Adhesive modelling in multi-material structures : Evaluating the strength and fatigue life of adhesive joints

Linköpings universitet SE–581 83 Linköping

Linköping University | Division of Solid Mechanics

Master’s thesis, 30 ECTS | Engineering Mechanics

20 | LIU-IEI-TEK-A--20/03648--SE

Adhesive modelling

in multi-material structures

–

Evaluating the strength and fatigue life of adhesive joints

Modellering av lim i multimaterialstrukturer

-Utvärdering av styrka och livslängd i limfogar

Nitin Narayanaswamy

Supervisor : Stefan Lindström Examiner : Robert Eriksson

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Abstract

Advancements in material science and manufacturing techniques are enabling the use of lightweight metal alloys and polymer composites in several combinations and shapes for producing more efficient and lightweight structures for automotive applications without compromising strength, stiffness and/or durability. When evaluating the strength of the structure, the joints are of importance. For multi-material structures adhesives are often the best type of joints. However, traditional finite element methods using stress criteria cannot accurately predict the failure of these adhesive joints under static loading.

In this thesis work a strength and fatigue model, formulated using energy release rate theory, is implemented in a post processing tool. Given a finite element model of an adhesive joint and a list of boundary elements and nodes this tool calculates the energy release rates in mode I and mode II, and if the fracture toughness of the adhesive is known, a prescribed mixed-mode failure index is calculated. To evaluate its predictions joint strength results are correlated to experiments. Specimens with combined shear and normal load forms the underlying experimental setup with change in strain rate and adhesive thickness as varying parameters. Methods for implementing the model for a car body structure with multiple adhesive joints is investigated, the tool proves to be scalable, however, the required finite element setup at the adhesive boundaries may not be present in a car body model and thus further work needs to carried out to accommodate irregularities like non-matching mesh in the car body finite element model. This model may be used for assessing the strength and durability of a car body structure comprising different materials joined together using adhesives.

Acknowledgements

The work presented in this thesis has been carried out in the months January to June 2019 at Volvo Cars, Gothenburg, Sweden. It is a part of my Masters Studies at the Division of Solid Mechanics at Linköping University, Sweden.

I would like to express my deepest gratitude towards my supervisors at Volvo Cars, Re-naud Gutkin, Åsa Sällström and Jonas Wessung, who have patiently guided me through the thesis work. Their genuine interest for the research subject and experience inspires me.

Many thanks to Niklas Jansson and Per-Anders Eggertsen for their work supporting this research, Högskolan i Skövde for performing tests as a part of Vinnova research project called PLUGG II.

I would also like to thank my supervisor Stefan Lindström and examiner Robert Eriksson at Linköping University, who have helped me prepare and plan my work. Their valuable feedback and experience has greatly helped improved my report and the outcome of this thesis. Further, I would like to extend many thanks to all my colleagues and professors at Linköping University.

I would like to thank my family and friends for supporting me and giving me the strength to fulfil this quest. Without your love and affection I would never have made it.

Contents

2 Failure prediction of adhesive joints 5 2.1 Energy release rate . . . 5

2.2 The J-integral . . . 7

2.3 Analytical model . . . 8

3 Implementation of failure prediction model 12 3.1 Implementation . . . 12

3.2 Verification . . . 14

3.3 Validation . . . 16

4 Results and discussion 18 4.1 Verification . . . 18

4.2 Validation . . . 19

5 Conclusion 23 Bibliography 25 6 Appendix 26 6.1 Path independency of the J-integral for adhesive joints. . . 26

1

Introduction

1.1

Motivations

A lightweight car body is typically made of several structures of different materials like alu-minium, mild steel, high strength steel, etc., often joined together using joining techniques like welding, riveting, adhesive bonding, etc. An example of this can be seen in Figure 1.1 which shows two sections of a car body which are optimised using multi-material structures. A complete car body is commonly made by joining several sections like the front floor, side wall, crash box, cross-members, see Figure 1.2a. These sections are made of thin-walled struc-tures, which are manufactured by first pressing sheet metal into the desired shape followed by trimming and cutting, to include features like stiffeners, mounting holes, etc. Multiple thin-walled structures are joined together to form the complete profile of a section. These sections can also be made using composites like carbon fiber reinforced plastics (CFRP) lam-inates.

(a)Top view of rear section. (b)View from the inside of rear section.

Figure 1.1:CFRP composite (black) and metal-alloy (gray) in Polestar 1 car body, Volvo Cars press.

1.1. Motivations

When joining two thin-walled parts made of similar metallic materials a conventional joining technique like welding or riveting is typically used. However, if they are dissimilar welding or riveting may not produce a mechanically stable structure due low joint stiffness and strength. If one part is made using polymer composite and the other using metal, conven-tional joining techniques cannot be used. These problems are overcome by using adhesives. An adhesive joint is commonly used alongside conventional joints which is known as hybrid joints. Adhesive lines on a typical car body are shown in Figure 1.2b. Using adhesive joints in a car body has the following advantages,

1. High joint stiffness as the joint is continuous. 2. Improved static fatigue strength.

3. Improved crash behaviour.

4. Less damage to adherends since they are not pierced. 5. Cost effective due to ease of manufacturing.

Determining the strength and durability of adhesive joints with cracks and singularities due to adherend corners, using critical stress values from finite element (FE) models is not accurate enough and may lead to a poor design or even failure. However a detailed non-linear FE analysis may improve the accuracy, but these simulations tend to be computational resource intensive which makes it undesirable for large FE-model analysis like that of a car body.

When designing components with cracks and singularities using fracture mechanics (FM), the stress-intensity factor (SIF) K is typically calculated. This value is compared to a property called fracture toughness or critical stress-intensity factor Kc, which may depend on the type of geometry, material properties and loading mode. Figure 1.3b illustrates the three fracture modes which may drive a crack. With respect to the crack plane (plane parallel to the crack surfaces) mode I is described as crack growth due to load acting in the perpendicular direc-tion such that it opens the crack, in mode II and mode III the load direcdirec-tion is parallel such that the two crack surfaces slide over each other, where in mode II the crack surfaces slide in the direction perpendicular to the leading edge of the crack while mode III is parallel.

However it is difficult to evaluate K from a global FE-model which makes use of rela-tively large elements for adhesive joints. This is because the stresses close to a crack-tip or singularity accelerates with decreasing distance. Accurate calculation of local stresses re-quires very small elements, and in some loading conditions stresses at a singularity may

(a)Multi-material car body. (b)Adhesive lines (turquoise) in a car body.

