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1 This article was downloaded by: [Blackmore, Robert][informa internal users] On: 4 August 8 Access details: Access Details: [subscription number ] Publisher Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: Registered office: Mortimer House, Mortimer Street, London W1T 3JH, UK International Journal for Computational Methods in Engineering Science and Mechanics Publication details, including instructions for authors and subscription information: Explicit FE-Formulation of Interphase Elements for Adhesive Joints Thomas Carlberger a ; K. Svante Alfredsson b ; Ulf Stigh b a SAAB Automobile AB, Trollhättan, Sweden b University of Skövde, Skövde, Sweden Online Publication Date: 1 September 8 To cite this Article Carlberger, Thomas, Alfredsson, K. Svante and Stigh, Ulf(8)'Explicit FE-Formulation of Interphase Elements for Adhesive Joints',International Journal for Computational Methods in Engineering Science and Mechanics,9:5,88 99 To link to this Article: DOI: 1.18/ URL: PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

2 International Journal for Computational Methods in Engineering Science and Mechanics, 9:88 99, 8 Copyright c Taylor & Francis Group, LLC ISSN: print / online DOI: 1.18/ Explicit FE-Formulation of Interphase Elements for Adhesive Joints Thomas Carlberger, 1 K. Svante Alfredsson, and Ulf Stigh 1 SAAB Automobile AB, Trollhättan, Sweden University of Skövde, Skövde, Sweden The potential of adhesive bonding to improve the crashworthiness of cars is attracting the automotive industry. Large-scale simulations are time consuming when using the very small finite elements needed to model adhesive joints using conventional techniques. In the present work, a D-interphase element formulation is developed and implemented in an explicit FE-code. A simplified joint serves as a test example to compare the interphase element with a straightforward continuum approach. A comparison shows the time-saving potential of the present formulation as compared to the conventional approach. Moreover, the interphase element formulation shows fast convergence and computer efficiency. Keywords Interphase Element, Dynamic Fracture, Adhesive Joint, Explicit FE, Execution Time NOMENLCATURE A metric matrix a 1 a 1 nodal degrees of freedom of interphase element a I (t) displacement and rotation of node I at time t B specimen width b volume load c I vector with arbitrary elements associated with node I t,t c time step and critical time step, body and outer surface δ ij Kronecker delta δ separation vector of adhesive layer E Young s modulus force vector associated with node I f I F applied load vector G matrix in the relation between δ and a g essential boundary conditions h thickness of adhesive layer H rotational inertia compensation number H 1,H thickness of lower and upper adherend, respectively I, J node numbers J cons, J lump moment of inertia from consistent (cons) and lumped (lump) mass matrix κ wave number ξ,ζ local coordinates of adhesive layer L,L b,l s element length, bonded length, and specimen length l distance between two zero-values of the peel stress M mass matrix m mass of adhesive in one element n outward normal vector n t, N DOF number of time steps, number of degrees of freedom N I shape function associated with node I. ω max largest eigenfrequency of the structure ω angular acceleration R virtual power associated with the adhesive layer r adherend number ρ i,ρ r material density for the interphase element (adhesive) and the adherends respectively. r {1, } S interface surface σ Cauchy stress tensor rise time T r, T s computer execution time and total time to simulate Received 6 September 7; accepted 8 April 8. The authors are grateful to Mr. Anders Biel and Dr. Kent t t ev time evaluation time for one degree of freedom and one Salomonsson for fruitful discussions during this study. The authors time step would also like to acknowledge the Swedish Consortium for Crashworthiness for funding this project. t prescribed traction vector Address correspondence to K. Svante Alfredsson, University adherend cross-sectional rotation of Skövde, P.O. Box 48, SE Skvöde, Sweden. u displacement vector svante.alfredsson@his.se v, w shear and peel deformation of adhesive layer 88 T c

3 EXPLICIT FE-FORMULATION OF INTERPHASE ELEMENTS FOR ADHESIVE JOINTS 89 v ν ν a,i x displacement and rotation vector of adherend midline weight function vector Poisson s ratio of adherend (index a) and adhesive (index i) coordinate 1. INTRODUCTION Car bodies consist of large shell structures connected with some joining technique. Traditionally, the most frequent technique is spot-welding. Although this method has many advantages, it is essentially limited to mono-material joints. Increasing demands on fuel efficiency and lowered emissions increase the need for further optimizations. To this end, a multi-material car body is a promising possibility. Recently the potential to use adhesive bonding has been identified. With this method it is not only possible to join dissimilar materials but also to improve both stiffness and strength in mono-material structures. In the product development phase, explicit FE-simulations are frequently used to evaluate the crashworthiness of prospective car bodies, cf. e.g. [1]. In these simulations, there is a need for efficient modelling technique of adhesive joints. Specifically it is necessary to model the structure in such a way that the simulation execution time, T r,iskept as short as possible, typically not exceeding 1 15 hours. An inherent limit with the explicit FE-method is the Courant limit. In order for an explicit FEsimulation to be numerically stable, the time step t must be smaller than /ω max, where ω max is the largest eigenfrequency of the structure, cf. e.g. []. The critical time step can be estimated as the shortest time it takes an elastic wave to travel the distance between two nodes. In practice, this critical time step sets a limit on the smallest element length for a given time step and total execution time. This critical element length is for an adhesive consisting mainly of epoxy about 1. mm with a given time step t = 1 µs. This element length should be compared to the typical adhesive layer thickness, h =. mm. Thus, introduction