DESIGN OF BI-ADHESIVE JOINT FOR OPTIMAL STRENGTH

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DESIGN OF BI-ADHESIVE JOINT FOR

OPTIMAL STRENGTH

Master Degree Project in Industrial Systems Engineering One year level 22.5 ECTS

Spring semester 2018 Siva Koti Reddy Vennapusa Supervisor: Ulf Stigh

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ABSTRACT

To support the trust in the design development of adhesively bonded joints, it is important to precisely predict their mechanical failure load. A numerical simulation model with a two-dimensional linear elastic cohesive zone model using a combination of a soft and a stiff adhesive is developed to optimize the strength of a lap-joint. Separation under mixed-mode conditions (normal and shear direction) is considered. By varying the length of the adhesives, the fracture load is optimized. The results obtained from the numerical experiments show an improvement in strength.

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Acknowledgements

I would like to first thank my thesis supervisor Prof. Ulf Stigh of the School of Engineering at the University of Skövde. He consistently guided me in the right way to finish thesis work. I wish to thank Daniel Svensson of School of Engineering at the University of Skövde for his valuable support when I struck with ABAQUS tool.

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1 Introduction

1.1 Background

The application of adhesively bonded joints found in aerospace, aeronautics, high technology industries and automotive field to traditional industries such as construction, sports and packing has been used considerably. Therefore, adhesives are becoming key research area due to its potential applications [1].

Adhesive joints used in structures repair often because of load carrying capabilities. Adequate design of joints used in structural parts and repair is required to get the desired performance of structure because they often undergo heavy loads and play a crucial role in establishing the efficiency of the structure. Despite advancements in bonding technology, the bonded joints also suffer from poor adhesion in case some substrates and less resistance to cleavage stresses [28]. The advantages of adhesive bond joints are high strength to weight ratio, stress distribution over a wide joint area with a continuous stress distribution, reduced corrosion, great design freedom, less finishing operation and simplicity with vibration damping are the sum of advantages of adhesively bonded joints. Bolts and rivets are often pointed out the high amount of stress at their joints which lead to having less static and fatigue strength compared to adhesive joints. However, adhesively bonded joints require extensive surface preparation in order to get a robust and durable joint [1].

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1.2 Problem Definition

The use of single adhesive for joining large parts suffers high stresses at the ends of the overlap and the remaining bond area experiences very small stresses. The effect of stresses can be improved using two adhesives, a large part of the bond line experiences large stress. In this work, the study of adhesive mechanical behaviour using a fracture mechanics approach using a combination of a soft adhesive and a stiff adhesive is developed to optimize the strength of a lap joint.

1.3 Aim and objectives

The aim of this master dissertation work is to obtain improvement in joint strength using bi-adhesive bonded joint which is evaluated by its fracture load. In addition, the aim is also to implement the methodology of the design process and to show how this can be done in an optimum manner.

The goal of the thesis is achieved through the following objectives

• The building of numerical models with different adhesive lengths.

• Assign element type for specified regions, mesh convergence study and simulation. • Validation of results with different adhesive length ratios.

1.4 Limitations

• No experimental investigations are made.

• Adhesives are considered as a linear elastic material. Failure is predicted to occur based on a fracture mechanics condition and no damage was considered in the material. • Only one specific adhesive bonded joint configuration and load case are studied using

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1.5 Research Methodology

The proposed plan for this research project to achieve the expected results consists of various stages. The methodology followed here similar to methods from the literature.

• Study of previous documents related to this work (different approaches for predicting the strength of the adhesive bonded joints, mixed adhesive joints, cohesive zone model (CZM) and different parameters affecting the strength).

• Selection of substrates, adhesive material, and their properties and calculation of Cohesive zone properties from adhesive properties.

• Implementation of linear elastic analysis in Abaqus.

