d Γ d Γ d Figure 4. Average position of the discontinuity in the deformed conguration. and (19). However, since the triads g i at both sides of the discontinuity are normally dierent, see Figure 4, it is not possible to assign a unique direction for the discontinuity in the deformed position. Therefore, the average direction is taken [11]. The base vectors in deformed position are then equal to g ; dis = ĝ g = E + û ; + 1 2ũ; + D ; + û 1; ũ 1; g 3; dis = ĝ g 3 = D + û ũ1 2û 2 ũ 2 (74a) (74b) which can be used to form the transformation matrix Q in a standard manner [18]: Q ij = cos(g i; dis ; G j ) (75) 8. LINEARIZATION OF THE GOVERNING EQUATIONS In order to solve the equations in a Newton Raphson process, a stiness matrix is needed: Ku = f ext f int (76) The stiness matrix can be found by dierentiating the internal force vector, Equation (68). Since both the stresses and the geometric matrices B u and B w are a function of the displacements, the stiness matrix can be decomposed in two terms, the so-called material part and the geometric part: Since K = K mat + K @b = @q = DB w (78) Copyright? 23 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 23; 58:213 24

16 228 J. J. C. REMMERS, G. N. WELLS AND R. DE BORST we obtain for the material part of the stiness matrix of the bulk material: B T udb u d B T udb u d B T udb w d B T udb w d + + B T udb u d B T udb u d B T udb w d B T udb w d Kbulk mat = B T wdb u d B T wdb u d B T wdb w d B T wdb w d + + B T wdb u d B T wdb u d B T wdb w d B T wdb w d (79) For the discontinuity, the following = QT C dis Q(N + d N 1 ) (8) Substitution there gives the contribution of the discontinuity to the material part of the stiness matrix: (N + d N 1 ) T Q T C dis Q(N + d N 1 )d K mat dis = d; (81) The total material part of the stiness matrix is formed by their sum; K mat = K mat bulk + Kmat dis. The geometric part of the stiness matrix is found by dierentiating the incremental change of the variational strains, Equation (42) with respect to the displacements: 2d( )=(û ; + H d; ũ ; ) (dû ; + H d; dũ ; ) +(û ; + H d; ũ ; ) (dû ; + H d; dũ ; ) (82a) 2d( 3 )=(û ; + H d; ũ ; ) (dû 1 + H d; dũ 1 ) +(û 1 + H d; ũ 1 ) (dû ; + H d; dũ ; ) 2d( 33 )=2(û 1 + H d; ũ 1 ) (dû 1 + H d; dũ 1 ) (82b) (82c) 2d( )=(û 1; + H d; ũ 1; ) (dû ; + H d; dũ ; ) +(û 1; + H d; ũ 1; ) (dû ; + H d; dũ ; ) +(û ; + H d; ũ ; ) (dû 1; + H d; dũ 1; ) Copyright? 23 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 23; 58:213 24

17 SOLID-LIKE SHELL ELEMENT ALLOWING FOR ARBITRARY DELAMINATIONS 229 +(û ; + H d; ũ ; ) (dû 1; + H d; dũ 1; ) G [(û ; + H d; ũ ; ) (dû ; + H d; dũ ; ) +(û ; + H d; ũ ; ) (dû ; + H d; dũ ; )] G [(û ; + H d; ũ ; ) (dû ; + H d; dũ ; ) +(û ; + H d; ũ ; ) (dû ; + H d; dũ ; )] (82d) 2d( 3 )=(û 1; + H d; ũ 1; ) (dû 1 + H d; dũ 1 ) +(û 1 + H d; ũ 1 ) (dû 1; + H d; dũ 1; ) (82e) 2d( 33 )= 8[(û 1 + H d; ũ 1 ) d (dŵ + H d; d w) +(ŵ + H d; w)d (dû 1 + H d; dũ 1 ) + w(û 1 + H d; ũ 1 ) (û 1 + H d; ũ 1 )] (82f) The incremental change of the virtual strains must be multiplied with the current stresses to yield the geometric part of the stiness matrix: A uu d A uu d A uw d A uw d + + A uu d A uu d A uw d A uw d K geo = (83) A wu d A wu d + A wu d A wu d + + where A uu, A uw and A wu are dened in Appendix B. Consistent linearisation of the surface integrals related to the tractions at the discontinuity, Equation (68b) will result in additional, non-symmetric, terms in the geometric part of the stiness matrix due to the transformation matrix Q, Equation (72). However, since these contributions are relatively small, they will have only a minor impact the iteration process [11]. Owing to the additional computational eort required to solve a non-symmetric system, these terms will be neglected. 9. IMPLEMENTATION ASPECTS 9.1. Enhancement of geometrical and internal nodes The magnitude of the displacement jump at the discontinuity is governed by an additional set of degrees of freedom which are added to the existing nodes of the model. Figure 5 shows Copyright? 23 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 23; 58:213 24

