IN
DEGREE PROJECT MECHANICAL ENGINEERING, SECOND CYCLE, 30 CREDITS
,
STOCKHOLM SWEDEN 2020
Material Properties of
Adhesive Joint
SHUYUE WANG
ROYAL INSTITUTE OF TECHNOLOGY
MASTER’S THESIS
Material Properties of Adhesive Joint
Degree Project in Mechanical Engineering, Second Cycle, 30 Credits
Solid Mechanics
Author: Shuyue Wang
Supervisor & Examiner: Industrial Supervisor:
Carl Dahlberg
Joakim Widell
Andreas Ankarbjörk
i
Abstract
The development of the commercial 5G network with high-frequency mmWave requires tighter base station grid, which increases the demand for smaller and more unobtrusive products. One way to connect materials of different properties and keeping the product size small is to use adhesive joints instead of screw joints. The thesis project is about understanding the material behaviour of adhesive joint and determining the material model.
Adhesive joints can be described as a highly temperature-dependent material, including both
hyperelastic and viscoelastic material behaviours. A relaxation test was carried out to evaluate
ii
Sammanfattning
Utvecklingen av den kommersiella 5G-nätverk med högfrekvent signaler sätter krav för en tätare basstationsnät, vilket i sin tur behöver mindre och mer diskreta produkter. Ett sätt att combinera material med olika egenskaper och samtidigt försöka minimera produktsstorleken är att ersätta skruvförband med limförband. Detta masterexamensarbete handlar om att undersöka materialbeteende hos limfog och att bestämma dess materialmodell.
Limfogd kan beskrivas som ett temperaturberoende material med både hyperelastiskt och viscoelastiskt materialbeteende. En relaxationstest genomfördes för att utvärdera limfogens beteende och dess temperaturberoende egenskaper. Testresultaten visade att Neo Hookean och
Generalized Maxwell material modellen kan användas för att beskriva de grundläggande
iii
Acknowledgements
I would like to express my gratitude to all the participants who provided me with assistance during the project. Without your support, this project would not have been possible. Particularly, I would like to thank my supervisors, Carl Dahlberg at KTH, Joakim Widell and Andreas Ankarbjörk at Ericsson, thank you for supporting and guiding me through my work. Additionally, I would like to thank Martin Öberg at KTH, Christofer Markou and Sofia Tran at Ericsson for helping me to set up and carry out the experiments. Finally, I would like to thank Andreas Rydin at ANSYS who helped me get through the difficulties of simulation.
Stockholm, June 2020
iv
List of Figures
Figure 1-1. Overview of 2G to 5G. ... 1
Figure 1-2. Definition of adhesive and adherends. ... 2
Figure 2-1. Adhesives with a) higher surface energy and b) lower surface energy than the surface tension of the adherend. ... 3
Figure 2-2. Volumetric and deviatoric stress effect. ... 5
Figure 2-3. Applied strain and stress response. ... 5
Figure 2-4. a) Spring element b) Dashpot element. ... 8
Figure 2-5. a) Kelvin-Voigt model b) Maxwell model. ... 9
Figure 2-6. Generalized Maxwell model. ... 9
Figure 2-7. Stress relaxation response. ... 10
Figure 2-8. Time-temperature superposition. ... 11
Figure 2-9. A schematic illustration of a master curve and the interpolation of the shift factor. ... 12
Figure 2-10. Motion in the Lagrange Description. ... 14
Figure 2-11. Engineering stress-strain curve containing 1, 2, and 3 inflection points. ... 16
Figure 2-12. Schematic reloading curve accounting for the Mullins effect. ... 18
Figure 2-13. a) Simple Lap Shear b) Double Lap Shear c) Lap Strap d) Double Lap Strap. ... 20
Figure 2-14. The stress-strain curve with the differently defined failure points. ... 21
Figure 2-15. Scarf joint. ... 22
Figure 2-16. T-peel test. ... 22
Figure 3-1. Simple lap shear test specimen design. ... 24
Figure 3-2. Specimen production process. ... 24
Figure 3-3. Simple lap shear test specimen curing process. ... 25
Figure 3-4. Fully-cured and cleaned specimen. ... 25
Figure 3-5. The machine setup of the simple lap shear test. ... 26
Figure 3-6. Definition of the engineering shear strain. ... 26
Figure 3-7. The setup pf the numerical simulation. ... 27
Figure 3-8. Manufacturing of the compression test sample. ... 28
Figure 3-9. Experiment setup for the repeated compression test. ... 28
Figure 3-10. Applied load in term of strain vs. time. ... 29
Figure 3-11. Numerical model for the simulation. ... 31
Figure 3-12. Plastic cover and aluminium base. ... 33
Figure 3-13. Joint specimen setup. ... 33
Figure 3-14. Applied load by prescribe the displacement. ... 33
Figure 3-15. The fixture setup of the cyclic loading test. ... 34
Figure 3-16. Tension machine setup. ... 35
Figure 3-17. Cyclic-loading machine setup. ... 35
Figure 3-18. The numerical setup of the adhesive joint during cyclic loading test. ... 36
Figure 4-1. Reaction force as function of time at different temperature. ... 37
Figure 4-2. Relaxation shear moduli and the master curve in loglog-scale together with the plot of the shift factors. ... 38
Figure 4-3. Relaxation shear moduli and the master curve in loglog-scale. ... 38
v Figure 4-5. Results from the relaxation tests compared to the corresponding numerical simulation
with iso- and hyper-viscoelastic material model. ... 40
Figure 4-6. WLF shift function vs. ANSYS build-in linear interpolation. ... 40
Figure 4-7. Force-displacement curve as output from the cyclic loading tests. ... 41
Figure 4-8. Determine and insert mean of un- and reloading curves. ... 41
Figure 4-9. Least square curve fitting considering Neo Hookean model. ... 42
Figure 4-10. Least square curve fitting considering 3rd order Ogden model. ... 43
Figure 4-11. Numerical simulations based on the three cases of boundary conditions. ... 44
Figure 4-12. Numerical simulations based on the three cases of boundary conditions. ... 44
Figure 4-13. Force-displacement graph of runout with 1 mm in displacement. ... 45
Figure 4-14. Force-displacement graph of unconnected adhesive joint. ... 45
Figure 4-15. Force-displacement graph of adhesive joint with introduced damage. ... 46
Figure 4-16. Force-displacement graph of total failed adhesive joint. ... 47
Figure 4-17. Force-displacement graph of adhesive joint damaged under loading. ... 47
vi
List of Tables
Table 1. Material data provided by the supplier. ... 23
Table 2. Boundary conditions of the top and bottom surface... 32
Table 3. A summary of the boundary conditions of the cyclic loading test simulation. ... 36
Table 4. Used material parameters in the relaxation simulations. ... 39
Table 5. Obtained material parameters according to Neo Hookean model. ... 42
Table 6. Obtained material parameters according to 3rd order Ogden model. ... 43
Table 7. Force-difference between undamaged and damaged joints. ... 46
vii
Abbreviations
VR Virtual Reality AR Augmented Reality mmWave Millimetre Wave RBS Radio Base Station
TTS Time-Temperature Superposition WLF William-Landel-Ferry
Contents
Abstract ... i Sammanfattning ... ii Acknowledgements ... iii List of Figures ... iv List of Tables ... vi Abbreviations ... vii 1. Introduction ... 1 1.1. Background ... 1 1.2. Aims... 2 2. Theoretic Background ... 3 2.1. Adhesive ... 3 2.1.1. Moisture Sensitive ... 3 2.1.2. Temperature Sensitive ... 42.2. Stress State in Adhesive Layer ... 4
2.3. Linear Viscoelasticity ... 5
2.3.1. Stiffness-Compliance Relation ... 6
2.3.2. Cyclic Loading Response ... 7
2.3.3. Rheological Operators ... 8
2.3.4. Generalized Maxwell Modell ... 9
2.3.5. Prony Series... 10
2.4. Time-Temperature Superposition ... 10
2.4.1. Williams-Landel-Ferry Shift Function ... 12
2.5. Hyperelasticity ... 13
2.5.1. Continuum Mechanics Foundations ... 13
2.5.2. Constitutive Law ... 14
2.5.3. Strain Energy Function ... 15
2.5.4. Neo Hookean Material Model ... 16
2.5.5. Mooney Rivlin Material Model ... 16
2.5.6. Ogden Material Model ... 17
2.5.7. Yeoh Material Model ... 17
2.6. Mullins Effect ... 17
2.7. Adhesive Test [24] ... 19
2.7.1. Lap Shear Tests ... 20
2.7.2. Tension and Peel Tests ... 21
3. Method ... 23
3.1. Elasticity and Viscoelasticity ... 23
3.1.1. Elastic Material Data ... 23
3.1.2. Relaxation Test ... 23
3.1.3. Verification of Material Model... 26
3.2. Mullins effect ... 27
3.2.2. Material Constants in Mullins Effect ... 29
3.2.3. Numerical Setup ... 30
3.3. Cyclic Loading Test ... 32
3.3.1. Experimental Setup ... 32
3.3.2. Numerical Setup ... 35
4. Results ... 37
4.1. Elasticity and Viscoelasticity ... 37
4.2. Mullins Effect ... 41
4.3. Cyclic Loading Test ... 44
5. Discussion ... 49
5.1. General ... 49
5.2. Relaxation Test ... 49
5.3. Compression Test ... 50
5.4. Cyclic Loading Test ... 51
6. Conclusion ... 52 7. Further Work ... 53 7.1. Experiments ... 53 7.2. Material Model ... 53 7.3. Additional Aspects ... 54 Appendix ... 57
I. Relaxation Test: Reaction Forces - Time: ... 57
II. Relaxation Test: Iso-viscoelasticity vs. Hyper-viscoelasticity: ... 58
III. Relaxation Test: William-Landel-Ferry shift function vs. ANSYS linear interpolation ... 59
IV. Cyclic Compression Test: Reaction Force - Displacement ... 60
V. Cyclic Compression Test: Simulation Results vs. Linear Regression (Based on material data) . 61 VI. Cyclic Compression Test: Stress Plot of Three Boundary Conditions ... 62
VII. Cyclic Loading Test: 𝛿 = 1 𝑚𝑚 – Runout ... 63
VIII. Cyclic Loading Test: 𝛿 = 2 𝑚𝑚 – Unconnected Joint ... 64
IX. Cyclic Loading Test: 𝛿 = 2 𝑚𝑚 – Introduced Damage ... 65
1
1. Introduction
Ericsson is one of the leading providers of Information and Communication Technology and is also one of the key players in the commercial 5G network. Previous generations of the mobile network can be conclude with voice and SMS communications in 2G, web-browsing in 3G, and streaming at a higher speed of data and video in 4G [1], illustrated in Figure 1-1.