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2 Literature Review

2.1 Overview of adhesive joints

Due to high advancements and requirements of bonding concepts and lightweight structures with high strength and design flexibility, the automotive and aeronautical industry is becoming more active in the application of adhesive joints [2]. Increase in use of composite materials, the technique of adhesive joints simplifies with ease, ability and suitable for any material with complex structures without the need for any fastening techniques. In addition, adhesive joints have advantages of combining thin-walled plates, which are very light structures but strong enough to carry high loads. The advantage of stress distribution along the joint is shown in figure 2. 1.

Figure 2. 1. Stress distribution for a riveted joint and an adhesive joint [3]

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2.2 Joint design criteria

The bonded area of the joint is large enough to resist the loads or forces acting due to service life. The evaluation of stresses in the joint is not a reliable method of deciding the exact dimensions required. Finding allowable stresses in a joint is a relatively difficult choice. The efficiency of the bonded joint is affected by the environmental condition, curing temperature, age, composition, and adherend size and bond line thickness. The stresses in the bond is normally a combination of many stress components. The relative stiffness of the adhesive with respect to the adherend has a noticeable effect on the stress distribution [4].

Figure 2. 2. Single adhesive joint deformed shape and stresses along overlap length [5].

As seen in figure 2. 2, the peel stresses and shear stresses are maximum at the ends of the bond line. The peel stresses develop at the ends due to the bending moment. If the length of the joint is doubled, the load carrying capabilities are not doubled. The improvement in strength is gained by increasing the width of the joint.

2.2.1 Effect of Geometric parameters on strength of adhesive joint

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Figure 2. 3. Effect of width and overlap length on strength of a single lap joint [4].

(a) (b)

Figure 2. 4. (a)Effect of adherend thickness on the failure load with low modulus adhesive SBT9245; (b) Effect of adherend thickness on the failure load with high modulus adhesive

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The effect of adherend thickness on failure load is illustrated in figure 2. 4(a), the increase in adherend thickness and overlap length affects the failure load carried by SLJ with SBT9245 which have low modulus. Similarly, in the case of AV119 adhesive with high elastic modulus have no effect of overlap length with respect to the strength of SLJ, shown in figure 2. 4(b). However, there is an effect in failure load due to an increase in adherend thickness for different lengths of the overlap [7]. As seen in figure 2. 5 (a), the load values are decreased in different cases of thickness 98 to 118μm; however, thickness reduction is limited to strength, not increases but decreases after a certain value of adhesive thickness. The decreases in strength are due to a weak boundary layer in case of thin adhesive thickness [5].

2.3 Mixed adhesive joints

As the stress distribution across the overlap in a single lap joint is not distributed equally, an ideal joint has a variation of flexibility and strength properties which accommodates created forces equally, referred as a functionally graded adhesive bonded joint. The best way to improve the strength of the joint is by using a ductile and a soft adhesive at the ends of the overlap ends to reduce the high-stress concentration, while a stiff adhesive is used between the softer adhesives. By combining both adhesives enlarge the application possibilities efficiently, different methods have been proposed: high temperature and low-temperature adhesives, using rubber softening the rigid adhesive and different combination of adhesives [3].

(a) _(b)

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According to [8], an ideal joint is a joint which the adhesive properties and strength vary across the bond line. Shear stress-free condition at the ends of the overlap, adhesive stress distribution is shown in figure 2.6.

Figure 2. 6. Stress distribution when shear stress-free condition at the ends of the overlaps

[3].

In the past, many researchers have proposed mixed-modulus adhesive joint with improvements in stress distribution and increased strength. The concept of mixed joints, where the soft adhesive is placed at the end of the joint which prevents failure. The method entitles the introduction of more soft adhesives at the overlap ends. While the central adhesive subjected to less deformation during loading. In a large temperature gradient condition, the bi-adhesive joints could yield improvements in mechanical strength [9].

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The use of composite adherends with epoxy adhesives with single lap joint achieved strength improvements [3]. The results with high modulus adhesive experienced low strength of the adherends across the thickness, which results in delamination of composite due to high stresses at the ends of overlap. In other cases, with mixed adhesives with the same composite adherends resulted in the same strength with cohesive failure [3].

