Structural analysis of thermal interface materials and printed circuit boards in telecom units - a methodology

Structural analysis of thermal interface materials and printed circuit boards in

telecom units - a methodology

Mattias Good

Engineering Physics and Electrical Engineering, masters level 2016

Luleå University of Technology

Department of Computer Science, Electrical and Space Engineering

Sammanfattning

En struktur analys p˚a Ericssons MINILINK-6352 har utf¨orts f¨or att unders¨oka sp¨anningar och deforma- tioner p˚a enheten, fr¨amst med fokus p˚a de termiska gr¨anskiktsmaterialen och buktningar av kretskortet.

Dessa ¨ar viktiga aspekter n¨ar man ¨overv¨ager om enheten ¨ar termiska l¨ampad ur en mekanisk synvinkel, d¨ar god ytkontakt mellan de olika kropparna ¨ar avg¨orande f¨or ordentlig kylning genom v¨armeledning.

Analysen kr¨aver tillr¨acklig materialdata till gr¨anskiktsmaterialen och kretskortet f¨or att kunna skapa l¨ampliga matematiska modeller.

Enaxliga kompressionstester har genomf¨orts f¨or att karakterisera de hyperelastiska och viskoelastiska lagar f¨or fyllda silikongummimaterial som anv¨ands som termiska gr¨anskiktsmaterial, som ibland kallas f¨or gappad. B¨ojning av ett kretskort simulerades och j¨amf¨ordes med ett tre–punkts b¨ojtest f¨or att verifiera om befintlig materialdata i ber¨akningsprogrammen var tillr¨acklig, j¨amf¨orelsen visade god ¨overensst¨ammelse.

Kretskortet med dess komponenter, som modellerades som styva block, med gappads ovanp˚a som komprimeras av en platta simulerades och ett svagt omr˚ade hittades. Detta omr˚ade var sedan tidigare k¨ant och har i ett senare skede eliminerats genom att tills¨atta ytterligare en st¨odpelare. D¨arav visar denna studie en metod f¨or att hitta intressanta regioner tidigt i konstruktionsfasen som l¨att kan ¨andras f¨or att uppfylla n¨odv¨andiga krav och undvika brister i konstruktionen. Arbetet har visat sig anv¨andbart genom att hitta detta svaga omr˚ade i exempel produkten, arbetet ger ¨aven tillr¨ackligt med information och exempeldata f¨or att ytterligare utreda liknande produkter. Kombinationen av erfarenhet och simulering m¨ojligg¨or smartare designval.

Abstract

A structural analysis on Ericssons MINILINK-6352 has been performed in order to investigate stresses and deformations of the unit, mainly focusing on the thermal interface materials and warpage of the printed circuit boards. These are important aspects when considering if the unit is thermally adequate from a mechanical point of view, where good surface contact between various bodies are critical for proper cooling through heat conductivity. The analysis requires sufficient materal data for the interface material and the circuit board in order to create suitable mathematical models.

Uniaxial compression tests have been conducted to characterise the hyperelastic and viscoelastic constitutive laws of a filled silicone rubber material used as a thermal interface material, commonly referred to as a thermal pad. Bending of a printed circuit board was simulated and compared to a three- point bend test on the circuit board in order to verify material data already available in the computational software, which showed good agreement.

The entire radio unit was mechanically analysed during its sealing process. The circuit board with attached components modelled as stiff blocks with thermal pads on top compressed by plates was sim- ulated and a weak area was found. This area in question was already known and has in a later stage been eliminated by adding an additional supporting pillar. Hence this study shows a methodology to find regions of interest at an early design phase which can easily be altered to fulfil necessary require- ments and eliminate design flaws. This work has proven useful in finding weak regions in the example product, it also provides enough information and example data to further investigate similar products.

The combination of experience and simulation allows for smarter design choices.

Acknowledgments

Firstly, I would like to express my gratitude to my advisor Peter Melin at Ericsson for supporting me through my work. Special thanks to J¨orgen Gustafsson and Karen Bruzda for providing me with all the necessary measurements making this work possible. Also thanks to Jonas Norlin for helping me with all the simulation difficulties throughout the project. I would also like to acknowledge the employees and consultants at Ericsson for providing interesting discussions during the weekly meetings. I am also grateful for the valuable comments by Professor Janis Varna at Lule˚a university of technology. Lastly, I would like to recognise Mikael Stallg˚ard for connecting me with Ericsson and supporting me during this time.