of an adhesive layer modelled with continuum finite elements (three elements across the adhesive layer) into a structure with an original critical time step of 1 µs will increase the execution time by a factor of about eighteen, cf. [1]. A trick, often used to circumvent the Courant limit, is mass scaling or a combination of mass scaling and a reduction of Young s modulus [3]; the material density is increased and/or Young s modulus of the adhesive is decreased for the too small elements until the Courant limit is fulfilled. This method is to be used with caution since the added mass or decreased stiffness will influence the results. Especially if the adhesive bond line is oriented along the crash direction, added mass may give too large influences on the results. In a recent study of the butt joint, a 4% reduction of the stiffness of the adhesive layer is shown to be tolerable [1]. These types of results should, however, be taken with some caution since they are confined to specific joint geometries and load cases. One way to avoid the use of small time steps or mass-scaling is to model the adhesive layer as a cohesive zone rather than as h v τ FIG. 1. Deformation modes of the adhesive layer with thickness h: peel, w, and shear, v. Conjugated stress components σ and τ. σ a continuum. The use of cohesive laws to model adhesive joints was proposed in [4] and [5]. In this approach, the deformation of the adhesive layer is assumed to be dominated by two deformation modes viz. peel and shear, cf. Figure 1. This assumption is supported by asymptotic analyses, cf. [6]. A number of experimental and numerical studies show that the cohesive law model predicts fracture accurately, cf. e.g. [7 9]. In a cohesive law, the traction acting on the adherends depends on the deformation of the adhesive. At small deformation, the response is elastic. At some critical state, the adhesive deforms inelastically. Eventually the traction tends to zero and the adhesive fractures. At this moment, a crack propagates in the adhesive layer. With a structural epoxy adhesive, the zone of inelastic behavior in front of the crack tip is very much larger than the thickness of the adhesive layer, cf. [8, 1]. A recent review of cohesive models is given in [11]. In a finite element context, cohesive models are traditionally used to model fracture in a continuum model. However, in many adhesive joint applications, large structures are modelled with shell elements. In [1] and [13], FE-methods to analyze delamination of composite shells are developed. Nodes on two sides of the anticipated delamination are connected by a generalized spring. The force-deformation relation for the spring represents the action of the cohesive surface. That is, each spring represents the adhesive in an area around the spring. Thus, it is necessary to consider the mesh size when assigning constitutive properties to the spring. In this paper a dedicated finite element for modelling adhesive bonds between shell elements is presented. The novel approach is to span the entire volume from the shell mid plane of the first adherend across the adhesive volume to the mid plane of the second adherend in one interphase element. As compared to the spring elements in [1, 13], no consideration to the mesh size has to be done to input constitutive behavior to the interphase element. The computational efficiency of the element is demonstrated by comparison with methods used in the automotive industry today, where continuum elements are used to model the adhesive layer.. INTERPHASE FORMULATION Figure shows a part of an adhesive joint consisting of two shells. In this case the shells are two of the parts creating σ τ w

4 9 T. CARLBERGER ET AL. FIG.. 3D-model of adhesively joined sheets (adherends). The adhesive layer is represented by a solid element connected to the shell elements by rigid bars. the B-pillar of a passenger vehicle. Typical dimensions are adhesive layer thickness h =. mm and thickness of shells H 1 H =.8 mm. Thus, we often find h < H 1 H. The bonded sheets are usually modelled with shell elements in the explicit FE-formulation. This means that the displacement field in the shells is governed by nodal displacements and rotations in the FE-model, cf. e.g. [14, 15]. The coupling of the rotational degrees of freedom to the deformation of the adhesive layer has to be considered in this case. For instance, assume a state of deformation where all displacements are zero but rotations exist, then the adhesive layer may be deformed in shear. Thus, in order to connect the thin adhesive layer correctly to the shells, the coupling of the rotational degrees of freedom of the shells to the deformation of the adhesive layer has to be considered. A simple method to achieve this coupling is to model the adhesive as a continuum using solid elements and to connect these to the shell elements by the use of rigid bars or stiff beam-elements, cf. Figure. This approach does, however, introduce some disadvantages as will be demonstrated here. The adhesive element is connected to the adherends by rigid or stiff beams. Although this is a tedious work, the process may be automated. In this case, the use of rigid beams is preferred, since it does not add degrees of freedom to the system. The stiff beams will add degrees of freedom to the system, but more severely, they will negatively influence the critical time step, t c,ofthe system. Since the adhesive is modelled as a continuum, the very thin thickness of the adhesive layer will, referring to the Courant limit, imply a correspondingly short time step. An attractive alternative method to avoid this tedious work and not severely influencing the critical time step is to use an interphase element, cf. Figure 3b. Obvious profits of modelling the adhesive with interphase elements are easier modelling and shorter execution time, as will be exemplified in this paper. The interphase formulation uses relative displacements. It may be noted that the use of relative displacements reduces the condition number of matrices, cf. [16]. Next, we will derive the governing equations in the FE-problem and the consistent mass matrix. FIG. 3. D-model of adhesively joined sheets (adherends). (a): Adhesive and adherends. (b): Interphase element.