Figure 2. 8. Schematic shear stress distribution at failure in the dual adhesive joint [3].

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3 Theory

3.1 Linear elastic fracture mechanics

One assumption of fracture mechanics based on continuum mechanics is that the material does not have defects. The presence of imperfections or geometric discontinuities violates the principle of continuum mechanics. Cracks, deboning, delamination and any other imperfections present in the specimen causes stress concentration and therefore initiations spots for a fracture to occur and propagate. This may lead to failure of the component.

The Fracture mechanics approach mainly relies on the stress intensity factor which is equivalent to an energy-based principle. The stress intensity factor (K) is a scalar parameter which is used to predict stress state at the crack tip, originated from the residual stresses or the remote load. The energy-balanced fracture mechanics criterion compares the strain energy release rate due to an applied load to the critical fracture energy (Gc) of the material [10].

Figure 3. 1. Illustration of three different modes of fracture [11].

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For mode-I loading, the stress intensity factor is often given by:

𝐾_𝐼= 𝑌 𝜎_𝑟 √𝜋𝑎 , (1)

Where, r is the remotely applied stress or externally applied stress perpendicular to the crack

face, ‘a’ is the crack length and Y is a non-dimensional geometric factor. The condition for a crack to propagate is given by

𝐾_𝐼 = 𝐾_𝐼𝐶 (2)

Where 𝐾_𝐼𝐶 is the material property, measuring the ability of a material to prevent crack growth which denotes the material fracture toughness. If the 𝐾𝐼 value is smaller than 𝐾𝐼𝐶, the crack tip

does not propagate.

The hypothesis of energy balanced criterion states that for an applied loading condition the crack propagates when the energy release rate is equal to the energy required for a crack to propagate. It can be expressed as

𝐺 = 𝐺_𝑐 (3)

If the applied loading is a combination of peel and shear stresses (mixed-mode), the effect of mixed modes occurs at the crack tip. [13] States that the energy balanced approach is preferable over the stress intensity approach because the energy release rate (G) has an important physical meaning related to crack energy absorption process, and the stress intensity factor value (K) is hard to obtain. However, both the methods are mathematically equivalent. For mixed-mode loading, the energy balanced crack propagation criterion is assumed

(𝐺𝐼 𝐺_𝐼𝐶) 𝛼 + (𝐺𝐼𝐼 𝐺_𝐼𝐼𝐶) 𝛽 = 1 (4)

Where GIC and GIIC are the critical energy release rate in mode-I and mode-II respectively and

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3.1.1 Separation of material

From a more fundamental point of view, crack propagation occurs due to the separation of material by the crack. Fig 3.2 illustrates a common understanding of how the stresses between two separating surfaces develop during the process.

Figure 3. 2. Potential energy and applied force as a function of separation [10].

From figure 3. 2, the equilibrium spacing occurs at minimum potential energy. The tensile force required to separate from the equilibrium position is that in order to sever the bond completely the tension force must overcome cohesive force. The bond energy is given by

𝐸𝑏 = ∫ 𝑃 𝑑𝑥 ∞

𝑥0

(5)

Where xo is the equilibrium distance shown in figure 3.2 and P is the applied force. The

behaviour of the curve from figure 3.2 is used to evaluate the cohesive strength of the bond by idealizing the force-displacement relationship as

𝑃 = 𝑃_𝐶 𝑆𝑖𝑛 (𝜋 𝑥 𝜆 )

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Where λ is the distance defined in figure 3.2. In the case of small displacements, the equation for force-displacements is linear:

𝑃 = 𝑃_𝐶 (𝜋 𝑥 𝜆 )

(7)

The bond stiffness is given by

𝑘 = 𝑃𝐶

𝜋 𝜆

(8)