1.2. Aims

Distance from crack-tip

Str

ess

Global FE Local FE

Real world

(a)FE and actual stress along crack-tip. (b)Fracture modes.

Figure 1.3

never converge. This is depicted in Figure 1.3a. It is also difficult to incorporate SIF methods for in-homogeneous, layered materials like a multi-material sandwich or CFRP laminates.

Energy release rate (ERR) methods have the potential to solve these problems. In ERR the balance of change in potential energy of a loaded joint and the change in energy required to propagate a crack is evaluated to determine weather a crack may grow and, if so, the rate of this crack growth. Far field stresses can be used to evaluate the strength of joints when confined to linear elastic fracture mechanics (LEFM) or when the plastic zone is relatively small. The loads associated to a car body can be complex and adhesive joints may fracture in several modes. Fracture criteria based on ERR can easily be generalised into a mixed-mode criterion. It is also relatively easy to determine fracture toughness from experimental tests when using the ERR method, which is why it is preferred over SIF when investigating adhesive joints in multi-material structures [7].

Analytical models based on ERR, combined with FE-analysis in some way, can be used in-stead of conventional methods. The combined methods are known to show good correlation with experimental tests [6]. They also have an added advantage of requiring less computa-tional power compared to a detailed FE-model. By using a combined method, the detailed FE-model of adhesive joints in a car body can be eliminated.

1.2

Aims

The aims of this thesis work are to:

• Develop a script within a post processing tool, which calculates the stresses and energy release rates in an adhesive joint based on energy release rate theory.

• Validate the analytical model with coupon tests.

The goal at Volvo Cars is to analyse an entire car body with multiple adhesive joints and non-matching meshes of the adherends under mixed-mode loading states.

1.3

Research questions

The main research questions investigated is,

How can the strength and fatigue life of adhesive joints in multi-material automotive structures be efficiently predicted using an energy based method?

1.4. Limitations

1. Can strength and fatigue life of adhesive joints in large car body structures be accurately predicted using an energy release rate method?

2. How can the geometrical and loading complexities of a car body be successfully incor-porated?

1.4

Limitations

The ERR method considered in this work is constrained to linear elastic fracture mechanics (LEFM). However under mixed-mode, impact and temperature dependant loading, adhesive joints may not fail due to small scale yielding.

Failure in adhesive joint may occur in two ways, cohesive and adhesive failure. Cohe-sive failure occurs when the bonds within the adheCohe-sive material break, this kind of failure is common in car body structure if they may occur. However if the joining surfaces are not prepared to a desired specification, it may fail at the interface between the adhesive and the adherend. In this work joints are assumed to be produced within the desired specification and thus adhesive failure is ignored.

The analytical model [6] to be used does not account for mode III failure which may be critical in mixed-mode load cases. The model is formulated using elementary beam theory for the adherends and thus adherends are limited to thin beam-like structures, i.e, structures with relatively low thickness. If the adherends are anisotropic like with composites, elementary beam theory is not sufficient to accurately capture its behaviour.

Non-matching mesh between the adhesive and the adherends is likely to be present in the global FE-model of the car body. This could be a limiting factor in terms of accuracy. Ad-herends are modelled using shell elements to capture bending. The method to be developed may have to be modified if the adherends are modelled using solid elements.

2

Failure prediction of adhesive

joints

The theoretical background of energy release rate and adhesive failure prediction method is presented. Relevant literature is suggested for further reading.

2.1

Energy release rate

Consider a simple rectangular plate with a crack along the symmetry line as shown in Figure 2.1. The plate is made of linear elastic material. When the plate is externally loaded un-der quasi-static conditions the crack may grow when the load reaches a critical load value. This crack propagation creates new surfaces. To create these new surfaces energy is needed. This energy is called the total surface energy Ws. The energy may become available in the following two ways [3],

1. Internal elastic strain energy U: Under load the plate contains some internal energy called strain energy which is associated to its elasticity. This strain energy is released if the crack propagates leading to new crack surfaces.

2. External work W: Load applied on the plate will perform some work when the crack grows, because the load application point will move. This work may be transmitted to the crack region for creation of new surfaces.

For crack propagation, the energy stored in the structure must be sufficient to overcome the surface energy needed to create new crack surfaces. At equilibrium the net change in total energy of the plate must be equal to zero. This total energy Etis given by,

Et=Π+Ws (2.1)

whereΠ is the potential energy written as the difference between internal energy and external work,

Π=U ´ W (2.2)

Under quasi-static conditions, an incrementally small change in displacement∆ of the load point will produce a small change in external work which is obtained as,

2.1. Energy release rate

Figure 2.1:Infinite plate with edge crack.

where P(∆)is force and d∆ is the differential displacement of load point. For a small change

in potential energy equation 2.2 gives,

dΠ=dU ´ dW (2.4)

In equation 2.4 dW =0 when constant displacement is applied as load since d∆ =0 and dW ‰ 0 for constant force since d∆ ‰ 0. These two load states are termed as displacement control and load control respectively. In Figure 2.1 the plate is under load control.

If A is the area of the crack surface then equation 2.1 gives, dEt dA = dΠ dA + dWs dA (2.5)

The change in total energy is zero, which gives, ´dΠ

dA = dWs

dA (2.6)

The energy release rate (ERR) G (in honour of Griffith’s work on brittle fracture) can now be defined as the energy available to create new crack surfaces,

G=´dΠ

dA (2.7)

Similar to the fracture toughness Kcin LEFM based on critical stress, the critical value of G is defined as Gc,

Gc= dWs

dA (2.8)

Fracture may occur when G ě Gc.

Gc is determined by experimental tests. It may be related to a combination of material properties and loading mode, but it is important to note that Gc may also be a function of change in crack length a [3]. This is also true for G (i.e. dG/da ‰ 0) which is evident from equation 2.7 since dA=tda where t is the thickness of the plate.

To calculate G, the potential energyΠ is to be evaluated which is equal to the elastic strain energy U for small-scale yielding in a linear elastic material under quasi-static conditions. One way to derive U is to use the J-integral method, which is described in the next section.

Since this work is limited to crack initiation only, no further explanation will be given on G and Gc, however to consider crack growth behaviour and large-scale yielding further reading may be of interest [3, 7, 2].

2.2. The J-integral

(a)Simple crack.

x y P3 P2 P1 0 1 No rm ali ze d st ress (b)Path independency. Figure 2.2:Test

For crack initiation the energy release rate G can be used to determine the first signs of fracture in a structure or the crack initiation point, provided that the loading mode and critical value of G are known.