5 EXPLICIT FE-FORMULATION OF INTERPHASE ELEMENTS FOR ADHESIVE JOINTS Governing Equations Consider a body occupying a domain with a boundary. Balance of linear momentum in the body is given by σ ji,j + ρb i ρü i = in. (1) where σ is the Cauchy stress tensor, ρ the density, b the body force per unit mass, u the displacement vector. Dots are used to Indicate differentiation with respect to time, t. Here, lowercase indices indicate Cartesian coordinates, i, j {1,, 3}; all components are taken with respect to a common inertial frame. Moreover, summation is to be taken over repeated indices. On the parts of the boundary where the traction is prescribed the following equation applies σ ji n j = t i on. () where t is the prescribed traction vector. On other parts of, the displacement and velocity might be prescribed. To obtain the weak form of Equation (1), we multiply with an arbitrary weight function ν and use the divergence theorem, cf. e.g. [17]. Omitting some intermediate steps, the result is ν i, j σ ji d ν i ρb i d ν i t i d( ) + ν i ρü i d =, (3) which is known as the principle of virtual power. In order to FEformulate this equation we introduce the conventional FE-ansatz u = N I a I, (4) where the indices in capital letters indicate the node numbers, i.e. I = 1,,... n nodes, and summation should be taken over repeated indices in capital letters. Here a I (t)isthe displacement/ rotation of node I and N I is the shape function associated with node I. The weight function ν is approximated according to the Galerkin-method, i.e. using the same shape functions as are used for the displacements ν = N I c I, (5) where c I is a vector with arbitrary elements associated with node I. Now, substituting Equation (5) into the principle-of-virtual power, Eq. (3) and noting that the c I :s are arbitrary we arrive at N I σ ji d N I ρb i d N I t i d( ) x j + N I ρü i d = (6) With Equation (4) we arrive at f int I f ext I + Mä I =, (7) where f int ii and N I σ ji d, fii ext x j M ijij δ ij N I ρb i d ρ N I N J d. N I t i d( ), (8a,b) (8c) Here, δ ij is the Kronecker delta. Equation (7) is a set of second order ordinary differential equations, f ext I (a, ȧ, t) f int I (a, ȧ, t) = Mä I f kin I, (9a) where the elements of f kin I f kin ii are described by N I ρü i d = ρ N I N J d ä ij. (9b) With a diagonal mass matrix, Equation (9a) constitutes the set of equations of the explicit FE-method. The procedure to diagonalize the generally non-diagonal mass matrix is known as lumping, cf. e.g. []. Essential boundary conditions complement Equation (9a). These are imposed on n c degrees of freedom, i.e. g I (a, ȧ, t) =, I = 1,,...,n c (1) where the nodal degrees of freedom a I have been collected in the vector a. The elements of a are usually nodal displacements and, for beam and shell formulations, nodal displacements and rotations. Similarly, the elements of f are associated with nodal forces, and for beam and shell elements, with nodal forces and moments. These equations constitute the foundation for the development of an interphase element in the next subsection... The Interphase Element In this section we will derive the nodal forces and the mass matrix associated with the interphase element. We will also discuss the properties of damping associated with an adhesive layer. Consider an arbitrary point on the adherend r, r {1, } according to Figure 4. Here r = 1 denotes the lower adherend and r = the upper one. Let a (r) denote the degrees of freedom of the adherend r. According to Figure 3, the vectors are given by a (1) = [ a 1 a a 3 a 4 a 5 ] T a 6, a () and = [ a 7 a 8 a 9 a 1 a 11 ] T a 1 (11) Furthermore, let v (r) denote the displacement and rotation of a point along the mid line of adherend r.for a beam element, this vector contains the two mid line displacements u (r) x and u (r) y in x- and y-direction, respectively and the cross-sectional rotation,

6 9 T. CARLBERGER ET AL. Adherend mid line v () interface surface S () u () Current configuration i.e. (r) z, Reference configuration y z v (r) = [ u (r) x FIG. 4. u (r) y x ξ u (1) v (1) node interface surface S (1) Adherend 1 mid line Beam degrees of freedom, v (r) and displacements u (r) on the interface surfaces. ] (r) T z. (1) The local degrees of freedom v (r) are interpolated from the element nodal degrees of freedom, a (r),by v (r) = N (r) a (r) (13) ForaMindlin beam element, the interpolations of the displacements and rotation of the middle line are independent, i.e. u (1) x N 1 N v (1) = u (1) y = N 1 N N 1 N (1) z a 1 a : a 6 = N(1) a (1) (14) where the nodal shape functions are given by N 1 = 1 ξ and N = ξ with ξ given in Figure 4. Now, let u (r) = [ u (r) x u (r) y ] T be the displacement on the interface between the adherend (r) and the adhesive. Figure 4 shows where [ ] 1 A (1) H1 / = 1 [ ] and A () 1H / =, (16) 1 are metric matrices. The displacement of the interface boundary, u (r),isobtained from Eqs. (13) and (15): u (r) = A (r) N (r) a (r). (17) The separation of the interface surfaces S (1) and S (), i.e. the deformation of the adhesive layer, δ,isobtained from Equation (17) as δ = u () u (1) = [ A (1) N (1) A () N ()] a Ga, where a = [ a (1)T a ] ()T T (18a,b) whereby the matrix G is defined. The total separation of the adherends is given by the components δ x = deformation of the adhesive layer in the x-direction and δ y = deformation in the y-direction. This forms the separation vector δ = [ δ x δ y ] T. By use of the orientation of the adhesive layer, the separation vector can be decomposed in the peel and shear components, respectively. Comparing with Equation (17), weight functions, ν (r), are chosen according to the Galerkin method as u (r) = A (r) v (r) (15) ν (r) = A (r) N (r) c (r) (19)