By converting k to Young's modulus (E) and Pc to cohesive stress (σc). Solving cohesive

stress gives

𝜎_𝐶 = 𝐸 𝜆 𝜋 𝑥₀

(9)

The surface energy can be written in the form of equation 10

𝛾_𝑠 = 1 2∫ sin ( 𝜋𝑥 𝜆 ) 𝜆 0 𝑑𝑥 = 𝜎_𝐶 𝜆 𝜋 (10)

Substituting equation 9 into equation 10 and solving for cohesive stress 𝜎_𝐶 gives

𝜎_𝐶 = √𝐸 𝛾𝑠 𝑥_𝑜

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3.2 CZM in ABAQUS

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can often be designed as a linear elastic in terms of degradation of stiffness under tensile or shear loading but unaffected by pure compression [15].

The cohesive surface is an element layer thick in between the adherends where the crack propagation takes place. The cohesive layer is modelled as a separate section of the model. The cohesive element subjected to linear displacement formulation and the third direction stresses does not affect the behaviour of the element [15]. According to the traction-separation law, the stresses and separation displacements are calculated in each increment at integration points. the connection between the cohesive zone and substrate material is made by using shared nodes to make orientation. The orientation defines the cohesive element thickness and consistent mesh throughout the adhesive layer.

Figure 3. 3. Cohesive zone interaction with surrounding material [15].

3.2.1 Traction-separation based modelling

The elastic behaviour can be expressed in terms of an elastic constitutive law, the nominal stress to nominal stain across the bonding interface. The nominal stresses are the ratio of force components to the original area at each integration point and the nominal stains are the separation divided by the original thickness at integration points [15].

The nominal traction stress vector t consists of two components in two-dimensional problem σ and τ, which represents normal stress and shear stress, and respective separation is denoted by

w and v.

The stains can be defined as,

𝜀_𝑛 = 𝑤

𝑇_𝑜; 𝜀𝑠 = 𝑣 𝑇_𝑜

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Where 𝜀_𝑛, 𝜀_𝑠 are strains along normal and shear direction, To is the original thickness of the

adhesive. The elastic behaviour can be expressed as,

𝑡 = {𝜎 𝜏} = [ 𝑘_𝑛𝑛 𝑘_𝑛𝑠 𝑘_𝑛𝑠 𝑘_𝑠𝑠] { 𝜀𝑛 𝜀𝑠} = 𝑘 𝜀 (13)

If traction-separation is specified, which ensures that nominal strains are equal to separation displacements. The default value of constitutive thickness (𝑇_𝑜) in Abaqus is taken as one. Therefore, the stress components from equation 17 becomes separation displacements, i.e. the elastic behaviour can be written as,

{𝜎_𝜏} = [𝑘𝑛𝑛 𝑘𝑛𝑠 𝑘𝑛𝑠 𝑘𝑠𝑠] {

𝑤 𝑣}

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The elasticity matrix k from equation 14 provides fully coupled behaviour between all the components of the stress vector and separation vector. If uncoupled behaviour between the normal and shear components is desired, set the off-diagonal terms in the stiffness matrix to zero. {𝜎 𝜏} = [ 𝑘𝑛𝑛 0 0 𝑘𝑠𝑠] { 𝑤 𝑣} (15)

The thickness dependent stiffness of the adhesive can be written as

Normal elastic stiffness, 𝑘_𝑛𝑛= 𝐸𝑎𝑑ℎ𝑒𝑠𝑖𝑣𝑒

𝑇𝑜

Shear elastic stiffness, 𝑘_𝑠𝑠 = 𝐺𝑎𝑑ℎ𝑒𝑠𝑖𝑣𝑒

𝑇𝑜

(16)

Where Eadhesive is Young's modulus and Gadhesive is the shear modulus of the adhesive. The matrix

K is stiffness parameters of the adhesive layer, for thin layers of thickness T0 = 1, 𝑘𝑛𝑛=

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Figure 3. 4. Load-displacement characteristics for a linear problem.