2.2

The J-integral

The J-integral is a line integral around a crack tip. It is widely used in linear elastic frac-ture mechanics (LEFM), non-linear fracfrac-ture mechanics (NLFM) and elastic-plastic fracfrac-ture mechanics (EPFM) to calculate the energy release rate and to study crack growth.

We introduce a coordinate system with x axis tangential to the crack, and z axis perpen-dicular to the crack (see Figure 2.2a). The definition of the J-integral as given by J.R. Rice [9] is, J= ż Γ  wdy ´ Ti Bui Bxds  (2.9) where ds is a infinitesimal arc element on the curveΓ surrounding the crack tip as indicated in Figure 2.2a, dz=nxds, n is the unit vector defined as the outward normal to ds, Ti =σijnj is the traction vector , u is the displacement vector and w is the strain energy density given as,

w=

ż_ε 0

σijεij (2.10)

The J-integral is path independent for any non-closed path around a crack-tip, provided that the plastic zone around the crack is relatively small [9, 2]. Figure 2.2b shows a contour plot of an arbitrary stress component for the plate with crack shown in Figure 2.1 above the symmetry line. If the J-integral is computed by calculating the elastic strain energy using stress and strain components along the three paths P1, P2 and P3 of Figure 2.2b in clockwise or anti-clockwise direction, the results are equal.

When calculating ERR from FE-models the path independent property of the J-integral is useful, since the local stress field is mesh dependent. It is also useful in analytical models that are limited to far field stress strain calculation, which is the case for the analytical model [6] used in the next section of this thesis work.

In ductile materials the plastic zone around a crack-tip is much smaller than the crack length. The energy required to propagate the crack is not critically dependant on the plastic zone. This means that a purely elastic solution may be used to calculate the energy available to fracture. In LEFM when a crack propagates as a result of loading in mode I, II, III or mixed-mode, the J-integral is equal to G which is equivalent to calculating the SIF K in that mode.

2.3. Analytical model

2.3

Analytical model

From the previous sections it is clear that G is to be evaluated to predict failure in adhesive joints. To calculate G using the J-integral, far field stress strain components are to be de-rived. This forms the foundation for the analytical model. Given material properties, bound-ary conditions and thickness of a non-symmetric joint (material and thickness of substrates are different), the analytical model should establish an expression for G in different fracture modes.

A solution for energy release rates and stresses in adhesive joints as described by Quantian Luo and Liyong Tong [6], forms the basis for this thesis work. Only a part of the solution is presented here. The complete solution and its derivation can be found in [6] and parts of the solution in Appendix 6.2 with some supporting solutions.

The analytical model described in [6] consists of two parallel beams (substrates) with an adhesive layer in between as seen in Figure 2.4. Substrates 1 and 2 are modelled as Tim-oshenko beams under plane strain conditions. The adhesive layer is formulated as a two parameter elastic medium, based on traction separation theory, in plane stress conditions. Two stiffness parameters are used for the adhesive, kσ in the transverse direction and kτ in

shear, which are,

kσ = Ea ta kτ = Ga ta (2.11)

Here Ea is the elastic modulus in transverse direction, Ga is the shear modulus and tais the adhesive thickness. The x axis is parallel to the substrates and at the crack tip x = 0, the y axis is parallel to the leading edge of the crack and the z axis is normal to the substrates as seen in Figure 2.3.

In Figure 2.4 Nn, Mnand Qn are the normal force, bending moment and shear force re-spectively at the beam end, where n=1 and n=2 for top and bottom substrates respectively, and n = 0 far away from the adhesive tip. Note that the forces and moments are given per unit width measured along the axis perpendicular to x-z plane in Figure 2.3.

Normal stress or peel stress in the transverse direction σ and shear stress τ in the adhesive are calculated using the displacement jumps between the two interfaces. The stresses are assumed to be constant through the adhesive thickness ta, hence,

σ=σ(x) τ=τ(x) ε= σ Ea γ= τ Ga (2.12)

The energy release rate G is calculated with the help of the J-integral method. Using results from Appendix 6.1 the J-integral for path 1 in Figure 2.3 can be calculated which gives energy release rates GIand GI Iin terms of peel and shear stresses as [6],

GI = żε(0) 0 taσdε= 1 2kσ [σ(0)]2= ta 2Ea[σ(0)] 2 GI I = ż_γ(0) 0 taτdγ= 1 2kτ [τ(0)]2= ta 2Ga[τ(0)] 2 (2.13)

where σ(0)and τ(0)are peel and shear stress, ε(0)and γ(0)are peel strain and shear strains at the adhesive edge, i.e. at x=0.

The path independency of the J-integral along an adhesive joint and the derivation of equation 2.13 from equation 2.9 is presented in Appendix 6.1.

2.3. Analytical model

The governing differential equation for peel stress and coupled shear stress is evaluated from equilibrium equations of substrate 1 and 2 (with the help of the free body diagram shown in Figure 2.4), and constitutive equations from the Timoshenko beam theory,

d3τ dx3 =k1 dτ dx+k2σ d4σ dx4 =k5 d2σ dx2 +k4σ ´ k3 dτ dx (2.14)

where k1=kτk1d, k2=kτk2d, k3=kσk2d, k4=kσk4dand k5=kσk5d are constants relating to stiffness and, k1d =  4αa1D1 ∆1 + 4αa2D2 ∆2  ´ t1C1 ∆1 ´ t2C2 ∆2  k2d =  t1Ad1 2∆1 ´ t2Ad2 2∆2  ´ C1 ∆1+ C2 ∆2  k4d = Ad1 ∆1 + Ad2 ∆2 , k5d= 1 Gk1 + 1 Gk2 αai = 1 +αki 4 , αki = Adit2i

4Di , ∆i= AdiDi´C 2 i

(2.15)

where i=1, 2 for substrate 1 and 2, respectively. Adi, Gki and Di are the elastic moduli in normal (x axis), shear (x-z plane) and bending (about axis perpendicular to x-z plane) respec-tively, Ciis the coupled extension-bending stiffness which is assumed to be equal to zero and t1, t2is the thickness of substrate 1 and 2 respectively.

To solve equations 2.14, the shear stress and peel stress are assumed to vary as [6],

τ=Ae ? kσβtx_=Aeβx σ=Be ? kσβtx_=Beβx (2.16)

The characteristic polynomial associated with equation 2.16 is [6],

λ3+a1λ2+a2λ+a3=0 or βt=0 (2.17) where λ=β2_t and, a1, a2and a3are constants which reflect the influence of stiffness.

2.3. Analytical model

Figure 2.4:(a) An adhesively bonded joint under general loading and (b) free body diagrams of the adhesive and substrates [6].