The demand for global data traffic is expected to increase exponentially. Thus, there is a great need for more efficient technology to support the world with higher data rates and spectrum utilization. New applications of e.g. virtual and augmented reality (VR and AR), 4K/8K video streaming, and Industry 4.0 require more advanced technology to provide higher bandwidth, greater capacity, better security, and lower latency. The transformation from 4G to 5G implies an evolutionary change for both individual consumers and multiple industries.[1]
Figure 1-1. Overview of 2G to 5G.
The coverage area of the 5G network is divided into areas represented by cells, each of which is served by a separate antenna. The 5G technology uses high-frequency millimetre wave (mmWave) network that requires a tighter base station grids to support outdoor telecommunication equipment. Consequently, it has become more common for outdoor telecommunication equipment to be installed in urban environments. This densification of equipment and the requirement of urban appearance force the development of smaller and more unobtrusive products. Simultaneously, the mmWave products require specific material that is transparent to electromagnetic radiation, which means that the equipment coverage has changed into a combination of polymer and metal components. One way of connecting materials of different properties and keeping the products size small is to use adhesive joints.
1.1. Background
2 eliminate the product size and retain the durability of the plastic-aluminium bonding. The radome and heatsink are thereby defined as the adherends, illustrated in Figure 1-2.
Adhesive has the properties that provide reliable bonding between dissimilar materials on account of preventing corrosion and accommodate for different thermal expansion of adherends. Different from the traditional mechanical joint, adhesive joint has the advantages of wide application range and positive influence on the load transmission paths, which results in lower stress gradient and a more uniformed stress field. [2]
Figure 1-2. Definition of adhesive and adherends.
The main ingredients of different adhesives are polymers with different material properties.[3] All adhesives have one characteristic in common, that is their sensitivity to temperature and humidity.[4] The outdoor RBS products are facing different environmental and weather challenges, including cyclic loading and fatigue damage caused by temperature changes; electronics corrosion in conjunction with ionic contamination caused by humidity variation. All these factors will affect the durability of adhesive joint and have significant impact on the life expectancy of equipment.
1.2. Aims
This thesis attempts to evaluate the life expectancy of adhesive joint used in the RBS products with dissimilar adherends. The aim of the project is to address the material model needed to understand and describe the material behaviour of adhesive joint during cyclic loading using experiments and numerical simulations.
The aims of the project are:
• identify and determine the material model of adhesive joint.
3
2. Theoretic Background
2.1. Adhesive
Adhesive joints have been used to connect two surfaces for many thousands of years, and the use of adhesives in primary structures has increased rapidly during the last decade. The increased use of adhesive joints instead of the traditional joins, e.g. screws, is due to their good properties such as corrosion-resistance, electrical and thermal insulation, easy and quick application etc.[5]
When working with adhesive it is important to understand its basic characteristics and the mechanisms that are creating joint effect. Surface science, which describes the physical and chemical phenomena at the interface of two surfaces, is one of the key properties of obtaining a suspected and durable adhesive joint. One guideline to be followed when choosing adhesive is to consider the surface energy of the adhesive and the adherend. In order to obtain a good bonding effect, the adherend should have higher surface energy that allows the adhesive to have good contact with the solid surface through a good wetting property. If the adherend has lower surface energy, then the wetting ability of the adhesive is very small. Consequently, the adhesive will keep its spherical shape when it comes into contact with the solid surface, as shown in Figure 2-1. However, if the adherend has a higher surface energy, the adhesive will have a better wetting capability and better bonding effect will be obtained.[3]
Figure 2-1. Adhesives with a) higher surface energy and b) lower surface energy than the surface tension of the adherend.
2.1.1. Moisture Sensitive
4
2.1.2. Temperature Sensitive
It is well known that the temperature variation affects the mechanical properties of polymers dramatically, due to the higher molecular mobility that is expected under elevated temperature.[7] Both long- and short-term exposure to high temperature usually leads to irreversible chemical and physical changes within adhesives, in which the bonding strength of adhesive is inversely proportional to the temperature.[4] The glass transition temperature 𝑇𝑔 for polymer materials defines the critical level of significant changes in mechanical properties. Polymers at temperature above 𝑇𝑔 have a rubbery behaviour and the matrix properties are permanently degraded, while polymers below 𝑇𝑔 have a glassy state. The polymer property
reduction is reversible upon dehydration. A significant loss is likely to occur at the temperature level around 𝑇𝑔− 20 ℃, which is considered as the limiting use temperature for the most
applications.[4,7] The thesis project focuses on the thermal impact on adhesives.
2.2. Stress State in Adhesive Layer
The stress state in adhesive joint layer can be defined as the Cauchy Stress Tensor, which describes the loading force and the loaded area in the spatial configuration. The stress tensor can be divided into two parts as
𝝈 = 𝝈𝑑𝑒𝑣 + 𝝈𝑣𝑜𝑙, (2.2-1)
where 𝝈𝑑𝑒𝑣 and 𝝈𝑣𝑜𝑙 denotes the deviatoric and volumetric stress tensor, respectively. Here, the deviatoric stress, also defined as isochoric stress, represents the constitutive information while the volumetric stress, also known as hydrostatic stress, enforces the incompressibility, illustrated in Figure 2-2. In particularly, the volumetric stress is defined as the mean of the three normal stresses as 𝜎𝑘𝑘 = 𝑡𝑟(𝝈) = 1 3(𝜎1+ 𝜎2+ 𝜎3), (2.2-2) 𝝈𝑣𝑜𝑙 = [ 𝜎𝑘𝑘 0 0 0 𝜎𝑘𝑘 0 0 0 𝜎𝑘𝑘 ], (2.2-3)
and it is capable to change the total volume of the material. The deviatoric stress is the Cauchy stress reduced by the volumetric part as
𝑆𝑖𝑗 = 𝜎𝑖𝑗 −1 3𝜎𝑘𝑘𝛿𝑖𝑗, (2.2-4) 𝝈𝑑𝑒𝑣 = [ 𝜎11 𝜎12 𝜎13 𝜎21 𝜎22 𝜎23 𝜎31 𝜎32 𝜎33] − [ 𝜎𝑘𝑘 0 0 0 𝜎𝑘𝑘 0 0 0 𝜎𝑘𝑘 ]. (2.2-5)
Here 𝜎𝑘𝑘 is the trace of the Cauchy stress, 𝑆𝑖𝑗 denotes the stress deviator, and 𝛿𝑖𝑗 is the
5
Figure 2-2. Volumetric and deviatoric stress effect.