(a) Mode-I (b) Mode-II

Figure 3. 5. Linear cohesive laws for loading in pure mode-I and mode-II.

For mixed-mode loading, the failure is predicted by using linear power law from the energy release in the pure modes.

𝐺𝐼 𝐺𝐼𝐶

+

𝐺𝐼𝐼 𝐺𝐼𝐼𝐶

= 1

(17)

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mechanics (LEFM), i.e. 𝜙_𝐼 = 𝐺_𝐼 and 𝜙_𝐼𝐼 = 𝐺_𝐼𝐼 [12]. For mixed-mode loading, the work performed by cohesive traction is the sum of integral over separation. It is given by,

𝜙 = 𝜙_𝐼+ 𝜙_𝐼𝐼 (18)

𝜙 = ∫ 𝜎 𝑑𝑤 + ∫ 𝜏 𝑑𝑣 (19)

By substituting stresses as a product of stiffness and separation from equation 15. The fracture energy is given by,

𝜙 = 1

2 𝑘𝑛𝑛 𝑤

2₊1

2 𝑘𝑡𝑡 𝑣

2 (20)

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4 Numerical Modelling

4.1 Joint geometry

The geometric configuration of the single lap joint with boundary conditions are shown in figure 4. 1. A finite element analysis is carried out using the numerical simulation tool Abaqus® software (Dassault Systèmes, France) to obtain the separation displacements values along normal and shear directions to predict the fracture energy. The simulation of a mono-adhesive lap joint with a stiff adhesive, a soft adhesive and a bi-adhesive using both adhesives are compared. The geometric parameters of the joint with a single adhesive and mixed adhesive are shown in figure 4. 1.

Figure 4. 1. Geometric parameters and boundary conditions of an adhesive joint

From figure 4.1 Lo is the overlap length, tA is the adhesive thickness, tp is the adherend thickness,

LT is the total length of the joint from its end, L1 is the soft adhesive length, L2 is the stiff

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Table 4. 1. Geometric parameters

Lo 12.5mm

tA 0.25mm

tp 1.62mm

 1mm

LT 62.5 mm

In a bi-adhesive joint, the single adhesive is replaced by two adhesives, which is highlighted in figure 4. 1.

For the demonstration of increasing the length of the soft adhesive at the ends of the overlap the 𝐿₁ can be written as 𝐿₁ = 𝛼 𝐿₂.

The overlap length of the bi-adhesive can be written as

𝐿𝑜 = 2𝐿1+ 𝐿2 = 𝐿2 (2𝛼 + 1) (21)

Thus, 𝐿₂ = 𝐿_𝑜/(2𝛼 + 1) and only one parameter is varied in the optimization, were 0 ≤ 𝛼 < ∞.

Material Properties

The adherend material for the joint is selected as Aluminum 2024 T3 alloy and two adhesives were selected, ESP110 (stiff adhesive) produced by Permabond and DP490 (soft adhesive) by 3M for all cases [16]. The adhesive and adherend material properties required for the linear elastic analysis are presented in table 4. 2. The thickness dependent traction properties of adhesives are given in table 4. 3.

Table 4. 2. Adherend and adhesive properties.

Parameters Al-2024 T3 ESP110 DP490

E (GPa) 70 2.8 0.28

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Table 4. 3. Adhesive thickness dependence cohesive properties for 0.25 mm thickness.

Parameters ESP 110 (Stiff adhesive) DP490 (Soft adhesive) knn (MPa/mm) 11200 1120 kss (MPa/mm) 4000 400 GIIC (N/mm) 2,1 5,1

4.2 Meshing of geometry

The mesh for adherends is constructed by using CPE4 (A 4-node bilinear plane strain quadrilateral) element. The element type assigned for the cohesive zone is a 4-noded two-dimensional cohesive element (COH2D4). The edges of adherend connecting adhesive ends are biased to create dense meshing near the bond ends are shown in figure 4.2.