7 EXPLICIT FE-FORMULATION OF INTERPHASE ELEMENTS FOR ADHESIVE JOINTS 93 where c (r) contains arbitrarily chosen nodal values. Now, form the virtual power associated with the action of the adhesive on the adherends, R, by R = S (J) ( ν () ν (1))T t ds () where t is the traction acting on the adhesive on the adhesive/adherend interface S (J), cf. Equation (3). With Equation (19) FIG. 5. Local thickness- and length-coordinate systems. ν () ν (1) = Gc where c = and we arrive at [ c (1)T c ()T ] T, (1a,b) R = c T S (J) G T t ds c T F () where the integral is identified as the contribution to the nodal force vector from the adhesive layer. We may express the integrand in this expression by the use of Equation (18a) as leading to F = [ N (1) T ] G T A (1)T t = t (3) N ()T A ()T [ F (1)T F ()T ] T, where F (I ) = S (I ) ( A (I ) N (I )) T t ds (4a,b) which are the forces used in the explicit FE-formulation. Up to this point, the derivation is applicable for general, nonlinear cohesive laws. For simplicity, we now assume linear elastic behavior of the adhesive in two dimensions. If we assume that the adhesive layer is oriented along the x-axis, the traction is given by [ ] G/h t = δ (5) E/h where E and G are the elastic modulii in pure uniaxial straining and shear, respectively. A non-zero value of the normal traction, t, implies that the lateral normal stress in the adherends, σ,isnon-zero, cf. Equation (). Strictly speaking, this stress leads to deformations of the adherends. However, most beam element formulations do not account for lateral normal stresses and corresponding deformations. Thus, when implementing the interphase element into an existing FE-code, this effect cannot be accounted for. Fortunately, due to the large mismatch in stiffness between the adherends and the adhesive, the thickness change of the adherends will only have a minor influence on the results in terms of the adhesive deformations. Hence the effect of adherend thickness change is neglected is the present study. The stable time step is calculated according to t c /ω max, where ω max is the maximum eigenfrequency of the structure. It should be noted that the calculation of ω max is not performed in commercial explicit codes due to the amount of work associated with this task and the requirements of computer memory. Instead, the approximation t c = l min c where c = E ρ (6a,b) and l min is the shortest element length in the model, is used. However, the approximation does not take care of contact stiffness, such as cohesive zone stiffness. Care has to be taken in choosing, e.g., contact penalty stiffness or cohesive zone stiffness, such that the influence on the time step does not exceed the scale factor. We will now derive the mass matrix, Equation (8c), associated with the interphase element. Let ζ denote a through the thickness coordinate in the adhesive layer with ζ = at the lower interface and ζ = 1atthe upper interface, cf. Figure 5. The displacement u in the adhesive at a point ζ is written u = [ A (1) N (1) (1 ζ ) A () N () ζ ] a = N el a, (7) and the element mass matrix, Equation (8c), is derived in a straightforward manner. The result is M = ρ i N T el N eldv = ρ i BhL 6 V 1 [ N (1)T A (1)T A (1) N (1) N (1)T A (1)T A () N () N ()T A ()T A (1) N (1) N ()T A ()T A () N () ] dξ (8) where ρ i is the density of the adhesive. Evaluation gives the consistent element mass matrix M = ρ i BhL 144

8 94 T. CARLBERGER ET AL. 16 8H 1 8 4H 1 4 H 8 4H H1 4H 1 H1 H 1 H 1 H 4H 1 H 1 H 16 8H 1 8 4H 4 H H1 4H 1 H 1 H H 1 H 1 H 16 8H 8 4H 16 8 S Y M. 4H 4H H 16 8H 16 4H Since we will use the element mass matrix in an explicit FEcode, this matrix has to be transformed to a diagonal matrix to allow the simple numerical scheme in the explicit FE-code. In most instances, this is a well-established procedure; the literature provides several methods for lumping, cf. e.g. [, 17]. In the present case, the mass of the adhesive layer has an offset to the nodes of the interphase element. Thus, if we simply distribute the mass of the adhesive layer, m = ρbhl,tothe nodes, the moment of inertia will be too large. In a crashworthiness analysis, essentially rigid body accelerations are anticipated in large parts of the structure. Thus, we aim at developing a method to lump the element mass matrix such that a rigid body motion gives the correct momentum. Accordingly, the adhesive mass is evenly distributed to the four nodes. By this procedure m/4 is placed at each of the four nodes and on the diagonal of the lumped mass matrix at the positions associated with the translational degrees of freedom. This procedure provides the correct inertia for rigid body translational motion. To determine a suitable diagonal element for the rotational degrees of freedom, we write the lumped element mass matrix as M lumped = ρ ibhl 11αH 1 11αH1 11αH 11αH 4 diag (3) where α is a number to be determined to give an appropriate moment of inertia. Now, we assume a rigid body rotation of the adhesive layer about its mass center, cf. Fig. 