Contents

1 Background & Theory 1

1.1 Introduction . . . 1

1.2 Thermal interface material - Filled silicone rubber . . . 1

1.3 Hyperelasticity . . . 2

1.3.1 Mooney–Rivlin . . . 3

1.3.2 neo–Hookean . . . 3

1.3.3 Ogden . . . 4

1.3.4 Yeoh . . . 4

1.3.5 Arruda–Boyce . . . 4

1.3.6 Gent . . . 4

1.4 Linear viscoelasticity . . . 5

1.4.1 Generalised Maxwell model . . . 6

1.5 Mullins effect . . . 7

1.6 Printed circuit boards . . . 8

2 Method 9 2.1 Determining the material parameters . . . 9

2.1.1 Hyperelasticity . . . 9

2.1.2 Viscoelasticity . . . 11

2.2 Pad test - uniaxial compression . . . 11

2.3 Pad modelling . . . 13

2.3.1 Contact surfaces . . . 14

2.3.2 Compressibility . . . 14

2.3.3 One–element model . . . 14

2.3.4 Single pad in product . . . 15

2.4 PCB - three–point bend test . . . 16

2.5 Complete model - the setup . . . 19

3 Results & Discussion 21 3.1 Pad data . . . 21

3.2 Material parameters . . . 21

3.2.1 Hyperelasticity . . . 21

3.2.2 Viscoelasticity . . . 23

3.3 One–element pad . . . 24

3.4 Mullins effect during cyclic loading . . . 25

3.5 PCB - comparing results . . . 26

3.6 Complete model - stress and deformation analysis . . . 27

3.7 Future work . . . 29

3.7.1 Temperature dependence . . . 29

3.7.2 Long term effects . . . 30

3.7.3 Plasticity consideration - permanent deformation . . . 30

3.7.4 Mullins effect . . . 30

3.7.5 Additional tests - biaxial, shear, tension . . . 30

3.8 Conclusions . . . 30

Bibliography 33

A Pad data 34

B Parameter fit 37

C Mullins effect 40

D Complete model 43

Chapter 1

Background & Theory

1.1 Introduction

Maintaining proper working temperature in various electronic devices is important as modern technology is implemented everywhere. Cooling solutions is a crucial necessity requiring smart solutions, commonly achieved by conducting heat away to a heat sink which in turn is cooled through convection. This study aims to create a methodology or a process to mechanically analyse thermal interface materials employed in Ericssons products. Currently the choice of material is dependent on previous experiences making it difficult to chose new alternative routes and solutions. The thermal properties of interface materials are well known and important but without good surface contact heat can not be transferred properly. A filled silicone pad is one of many solutions, a soft material minimising damage to delicate components while still maintaining good thermal properties.

The goal is to utilise advanced computer softwares to mathematically model Ericssons MINILINK–

6352 focusing on the interface between the frame (heat sink) and the circuit board with its hot compo- nents. A mechanical analysis will allow stresses and deformations to be extracted, information which may act as guidelines on how to structurally improve the product. Possible even help the designers perform their very own analysis using mechanical modules in their design tools. All simulations will however require proper material models and acquiring these is a significant part of this work.

The mechanical behaviour of these silicone materials are relatively complex and can be somewhat characterised through inexpensive simple methods, such as the uniaxial compression test. The nonlinear behaviour can be described by a time-independent hyperelastic constitutive law, and a time-dependent viscoelastic law. These two different material laws are the key components necessary to describe the process of sealing a radio product and letting it rest. Pressure is applied on different components causing the circuit board to warp depending on how it is supported which may alter the contact between pad and component. The printed circuit board consists mainly of two materials, copper and glass-refinforced epoxy (FR-4).

The specified radio unit has been simulated during a sealing process to determine the forces required to compress the thermal interface materials. Hence giving some indication on where good contact is ensured, and where there are apparent weak areas due to the circuit board being deformed.

1.2 Thermal interface material - Filled silicone rubber

Due to the low heat conductivity of air (0.026 W/mK) a thermal interface material (TIM) is necessary to conduct generated heat away from electric components. There are several types of different TIMs avail- able such as greases, phase change materials, filled polymer matrices, and carbon materials. Important characteristics of the materials are the thermal- conductivity and resistance properties. The different types of TIMs has different advantages, and different ways of being applied [1]. A TIM with the highest thermal conductivity does not automatically make it the best interface material. Good surface contact between the material and various components is important as small gaps of air may cause overheating in the product [2].