Solving for real and complex roots for the polynomial equation 2.17, gives the solution for stresses as,

τ=A1eβ1x+A2e´β1x+eβ2x(A3sin β3x+A4cos β3x) +e´β2x(A5sin β3x+A6cos β3x) +A7 σ=B1eβ1x+B2e´β1x+eβ2x(B3sin β3x+B4cos β3x) +e´β2x(B5sin β3x+B6cos β3x)

(2.18) where βi = ?kσβti (i = 1 ´ 3) are the eigenvalues, Aj (j = 1 ´ 7) and Bk (k = 1 ´ 6) are integration constants.

Boundary conditions, peel and shear stress are assumed to be equal to zero at the far end as x approaches infinity,

x ÝÑ 8: τ=0, σ=0 (2.19)

Equation 2.18 in 2.19 gives A1=A3=A4=A7=0 and B1=B3=B4=0, so that,

τ=A2e´β1x+e´β2x(A5sin β3x+A6cos β3x) σ=B2e´β1x+e´β2x(B5sin β3x+B6cos β3x)

(2.20)

At the adhesive crack, x=0, equation 2.20 simplifies to,

τ(0) =A2+A6+A7, where A7=0

σ(0) =B2+B6 (2.21)

where Aiand Bi (i= 2, 5, 6) are dependent constants and their relationship can be found by substituting 2.21 into 2.14,

B2=K1A2

B5=K2A5+K3A6 B6=´K3A5+K2A6

2.3. Analytical model where, K1= ? kσβt1[(kτk1d/kσ)´ β2t1] (kτk2d/kσ) =akσKt1 K2= ? kσβt2[(kτk1d/kσ)´(β2t2´3β2t3)] (kτk2d/kσ) =akσKt2 K3= ? kσβt3[(kτk1d/kσ)´(β2t3´3β2t2)] (kτk2d/kσ) =akσKt3 (2.23) and, A2=´ ∆´ 11 ∆´ kτHnd ? kσ +∆ ´ 21 ∆´ Hmd ? kσ ´∆ ´ 31 ∆´ Hqd ? kσ A5=´∆ ´ 12 ∆´ kτHnd ? kσ +∆ ´ 22 ∆´ Hmd ? kσ ´∆ ´ 32 ∆´ Hqd ? kσ A6=´∆ ´ 13 ∆´ k_?τHnd kσ +∆ ´ 23 ∆´ Hmd ? kσ ´∆ ´ 33 ∆´ Hqd ? kσ (2.24)

where ∆´ _{is the determinant and ∆}´

ij (i = j = 1, 2, 3) are the cofactors of the T´ matrix. Expressions for Hnd, Hmd, Hqdand the T´matrix are found in Appendix 6.2.

In equation 2.19, the shear stress at infinity is assumed to be equal to zero and hence the constant A7 in equation 2.18 is set to zero. However, if instead we only require that σ(x)

and τ(x)must be bound i.e, not approach to infinity, we find that A1 = A3 = A4 = 0 and B1=B3=B4=0, while A7becomes an unknown constant, particularly for a case where the shear force Q0‰0 the shear stress cannot be zero. This additional shear stress is calculated using equation 2.25 and included in equation 2.18 by substituting its value for A7.

τa= E1Q EI t1  t1 2 +y1  (2.25) Equation 2.25 is formulated by simplifying the adhesive joint model into a composite beam consisting of substrate 1 and 2 which is subjected to shear force Q=Q1+Q2. The weighted stiffness method is used to calculate y1and EI which are,

y1=

´Ad1t1/2+Ad2(t2/2+ta) Ad1+Ad2

EI=D1+D2+ (y1+t1/2)2Ad1+ (´y1+ta+t2/2)2Ad2

(2.26)

Due to mixed-mode behaviour, a failure (crack initiation) criterion is to be defined which can combine failure in mode I and mode II into a single criterion. If GIcand GIIcare the fracture toughness in mode I and mode II respectively, then a mixed-mode ERR failure index is here defined as [2], Gmixed´mode=  GI GIc 2 + GII GIIc 2 (2.27) if Gmixed´mode ě1 then a crack has initiated. The total energy release rate is also defined as Gtotal=GI+GII.

Summarising, to determine the stresses at the adhesive end, equations 2.21 along with 2.25 are used, and to calculate energy release rate in mode I and mode II equation 2.13 will be used. For crack initiation Gmixed´modeě1.

3

Implementation of failure

prediction model

3.1

Implementation

The analytical model described in Section 2.3 requires material properties of the adhesive and substrates, and the forces and moments applied to the substrates as input. Stresses and en-ergy release rates at the crack will be the output of the calculations. Computing σ(0), τ(0), GI and GII, and extraction of data from an FE-model is here implemented in a Python environ-ment [5], using standard open-source libraries like math [4] and NumPy [8]. To enable data flow between the Python environment and the Meta post processing environment, where the input for the analytical model is available as an FE-model, the Meta library for Python is used as an API (Application Programming Interface). The workflow is shown in Figure 3.2. In addition to the two environments, the input to Meta is also shown. For this work Abaqus FE-models will be used.

For accurate calculation, an FE-model has to be setup so that it is analogous to the ana-lytical model, an example is shown in Figure 3.1. The substrates (light and dark gray) are

3.1. Implementation

Calculate material constants

Generate local co-ordinate Elements

Extract local force & moments Calculate N1 ,M1, Q1, N2, M2 & Q2 Calculate σ(0), τ(0), G_I& G_II Print element results Input file Result file Toolbar GUI Plot element results User input

Figure 3.2:Flow chart showing data flow between the three environments.

modelled as shell elements (S4 in Abaqus) and the adhesive (turquoise) using solid element (C3D8 in Abaqus). Circled (black) nodes of the respective substrate elements represent the adhesive boundary or adhesive end and the distance between the nodes gives the width. Ad-hesive elements are connected to the substrates using rigid elements (red) with all six DOFs constrained. The rigid elements length is equal to the distance between the mid plane of a substrate and its interface with the adhesive. Force and moment inputs for the analytical model are calculated using reaction force and moment data from the circled nodes. To match units nodal reaction forces and moments at the respective substrate are added and divided by the width.

In Abaqus, elemental reaction force and moment values in global coordinate system at nodes can be requested by including the code given below, in the input file. In Figure 3.1 two shell elements are shown for the top and bottom substrates, the reaction forces are equal and opposite in direction for the two elements at the top or bottom substrate. To match the analytical model, the outside (left in Figure 3.1) shell element for the top and bottom substrate are chosen, using these element numbers the nodal force and moment values in the correct direction are extracted. In Abaqus, the nodal values for each element are extrapolated from integration points. This data is stored in the result file upon solving the input file. The input and result files are read into Meta by the user, making them available in the Meta environment.