6. This motion gives the kinetic energy T = 1 4 ρ ibhl(h + L )ω, (31) where ω is the angular velocity. With the suggested lumped element mass matrix, Equation (3), the kinetic energy for the (9) interphase element is T = 1 ȧt M lumped ȧ = ρ ibhl 8 [ (H1 + h) + (H + h) ] + L αh1 αh ω. (3) With α = 3( H1 + H ) ( ) ( + 6h H1 + H + 4 L + h ) 1 ( H1 + H ), the lumped mass matrix gives the same kinetic energy as the adhesive layer. It may be noted that α is negative, i.e. the d.o.f. corresponding to rotations of the nodes are given negative values for the moment of inertia to compensate for the excessive rotational inertia caused by the lumping. The need for an improvement of the element mass matrix is dependent on the relation between the mass matrix of the adherend and the mass matrix of the adhesive. The lumped element mass matrix for a Mindlin beam element is M r Mindlin = ρ rbh r L 1 1 H 4 r 1 1 Hr diag, r {1, } (33) This beam element mass matrix can be compared to the mass matrix of the interphase element. To this end, form the collected element mass matrix of the two beam elements connecting to the interphase element, i.e. [ ] M 1 M Mindlin = Mindlin M (34) Mindlin To study the importance of modelling the interphase mass matrix correctly, we form the ratio between the inertia of the adherend and the interphase. With typical values the relations in Table 1 result. As Table 1 clearly shows, the elements of the mass matrix of the interphase element are much smaller than those of the TABLE 1 Ratio between lumped rotational inertia, J lumped adhes., and consistent rotational inertia, Jadhes. cons, for the interphase element and ratio between adherend lumped inertia, J lumped adher., and interphase element lumped inertia, J lumped adhes., with typical data for an automotive application FIG. 6. Adhesive, as rigid body, rotating around center. Adhesive thickness h =. mm Adherend thickness H 1 =H =.8 mm Adhesive density ρ i = 135 kg/m 3 Adherend density ρ 1 = ρ = 78 kg/m 3 Element length L = 5mm J lumped adhes. J cons adhes. = 4.1 J lumped adher. = 46 J lumped adhes.

9 EXPLICIT FE-FORMULATION OF INTERPHASE ELEMENTS FOR ADHESIVE JOINTS 95 mass matrix of the beam element. For numerical stability of the explicit method, increased mass is positive, and thus we will not ignore the adhesive mass completely. Therefore we will use the lumped mass matrix of Equation (3) for the interphase element. Material damping is negative for the numerical performance of the explicit FE-method, since it decreases the stable time step, cf. e.g. [17]. Damping in structures is due to mechanisms such as hysteresis in the material and slip in connections. These mechanisms are not well understood. A popular assumption is damping due to viscous mechanisms. However, the amount of dissipated energy due to viscous damping is generally negligible as compared to the energy dissipated by the plastically deforming adherends in a crash analysis. The main task for the adhesive is not to dissipate energy, but keep the adherends together. In this way, the structure is able to dissipate a much larger amount of energy. Since this is a frequent question regarding fracture of adhesives, let us do the following observation. Assume a vehicle with the mass 1 kg is travelling at the velocity m/s. The kinetic energy is then, J. The length of all joints in a car is approximately 4 m. If all joints were adhesively bonded and broke at the moment of impact with a typical joint width 15 mm and typical fracture energy 1 J/m, the total fracture energy would be 6 J. Thus, it is obvious that the kinetic energy is dissipated by other, more important, mechanisms than the fracture energy of the adhesive. The most important energy dissipation mechanism is plastic work by the adherends. 3. NUMERICAL STUDIES The interphase formulation is tested in a D explicit FEmodel of a Double Cantilever Beam, DCB, structure subjected to symmetric dynamic loading, cf. Figure 7. The purpose of the study is to evaluate the convergence of the simplified interphase formulation. The beams are modelled with linear elastic, linearly interpolated Mindlin beam elements and the adhesive with linear elastic, linearly interpolated interphase elements, cf. Eqs. (4, 3), and analyzed using an explicit Matlab R FE-code. This model, shown in Figure 7a, is denoted the beam/interphase model in the following. Due to the constraint imposed by the stiff adherends, an effective Young s modulus, Ē, of the adhesive layer is used in the interphase element, Ē = E i (1 ν i ) (1 + ν i )(1 ν i ), (35) cf. [18]. For comparison, an alternative DCB-model is developed using conventional linearly elastic, linearly interpolated Mindlin beam elements for the adherends and linear elastic, four-node rectangular continuum elements for the adhesive, cf. Figure 7b. The continuum elements of the adhesive are connected to the beam elements of the adherends by means of rigid connections, cf. Figure 7b. The adhesive layer is modelled with three elements over the thickness to provide a possibility for some stress distribution through the thickness of the adhesive layer. This model, which is denoted the beam/continuum model, is evaluated with the commercial FE-software ABAQUS/Explicit, v. FIG. 7. (a) DCB modelled with interphase elements; (b) DCB modelled with continuum adhesive and rigid connections. The layer thickness, h, is. mm.