2.3. Linear Viscoelasticity
When working with adhesive, isotropic linear viscoelasticity with elastic and viscous characteristics is adequate to be appliedaiming to describe the time-dependent response.[9] One approach to studying viscoelasticity is to perform an experiment where the material is exposed to a prescribed strain, which can be described by the step function as
𝜀(𝑡) = 𝜀0𝐻(𝑡), (2.3-1)
where the 𝐻(𝑡) denotes the Heaviside step function defined by
𝐻(𝑡) = {
0, 𝑖𝑓 𝑡 < 0 1/2, 𝑖𝑓 𝑡 = 0 1, 𝑖𝑓 𝑡 > 0
. (2.3-2)
The stress 𝜎(𝑡) can be measured as the response to the instantaneously applied strain, and the stress will decrease with time and approach a constant value asymptotically.[10] This stress response behaviour is shown in Figure 2-3 a), which is defined as the Stress Relaxation of viscoelastic material. The time it takes to reach the constant value and the constant value itself are dependent on the material properties at a given temperature. The stress response curve can be used to obtain the Stress Relaxation Modulus 𝐸(𝑡) which is defined by[9]
𝐸(𝑡) =𝜎(𝑡)
𝜀0 . (2.3-3)
6 The strain history can be decomposed into a sum of infinitesimal strain steps as
𝜀(𝑡) = ∑∞𝑖=1∆𝜀𝑖𝐻(𝑡 − 𝜏𝑖), (2.3-4)
in order to address the model predicting the stress response, where ∆𝜀𝑖 is the strain increment applied at time 𝜏𝑖. Thus, the stress response obtained with the linear superposition principle
becomes
𝜎(𝑡) = ∑∞𝑖=1∆𝜀𝑖𝐸(𝑡 − 𝜏𝑖), (2.3-5)
where 𝐸 is the stress relaxation modulus. When the number of strain increments reaches infinity, the summation expression can be further derived into an integral expression as [9]
𝜎(𝑡) = ∫ 𝐸(𝑡 − 𝜏)𝑑𝜀(𝑡) = ∫ 𝐸(𝑡 − 𝜏)𝑑𝜀(𝜏) 𝑑𝜏 𝑑𝜏 𝑡 −∞ 𝑡 −∞ . (2.3-6)
Similarly, when constant stress increments are applied, the corresponding strain response can be derived into 𝜀(𝑡) = ∫ 𝐷(𝑡 − 𝜏)𝑑𝜎(𝑡) = ∫ 𝐷(𝑡 − 𝜏)𝑑𝜎(𝜏) 𝑑𝜏 𝑑𝜏 𝑡 −∞ 𝑡 −∞ , (2.3-7)
where 𝐷 is the Creep compliance of linear viscoelastic material. The phenomenon is defined as Creep, as shown in Figure 2-3 b). The integrals in Equation (2.3-6) and (2.3-7) are defined by the Boltzmann Superposition Principle, which are the fundamental basis of linear viscoelastic material.[7]
The model description presented so far is based on one-dimensional problems, and needs to be extended to three-dimensional structures. It is convenient to divide the stress and strain into deviatoric and volumetric parts.[9] The stress response becomes
𝝈(𝑡) = ∫ 2𝜇(𝑡 − 𝜏)𝜺̇𝑑𝑒𝑣𝑑𝜏 𝑡
0 + ∫ 𝜅(𝑡 − 𝜏)𝜺̇𝑣𝑜𝑙𝑑𝜏 𝑡
0 , (2.3-8)
where 𝝈 is the Cauchy stress tensor, 𝜇 and 𝜅 are the relaxation shear modulus and bulk modulus, respectively. The strain tensor is divided into deviatoric 𝜺𝑑𝑒𝑣 and volumetric parts 𝜺𝑣𝑜𝑙, and the time derivate of the strain tensors are denoted by 𝜺̇𝑑𝑒𝑣 and 𝜺̇𝑣𝑜𝑙. For polymers, the volumetric relaxation is generally much smaller and less influential than the deviatoric relaxation. Thus, the volumetric contribution is negligible. If the material is nearly incompressible meaning 𝜅 ≫ 𝐸, as in the case of most rubber-like materials, the shear relaxation modulus can be further simplified to
𝜇(𝑡) ≈ 𝐸(𝑡)
3 . (2.3-9)
2.3.1. Stiffness-Compliance Relation
The fundamental basis of linear viscoelasticity is the Boltzmann Superposition principle, which describes the stress and strain response of a material. Laplace transform can be used to relate Equation (2.3-6) and (2.3-7) as
∫ 𝐷(𝑡 − 𝜏)𝐸(𝜏)𝑑𝜏 = ∫ 𝐸(𝑡 − 𝜏)𝐷(𝜏)𝑑𝜏 = 𝑡0𝑡 0𝑡 . (2.3-10) The explicit relation between material stiffness and compliance can be further expressed by
𝐷(𝑡) =sin (𝑚𝜋)
𝑚𝜋 1
7 where 𝑚 describes the slope of the creep compliance curve as function of log-time. In general, 𝑚 → 0 as the time approaches to zero or infinity (𝑡 → 0 or 𝑡 → ∞). In order to satisfy these two conditions, the explicit expression can be approximated by the L’Hopital’s Rule as
lim 𝑚→0 sin (𝑚𝜋) 𝑚𝜋 = 𝜋∙cos (𝑚𝜋) 𝜋 = cos(𝑚𝜋) = 1. (2.3-12)
Thus, the compliance and stiffness of a material are related to each other by 𝐷(𝑡) = 1
𝐸(𝑡) (2.3-13)
only when 𝑡 → 0 or 𝑡 → ∞. In the remainder of the time domain, the relation between compliance and the stiffness should satisfy the convolutional integral given in Equation (2.3-10).[7]
2.3.2. Cyclic Loading Response
Small-strain dynamic mechanical test under cyclic loading can be used to study the viscoelastic behaviour. The response to a sinusoidal applied strain given by
𝜀(𝑡) = {𝜀0sin(𝜔𝑡) , 𝑖𝑓 𝑡 ≥ 0
0, 𝑖𝑓 𝑡 < 0 (2.3-14) is considered, where 𝜀0 is the strain amplitude and 𝜔 is the angular frequency. The stress response with respect to time 𝑡 can be derived using Equation (2.3-6) as
𝜎(𝑡) = ∫ 𝐸(𝑠)𝜔𝜀0cos[𝜔(𝑡 − 𝑠)]ds ∞
0 , (2.3-15)
where 𝑠 = 𝑡 − 𝜏. The stress response to the sinusoidal strain will eventually reach steady state and become sinusoidal with the same angular frequency 𝜔, but the phase is delayed by angle 𝛿. The cyclic stress response is expressed by
𝜎(𝑡) = 𝜎0sin (𝜔𝑡 + 𝛿), (2.3-16) where 𝜎0 is the stress amplitude when the strain is applied immediately and the phase shift 𝛿 is between 0 and 𝜋/2. By using the trigonometric functions, the stress response can be further rewritten as a combination of an in-phase and an out-of-phase response as
𝜎(𝑡) = 𝜎0cos(𝛿) sin(𝜔𝑡) + 𝜎0sin(𝛿) cos(𝜔𝑡). (2.3-17)
Complex variables are usually used to describe the sinusoidal response of viscoelastic material, and the applied strain in Equation (2.3-14) can be expressed by
𝜀∗(𝑡) = {𝜀0𝑒𝑖𝜔𝑡, 𝑖𝑓 𝑡 ≥ 0
0, 𝑖𝑓 𝑡 < 0, (2.3-18) where 𝑖 is the solution of 𝑖2 = −1. The complex stress response is
𝜎∗(𝑡) = 𝜎
0𝑒𝑖(𝜔𝑡+𝛿), (2.3-19)
and can be rewritten as
𝜎∗ = 𝐺∗𝜀∗, (2.3-20)
where 𝐺∗ is the Complex Dynamix Modulus. The complex dynamic modulus is derived from the stress-strain response as
𝐺∗ =𝜎0 𝜀0𝑒 𝑖𝛿 =𝜎0 𝜀0cos(𝛿) + 𝑖 𝜎0 𝜀0sin (𝛿). (2.3-21)
The real part of 𝐺∗ is denoted by 𝐺′ and the imaginary part by 𝐺′′, and 𝐺∗ becomes
8 { 𝐺′=𝜎0 𝜀0cos(𝛿) 𝐺′′ = 𝜎0 𝜀0sin (𝛿) , (2.3-23)
where 𝐺′ is the Storage Modulus for measuring the energy stored and recovered in each cycle,
and 𝐺′′ is the Loss Modulus defining the energy dissipation caused by internal material damping.[12]
2.3.3. Rheological Operators
Viscoelastic material is a combination of elastic and viscous properties, in which the stress-strain relation is time-dependent. Two fundamental element types, a spring and a dashpot shown in Figure 2-4, are used to describe elastic and viscous behaviour respectively. The constitutive equation of the spring and dashpot model can be mathematically expressed by using the Rheological Operators.[13] The elastic component is described by
𝜎 = 𝐸𝜀 or 𝑑𝜀 𝑑𝑡 = 1 𝐸 𝑑𝜎 𝑑𝑡, (2.3-24)
where 𝐸 is the elastic modulus. The viscous component is expressed by 𝜎 = 𝜂𝑑𝜀
𝑑𝑡, (2.3-25)
where 𝜂 is the viscosity of the material.[13]
Figure 2-4. a) Spring element b) Dashpot element.