*ELEMENT OUTPUT, POSITION=NODES NFORC

The calculation starts by manually identifying the boundary nodes, the outside elements for the top and bottom substrates and one adhesive element by their ID numbers. Material properties E, ν and t are identified and extracted using appropriate functions in the Meta

3.2. Verification

library. However for the solid elements representing the adhesive there is no thickness at-tribute in its section card data. Therefore a subroutine which takes one adhesive element as input and returns the smallest distance between all pairs of nodes is created. This method has two problems, one is when there are more than one element through the adhesive thickness and the other is when the element length along the width is smaller than the thickness of the adhesive. However this problem is not present in the FE-models used for this work. The lim-itations are not addressed here. Instead the user must enter the correct thickness manually in the Python script.

Constants in equation 2.21 are calculated next. The constants along with N1,2, M1,2and Q1,2are the inputs for another subroutine which calculates peel stress σ(0), shear stress τ(0)

and the energy release rate GIand GI Ihence completing the calculation. The constants N1,2, M1,2 and Q1,2 are calculated by converting the global force and moments into local values using a coordinate transformation subroutine.

When an FE-model consisting of an adhesive joint between two components as seen in Figure 3.3 is analysed, the boundary around the joint lines must be divided into small seg-ments. Each segment consists of one element for the adhesive and one each for the substrates along the width, like in Figure 3.1. With each individual segment the material properties remain the same, but the orientation of the local coordinate system and nodal forces and mo-ments changes. Thus the constants in equation 2.21 are calculated once and the subroutine which calculates stresses and energy release rates is called for every segment on the bound-ary, where N1,2, M1,2and Q1,2are also calculated for each segment.

The co-ordinate transformation subroutine calculates a 3X3 local unit matrix with respect to the global co-ordinate system. The matrix elements represent the contribution of any vari-able with components in global co-ordinate system with respect to a new co-ordinate system. The new co-ordinate system is orientated as shown in Figure 3.1, that is, the y axis is parallel to the line joining the two nodes on the top element, z axis is parallel to the line joining the mid points of the top and bottom nodes along the width, and the x axis is mutually perpen-dicular to axis y and z.

Distance between the two nodes on the top element or the segment width L, stresses

σ(0)and τ(0), ERR GI, GII, Gtotal, Gratio and Gmixed´mode are stored element wise for each step. Selected results can be plotted by the user. A Gmixed´modecontour plot may be used to evaluate failure. These results are displayed in Meta as elemental centroid values using a separate subroutine and Meta function.

For practical use, a Meta user toolbar is created, this toolbar contains buttons and data input fields for element and node ID numbers. The toolbar is used to link user inputs and the Python environment. It is also used to read the generated result data file.

3.2

Verification

The new method was verified by comparing its results to the results from an earlier method implemented at Volvo Cars, which employs the same analytical model [6]. Data from a fa-tigue test of a loudspeaker bracket structure was used. However no detail regarding the test setup is available. The FE-model consists of multiple elements with single nodes on the adhesive boundary. The adhesive experiences a mixed-mode condition due to geometrical complexity and a non-symmetric joint, making it a good example to test the new method and its limitations.

The structure consists of a bracket made of a CFRP composite and a steel substrate (analo-gous to substrate 1 and 2 in section 2.3), which are joined together with a Sika multi-material adhesive. The steel substrate is spot welded to side plates in the test setup. This is simu-lated by selecting a few nodes on the spot welded edge of the steel substrate, see Figure 3.3b, and constraining them in the three translation DOF. A load is applied on the CFRP substrate

3.2. Verification

(a)Load direction indicated by the green arrow. (b)Pink circles show the location of spot welds.

Figure 3.3: FE-model of loudspeaker bracket, CFRP composite bracket, steel substrate and adhesive in light gray, dark gray and turquoise respectively.

such that it opens the joint, the direction of load and its location is described in the following paragraph.

The geometry and the mesh of the FE-model is shown in Figure 3.3. Both the bracket and substrate are modelled using shell elements (S4 and S3R in Abaqus) of 3.5 mm size and the ad-hesive is modelled using solid elements (C3D8 and C3D6 in Abaqus). Elements of the bracket and substrate that are adjacent to the adhesive elements are matched. This aligns adhesive boundary nodes with the immediate nodes on the bracket and substrate, such that all the edge segments are similar to Figure 3.1. The adhesive elements are connected to the bracket and substrate elements through these aligned nodes using rigid elements, using multi-point constraint in Abaqus, which links both the translation and rotation DOFs. A load of 700 N in the direction shown in Figure 3.3a is applied to a node at the centre of the beam structure (pink lines) to simulate the joint opening, this node is also constrained to translate along the load direction only.

Result calculation starts at edge segment 1, see Figure 3.4, followed by 2, 3, and so on around the adhesive till segment n. Using L and its summation, results along the boundary can be plotted.

Figure 3.4:View normal to adhesive areas of loudspeaker bracket FE-model showing the two areas and the direction of result computation around the adhesive boundary.

3.3. Validation

In the Volvo Cars method, analytical results are computed using MATLAB. Elemental forces at the adhesive boundary nodes and the nodal coordinates are output from the FE-simulation. These forces and coordinates are the inputs for the MATLAB script. The same FE-model is analysed using the new method. Both the methods employ the same analytical model with the difference being the implementation of nodal data and the programming environment.

Analytical σ(0), τ(0), GI and GIIalong the adhesive boundary of area 1, see Figure 3.4, from the two methods are compared to verify the implementation of the new method.

3.3

Validation

The analytical model is validated with respect to change in strain rate and effect of adhesive thickness in test series 1 (TS1) and 2 (TS2), respectively. The experimental test TS1 was per-formed at the Högskolan i Skövdeas as a part of a Vinnova research project called PLUGG II and TS2 was performed at Volvo Cars. Test results and FE-models are used for validation of the new method.

In TS1 and TS2, a tensile-shear specimen called KS2 is used, shown in Figure 3.5. Using a KS2 specimen and by varying the load direction, mixed-mode behaviour can be captured. In this work, KS2 specimens with load angle c equal to 0, 30, 60 and 90 degrees are investigated, where 0 degree and 90 degree corresponds to pure shear and peel, or mode II and mode I loading respectively. Both the top and bottom substrates are made of steel and a SIKA multi-material adhesive is used to join them.