Figure 4. 2. Mesh defining element types.

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analysis. From mesh convergence, the number of elements required in a model to ensure the results of simulation does not affect by the size of the element.

Figure 4. 3. Mesh convergence: element size along the cohesive zone.

The mesh convergence study enables to obtain accurate values of stresses and can be performed at local surfaces depending on interest. A 4-point convergence curve of cohesive zone shear stress is plotted in figure 4. 3. At least three convergence simulations are required to confirm that the convergence is achieved [17]. The model with the element size 0.75mm across the cohesive zone gives shear stress value of 46.6 MPa at the crack tip. The mesh is constructed with 250 elements the shear stress increased by 27.8%. The last two points of convergence curve with element size 0.25mm and 0.1mm give the difference of 0.14%. The convergence occurs at element size 0.1mm with shear stress value 89.7 MPa.

0 10 20 30 40 50 60 70 80 90 100 0.0625 0.125 0.25 0.5 1

τ

(MP a)

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5 Results and Discussion

In this section, the results from the numerical simulations are presented. The different scenarios of adhesive joints with different design parameter α, increasing the portion of soft adhesive to study the behaviour of the joint. An adhesive joint with a stiff adhesive, a soft adhesive and a mixed adhesive are designed to make a comparison of capabilities. The boundary conditions for all the simulations used are the same shown in figure 4. 1. The one end of the lap-joint fixed and another end is prescribed with a displacement value of 1mm.

5.1 Shear stress distribution in single and mixed-adhesive lap joints

In this part of the work, several linear elastic simulations are carried out to predict shear stress distribution along the overlap. For this analysis, joint with a stiff adhesive (ESP110), a joint with soft adhesive (DP490) and a bi-adhesive joint (combination of ESP110 and DP490) with α = 0.5 are used to study the stress distribution along the overlap.

(a) (b)

(c)

Figure 5. 1. Shear stress contours of adhesively bonded joint showing only the adhesive section, a) Soft adhesive DP490, b) Stiff adhesive ESP110, and c) Bi-adhesive combination

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Figure 5. 2. Shear stress vs position of overlap for different cases.

The shear stress contours of three joints with ESP110 adhesive (E= 2.8GPa), DP490 adhesive (E=0.28GPa) and mixed adhesive are shown in figure 5. 1. The plots of shear stress obtained from the simulations are shown in figure 5.2. It can be observed that the shear stress pattern along the overlap when using adhesive DP490 is more uniformly distributed compared to joint with a soft adhesive ESP110. The peak shear stress at the end of the overlap for a mixed adhesive joint is lower than the other two cases with a stiff adhesive and a soft adhesive used alone. The shear stress values at the ends of the overlap for mixed adhesive are three times smaller than the value of shear stress for the joint made using stiff adhesive and the shear stress value of the joint made using a soft adhesive is 40% higher than the shear stress of a bi-adhesive joint.

5.2 Traction versus separation

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b)

c)

Figure 5. 3. Traction–separation curves (Mode-I and II) for the above three cases, a) Soft adhesive alone, b) Stiff adhesive alone and c) Mixed adhesive.

From stress-separation plots shown in figure 5. 3 are the values of stresses and separation displacements at the crack tip for different joints. The separation strengths in terms of stress reach a maximum value at certain bearable displacement. For the adhesive joint with soft adhesive DP490 reaches the shear stress value of 49.63MPa and normal stress reaches to

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3.73MPa at respective separation displacements in both directions. The area under τ-v curves gives mode-II energy release rate (GII) and similarly, mode-I energy release rate (GI) is given

by the area under σ-w plot. The observation from figure 5.3, the area under σ-w plot is smaller than τ-v curves. The energy release rate in mode-I (GI) is smaller than the energy release rate

in mode-II (GII). Since NLGEOM feature in Abaqus is turned off for all the simulations (no

large deformations), the effect of deformation in peel direction cannot be captured well. The energy release rate at the crack tip for the joint with a stiff adhesive is 1.8602 N/mm and soft adhesive fractures more with 3.0853 N/mm. For the mixed adhesive joint with α = 0.5, the energy release rate at the crack tip is half the value of the energy release rate of joint with soft adhesive.