Two simple models used to represent linear viscoelastic material are Kelvin-Voigt and Maxwell models, presented in Figure 2-5.[12] The Kelvin-Voigt model consists of a parallel-connected
elastic spring and viscous dashpot. Since the two elements are arranged in parallel, the model is constrained by the strain in each element. Thus, the total stress is equal to the sum of the stress of the spring and dashpot as
𝜎 = 𝐸𝜀 + 𝜂𝑑𝜀
𝑑𝑡. (2.3-26)
For dynamic response, the storage modulus 𝐺′ and the loss modulus 𝐺′′ are defined by 𝐺′= 𝐸 and 𝐺′′ = 𝜂𝜔. (2.3-27) The Maxwell model is defined with a series-connected spring and dashpot, and the model is constrained with identical stress in each element. Thus, the total strain is equal to the sum of the strain of the spring and dashpot that
𝑑𝜀 𝑑𝑡= 𝜎 𝜂+ 1 𝐸 𝑑𝜎 𝑑𝑡. (2.3-28)
The storage and loss modulus become 𝐺′= 𝐸(𝜂𝜔)2
𝐸2+(𝜂𝜔)2 and 𝐺
′′ = 𝐸2𝜂𝜔
9
Figure 2-5. a) Kelvin-Voigt model b) Maxwell model.
2.3.4. Generalized Maxwell Modell
Generalized Maxwell model is commonly used to describe linear viscoelastic material, that
consists of several parallel-connected Maxwell models together with a single spring element, shown in Figure 2-6. Consider a uniaxial stress relaxation where a constant strain 𝜀0 is applied, the Equation (2.3-3) is used to determine the relaxation modulus. The asymptotic limits of the relaxation modulus are defined by
{ 𝐸(𝑡 = 0) =𝜎0 𝜀0 = 𝐸0 𝐸(𝑡 → ∞) =𝜎∞ 𝜀0 = 𝐸∞ , (2.3-30)
and the stress relaxation response is illustrated in Figure 2-7. The total stress response can be expressed as a sum of the stress of the spring and the Maxwell models as
𝜎(𝑡) = 𝜎∞+ ∑𝑛𝑖=1𝜎𝑖𝑒−𝑡/𝜏𝑖 = 𝜀0[𝐸∞+ ∑𝑛𝑖=1𝐸𝑖𝑒−𝑡/𝜏𝑖], (2.3-31)
where 𝜏𝑖 is the relaxation time for each element defined by 𝜏𝑖 = 𝜂𝑖/𝐸𝑖. The definition of the
time constant is that the stress response becomes 1/𝑒 ≈ 37 % when 𝑡 = 𝜏𝑖.[14]
10
Figure 2-7. Stress relaxation response.
2.3.5. Prony Series
Prony series is commonly used to describe the Generalized Maxwell model. The relaxation
modulus represented by the Prony series becomes
𝐸(𝑡) = 𝐸∞+ ∑𝑛 𝐸𝑖𝑒−𝑡/𝜏𝑖
𝑖=1 , (2.3-32)
which can be further normalized. The normalized relative modulus 𝛼𝑖 is 𝛼𝑖 =
𝐸𝑖
𝐸0, (2.3-33)
and the relaxation modulus is rewritten as
𝐸(𝑡) = 𝐸0+ (𝛼∞+ ∑𝑛 𝛼𝑖𝑒−𝑡/𝜏𝑖
𝑖=1 ). (2.3-34)
Here, the sum of the relative modulus should be less than or equal to 1.[11]
The descriptions obtained above can be further extended to three-dimensional isotropic viscoelastic material, in which the stress tensor is divided into deviatoric and volumetric part as presented in Equation (2.3-8). In the hereditary integral used in Equation (2.3-8), the previous history of the material is considered by including the shear 𝜇(𝑡) and bulk relaxation modulus 𝜅(𝑡). For an isotropic material, the shear and bulk relaxation modulus can be written as
𝜇(𝑡) = 𝐸(𝑡)
2(1+𝑣) (2.3-35)
𝜅(𝑡) = 𝐸(𝑡)
3(1−2𝑣), (2.3-36)
where the Poisson’s ratio for incompressible material is 𝑣 ≈ 0.5 and implies 𝜅 → ∞.[15]
2.4. Time-Temperature Superposition
Temperature dependence of viscoelastic material can be described by Time-Temperature
Superposition (TTS). Increased temperature implies a more rapid molecular diffusional motion
in viscoelastic material.[7] Consequently, the relaxation and creep will be reduced when studying the material behaviour at higher temperature. As shown in Figure 2-8, the temperature dependence can be obtained by using TTS to scale the response time.
11 relaxation modulus curves at different temperatures can be used to establish a master curve and determine the corresponding shift factors. A schematic illustration is shown in Figure 2-9.[9] The horizontal time shift of relaxation modulus is described by[16]
log(𝑡) − log(𝑡𝑟𝑒𝑓) = log (𝑎𝑇), (2.4-1) where 𝑡 is the time at temperature 𝑇, and 𝑡𝑟𝑒𝑓 is the time at the reference temperature 𝑇𝑟𝑒𝑓. The shift factor can be further rewritten as
𝑎𝑇 = 𝑡
𝑡𝑟𝑒𝑓, (2.4-2)
and has the properties of
{
𝑇 < 𝑇𝑟𝑒𝑓, 𝑎𝑇 > 1 𝑇 > 𝑇𝑟𝑒𝑓, 𝑎𝑇 < 1 𝑇 = 𝑇𝑟𝑒𝑓, 𝑎𝑇 = 1
. (2.4-3)
Hence, the behaviour at temperature 𝑇 becomes predictable and thus is similar to the behaviour at the reference temperature 𝑇𝑟𝑒𝑓 when the time is accelerated by the factor 𝑎𝑇. The relaxation modulus at temperature 𝑇 is expressed by[13]
𝐺(𝑡, 𝑇) = 𝐺(𝑎𝑇𝑡, 𝑇𝑟𝑒𝑓). (2.4-4) Another common application of TTS is to use experimental data obtained at several temperatures to estimate the stress-strain response at a longer time period.[9] Many applications of viscoelastic material are designed to have a long life expectancy. Thus, it is inconvenient to conduct experiments to study the material behaviour for years. In fact, it is more practical to apply TTS and predict the development of material behaviour in a longer time range.
12
Figure 2-9. A schematic illustration of a master curve and the interpolation of the shift factor.