In TS1, strain rates of 10 mm/min and 1000 mm/min are tested. Substrate and adhesive thicknesses are 1.2 mm and 0.5 mm, respectively. In TS2, a substrate thickness of 1.4 mm and adhesive thicknesses 0.2, 0.5 and 1.0 mm are used.

A linear elastic quasi-static FE-model is used for the specimen and part of the test rig setup as shown in Figure 3.6b. The substrates are modelled using 3 mm shell elements (S4 in Abaqus) and the adhesive layer is modelled with 1.5 mm solid elements (C3D8 in Abaqus). Contacts in the assembly are modelled using multiple point constraints (MPC). The top fix-ture is subjected to 5 mm displacement, and the bottom fixfix-ture is fixed. This is achieved by using a single reference node and an MPC to connect the face of the fixture end to the reference node. Boundary conditions are assigned to these reference nodes. The reference nodes are offset by a small distance from the face of the fixture end. At the top, a single beam element is modelled between the reference node and the fixture. The stiffness of this

(a)Load direction represented by dashed arrow.

(b) KS2 specimen loaded in mode I, Högskolan i Skövde.

Figure 3.5: KS2 specimen used for TS1 and TS2 and a picture showing 90 degree test setup taken at University of Skövde.

3.3. Validation

(a) GIand GIIin light and dark gray respectively. (b)Specimen holding fixtures in yellow.

Figure 3.6: Fracture toughness variation with thickness and FE-model of test setup showing models with 90, 60, 30 and 0 degree load direction from left to right.

beam is varied to match the elastic slope of the model with test result. This stiffness accounts for the remaining parts of the test rig which is not modelled here. This stiffness calibration was performed for the 0 degree model and the same beam stiffness is used for the remaining models.

The FE-models are solved and the results are loaded into the Meta post processing tool. The top and bottom element ID and the boundary node ID are entered into the toolbar and the new method is run. For failure initiation calculation using equation 2.27, fracture tough-ness for different thicktough-ness are used which are provided by Sika, see Figure 3.6a. Previous records within Volvo Cars shows a coefficient of standard deviation of 25 % in the fracture toughness and this deviation is accounted for in both the tests by calculating a minimum and a maximum value of fracture toughness and recalculating Gmixed´mode. Results along the boundary are monitored to find the critical segment, when failure is initiated at the critical segment, reaction force in the beam element at the top fixture and the applied displacement to the reference node are recorded and plotted against the test results.

4

Results and discussion

4.1

Verification

Results from the verification of the loudspeaker bracket are shown in Figure 4.1. Peel and shear stresses and energy release rates predicted by the new method are comparable with those of the Volvo cars method, however at some segments there are deviations. These seg-ments are observed to include a triangular element sharing a node with the top and bottom element. In the new method, the contribution from a triangular element sharing a node with a segment, as seen in Figure 4.2a, is included by adding half the force and moment to every edge segment sharing the same node. In Volvo Cars method, it is not completely clear how this is implemented. The new implementation adds more detail to the model and is hence assumed more accurate.

In Figure 4.1, the ERR in mode I and mode II at the critical segment is approximately 0.58 and 0.26 kJ/m2. Mixed-mode ERR failure index is calculated to be around 0.005. Failure does

0.0 0.2 0.4 0.6 0.8 1.0 Normalized distance −10 −5 0 5 10 15 20 25 30 Str ess, M Pa Peel Shear

(a)Stress in mode I and II.

0.0 0.2 0.4 0.6 0.8 1.0 Normalized distance 0.0 0.1 0.2 0.3 0.4 0.5 0.6 En erg y r ele ase ra te, kJ /m 2 G I G II

(b)Energy release rate in mode I and II.

Figure 4.1:Comparison of Volvo Cars method in gray and the new method in black along the adhesive boundary of area 1, where x axis corresponds to the red arrow in Figure 3.4

4.2. Validation

(a)Segments sharing a node of a tria element.

(b) Deformed 0 degree specimen, Högskolan i Skövde.

Figure 4.2

not occur since mixed-mode ERR is well below 1. However, failure may occur due to cyclic loading. However since no fatigue data is available the fatigue strength remains an unknown.

4.2

Validation

Force-displacement curves from TS1 and TS2 are shown in Figure 4.3, where the applied dis-placement is shown on the x axis and the recorded reaction force at the top fixture is shown on the y axis. Using experimental curves, maximum reaction force and corresponding dis-placement are plotted as x-marks. Results from the new method or analytical solution are plotted using solid circles. These points are recorded at critical segments when the mixed-mode ERR failure index is equal to 1. Two additional solid circles are plotted for the 25% in scatter observed for the fracture toughness.

In simulation failure starts at critical segments. Limited data of TS1 and TS2 failure sug-gests that failure starts and progresses on the longest edges.

Test results clearly show scatter. It is more distinct in TS1 compared to TS2. When compar-ing strength in TS1 quasi-static and dynamic loadcompar-ing, both show comparable results, gocompar-ing from 90 to 0 degree the effect of strain rate diminishes. However high strain rates may pro-duce different results. Analytical prediction overestimates the strength in all four loading angles, however analytical predictions considering the negative 25% deviation in fracture toughness shows good correlation. The 0 degree model overestimates the strength signifi-cantly. From Figure 4.3, it can be seen that the experimental curves on 90, 60 and 30 degree tests show a clear peak strength or failure initiation point while the 0 degree test is similar to a hardening curve. This behaviour maybe due to plastic deformation in the sheet metal of the test specimens. Figure 4.2b shows a picture of one such specimen, large plastic deforma-tion can clearly be seen around the mounting holes. It may also be due to adhesive material failure under pure shear.

Failure prediction in the 0 degree model may be improved if the fracture toughness at the end of linearity in Figure 4.3g or at the beginning of the hardening type behaviour is known. Using this new fracture toughness and recalculating the mixed-mode ERR failure index to be equal to 1 a more accurate strength prediction can be made and thus eliminate modelling of the non-linearity behaviour. Another approach would be to calculate the energy associated with the plastic deformation or non-linearity and subtract it from the energy calculated by the

4.2. Validation 0 1 2 3 4 5 Displacement, mm 0 2 4 6 8 10 12 14 Re ac tio n fo rce , k N Experiment Linear FE Exp. failure Analytical

(a)TS1 90, Högskolan i Skövde.