10 96 T. CARLBERGER ET AL. 5 σ (MPa) FIG. 8. Continuum D model for comparison. Adhesive starts at x =. Adhesive middle layer at y = Additionally, we compare these two approximate models to a fully converged D-model where both the adherends and the adhesive layer are modelled using continuum elements, cf. Figure 8. This model is denoted the continuum model. All models are in-plane deformation. The DCB structure is given typical automotive dimensions: specimen length, L s = 3 mm, specimen width, B = 3 mm, adherend height H 1 = H = H =.8 mm, adhesive layer thickness h =.mmand the bonded length L b = 15 mm, cf. Figure 7. The material of the adherends is steel with Young s modulus E r = 1 GPa, density ρ r = 78 kg/m 3, and Poisson s ratio ν r =.3. The adhesive material is epoxy with Young s modulus E i =. GPa, density ρ i = 135 kg/m 3, and Poisson s ratio ν i =.4. A symmetric force F = 1 N is applied in a smooth manner, such that F = 5(1 cos(π(t/t c )) N for t < T c, where T c corresponds to a fraction of the primary eigenfrequency in bending; the corresponding period is T = 4 µs. For t T c, the force F = 1 N, is held constant w (µm) FIG. 1. Traction-separation behavior for the engineering epoxy adhesive DOW Betamate XW144-3 with layer thickness h =. mm. Data from [1]. Depending on the rise time of the applied force, the deformation response at the adhesive tip will be influenced by wave propagation. This is shown in Figure 9 for the beam/interphase model. With a long rise time, longer than about.1t, the response is mainly characterized by the lowest eigenmode and the influence of T c is marginal. With a gradually increasing rise time, the response approaches that of a quasi-static solution. For relevance in comparison, the rise-time, T c,ishere chosen to.1t, since this gives both a dynamic and a relatively smooth stress response. The comparison is designed to reveal whether or not the interphase element formulation works in the explicit formulation. In this sense, the comparison is believed to be relevant. In an explicit FE-analysis, the elastic properties are crucial for the critical time step [19]. Hence, we focus attention to the linear elastic part of the solution. However, the linear elastic part represents only a minor part of the cohesive law, cf. Figure 1. The solution in the linear region will affect the initiation of nonlinear response, but will hardly influence the response in the nonlinear region. The variation of the peel deformation, δ y, along interphase, T c =.1T interphase, T c =.1T interphase, T c =.1T interphase, T c =.5T interphase, static peel stress (MPa) normalised time ( t/τ ) FIG. 9. Load rise time, T c, influence on peel stress response at adhesive tip, calculated with the interphase model.