2.4.1. Williams-Landel-Ferry Shift Function
Several empirical models are describing the TTS, and one of the most used formulations of the shift factor for polymer material is the William-Landel-Ferry Shift Function (WLF). The WLF shift function is expressed by
log(𝑎𝑇) = − 𝐶1(𝑇−𝑇𝑟𝑒𝑓)
𝐶2+(𝑇−𝑇𝑟𝑒𝑓), (2.4-5)
where 𝐶1 and 𝐶2 are the empirical constants and have the unit of [-] and [℃] respectively.[13] Equation (2.4-5) can be rewritten as a linear function of
{ 𝑦 = 𝑘𝑥 + 𝑚 1 log (𝑎𝑇)= − 𝐶2 𝐶1 1 (𝑇−𝑇𝑟𝑒𝑓)− 1 𝐶1 , (2.4-6)
where the parameters are defined by
{ 𝑦 = 1 log (𝑎𝑇) 𝑥 =(𝑇−𝑇1 𝑟𝑒𝑓) 𝑘 = −𝐶2 𝐶1 𝑚 = − 1 𝐶1 . (2.4-7)
This linear expression together with the obtained shift factors can be used to estimate the constants 𝐶1 and 𝐶2.[14]
The WLF shift function is considered as a general equation, which is used to determine shift factors of all amorphous glass-forming polymers above 𝑇𝑔. The definition of the glass transition
13 polymers change from a relatively hard and brittle state to a more viscous and ductile state with the increase of temperature. However, shift factors below 𝑇𝑔 can be estimated by using the
Arrhenius Activation Energy Equation.[14]
2.5. Hyperelasticity
Hyperelastic material tends to give an elastic response when exposed to large deformations and shows nonlinear material behaviour at large shape change. It has characteristics similar to ideal elastic solid, which means that when undergoing adiabatic or isothermal deformation, the stress only depends on the current strain and is independent of the loading history or loading rate. Meanwhile, the large deformation of the material is reversible during the underloading process.
Another characteristic property of hyperelastic material is the resistance to volume change, i.e. the material has very high bulk modulus and becomes nearly incompressible. Additionally, the shear modulus of hyperelastic material is temperature-dependent and becomes stiffer if heat is introduced.
Hyperelastic material model is the simplest model representation that is able to capture micro-mechanisms that drive the deformation behaviour of polymers. Four of the most commonly used hyperelastic material models, Neo Hookean, Mooney Rivlin, Ogden and Yeoh, are presented in this section.
The constitutive law describing the stress-strain relation of hyperelastic material can be derived from the Helmholtz free energy. Furthermore, an essential tool for describing hyperelastic model is Continuum Mechanics.[9]
2.5.1. Continuum Mechanics Foundations
The mathematical definition of a continuum is an approximation of a material, in which the mass of the material ∆𝑀 is continuously distributed on an arbitrarily small volume ∆𝑉 such that lim ∆𝑉→0 ∆𝑀 ∆𝑉 = 𝑑𝑀 𝑑𝑉 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡. (2.5-1)
The constant is further defined as the density of a continuum, which is denoted by 𝜌.[14]
In continuum mechanics theory, two different configurations of continuum are observed, as shown in Figure 2-10. Referential Configuration contains the undeformed continuum in the reference space and time, and Spatial Configuration with the continuum whose physical properties are changed when moving in space and time. To clarify, the referential configuration is described in uppercase letters and the spatial configuration is represented in lowercase letters. In continuum mechanics, the motion of a continuum in the Lagrangian description is represented by a mapping function 𝝌 given by
14 where 𝑡 is the time, 𝑿 and 𝒙 denotes the position of the continuum in the referential and spatial configuration, respectively.[12]
Figure 2-10. Motion in the Lagrange Description. [8]
The Cauchy stress tensor presented in section 2.3. is a symmetric stress tensor defined by
𝝈 = [
𝜎11 𝜎12 𝜎13 𝜎21 𝜎22 𝜎23
𝜎31 𝜎32 𝜎33], (2.5-3) where 𝜎𝑖𝑗 = 𝜎𝑗𝑖. The Cauchy stress is commonly known as the True Stress, in which the load force and the considered surface are defined in the spatial configuration. However, in some circumstances it is more convenient to consider stress state defined in the referential configuration, which is expressed by the Second Piola Kirchhoff Stress Tensor 𝑺. Or sometimes, stress state depending on both referential and spatial configurations could be of interest, which is expressed by the First Piola Kirchhoff Stress Tensor 𝑷.
The relation between the first and second Piola Kirchhoff stress and the Cauchy stress can be defined by using the Deformation Gradient 𝑭 and the Volume Ratio 𝐽.[17] The deformation
gradient is expressed as the gradient of the motion of the continuum as
𝑭 = 𝜕𝒙 𝜕𝑿= [ 𝜕𝑥1 𝜕𝑋1 𝜕𝑥1 𝜕𝑋2 𝜕𝑥1 𝜕𝑋3 𝜕𝑥2 𝜕𝑋1 𝜕𝑥2 𝜕𝑋2 𝜕𝑥2 𝜕𝑋3 𝜕𝑥3 𝜕𝑋1 𝜕𝑥3 𝜕𝑋2 𝜕𝑥3 𝜕𝑋3] , (2.5-4)
and the volume ratio is defined by the determinant of the deformation gradient as
𝐽 = det 𝑭. (2.5-5)
Consequently, the first Piola Kirchhoff stress can be expressed in terms of the Cauchy stress as
𝑷 = 𝐽𝝈𝑭−𝑇, (2.5-6)
and the second Piola Kirchhoff stress becomes
𝑺 = 𝑭−1𝑷 = 𝐽𝑭−1𝝈𝑭−𝑇. (2.5-7)
2.5.2. Constitutive Law
15 strain energy function is a scalar function of the deformation gradient, which describes the strain energy per unit volume. Thus, the constitutive law can be expressed by[12]
𝑷 = 𝑭∂Ψ ∂𝐄, (2.5-8) 𝑺 =∂Ψ ∂𝐄, (2.5-9) 𝝈 =1 𝐽𝑭 ∂Ψ ∂𝐄𝑭 𝑇, (2.5-10)
where 𝑬 is the Green Lagrange Strain defined by 𝑬 = 1
2(𝑪 − 𝑰), (2.5-11)
and 𝑪 is the Right Cauchy Green Strain defined by
𝑪 = 𝑭𝑇𝑭. (2.5-12)
Consequently, the stresses can also be expressed as a derivate with respect to 𝑪 as[17]
𝑷 = 2𝑭∂Ψ ∂𝐂, (2.5-13) 𝑺 = 2∂Ψ ∂𝐂, (2.5-14) 𝝈 =2 𝐽𝑭 ∂Ψ ∂𝐂𝑭 𝑇, (2.5-15)
2.5.3. Strain Energy Function
The strain energy function is commonly expressed as a function of principle stretches 𝜆𝑖 or
strain invariants 𝐼𝑖 as
Ψ = Ψ(𝜆1, 𝜆2, 𝜆3) = Ψ(𝐼1, 𝐼2, 𝐼3). (2.5-16)
Here, it is common to use the invariants of 𝑪 that are defined by[17]
𝐼1 = 𝑡𝑟(𝑪) = 𝜆12+ 𝜆22+ 𝜆32, (2.5-17) 𝐼2 = 1 2[𝑡𝑟(𝑪) 2− 𝑡𝑟(𝑪2)] = 𝜆 1 2𝜆 2 2+ 𝜆 2 2𝜆 3 2+ 𝜆 3 2𝜆 12, (2.5-18) 𝐼3 = det(𝑪) = 𝜆12𝜆 2 2𝜆 3 2. (2.5-19)
For incompressible material, it is convenient to divide the strain energy function into deviatoric and volumetric part. The volumetric part is constrained by incompressibility that is defined by the volume ratio. For an incompressible material, the volume should be kept constant. Thus, the volume ratio becomes
𝐽 = 𝜆1𝜆2𝜆3 = 1. (2.5-20)
Consequently, the strain energy function becomes
Ψ = Ψ𝑑𝑒𝑣(𝜆1, 𝜆2, 𝜆3) + Ψ𝑣𝑜𝑙(J), (2.5-21)
Ψ = Ψ𝑑𝑒𝑣(𝐼1, 𝐼2) + Ψ𝑣𝑜𝑙(J). (2.5-22) However, the volumetric strain energy function is equal to zero for incompressible material. Conversely, if the material is nearly incompressible, the volumetric strain energy includes a compressibility parameter 𝐷𝑖.[15]
The polynomial expression of the strain energy function based on the first and second invariants of the Left Cauchy Green Deformation Tensor can be derived as
16 where 𝐶𝑖𝑗 are the material constants, and the invariants are defined by
𝐼̅1 = 𝐽−2/3𝐼1, (2.5-24)
𝐼̅2 = 𝐽−4/3𝐼2. (2.5.25) The initial shear and bulk modulus are included in the polynomial representation through
𝜇0 = 2(𝐶10+ 𝐶01), (2.5-26) 𝜅0 = 2
𝐷1. (2.5-27)
However, if an incompressible material is considered, the second term in Equation (2.5-23) becomes zero.[18]
The general guideline of the order of the polynomial representation depends on the number of inflection points captured in engineering stress-strain curve, illustrated in Figure 2-11. When large deformation is involved, more data and thus a larger range of stress-strain curve should be considered. General guidelines with polynomial expression of order 𝑁 = 2 or 𝑁 = 3 should be used for strains up to 100 − 300 %.[18]
Figure 2-11. Engineering stress-strain curve containing 1, 2, and 3 inflection points.