5.3 Effect of α values on work of separation

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Table 5. 1. Values of stresses at the ends of adhesive for all configurations.

(𝐿1

𝐿2) = 𝛼

Shear stress at the crack tip

(MPa)

Normal stress at the crack tip

(MPa)

Shear stress at the ends of the middle adhesive

(MPa)

Normal stress at the ends of the middle adhesive (MPa) ESP110 only 119.48 41.32 _ _ DP490 only 49.63 3.73 _ _ 0.5 35.04 3.64 89.58 29.46 0.63 36.21 3.61 85.46 29.61 0.75 36.73 3.57 83.23 29.55 1 38.22 3.60 79.66 29.25 1.46 39.71 3.61 79.06 29.18 2 40.93 3.64 81.83 29.36 2.625 41.5322 3.62 87.54 29.3015

Table 5. 2. Separation displacements and energy release rate at the ends of adhesives.

α

Ends of adhesive (Crack tip)

Mid adhesive ends Energy release rate at

end overlap (N/mm)

Work of separation at

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The stress versus separation displacement curves for all ‘α’ cases are presented in Appendix A and the values of stresses are shown in table 5.1 and separation displacements are presented in table 5.2. From the discussion of energy release rate from section 5.2, the total energy release rate at the crack tip for α = 0.5 has less value when compared to adhesive joints with an individual stiff and a soft adhesive alone. The energy release rate can be further improved by increasing α value. The sustainability of CZM modelling can capture the features of the failure process such as the critical displacement, stiffness and maximum load value sustained by the structure [18]. From figure 5.4 the energy release rate at the crack tip for different α values are shown, the energy release rate increases with increasing length of DP490 soft adhesive.

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Figure 5. 5. Work of separation at the ends of the middle adhesive (ESP110) in a bi-adhesive joint.

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The work of separation at the ends of the middle adhesive by increasing the length of the softer adhesive was shown in figure 5.5. From the values of numerical simulations, the energy release rate at the end of the stiff adhesive ESP110 decreases from α = 0.5 to 1.46 and starts increasing further. The value of ϕ at the ends of the stiff adhesive is limited to certain α value. The energy release rate is normalized with respect to the critical energy rate (GIIC) to plot the distribution

of fraction values along overlap shown in figure 5.6. The length of the soft adhesive increased further to achieve an optimum ϕ/GIIC value at ends of adhesive. when α is increased from 0.5

to 0.63, the calculated normalized ϕ is increased by 6.36%. The normalized fracture energy (ϕ/GIIC) along the adhesive layer for different α values are plotted in figure 5.6. From figure 5.6

the value of normalized ϕ with the optimum condition was obtained at α = 2 which shows both soft and stiff adhesive in a mixed adhesive joint have the same value of normalized fracture energy at their ends. For linear criteria, the condition for crack propagation is assumed,

(𝐺𝐼 𝐺_𝐼𝐶) + (

𝐺𝐼𝐼

𝐺_𝐼𝐼𝐶) = 1

(22)

Since Nlgeom feature is turned off, the overall ϕ value is highly dominated by work of separation in mode-II (𝜙_𝐼𝐼). From the condition fracture mechanics equation 22, the failure is predicted. The normalized ϕ calculated from the analysis satisfies the condition of fracture mechanics.

Figure 5. 7. The significance of α value from a production view.

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From the production view, the value of αoptimum shown in figure 5.9 has robust nature while

manufacturing a joint. If the value of α is altered slightly by ∆error, i.e. α = 2.1 or 2.2 or 1.9 still

the joint performs efficiently.