18 23 J. J. C. REMMERS, G. N. WELLS AND R. DE BORST Figure 5. Enhanced nodes (black) whose support contains a discontinuity (grey surface). The nodes on the edge of the discontinuity are not enhanced in order to assure a zero delamination opening here. the activation of these additional sets of degrees of freedom for a given (static) delamination surface in the model. Both the geometrical and the internal nodes are enhanced when the corresponding element is crossed by the discontinuity. This implies that each geometrical node now contains three additional degrees of freedom next to the three regular ones, giving six degrees of freedom in total. Each internal node has one extra degree of freedom added to the single regular degree of freedom. The discontinuity always stretches through an entire element. This avoids the need for complicated algorithms to describe the stress state in the vicinity of a delamination front within an element. As a consequence, the discontinuity touches the boundary of an element. The geometrical and internal nodes that support this boundary are not enhanced in order to assure a zero crack tip condition [8]. At variance with conventional interface elements, a criterion is needed for the placement of the discontinuity upon propagation. This criterion is based on the stress state at the delamination front, which can be monitored by adding temporary sample points. When the criterion exceeds a threshold value, the discontinuity is extended into the new element. The corresponding nodes of this element are enhanced with an additional set of degrees of freedom Condensation of the internal degrees of freedom Since the internal degrees of freedom are not able to support an external loading, it was suggested by Parisch to eliminate them on element level by condensation [14]. The complete set of the linearised equilibrium equations for a single element can be written as K aa K ab K ap K aq da fa ext fa int K ba K bb K bp K bq db f ext b f int b = K pa K pb K pp K pq dp f int (84) p K qa K qb K qp K qq dq f int q where da etc. denote the incremental degrees of freedom. Since the applied force at the internal degrees of freedom are equal to zero, we can write the incremental internal degrees Copyright? 23 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 23; 58:213 24

19 SOLID-LIKE SHELL ELEMENT ALLOWING FOR ARBITRARY DELAMINATIONS 231 of freedom dp and dq as follows: [ ] [ ] 1 ( )[ ] dp Kpp K pq Kpa K pb da = + dq db K qp K qq K qa K qb f p int f int q (85) The internal degrees of freedom can be eliminated by substituting this relation into Equation (84). Further derivation yields the condensed stiness matrix: [ ] [ ][ ] 1 [ ] Kaa K ab Kap K aq Kpp K pq Kpa K pb K = (86) K ba K bb K bp K bq and the condensed internal forces vector [ f int ] [ ][ ] 1 f int a Kap K aq Kpp K pq = f p int (87) f int b K bp K bq It is emphasized that the additional degrees of freedom at geometrical nodes that describe the displacement jump cannot be condensed. The magnitude of the displacement jump must be continuous across element boundaries. The degrees of freedom that describe this jump are therefore global and cannot be solved on the element local level. An exception is made for the additional internal degrees of freedom q. Since the regular internal degrees of freedom p are already discontinuous across element boundaries, there is no need for the additional internal degrees of freedom to be continuous. In practice, an enhanced eight-node solid-like shell element has a total of 56 degrees of freedom; two times three in each geometrical node plus two times four internal degrees of freedom. These last two sets are eliminated by condensation reducing the contribution of the element to the global solution vector to 48 degrees of freedom Shear locking It was observed by Parisch [14] that the eight-node element suers from the same shear locking phenomenon as general four-node shell elements. A remedy to this problem is the assumed natural strain approach [19]. Here, the transverse shear strains 3 in Equation (32b) are evaluated at four alternative positions, the tying points A D in Figure 6 along the edges K qp K qp K qq K qq K qa f int q K qb η C ζ 7 B ξ 8 IV 4 D 5 A 6 II 2 geometrical node internal node I tying point 1 Figure 6. Location of tying points in eight-node solid-like shell element. Copyright? 23 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 23; 58:213 24