0 1 2 3 4 5 Displacement, mm 0 2 4 6 8 10 12 14 16 18 20 Re ac tio n fo rce , k N Experiment Linear FE Exp. failure Analytical (b)TS2 90, Volvo Cars. 0 1 2 3 4 5 Displacement, mm 0 2 4 6 8 10 12 14 16 Re ac tio n fo rce , k N Experiment Linear FE Exp. failure Analytical (c)TS1 60, Högskolan i Skövde. 0 1 2 3 4 5 Displacement, mm 0 2 4 6 8 10 12 14 16 18 20 22 Re ac tio n fo rce , k N Experiment Linear FE Exp. failure Analytical (d)TS2 60, Volvo Cars. 0 1 2 3 4 5 Displacement, mm 0 4 8 12 16 20 24 28 Re ac tio n fo rce , k N Experiment Linear FE Exp. failure Analytical

(e)TS1 30, Högskolan i Skövde.

0 1 2 3 4 5 Displacement, mm 0 4 8 12 16 20 24 28 32 36 Re ac tio n fo rce , k N Experiment Linear FE Exp. failure Analytical (f)TS2 30, Volvo Cars. 0 1 2 3 4 5 6 7 8 9 Displacement, mm 0 8 16 24 32 40 48 56 64 Re ac tio n fo rce , k N Experiment Linear FE Exp. failure Analytical (g)TS1 0, Högskolan i Skövde. 0 1 2 3 4 5 6 7 8 9 Displacement, mm 0 8 16 24 32 40 48 56 64 72 80 Re ac tio n fo rce , k N Experiment Linear FE Exp. failure Analytical (h)TS2 0, Volvo Cars.

4.2. Validation 0.0 0.5 1.0 90 G I/G total 0.0 0.5 1.0 60 0.0 0.5 1.0 30 0.0 0.2 0.4 0.6 0.8 1.0 Normalized distance 0.0 0.5 1.0 0

(a)TS1, Högskolan i Skövde.

0.0 0.5 1.0 90 G I/G total 0.0 0.5 1.0 60 0.0 0.5 1.0 30 0.0 0.2 0.4 0.6 0.8 1.0 Normalized distance 0.0 0.5 1.0 0 (b)TS2, Volvo Cars.

Figure 4.4: ERR ratio along adhesive boundary, red, black and blue correspond to adhesive thickness 0.2, 0.5 and 1.0 mm.

analytical model. However if complete failure of the joint is to be captured then a non-linear plastic FE-model may be required.

Strength prediction using the analytical model for TS2 is sensitive to the adhesive thick-ness by a margin greater than observed in experimental tests. An increase in thickthick-ness shows an increase in the model predicted strength, see TS2 results in Figure 4.3. In the 0 degree model the predicted strength is highest for the smallest thickness. This may be due to shear locking of solid elements of the adhesive layer. For 0.2 mm adhesive layer, high aspect ra-tio solid elements are produced which imparts addira-tional stiffness and hence increases the reaction force. Cohesive elements may be used to overcome this result.

Figure 4.4 shows the ratio of ERR in mode I to the total ERR for all four load angles along the adhesive boundary. As the load angle decreases the contribution from mode I to the ERR clearly decreases, except at the corners. This shows that the mixed mode ERR is captured in TS1 and TS2.

Summarising the results of TS1 and TS2, see Figure 4.5 where the maximum reaction force recorded in tests are compared with predicted reaction forces against specimen loading angle, scatter is present in the experiments. Strength is clearly over-predicted for all four load angles. The prediction improves when going from mode II to mode I. Mixed-mode ERR is captured. Additional experimental data or a modified experimental setup with stiffer steel plates, for the 0 degree model may reduce over prediction of strength.

4.2. Validation

90 60

30 0

Specimen angle, degree 10 20 30 40 50 60 Re ac tio n fo rce , k N Exp. failure Analytical

(a)TS1, Högskolan i Skövde.

90 60

30 0

Specimen angle, degree 10 20 30 40 50 60 70 80 Re ac tio n fo rce , k N Exp. failure Analytical (b)TS2, Volvo Cars.

Figure 4.5:Strength prediction against load angle for TS1, quasi-static and dynamic loading in black and blue, and TS2, red, black and blue correspond to adhesive thickness 0.2, 0.5 and 1.0 mm.

5

Conclusion

An analytical model to predict failure in adhesive joints based on energy release rate theory as described in Section 2 and [6] is implemented in the post processing tool Meta, using Python script supported by finite element models and user input. The implementation supports finite element models with triangular element contribution but is limited to models having matching mesh at the adhesive boundary. Stresses and energy release rates in mode I and mode II are calculated by the script and, given the fracture toughness, failure strength is predicted using a mixed-mode criterion. The implementation is successfully verified with an older method, and it is much more efficient as it eliminates manual calculation and input.

The new method is validated using results from two tests. The results show large scatter in test one and comparably less scatter in test two. Test one showed no clear distinction in strength values for the two strain rates that are tested. In test two, three different adhesive thicknesses are studied. The results show comparable strength values for all three thickness. The new method over predicts the strength. The prediction is less accurate under mode II and improves as the load is more in mode I. In pure mode II or for the 0 degree specimen, the prediction may be improved by determining the fracture toughness at failure initiation and using this in the implementation, another possibility to modify the experimental setup such that no plastic deformation occurs in the steel plates. Shear locking may have increased the predicted strength when using thin solid elements in test two, this may be overcome by using cohesive elements. Results show that the new method is sensitive to load direction in two dimensions.

No work is done on analysing a complete car body. However the presented implementa-tion allows investigaimplementa-tion of multiple adhesive joints provided that their element and bound-ary node numbers are know, and that they have matching mesh. Mode III failure, non-matching mesh and automation of adhesive boundary recognition remain as future work.

Improved adhesives and adhesive analysis methods in the automotive industry may bring about lightweight construction. Lightweight cars can be engineered to reduce carbon emissions and they also help in electrification. This improves the world climate and thus improve the well being of the society. Lightweight cars may also be cheaper to manufacture.

Future work

Although joints typically fail in mode I and mode II, modelling mode III failure may help reduce complexities in car body analyses with respect to orientation of joint in global coordi-nate system. However modelling mode III failure will require derivation of a 3 dimensional problem and it may be more complex to solve.

Some kind of plastic correction can be included into the script. This will of course require additional user and material input but may improve the results, like in the case of 0 degree model where some plastic behaviour is seen.

In [6], the analytical model is extended to delamination problems i.e. the adhesive thick-ness tends to zero. This can be used to predict failure in composite laminates based on energy release rate.

Adhesive strength and stiffness varies with temperature change as well as physical age-ing. This may be incorporated using simple empirical methods or by sub-modelling critical regions to perform more detailed transient analysis.

Non-matching mesh is often seen in car body models. This may be solved by extrapolat-ing force and moment values from nodes close to the adhesive boundary. If successful this method also has an added advantage of not considering triangular element contribution.