11 EXPLICIT FE-FORMULATION OF INTERPHASE ELEMENTS FOR ADHESIVE JOINTS 97 deflection (µm) continuum beam-interphase beam-continuum distance from adhesive tip (mm) FIG. 11. Displacement at adhesive tip vicinity. Results from continuum, beam/interphase and beam/continuum models. the adhesive layer is evaluated at the moment the first maximum deflection of the loading points is passed. The total deformation of the adhesive layer is used since, for the beam/interphase model, it suffices to capture the deformation in the part of the adhesive layer where non-linear deformation takes place. Converged results of all three models are shown in Figure 11. The results show minor differences between the different modelling techniques. Convergence is studied as the mesh size is decreased. A relevant length parameter in this context is the wave number, κ 4 6Ē/Ei H 3 h, which emerges from a quasi-static analysis, cf. e.g. []. The wave number has a simple interpretation; with l denoting the distance between two neighboring zero-values of the peel stress, κl = π. Thus, it is expected that an element length, L, smaller than l = π/κ 3mmshould be necessary for convergence. That is, a number of elements span l. The deformation profiles obtained with different element sizes are plotted in Figure 1 for both the beam/interphase and the beam/continuum models. The convergence is similar for the two models. Figure 13a shows that the converged value of the maximum peel deformation is achieved already with κl =.4 with the beam/interphase model. Thus, with the element length.3 mm a correct maximum peel deformation is achieved. However, although the deformation profile resembles the converged one, it appears too coarse. Using half that element length, κl = 1. gives a result that appears more adequate. Thus, using more than about three elements to resolve the deformation profile between two zero-values of the peel deformation appears adequate for many applications. For the beam/interphase model, convergence of the deformation implies that also the stress is converged since the stress is proportional to the deformation, cf. Equation (5). However, for the beam/continuum model the stress in the adhesive layer is computed from the strain field, which converges considerably slower than the displacement field. Convergence of stress at the middle layer of the adhesive tip is shown in Figure 13b for the beam/continuum model. The beam/continuum model requires an element length κl =.31, for convergence of the stress at the middle of the adhesive tip. That is one fourth of the element length required by the beam/interphase model, which for the 3D case means that the number of elements required for simulation of an adhesive joint with continuum elements is 16 times larger. The material data needed for simulation depends on the theory used for the modelling. Thus, if a beam/continuum model is to be used, the experiments used to measure the material behavior should be evaluated considering the full field of stress and deformation in the adhesive layer. On the other hand, if the beam/interphase model is to be used, only the peel and shear stress and deformation of the layer should be considered. This latter method appears adequate and preferable, cf. e.g. [7]. The simulations using the beam/continuum model suffer from longitudinal stress waves overlaying the peel stress and also, to some extent, from hour-glassing. Neither of these deficits plagues the beam/interphase model, since the interphase element is fully integrated and lacks stiffness in the longitudinal direction. Apart from providing an easier geometrical modelling as compared to the beam/continuum model, the beam/interphase model gives a shorter execution time. With T s denoting the total time to be simulated, the number of time steps n t = T s /t = T s c/l, where the time step is related to the wave speed c and the element length according to the estimated Courant limit, cf. e.g. [1]. Thus, the smaller elements, which are necessary using the beam/continuum model, lead to an increased number of time steps. With a specific object to be analyzed, the length of the elements and the number of nodes are related to (a) 1.5 separation (µm) (b) 1.5 separation (µm) distance from adhesive tip (mm) distance from adhesive tip (mm) FIG. 1. Displacement profiles acquired with consecutively smaller elements with (a) beam/interphase formulation and (b) beam/continuum formulation. Captions correspond to element size κl.

12 98 T. CARLBERGER ET AL. (a) Adhesive tip displacement (µm) beam/interphase continuum beam/continuum Relative element size, κl, (mm) (b) Peel stress (MPa) Relative element size,κ L, mm FIG. 13. Convergence study. (a) Peel displacement at the adhesive tip. (b) Peel stress at the adhesive middle layer for the beam/continuum model. the size of the object. For the models of the DCB-specimen analysed here, the number of pairs of beam elements along the bond line equals L b /L, cf. Figures 7b and 14. With the beam/interphase model, each pair of beam element nodes contributes with six degrees of freedom (DOF) to the total number of DOF. With the beam/continuum model, using three elements through the thickness of the adhesive layer, each pair of beam element nodes contributes with ten DOF, cf. Figure 14. The total number of DOF for an analysis with the beam/interphase model is N DOF = 6(L b /L + 1) 6L b /L. The factor 6 is changed to 1 for the beam/continuum model. The element length L for a converged analysis for the beam/continuum model is one fourth of that for the beam/interphase model, which implies the same relation between the required time steps for the two models. With t ev denoting the evaluation time for one degree of freedom and for one time step, the total execution time for an analysis T r = 6t ev T s c/l(l b /L+1) 6t ev T s cl b /L, where the factor 6 is changed to 1 for the continuum model. Thus, the relation between the execution time with the beam/continuum model to the execution time for the beam/interface model is about 7. A similar analysis for a 3D shell/interphase model gives T r 1t ev T s c(l b ) /L 3 cf. Figure 15. The factor 1 is changed to 18 (a) 3 6 degrees of freedom beam adherends for a shell/continuum model. Thus, the relation between the execution time with a shell/continuum model to the execution time for a shell/interface model is about 96. These numbers apply to structures constituting the adhesive joint. In automotive structures where the adhesive joints constitute a minor part of the complete structure, the numbers are smaller. Considering a structure where one percent of the total degrees of freedom (DOF) connect to adhesive joints modelled with interphase elements. If we substitute the interphase elements for continuum elements, the total number of DOF will have to increase due to the slower convergence of this element type. In the part of the structure constituting adhesive joints, the number of DOF will increase by a factor of 4, i.e. the total number of DOF will increase by 3%. The time step is four times smaller when continuum elements are used instead of interphase elements. This smaller time step is used for the complete structure. Altogether this renders an execution time, which is almost five times the execution time when interphase elements are used. This is not acceptable for engineering use. 3 L interphase elements (b) degrees of freedom L continuum elements beam adherends rigid connections FIG. 14. Principal sketches of D adhesive joints modelled by (a) beam/interphase elements and by (b) beam/continuum elements. FIG. 15. Principal sketches of 3D adhesive joints modelled by (a) interphase elements and by (b) continuum elements.