20 232 J. J. C. REMMERS, G. N. WELLS AND R. DE BORST of the elements: 23 = C ˆ C 23 + A ˆ A 23 + H d; ( C C 23 + A A 23) 13 = B ˆ B 13 + D ˆ D 13 + H d; ( C C 23 + A A 23) (88) In this equation S denote the corresponding linear shape functions. The contributions of the assumed natural strain terms to the discretised variational strain and the geometric part of the stiness matrix can be found in Appendix C Numerical integration The construction of the element internal force vector and stiness matrix requires a proper integration of three domains;, + and d;. In this case, all three domains are standard geometrical entities (two six-sided volumes and a rectangular surface) and can therefore be integrated numerically with standard Gauss integration schemes. 1. NUMERICAL EXAMPLES Two examples are presented to demonstrate the performance of the element. In the rst example, attention is focused on the accuracy of the element for a decreasing thickness in geometrically linear applications. In the second example, the performance of the element in a geometrically and physically nonlinear analysis is illustrated by means of the simulation of a delamination buckling test A peel test To test the performance of the new element for a decreasing thickness, a peel test has been performed. The double cantilever beam shown in Figure 7 consists of two layers of the same material with Young s modulus E = 1: N=mm 2 and Poisson ratio =:. The beam has delaminated over its entire length. The test is performed for two dierent models. The rst model contains ten eight-node enhanced solid-like shell elements (SLS+8); the second model is built with just ve 16-node enhanced solid-like shell elements (SLS+16). In both cases, only one element in thickness direction is used. The delamination is modelled by a traction free discontinuity. Figure 7. Geometry of a delaminated double cantilever beam under peel loading. Copyright? 23 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 23; 58:213 24

21 SOLID-LIKE SHELL ELEMENT ALLOWING FOR ARBITRARY DELAMINATIONS 233 Figure 8. Double cantilever beam loaded by peel force. Left: Normalized linear solution as a function of the length over layer thickness ratio. Right: deformed mesh. Figure 9. Geometry of double cantilever beam with initial delamination a under compression. The linear out-of-plane displacements are given as functions of the ratio of layer thickness and beam length in Figure 8. The results are normalised by the exact solution that follows from the theory of beam deections. The eight-node enhanced solid-like shell element gives nearly exact results for aspect ratios up to 2. The performance of the sixteen-node solidlike shell element is even better. Indeed, it appears that the bending properties of the enhanced element are not aected by the enhancement and give the same performance as the underlying element [14]. The enhanced element eectively acts as two elements Delamination buckling of a cantilever beam Next, a combination of delamination growth and structural instability is considered [12, 2]. The double cantilever beam in Figure 9 has an initial delamination length of a = 1 mm and is subjected to an axial compressive load 2P. Two small perturbation forces P are applied to trigger the desired buckling mode. Both layers are made of the same material with Young s modulus E = 135 N=mm 2 and Poisson s ratio =:18. Owing to symmetry in both the geometry of the model and the applied loading, delamination propagation can be modelled with an exponential mode-i decohesion law: ( tdis n = t ult exp t ) ult vdis n (89) G c where tdis n and vn dis are the normal traction and displacement jump, respectively. The ultimate traction t ult is equal to 5 N=mm 2, the fracture toughness is G c =:8N=mm. The shear tractions Copyright? 23 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 23; 58:213 24