2.5.4. Neo Hookean Material Model
The Neo Hookean material model is obtained by considering the polynomial representation of first order, i.e. 𝑁 = 1, and 𝐶10= 𝐶11= 0. Together with Equation (2.5-26), the Neo Hookean model is described by
Ψ𝑁𝐻 = 𝜇0
2 (𝐼̅1− 3) + 1
𝐷1(𝐽 − 1). (2.5-28)
Based on statistical thermodynamics, the Neo Hookean model describes linear stress-strain relation at the initial deformation stage, and becomes nonlinear after a certain point. The Neo Hookean model is one of the simple models that describes hyperelastic material. It has the limit of inaccurate prediction at large strain, thus it is commonly used for moderate strain deformations with tension up to 30 − 40 % and pure shear up to 70 − 90 %.[18, 19]
2.5.5. Mooney Rivlin Material Model
17 Ψ𝑀𝑅 = 𝐶10(𝐼̅1− 3) + 𝐶01(𝐼̅2− 3) + 1
𝐷1(𝐽 − 1). (2.5-29)
The Mooney Rivlin model allows a simple definition of quasi-static temperature dependence and is sufficient to capture the temperature-dependent shear modulus. It is capable of predicting hyperelastic material where the maximum local strain is about 200 %.[18, 19]
2.5.6. Ogden Material Model
The Ogden material model is expressed with principle stretches instead of strain invariants as Ψ𝑂 = ∑ 𝜇𝑖 𝛼𝑖(𝜆1 𝛼𝑖+ 𝜆 2 𝛼𝑖+ 𝜆 3 𝛼𝑖− 3) + 𝑁 𝑖=1 ∑ 1 𝐷𝑖(𝐽 − 1) 2𝑖 𝑁 𝑖=1 , (2.5-30)
where 𝜇𝑖 and 𝛼𝑖 are empirically determined material constants. The initial shear modulus is included in the strain energy function as
𝜇0 = 1
2∑ 𝜇𝑖𝛼𝑖 𝑁
𝑖=1 . (2.5-31)
Thus, the Ogden model can be derived as polynomial expressions with different orders. If 𝑁 = 1, then the Neo Hookean model is obtained with
{𝜇𝛼1 = 𝜇0
1 = 2 , (2.5-32)
and the two-term Mooney Rivlin is obtained with { 𝜇1 = 2𝐶10
𝜇2 = −2𝐶01 𝑎𝑛𝑑 {
𝛼1 = 2
𝛼2 = −2. (2.5-33) Because the Ogden model uses principle stretches instead of invariants, it has the advantage that it can directly use the obtained test data. For 𝑁 = 3, the Ogden model with six parameters is obtained, which gives good agreement for tension strain up to 700 %. However, it requires huge experimental database to maintain its accuracy.[15, 18, 19]
2.5.7. Yeoh Material Model
The Yeoh material model is based on the first invariant and the volume ratio as Ψ𝑌 = ∑ 𝐶𝑖0(𝐼̅1− 3)𝑖 + ∑ 1 𝐷𝑖(𝐽 − 1) 2𝑖 𝑁 𝑖=1 𝑁 𝑖=1 , (2.5-34)
since the earlier studies have shown that the change of strain energy is less sensitive to the change in the second invariant than to the first one. Also, it has been shown that better prediction is obtained when ignoring the second invariant if limited experimental data is available. The initial shear modulus is related to the strain energy function through
𝜇0 = 2𝐶10. (2.5-35)
The Yeoh model can be derived into polynomial expressions with different orders. If 𝑁 = 1, the expression equals the Neo Hookean model. However, the third order Yeoh model is usually used to describe large strain deformations.[15, 19]
2.6. Mullins Effect
18 virgin material. The behaviour is commonly known as Mullins Effect.[20] The Mullins effect
describes how the link between the reinforcement particles and the rubber matrix undergo progressive breaks, and it also defines the structural changes of the rubber matrix itself.[21] An example of stress softening in uniaxial tension is presented in Figure 2-12, and it gives a schematic illustration of stress-stretch curve under loading-unloading. One should be aware that Figure 2-12 is an ideal representation of the Mullins effect. The recorded stress-stretch relation from physical experiments may differ from the ideal case due to e.g. plastic deformation or the presence of residual stresses. The primary load curve without any softening is shown in yellow and the 1st unloading process is shown in red. When the material is reloaded, the loading curve will follow the red curve again. If the applied load exceeds the initiation point of the 1st unloading, then the yellow load curve will be followed. Similarly, the 2nd un- and
reloading follow the green curve, and when larger load is applied, the stress-stretch curve returns to the yellow curve. Consequently, for the same stretch, the stress softening appears in the unloaded part where the stress value is smaller than the corresponding stress on the primary loaded part.
Figure 2-12. Schematic reloading curve accounting for the Mullins effect.
The Ogden-Roxburgh model is a simple pseudo-elastic phenomenological model commonly used when addressing the Mullins effect. The material parameters in the model can be determined under simple test conditions including uniaxial tension/compression test and shear test.[21] The model is based on the theory and assumption of incompressible isotropic elastic material. The general Ogden-Roxburgh model is a modification of the standard thermodynamic formulation of hyperelastic material, which is expressed by
19 becomes 𝜂 = 1. On the contrary, an active damage variable corresponds to material softening, and its limit is defined by 0 < 𝜂 ≤ 1. The last term in Equation (2.6-1) refers to the damage function, which is defined by[22]
𝜙(𝜂 = 1) = 0, (2.6-2)
−𝜙′(𝜂) = Ψ̃ (𝐹
𝑖𝑗). (2.6-3)
Consequently, the softening behaviour of the material can be described by the stress response as
𝝈 = 𝜂𝝈̃, (2.6-4)
where 𝝈̃ is the stress state of the virgin material, i.e. with inactive damage variable 𝜂 = 1.[23]
Based on the assumption of material incompressibility, a modified Ogden-Roxburgh model is defined through the damage variable as
𝜂 = 1 −1
𝑟erf ( Ψ𝑚−Ψ̃
𝑚+𝛽Ψ𝑚), (2.6-5)
where 𝑟, 𝛽 and 𝑚 are positive material parameters, erf(∙) denotes the error function and Ψ𝑚 is the strain energy corresponds to the initiation point of unloading, represented with magenta stars in Figure 2-12. The material parameters are all unit-free constants except 𝑚, which has the dimension of energy density. Although the constants can be zero, 𝛽 and 𝑚 cannot be zero simultaneously.[21, 23]
2.7. Adhesive Test [24]
Compared with the wide use of adhesives today, there are few available and validated material data of adhesives. Therefore, the most common situation is that it is necessary to determine the material data for the adhesive under considered through experiments. The test methods can be divided into two categories; Bulk Specimen Tests and Joint Specimen Tests. The bulk specimen is completely made of the material under consideration, and many bulk specimens of adhesive can be obtained through casting. The bulk specimen test uses tensile, shear and compression data to obtain the yield and flow parameters. On the contrary, the joint specimen test is carried out by testing a structure of adherends connected by adhesive, and it provides validation of multi-axial yield prediction and failure criteria.
20
2.7.1. Lap Shear Tests
Lap shear test is the most common joint test method to determine the mechanical properties of adhesives. There are several different types of lap shear tests, such as Simple Lap Shear, Double
Lap Shear, Lap Strap and Double Lap Strap, as illustrated in Figure 2-13. The quantity
recorded in the shear tests is the average shear stress defined by the applied force divided by the overlapped bonding area. However, failure could also be defined by the yield stress, the maximum load point, the initiation of failure i.e. at the first load drop, or the point where catastrophic failure occurs, as shown in Figure 2-14.
Simple lap shear is widely used because of its low cost and easy manufacture. The test can be carried out with standard tension machine with suitable self-aligning clamp pairs to fix the specimen. Consequently, the long axis of the specimen will coincide with the loading direction passing through the centre line of the fixture assembly. It should be noted that the results carried out from the simple lap shear test are highly dependent on the mechanical stiffness of adherends. Because the eccentricity of load path may lead to out-of-plane bending, resulting in high peel stresses and non-uniformed shear stress in the adhesive layer. However, this could be reduced by increasing the modulus or thickness of the adherends, or by considering the use of double lap shear or lap trap specimens.