5.4 Fracture load on the joint

As a validation of cohesive zone modelling of bi-adhesive single lap joint, the reaction force (P) on the joint for different α cases are plotted in figure 5.8. The reaction force on joint decreases with increasing the length of the softer adhesive in a bi-adhesive joint. The force on the joint in each case is slightly differed by 0.3% or less in some cases. The trend showing a decrease in forces on the joint for different cases shown in figure 5.8.

Figure 5. 8. Variation of forces on a bi-adhesive lap joint with increasing soft adhesive.

The results obtained from the analysis when the displacement is applied gives the values of work of separation along the bondline and reaction forces on the joint. The critical load carried by the joint before crack initiation takes place can be scaled using equations 22.

From LEFM, the critical load is given by

𝑃_𝑐 = 𝑃 √𝐺𝐼𝐼𝐶 𝜙

(22)

Where 𝑃 is the reaction force on the joint, 𝜙 is work of separation along the bondline and GIIC

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𝑃_𝑐 = 𝑃 √𝐺𝐼𝐼𝐶 𝐺

(23)

Where G is the energy release rate.

Table 5. 3. Fracture load for the joint using a stiff adhesive and soft adhesive alone.

Joint with stiff adhesive (ESP110) Joint with soft adhesive (DP490)

P (N) GIIC (N/mm) G (N/mm) Pc (N) P (N) GIIC (N/mm) G (N/mm) Pc (N) 593.0 2.1 1.86 629.7 574.0 5.1 3.085 688.8

Table 5. 4. Fracture load for the soft adhesive and stiff adhesive in an optimized bi-adhesive joint.

Load on bi-adhesive joint

P (N)

Crack tip (Soft adhesive DP490) Stiff adhesive ends (ESP110)

GIIC (N/mm) G (N/mm) Pc (N) GIIC (N/mm) ϕ (N/mm) Pc (N) 566.7 5.1 2.1 883.1 2.1 0.87 877.9N

Figure 5. 9. Fracture load carried by the lap joint with different adhesives

0 200 400 600 800 1000

Stiff adhesive joint Soft adhesive joint Optimized bi-adhesive

joint

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The fracture load on the single adhesive lap joint made using a stiff adhesive and a soft adhesive are shown in table 5.3. The energy release rate in the stiff adhesive (G = 1.86 N/mm) is compared with the fracture energy of the stiff adhesive (GIIC = 2.1 N/mm) and similarly the

right side of table 5.3 shows the comparison of fracture energy of the soft adhesive (GIIC = 5.1

N/mm) and energy release rate (G = 3.08 N/mm) to obtain the fracture load. The fracture load obtained for the joint made using a stiff adhesive is 629.7N and soft adhesive fractures at the load value 688.8N.

The load carried by the bi-adhesive joint before crack initiation takes place is shown in table 5.4. Here, the fracture load on the optimized bi-adhesive joint is taken in to account to make a comparison with individual adhesives. The bi-adhesive uses a soft adhesive at the ends of overlap that means at the crack tip soft adhesive is present. The energy release rate of the soft adhesive (G = 2.1 N/mm) is compared with fracture energy of the soft adhesive (GIIC = 5.1

N/mm) to obtain fracture load. The critical load carried by the stiff adhesive in a bi-adhesive is obtained by relating work of separation at the ends of the stiff adhesive (ϕ = 0.87 N/mm) and fracture energy of the stiff adhesive (GIIC = 2.1 N/mm).

The values obtained from the comparison shows that the soft adhesive fractures at the load value 883 N and the critical load at the ends of the stiff adhesive is 877.9 N. It can be observed that the Pc at both ends of the adhesive is almost equal, differed by 0.6%. Figure 5.9 shows the

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6 Conclusion

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40 g) α = 2.625

Stress versus separation displacement curves at crack tip for the joint with different α values

Appendix B:

Reaction force on the adhesive joints for different cases.

α Force (N)

ESP110 (stiff adhesive) 593.0

DESIGN OF BI-ADHESIVE JOINT FOR OPTIMAL STRENGTH