22 234 J. J. C. REMMERS, G. N. WELLS AND R. DE BORST Figure 1. Delamination buckling test. Left: tip displacement as a function of the applied axial load P. Right: nal deformation. are equal to zero. The delamination surface is extended when the normal stress in thickness direction ahead of the delamination front ( 33 ) exceed the ultimate traction t ult. The critical load for local buckling of the beam, prior to delamination growth can be calculated analytically using the equation for a single cantilever beam with length a and thickness h. For the given material parameters, the buckling load is P cr = 2 Eh 3 =2:22 N (9) 48a 2 The nite element mesh used for the analysis is composed of eight-node enhanced solid-like shell elements and is shown in Figure 1. Again, it consists of only one element in thickness direction. In order to capture delamination growth correctly, the mesh is locally rened. Figure 1 shows the lateral displacement u of the beam as a function of the external force P. The load-displacement response for a specimen with a perfect bond (no delamination growth) is given as a reference. The numerically calculated buckling load is in agreement with the analytical solution. Steady delamination growth starts at a lateral displacement u = 4 mm, which is in agreement with previous simulations [2]. 11. CONCLUSIONS In this contribution, a new concept for the simulation of delamination growth in thin layered composite structures is presented. The delamination crack is incorporated in a solid-like shell element [14] by means of the partition-of-unity property of nite element shape functions. The approach has a number of advantages. First, the displacement jump is only activated as the delamination propagates, which already results in a reduction of the total number of degrees of freedom. Moreover, it is possible to model a laminate and capture delamination, with just one element in the thickness direction. With conventional techniques, at least double the number of elements is needed. These properties allow the use of coarse meshes and to analyse delamination phenomena on a structural level. A delamination buckling example underlines the excellent performance of the element in combined geometrically and physically non-linear analyses. Copyright? 23 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 23; 58:213 24

23 SOLID-LIKE SHELL ELEMENT ALLOWING FOR ARBITRARY DELAMINATIONS 235 APPENDIX A. DERIVATION OF THE B MATRIX The variational strains as derived in Equation (42) can be written in the following matrix notation: S = H 1 û 1 + H 1û ; 1 + H 1 1û 1; 1 + H 2û ; 2 + H 1 2û 1; 2 + H w ŵ + H d; (H 1 ũ 1 + H 1ũ ; 1 + H 1 1ũ 1; 1 + H 2ũ ; 2 + H 1 2ũ 1; 2 + H w w) (A1) where H 1 = d T e T 2 e T 1 + 4wd T d T ; 2 d T ; 1 (A2) H 1 1 = e T 1 e T 2 d T ; H 1 2 = e T 2 e T 1 d T ; H w = 2d d (A3) H 1 1 = e T 1 e T 2 d T + d T ; 1 [2e T 1 G e T 2 G 2 1] e T 2 G 1 2 d T ; 2 [e T 2 G e T 1 G e T 2 G 2 2] (A4) Copyright? 23 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 23; 58:213 24

24 236 J. J. C. REMMERS, G. N. WELLS AND R. DE BORST e1 T G 2 1 e T 2 d; T 2 [e1 T G e T 2 G 2 2] H2 1 = + e1 T d; T 1 [e1 T G e2 T G e T 1 G 2 2] d T (A5) The variational displacement vectors can be obtained from the nodal degrees of freedoms via the shape functions as dened in Equations (53) (59). Substituting these equations into (A1) gives û 1 = N 1 a; ũ 1 = N 1 b (A6) û ; = N ; a; ũ ; = N ; b (A7) û 1; = N 1; a; ũ 1; = N 1; b (A8) ŵ = N w p; w = N w q (A9) where B u and B w are dened as S = B u a + B w p + H d; (B u b + B w q) (A1) B u = H 1 N 1 + H1N ; 1 + H1N 1 1; 1 + H2N ; 2 + H2N 1 1; 2 (A11) B w = H w N w Note that the matrices B u and B w are identical for the regular displacement terms a and p as well as the additional terms b and q. The matrices B u and B w are still dened in the isoparametric frame of reference and must be transformed in the element local system l j, using the transformation tensor Tk i in Equation (33). APPENDIX B. STRESS DEPENDENT PART OF THE STIFFNESS MATRIX The relations for the incremental change of the virtual strains in Equation (82) can be written as a function of the nodal degrees of freedom a and b and the internal degrees of freedom p and q by using the shape functions as dened in Equations (53) (55), (58), and (59). 2d( )=(a + H d; b) T N T ;N ; (da + H d; +(a + H d; b) T N T ;N ; (da + H d; (B1a) Copyright? 23 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 23; 58:213 24