Bibliography

[1] T. Carlberger. “Adhesive joining for crashworthiness : material data and explicit FE-methods”. In: (2008). URL: http : / / www . his . se / PageFiles / 22209 / 2008 / Adhesive_joining_for_crashworthiness.pdf.

[2] Z. Chen, R.D. Adams, and Lucas F.M. da Silva. “The use of the J-integral vector to anal-yse adhesive bonds with and without a crack”. In: International Journal of Adhesion and Adhesives 31 (2011), pp. 48–55.ISSN: 0143-7496.DOI: https://doi.org/10.1016/j. ijadhadh.2010.11.005.

[3] T. Dahlberg and A. Ekberg. Failure, Fracture, Fatigue: An Introduction. Studentlitteratur, 2002.

[4] Python Software Foundation. Mathematical functions. Version 2.7.URL: https://docs. python.org/2.7/library/math.html.

[5] Python Software Foundation. Python Language Reference. Version 2.7. URL: http : / / www.python.org.

[6] Q. Luo and L. Tong. “Fracture Prediction of Adhesively Bonded Structures Using Energy Release Rates”. In: Journal of Adhesion Science and Technology 23 (2009), pp. 1415–1440. DOI: 10.1163/156856109X433009.

[7] J.A. Pascoe, R.C. Alderliesten, and R. Benedictus. “Methods for the prediction of fatigue delamination growth in composites and adhesive bonds – A critical review”. In: Journal of Engineering Fracture Mechanics 112-113 (2013), pp. 72–96.ISSN: 0013-7944.DOI: https: //doi.org/10.1016/j.engfracmech.2013.10.003.

[8] Community project. NumPy. Version 1.1.URL: https://www.numpy.org/l.

[9] J.R. Rice. “A Path Independent Integral and the Approximate Analysis of Strain Concen-tration by Notches and Cracks.” In: Journal of Applied Mechanics 35 (1968), pp. 379–386. ISSN: 0013-7944.DOI: 10.1115/1.3601206.

6

Appendix

6.1

Path independency of the J-integral for adhesive joints.

Consider an adhesive joint loaded on one end as indicated in Figure 6.1. The adhesive layer with thickness t is modelled using traction separation (σ, w and τ, v in mode I and mode II respectively, see Figure 6.1). Let the line joining points 1 and 4 be path A and the polyline joining points 1, 2, 3 and 4 be path B. For path independency the J-integral along path A and B must be equal [1].

J-integral defined by J.R. Rice [9] for an arbitrary pathΓ as, J= ż Γ  wdy ´ TiBui Bxds  (6.1) where w is the strain energy density given as,

w=

żε

0

σijεij (6.2)

J-integral along path A:

JA =J1´4 (6.3) since Bu/Bx=0, JA = ż A wdy (6.4)

JAcan be split into mode I and mode II as, J_IA = że 0 tσde J_{I I}A = żγ 0 tτdγ (6.5)

J-integral along path B:

JB=J1´2+J2´3+J3´4 (6.6)

since dy=0 along line 1-2, J1´2 = ż2 1  ´Ti Bui Bxds  = ż2 1 ´  τBux Bx +σ Buy Bx  (+dx) (6.7)

6.1. Path independency of the J-integral for adhesive joints.

Figure 6.1:Caption

similarly, dy=0 along line 3-4,

J3´4= ż4 3 ´TiBu 1 i Bxds ! = ż4 3 ´ ´τBu 1 x Bx ´ σ Bu1y Bx ! (´dx) (6.8)

where u1 is the displacement vector of the top adhesive-substrate interface. Since line 2-3 is far away from the adhesive tip we assume w=0 and T=0 which gives,

J2´3=0 (6.9)

equation 6.7, 6.8 and 6.9 in equation 6.6 gives,

JA = ż2 1 ´  τBux Bx +σ Buy Bx  (+dx) + ż4 3 ´ ´τBu 1 x Bx ´ σ Bu1y Bx ! (´dx) (6.10)

integrating from left to right (positive x-direction),

= żR L τ Bu 1 x Bx ´ Bux Bx ! dx+ żR L σ Bu1y Bx ´ Buy Bx ! dx = żR L τBv Bxdx+ żR L σBw Bxdx (6.11) since v=v(x)and w=w(x) JB= żR L τdv+ żR L σdw = żγ 0 tτdγ+ że 0 tσde (6.12)

or in mode I and mode II as,

JB_I = że 0 tσde J_{I I}B = żγ 0 tτdγ (6.13)

since equation 6.5 and 6.13 are equal path independency of the J-integral for an adhesive joint as illustrated in Figure 6.1 is proved.

6.2. Solution for shear and peel stresses for asymmetric substrates

6.2

Solution for shear and peel stresses for asymmetric substrates

To calculate the constants A2, A5and A6the following matrix expression is solved. $ & % L1 L2 L3 , . -=   T₁₁´ T₁₂´ T₁₃´ T₂₁´ T₂₂´ T₂₃´ T₃₁´ T₃₂´ T₃₃´   $ & % A2 A5 A6 , . -where L1= kτHnd ? kσ , L2= Hmd ? kσ and L3=´Hqd kσ (6.14)

and Hnd, Hmdand Hqd which are expressed in terms of substrate properties, forces and mo-ments as, Hnd=  D2+t2B2 2  N2 ∆2 ´  D1+t1B1 2  N1 ∆1 ´ 1 2  Ad2+2B2 t2  t2M2 ∆2 +  Ad1´2B1 t1  t1M1 ∆1  Hmd=´  A_d2M2 ∆2 ´ Ad1M1 ∆1  + B2N2 ∆2 ´ B1N1 ∆1  Hqd=´  A_d2Q2 ∆2 ´ Ad1Q1 ∆1  (6.15) and the coefficients of T´_{matrix are given as,}

T´ 11 =βt1, T12´=´βt3, T13´=βt2, T21´=Kt1  β2_t1´k5d  , T´ 22 =´2βt2βt3Kt2+Kt3 h β2_t2´ β2_t3  ´k5d i T₂₃´=´2βt2βt3Kt2´Kt2hβ2_t2´ β2_t3  ´k5d i T₃₁´=Kt1βt1  β2_t1´k5d  ´k2d kσ T₃₂´=Kt2βt3 h β2_t3´3β2t2  +k5di´Kt3βt2 h β2_t2´3β2t3  ´k5di T´ 33 =Kt3βt3 h β2_t3´3β2_t2+k5d i +Kt2βt2 h β2_t2´3β2_t3´k5d i ´k2d kσ (6.16)