13 EXPLICIT FE-FORMULATION OF INTERPHASE ELEMENTS FOR ADHESIVE JOINTS RESULTS AND CONCLUSIONS The governing equations for impact simulation and the finite element formulation are derived for adhesively bonded structures. An interphase formulation is presented for the D case of two beams joined by a thin adhesive layer. The interphase element mass matrix is derived and lumped. It is argued that the element material damping matrix is of little importance relative to other dissipative mechanisms. Finally, a D verification simulation is performed on a pure peel DCB specimen using the interphase elements representing adhesive joining of the two beam adherends. This model is compared to an identical structure where the adhesive is modelled with continuum elements connected to the two beams by rigid connections. The comparison not only involves the convergence of the two techniques but also their respective total execution times. The simulation using the interphase element formulation converges faster and can be evaluated using larger time steps and fewer degrees of freedom, thus rendering a significantly reduced execution time. The interphase formulation is an attractive technique of modelling adhesive joints during impact of shell structures. REFERENCES 1. T. Carlberger, U. Stigh, An Explicit FE-model of Impact Fracture in an Adhesive Joint, Engineering Fracture Mechanics, vol. 74, issue 14, pp. 47 6, 7.. R.D. Cook, D.S. Malkus, M.E. Plesha, R.J. Witt, Concepts and Applications of Finite Element Analysis, 4th ed., John Wiley & Sons, New York,. 3. P.A. dubois, Engineering consultant. Private communication, P.J. Gustafsson, Analysis of Generalized Volkersen Joints in Terms of Nonlinear Fracture Mechanics, Mechanical Behaviour of Adhesive Joints, Proceedings of European Mech. Colloquium 7, Pluralis, Paris, pp , U. Stigh, Damage and Crack Growth Analysis of the Double Cantilever Beam Specimen, International Journal of Fracture, vol. 37, pp. R13 R18, P. Schmidt, Modelling of Adhesively Bonded Joints Asymptotic Analysis, submitted for publication. 7. T. Andersson, U. Stigh, The Stress-elongation Relation for an Adhesive Layer Loaded in Peel Using Equilibrium of Energetic Forces, International Journal of Solids and Structures, vol. 41, pp , K. Leffler, K.S. Alfredsson, U. Stigh, Shear Behaviour of Adhesive Layers, International Journal of Solids and Structures, vol. 44, pp , S. Li, M.D. Thouless, A.M. Waas, J.A. Schroeder, P.D. Zavattieri, Mixedmode Cohesive-zone Models for Fracture of an Adhesively Bonded Polymer matrix Composite, Engineering Fracture Mechanics, vol. 73, pp , T. Andersson, A. Biel, On the Effective Constitutive Properties of a Thin Adhesive Layer Loaded in Peel, International Journal of Fracture, vol. 141, pp. 7 46, R. de Borst, J.J.C. Remmers, A. Needleman, Mesh-independent Discrete Numerical Representations of Cohesive-zone Models, Engineering Fracture Mechanics, vol. 73, pp , E.D. Reedy, F.J. Mello, Modeling the Initiation and Growth of Delaminations in Composite Structures, Journal of Composite Materials, vol. 31, no. 8, pp , R. Borg, L. Nilsson, K. Simonsson, Simulating DCB, ENF and MMB Experiments Using Shell Elements and a Cohesive Zone Model, Composites Science and Technology, vol. 64, pp , ABAQUS Analysis User s Manual, Version 6.5, ABAQUS Inc., Providence, RI, LS-Dyna Keywords User s Manual, Version 97, Livermore Software Technology Corporation, Livermore, April K.J. Bathe, Finite Element Procedures, Prentice Hall, Englewood Cliffs, NJ, T. Belytschko, W.K. Liu, B. Moran, Nonlinear Finite Elements for Continua and Structures, John Wiley & Sons, Chichester,. 18. U. Edlund, A. Klarbring, Analysis of Elastic and Elastic-plastic Adhesive Joints Using a Mathematical Programming Approach, Computer Methods in Applied Mechanics and Engineering, vol. 78, issue 1, pp , K.V.N. Rao, R. Krishna Kumar, Simulation of Tire Dynamic Behavior Using Various Finite Element Techniques, International Journal for Computational Methods in Engineering Science and Mechanics, vol. 8, issue 5, pp , 7.. K.S. Alfredsson, J.L. Högberg, A Closed-Form Solution to Statically Indeterminate Adhesive Joint Problems Exemplified on ELSspecimens, International Journal of Adhesion and Adhesives, to appear. DOI:1.116/j.ijadhadh.7.1., 7.