25 SOLID-LIKE SHELL ELEMENT ALLOWING FOR ARBITRARY DELAMINATIONS 237 2d( 3 )=(a + H d; b) T N T ;N 1 (da + H d; +(a + H d; b) T N T 1 N ; (da + H d; 2d( 33 )=2(a + H d; b) T N T 1 N 1 (da + H d; (B1b) (B1c) 2d( )=(a + H d; b) T N1;N T ; (da + H d; +(a + H d; b) T N;N T 1; (da + H d; +(a + H d; b) T N;N T 1; (da + H d; +(a + H d; b) T N1;N T ; (da + H d; (a + H d; b) T C T N 1; (da + H d; (a + H d; b) T C T N ; (da + H d; (a + H d; b) T N;C T (da + H d; (a + H d; b) T N T ;C (da + H d; (B1d) 2d( 3 )=(a + H d; b) T N T 1;N 1 (da + H d; +(a + H d; b) T N T 1 N 1; (da + H d; (B1e) 2d( 33 )=(a + H d; b) T N T 1 dn w (da + H d; +(a + H d; b) T N T wd T N 1 (da + H d; + w(a + H d; b) T N T 1 N 1 (da + H d; (B1f) where C = G 1 N ; 1 + G 2 N ; 2 (B2) The incremental change of the virtual strains still refers to the covariant components. However, the stress components are referred to the local system l j. Hence, for the set-up of the stress dependent part of the stiness matrix K geo the factor! ij is introduced as! mn = T m k T n l kl (B3) Elaboration of Equation (B3) and Equations (B1a) (B1d) give the following matrices: A uu =! 11 N T ; 1N ; 1 +! 22 N T ; 2N ; 2 +! 33 N T 1 N 1 +! 12 (N T ; 1N ; 2 + N T ; 2N ; 1 ) +! 23 (N T ; 2N 1 + N T 1 N ; 2 )+! 13 ((N ; 1 ) T N 1 +(N 1 ) T N ; 1 ) Copyright? 23 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 23; 58:213 24

26 238 J. J. C. REMMERS, G. N. WELLS AND R. DE BORST +! 11 (N T 1; 1N ; 1 + N T ; 1N 1; 1 C T 1N ; 1 N T ; 1C 1 ) +! 22 (N T 1; 2N ; 2 + N T ; 2N 1; 2 C T 2N ; 2 N T ; 2C 2 )! 33 4wN T 1 N 1 +! 23 (N T 1; 2N 1 + N T 1 N 1; 2 )+! 13 (N T 1; 1N 1 + N T 1 N 1; 1 ) +! 12 (N T 1; 1N ; 2 + N T 1; 2N ; 1 + N T ; 1N 1; 2 + N T ; 2N 1; 1 C T 1N ; 2 C T 2N ; 1 N; T 1C 2 N; T 2C 1 ) A wu =! 33 4Nwd T T N 1 A uw =! 33 4N1 T d T N w A ww = (B4a) (B4b) (B4c) (B4d) Integration yields of these terms over the domain yields the geometric part of the stiness matrix: A uu d A uu d K geo + = A wu d A wu d + A uu d A uw d A uw d + + A uu d A uw d A uw d A wu d + A wu d + (B5) APPENDIX C. ASSUMED NATURAL STRAINS It was noted by Parisch [14] that the eight-node shell element, which can be compared with an original four-node shell element suers from shear locking: the shear stiness is overestimated. As proposed by Bathe et al. [19], an assumed natural strain approach is applied to overcome this problem. Here, the strain terms a3 are replaced by an alternative strain, determined in four tying points located at the mid-surface of the element as shown in Figure 6. The strains are dened as 23 = C ˆ 23 C + A ˆ A 23 + H d; ( C 23 C + A A 23) (C1) 13 = B ˆ 13 B + D ˆ 13 D + H d; ( C 23 C + A A 23) Copyright? 23 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 23; 58:213 24