Double lap shear is symmetrical about the mid-plane of the specimen and hence can provide load eccentricity. Therefore, compared with the equivalent single lap shear test, the bond rotation and peeling stress are significantly reduced.
21
Figure 2-14. The stress-strain curve with the differently defined failure points.[24]
2.7.2. Tension and Peel Tests
Scarf joint uses relatively thick adherends that can be easily manufactured from pre-tapered adherends, e.g. from bars that are cut or sparked eroded at a desired scarf angle 𝜃. As shown in Figure 2-15, 𝜃 is the angle between the axis of adhesive layer and the axis of adherends that should be in the range of 0° < 𝜃 < 90°. The load applied at the end of the adherends will result in a mixed mode of normal and shear stress. The ratio between normal 𝜎 and shear stress 𝜏 within the bonding line can be controlled by varying the scarf angle as
𝜏
𝜎 = 𝑡𝑎𝑛𝜃, (2.7-1)
and it can be used to determine the yield and failure criteria. Ideally, when analysing stronger joints, the scarf angle should be set as large as possible.
22
Figure 2-15. Scarf joint.
23
3. Method
3.1. Elasticity and Viscoelasticity
The adhesive under consideration is a one-component adhesive based on silane-modified polymer. The time of skin formation and curing depends on humidity, temperature and joint depth. When the adhesive joint is completely cured, it will be transformed into an elastic solid. The in-service temperature of the adhesive ranges from −40 °C to 100 °C. Because the adhesive under consideration is polymer-based, the material model is assumed to be
Hyper-Viscoelastic.
3.1.1. Elastic Material Data
The temperature-dependent material data describing the elastic behaviour of the adhesive can be expressed by Young’s modulus 𝐸 and Poisson’s ratio 𝑣. The bulk properties of the material are provided by the supplier and are listed in Table 1. According to the supplier, the Poisson’s ratio is constant over temperature. Polymer material is generally assumed to be incompressible, therefore the Poisson’s ratio should be 𝑣 → 0.5. However, the Poisson’s ratio is set to 0.49 in the following analyses to avoid possible numerical instabilities.
The Neo Hookean model is the simplest hyperelastic material model and can be directly derived from the isoelastic material parameters. The two Neo Hookean parameters in the constitutive law presented by Equation (2.5-28) can be expressed by the initial shear and bulk modulus shown in Equation (2.5-26) and (2.5-27). Additionally, the shear and bulk modulus can be expressed by the Young’s modulus and Poisson’s ratio with Equation 35) and (2.3-36).
Table 1. Material data provided by the supplier.
Temperature [̊C] −𝟒𝟎 −𝟐𝟎 𝟎 𝟐𝟑 𝟓𝟎 𝟖𝟎 𝟏𝟎𝟓 Youngs Modulus [MPa] 14.4 3.2 2.5 2.2 2.2 1.8 1.7
Tensile Strength [MPa] 8.2 4.7 3.8 3.2 2.9 2.6 2.4 Elongation at tensile strength [%] 266 502 495 339 242 193 178
Shear Modulus [MPa] 4.83 1.07 0.84 0.74 0.74 0.60 0.57 Bulk Modulus[MPa] 240 53.3 41.7 36.7 36.7 30.0 28.3
3.1.2. Relaxation Test
24 The specimens were prepared by fixing two overlapped plates on a plastic sheet at a predetermined distance to ensure that the specimens can be demoulded after curing. Thereafter, a tape was used to form a barrier to avoid spread of the adhesive. The manufacturing process is illustrated in Figure 3-2. However, this arrangement requires several supports to hold the plates, which may be unavailable when several specimens need to be prepared at the same time. Thus, an alternative setup using double-sided tape to fix the two plates was used, as shown in Figure 3-3. The rest of the procedure was the same as the previous setup. Thereafter, according to the supplier's suggestion, the adhesive was filled in the eliminated space, and cured within at least 72 hours. After curing, the adhesive abundances was removed. Figure 3-4 presents the specimens in the curing process, which contained adhesive abundances that was removed afterwards. Figure 3-5 shows the fully cured and cleaned specimens, in which the voids appeared when filling the adhesive.
Figure 3-1. Simple lap shear test specimen design.
25
Figure 3-3. Simple lap shear test specimen curing process.
Figure 3-4. Fully-cured and cleaned specimen.
The relaxation test was performed in the Solid Mechanics laboratory at Royal Institute of
Technology (KTH). The testing machine for performing the relaxation test is shown in Figure
3-5, which includes a tension machine and a temperature chamber surrounding the test specimen in order to control the temperature. The predetermined boundary conditions were one fixed end that does not allow any movement, and the other one was loaded with prescribed displacement.
The equipment used can record the reaction force as a function of time, and the sampling frequency was 0.022 Hz, which was once every 45 seconds. The recorded reaction force was converted to shear stress via
𝜏(𝑡) =𝐹(𝑡)
𝑙𝑤, (3.1-1)
which was further manipulated to obtain the desired relaxation shear modulus by using Equation (2.3-3). The relaxation shear modulus is defined by
𝐺(𝑡) = 𝜏(𝑡)
𝛾 , (3.1-2)
where 𝛾 is the engineering shear strain illustrated in Figure 3-6 and defined by 𝛾 = 𝛿0
ℎ. (3.1-3)
26 The relaxation shear modulus at the five temperatures were plotted on a log-log-scale and shifted toward the reference temperature to build a master curve. Thereafter, the two experimental parameters 𝐶1 and 𝐶2 in WLF shift function was determined.
Figure 3-5. The machine setup of the simple lap shear test.
Figure 3-6. Definition of the engineering shear strain.
3.1.3. Verification of Material Model
The experiment was reconstructed with a Finite Element Model (FEM) utilizing the obtained material descriptions to check the accuracy of the experiment results. The commercial software
ANSYS Mechanical was used to perform numerical simulation using static analyses. The model
27
Figure 3-7. The setup pf the numerical simulation.
Two possible material models were considered to understand the adhesive behaviour. The viscose behaviour was always described by the Generalized Maxwell model, while the elastic behaviour was evaluated between Isotropic linear elasticity and Neo Hookean hyperelasticity. The two combinations, iso- and hyper-viscoelastic material model, were compared with the test results to find the model that best capture the adhesive behaviour and can be used for further analyses.
Numerical simulations aim to predict the material behaviours at unmeasured temperatures were carried out in two different ways. The first method was to introduce the WLF shift function into the material model with the two constants 𝐶1 and 𝐶2. The second method was to consider the build-in linear interpolation in ANSYS, in which the program linearly interpolated the material behaviours at the measured temperatures to predict the behaviours at unmeasured temperatures. Both methods were evaluated, and the results will be discussed in Section 5.2.
3.2. Mullins effect
Simple conditioned tests, such as uniaxial tension or compression, can be used to determine the material parameters 𝑟, 𝛽 and 𝑚. The classic dog-bone tensile specimen is a commonly used bulk specimen for determining the material properties. However, as described in Section 2.7, the preparation of dog-bone specimen may be time- and cost-consuming. Instead, a tension test with joint specimen can be considered. In the uniaxial tension test, the adhesive joint undergoes large deformation and can delaminate from the adherends, which results in adhesive failure. Therefore, a simple uniaxial compression test with cuboid bulk specimen was used to determine the softening behaviour of the material.
3.2.1. Experimental Setup
Cuboid specimens with a dimension of 2 × 2 × 1.5 cm3 were prepared through casting. The
28 one surface was in contact with air for curing, the specimens were rested for more than one week in order for the adhesive to be fully cured.
Figure 3-8. Manufacturing of the compression test sample.
The compression test was carried out at RT in Ericsson’s laboratory with a tension/compression machine. As presented in Figure 3-9, the boundary condition was to fix the bottom of the cuboid on a printing paper. The load was applied through a solid pressing against the top surface with prescribed displacement with the velocity of 1mm/min. A graphic description of the applied load is shown in Figure 3-10. Reaction force and displacement were obtained from the test, which were converted to principal stress and stretch. The conversion from force-displacement to stress-stretch is based on the assumption that a homogeneous and incompressible cuboid was considered, which will affect the setup of numerical simulation in Section 3.2.3.
29
Figure 3-10. Applied load in term of strain vs. time.