27 SOLID-LIKE SHELL ELEMENT ALLOWING FOR ARBITRARY DELAMINATIONS 239 Table CI. Position of tying points and the corresponding node numbers. Co-ordinates Corresponding nodes Tying point J K L M A. 1: B C D 1: The virtual strains in tying point S can be written directly in terms of the regular and enhanced degrees variational degrees of freedom a and b: a J b J a K b K 23 S = B ANS u + H d; B ANS u (C2) a L b L a M b M where the superscripts J; K; L and M represent the nodes which span the side on which the sampling point S is located, see also Table CI. The matrix B ANS u replaces the terms that are related to the membrane strains 13 and 23 in Equation (A11) and is equal to where B ANS u = 1 4 [(e d); ( e d); (e + d); ( e + d)]s (C3) e S = 1 4 ( x J x K + x L + x M ); d S = 1 4 (x J x K + x L x M ) (C4) where for example, x J is the position of node J in deformed conguration: x J = X J + a J + H d; b J (C5) Furthermore, the terms in the geometric part of the stiness matrix K geo that corresponds to the shear stresses! 13 and! 23 are replaced by Auu ANS d Auu ANS d + K geo; ANS = A uu ANS d A ANS uu d (C6) + + where A ANS uu =! 23 ( B H B 23 + D H D 23)+! 13 ( A H A 13 + C H C 13) (C7) Copyright? 23 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 23; 58:213 24

28 24 J. J. C. REMMERS, G. N. WELLS AND R. DE BORST where H S 3 represents the mapping matrices and is obtained by expanding the matrix H3 S to the correct nodal degrees of freedom according to Table CI [14]. I 3 I 3 H3 S = 1 I 3 I 3 (C8) 8 I 3 I 3 I 3 I 3 ACKNOWLEDGEMENT GNW gratefully acknowledges nancial support from the Netherlands Technology Foundation (STW). REFERENCES 1. Allix O, Ladeveze P. Interlaminar interface modeling for the prediction of delamination. Composite Structures 1992; 22(4): Schellekens JCJ, de Borst R. A nonlinear nite-element approach for the analysis of mode-i free edge delamination in composites. International Journal of Solids and Structures 1993; 3(9): Alfano G, Criseld MA. Finite element interface models for the delamination analysis of laminated composites: mechanical and computational issues. International Journal for Numerical Methods in Engineering 21; 5(7): Allix O, Corigliano A. Modeling and simulation of crack propagation in mixed-modes interlaminar fracture specimens. International Journal of Fracture 1996; 77(2): Babuska I, Melenk JM. The partition of unity method. International Journal for Numerical Methods in Engineering 1997; 4(4): Belytschko T, Black T. Elastic crack growth in nite elements with minimal remeshing. International Journal for Numerical Methods in Engineering 1999; 45(5): Moes N, Dolbow J, Belytschko T. A nite element method for crack growth without remeshing. International Journal for Numerical Methods in Engineering 1999; 46(1): Wells GN, Sluys LJ. A new method for modelling cohesive cracks using nite elements. International Journal for Numerical Methods in Engineering 21; 5(12): Moes N, Belytschko T. Extended nite element method for cohesive crack growth. Engineering Fracture Mechanics 22; 69(7): Schellekens JCJ, de Borst R. On the numerical integration of interface elements. International Journal for Numerical Methods in Engineering 1993; 36(1): Wells GN, de Borst R, Sluys LJ. A consistent geometrically non-linear approach for delamination. International Journal for Numerical Methods in Engineering 22; 54(9): Wells GN, Remmers JJC, de Borst R, Sluys LJ. A large strain discontinuous nite element approach to laminated composites. IUTAM Symposium on Computational Mechanics of Solid Materials at Large Strains 23; Bischo M, Ramm E. Shear deformable shell elements for large strains and rotations. International Journal for Numerical Methods in Engineering 1997; 4(23): Parisch H. A continuum-based shell theory for non-linear applications. International Journal for Numerical Methods in Engineering 1995; 38(11): Hashagen F, de Borst R. Numerical assessment of delamination in bre metal laminates. Computer Methods in Applied Mechanics and Engineering 2; 185(2 4): Ogden RW. Non-Linear Elastic Deformations. Ellis Horwood Limited: Chichester, U.K., Belytschko T, Liu WK, Moran B. Nonlinear Finite Elements for Continua and Structures. Wiley, New York, Bathe KJ. Finite Element Procedures. Prentice-Hall, Inc., Upper Saddle River, NJ, USA, Bathe KJ, Dvorkin EN. A formulation of general shell elements the use of mixed interpolation of tensorial components. International Journal for Numerical Methods in Engineering 1986; 22: Allix O, Corigliano A. Geometrical and interfacial non-linearities in the analysis of delamination in composites. International Journal of Solids and Structures 1999; 36(15): Copyright? 23 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 23; 58:213 24