3.2.2. Material Constants in Mullins Effect
Under a uniaxial compression test, the incompressible structure with homogeneous stress and strain state has the principal stress state of 𝜎1 = 𝜎 and 𝜎2 = 𝜎3 = 0, which results in the
principal stretch of 𝜆1 = 𝜆 and 𝜆2 = 𝜆3 = 𝜆−1/2. Based on Equation (2.5-20), the strain energy function of a specimen under uniaxial loading is expressed by
Ψ(𝐹𝑖𝑗, 𝜂) = Ψ(𝜆1, 𝜆2, 𝜆3, 𝜂) = Ψ(𝜆, 𝜆−1/2, 𝜂). (3.2-1)
The principal Cauchy stress is given by
𝜎𝑖 = 𝜆𝑖𝜕Ψ(𝜆1,𝜆2,𝜆3,𝜂)
𝜕𝜆𝑖 − 𝑝, (3.2-2)
where the index 𝑖 = 1, 2, 3 and 𝑝 is the hydrostatic pressure. The unknown hydrostatic pressure 𝑝 can be eliminated by considering the subtraction between the principal stresses. The subtraction of stresses under the uniaxial test gives
{𝜎1− 𝜎2 = 𝜎1 = 𝜆1
𝜕Ψ 𝜕𝜆1
𝜎2− 𝜎3 = 0
. (3.2-3)
The stress-stretch relation derived from the hyperelastic strain energy and the experimental data from the compression test was used to determine the material parameters in the modified Ogden-Roxburgh model. First, a nonlinear least-squares best-fit was carried out to describe the primary loading curve based on the strain energy function Ψ̃ . The next step was to use regression to find a suitable 𝑟, 𝛽 and 𝑚, which describe the un-/reloading curves based on Equation (2.6-4) and (2.6-5). In the equations, 𝝈 is the stress state on the un-/reloading part and 𝝈̃ is the stress state on the primary loading part at the same stretch.[19]
30 Ψ𝑁𝐻 = 𝜇0 2 (𝜆1 2 + 𝜆 2 2+ 𝜆 1 −2𝜆 2 −2− 3). (3.2-4)
Accounting for uniaxial loading, Equation (3.2-4) can be further reduced to Ψ𝑁𝐻 = 𝜇0
2 (𝜆
2 + 2𝜆−1− 3). (3.2-5)
Furthermore, the first principal stress can be derived from the strain energy function as
𝜎1 = 𝜇0(𝜆2− 𝜆−1), (3.2-6)
together with Equation (2.6-4) and (2.6-5) the stress state under the primary loading is
𝜎̃1 = 𝜇0(𝜆2− 𝜆−1) (3.2-7)
and under the un-/reloading is
𝜎1 = 𝜂𝜎̃1 (3.2-8)
where 𝜂 is the damage variable described by Equation (2.6-5). The material bulk modulus 𝜇0
was determined by the best-fit of the primary load curve with Equation (3.2-7). Thereafter, based on Equation (3.2-8), the material parameters 𝑟, 𝛽 and 𝑚 describing 𝜂 were determined with the best-fit of the un-/reloading curves.
The Neo Hookean model was preferred due to its simple constitutive model, but the model is limited for moderate deformation as discussed in Section 2.5.4. When working with deformation larger than 30%, higher-order constitutive models should be considered. For example, one can use the strain energy function described by 3rd order Ogden model. Similar to the Neo Hookean model, the strain energy function of the uniaxial compression test becomes
Ψ𝑂 = ∑ 𝜇𝑖 𝛼𝑖(𝜆 𝛼𝑖+ 2𝜆−𝛼𝑖/2− 3) 3 𝑖=1 , (3.2-9)
and corresponding stress becomes
𝜎1 = ∑3 𝜇𝑖(𝜆𝛼𝑖 − 𝜆−𝛼𝑖/2)
𝑖=1 . (3.2-10)
3.2.3. Numerical Setup
31
Figure 3-11. Numerical model for the simulation.
The simulation of the compression test was complicated because the material was highly incompressible, which led to numerical errors and difficulties in running the simulation. In this case, the adhesive cuboid experienced large deformation up to 50 % which led to highly distorted elements resulting in un-converged solution. The problem with distorted elements was solved by using larger elements that were less flexible. However, larger elements means that a stiffer model was considered, which generally provides higher stress responses.
Additionally, the compression test contained a fixed bottom surface and a top surface with large friction that should be approximately defined as fixed in the plane. This resulted in over-constrained elements on the surface layers, which caused singularities at the corners and surface elements. When using finer mesh to capture the deformation, the elements tended to become distorted resulting in convergence problems.
32
Table 2. Boundary conditions of the top and bottom surface.
Case
Top surface
Bottom Surface
1 Free 𝛿𝑥 ≠ 0 Frictionless support 𝛿𝑥 ≠ 0 𝛿𝑦 ≠ 0 𝛿𝑦 ≠ 0 𝛿𝑧= 𝑝𝑟𝑒𝑠𝑐𝑟𝑖𝑏𝑒𝑑 𝑙𝑜𝑎𝑑 𝛿𝑧 = 0 2 Free 𝛿𝑥 ≠ 0 Fixed 𝛿𝑥 = 0 𝛿𝑦 ≠ 0 𝛿𝑦 = 0 𝛿𝑧= 𝑝𝑟𝑒𝑠𝑐𝑟𝑖𝑏𝑒𝑑 𝑙𝑜𝑎𝑑 𝛿𝑧 = 0 3 Fixed 𝛿𝑥 = 0 Fixed 𝛿𝑥 = 0 𝛿𝑦 = 0 𝛿𝑦 = 0 𝛿𝑧= 𝑝𝑟𝑒𝑠𝑐𝑟𝑖𝑏𝑒𝑑 𝑙𝑜𝑎𝑑 𝛿𝑧 = 0
3.3. Cyclic Loading Test
A cyclic loading test was designed to study the material behaviour and joint damage of the adhesive joint under repeated cyclic loading. The cyclic loading test attempted to imitate the real application of the adhesive joint to obtain valuable results.
3.3.1. Experimental Setup
The cyclic loading test includes joint specimen with a plastic cover and an aluminium base, which were of the same material as those used in an Ericsson RBS product. The geometry of the adherends are shown in Figure 3-12, in which the specific dimensions are less of interest since the only purpose was to be installed in the loading equipment. However, the aluminium base had one important dimension to control the joint thickness, which is the height of the track. In the RBS product, the joint thickness is approximately 1 mm that leads to the track height of the aluminium base was set to 1 mm. Consequently, when the cover was attached to the base, the applied adhesive spread out and it obtained a thickness of 1 mm.
The plastic cover was attached to the aluminium base using an additional device to ensure that the cover was centred with respect to the base. The manufacturing process is presented in Figure 3-13. The adhesive was manually attached using an adhesive gun with a nozzle with the diameter of 2 mm. However, the dimension and shape of the joint were difficult to control, and the final joint width should be measured to carry out the numerical simulation.
33
Figure 3-12. Plastic cover and aluminium base.
Figure 3-13. Joint specimen setup.
Figure 3-14. Applied load by prescribe the displacement.
34 First, the specimen was loaded in a tension machine which can record the reaction force and displacement during loading, as shown in Figure 3-16. However, the tension machine was limited by its program which can only repeat 5 cycles at time, but the joint should be exposed to thousands of cycles for possible joint damage to appear. Thus, the first 25 cycles were carried out in the tension machine at a speed of 15 mm/min to obtain the reaction force and displacement. Thereafter, the joint specimen was loaded with a cyclic loading equipment at a speed of 16 rpm as shown in Figure 3-17. The cyclic loading equipment had a similar fixture setup as for the tensile machine, in which the only difference was that the fixture was laid-down and the attached load became horizontal. The cyclic loading equipment had the advantage of being able to expose the joint to unlimited load cycles, but the disadvantage was that it cannot record any data. Therefore, after the specimen was loaded in the cyclic loading equipment, it was moved to the tension machine again to measure the change of the mechanical properties in the adhesive joint. This loading process was repeated until a suspected joint damage was observed.
35
Figure 3-16. Tension machine setup.
Figure 3-17. Cyclic-loading machine setup.
3.3.2. Numerical Setup
36 Two different load scenarios were investigated through simulation; an undamaged and a damaged adhesive joint. The damage was introduced by cutting off 17 mm of the joint on each end. The undamaged joint was modelled by inserting a fixed support on the entire bottom surface, and the top surface prohibited all movement except the load applied in x-direction. The damaged joint had the same boundary conditions in the middle part, which is denoted by 𝑏), while the top and bottom surfaces at the ends 𝑎) and 𝑐) were defined with frictionless supports. The boundary conditions are summarized in Table 3.
Figure 3-18. The numerical setup of the adhesive joint during cyclic loading test. Table 3. A summary of the boundary conditions of the cyclic loading test simulation.