Silicone rubber filled with ceramics to alter its thermal properties are commonly employed in Ericsson

products. Thermal and mechanical characteristics of these silicone pads is of interest, and also the long term reliability should be assessed [3]. The ceramic filler not only alters the thermal properties of the silicone rubber but also the stiffness, hence the reason to not only consider heat conducting properties.

Silicone rubbers are highly nonlinear temperature-dependent materials. They exhibit stress softening effects, especially in filled rubber [4]. Hysteresis may also occur in these types of material as well as stress relaxation and creep effects. Hyperelastic models, viscoelasticit models, and Mullins effect are of interest when modelling filled silicone rubbers. Plastic effects also occurs as these particular materials tend to deform permanently when compressed. A plastic model would allow analysis of this irreversible permanent deformation even when the load is removed.

The typical look of these silicone pads are shown in figure 1.1, where the pads have been compressed to some extent.

Figure 1.1: Silicone pads compressed with the same initial thickness. The coloured area represent the original size of the pads.

1.3 Hyperelasticity

Hyperelastic models are commonly used to describe the mechanical behaviour of rubber-like materials and elastomers. A strain energy density function (W ) describes the model and can either be a function of principal stretch ratios λi or strain invariants Ii such that

W = W (λ1, λ2, λ3) = W (I1, I2, I3). (1.1) Because of material incompressibility the strain energy function is split into a deviatoric (Wd) and volumetric (Wb) part.

W = Wd(I1, I2) + Wb(J), (1.2a)

W = Wd(λ1, λ2, λ3) + Wb(J), (1.2b) where J = λ1λ2λ3 is the volume ratio, that is J²= I3. The strain invariants are defined as

I1= λ²₁+ λ²₂+ λ²₃, (1.3a)

I2= λ²₁λ²₂+ λ²₂λ²₃+ λ²₃λ²₁, (1.3b)

I3= λ²₁λ²₂λ²₃. (1.3c)

For a fully incompressible material the volumetric term Wb = 0. For a compressible material Wb is nonzero and a compressibility parameter d is included and determined from the initial shear modulus (µ0) and initial bulk modulus (κ0).

µ0= E

2(1 + ν) (1.4)

κ0= E

3(1− 2ν) (1.5)

Combining the two equations leads to

κ0= 2µ0(1 + ν)

3(1− 2ν), (1.6)

where µ0 is provided in the following sections for each model. The compressibility parameter is then determined by

The second Piola–Kirchoff stress tensor is determined from the strain energy function by Sij = ∂W

∂Eij = 2∂W

∂Cij (1.8)

where

Eij= 1

2(FkiFkj− δ^ij), (1.9)

Cij= FkiFkj. (1.10)

Eij and Cij is the Lagrangian–Green strain tensor and right Cauchy–Green deformation tensor respec- tively with δij being the Kronecker delta, and Fij is the deformation tensor. The second Piola–Kirchoff stress Skl is related to Cauchy stress σij through

σij =J⁻¹FikSklFjl, (1.11)

where the JacobianJ = det(F^ij) [5]. When investigating different tests the deformation tensor usually take the following matrix form:

F =





λ1 0 0 0 λ2 0 0 0 λ3



. (1.12)

In [6] two classical sets of data [7, 8] are used to compare and rank twenty different hyperelastic models, the Ogden model is highly ranked but requires a large experimental database for its six model parameters. The eight–chain model was shown to be inefficient in predicting the biaxial response if the parameters are determined with uniaxial data, it is however considered suitable when only using uniaxial data. For small strains the neo–Hookean model efficiently showed good accuracy considering its simplicity. In [9] however the neo–Hookean model was unable to capture the nonlinearity of soft tissues and silicone rubber during unaxial tension tests.

1.3.1 Mooney–Rivlin

Mooney [10] was the first to introduce a strain energy function in 1940 based on the first two strain invariants I1, I2 under the assumption of incompressibility, that is I3 = 0. 1948 Rivlin [11] generalised Mooney’s work to include higher-order terms turning the strain energy function into

WM R=

l

X

i=0 m

X

j=0

Cij(I1− 3)ⁱ(I2− 3)^j+1

d(J− 1)², (1.13)

where l and m are natural numbers, and Cij are model parameters and C00 is usually set to zero. This is known as the Mooney–Rivlin model, and is not only important because of historical reasons but also its high accuracy for isotropic rubber-like materials [9]. The initial shear modulus is determined by

µ0= 2(C10+ C01). (1.14)

1.3.2 neo–Hookean

If l = m = 1 and C11= C01= 0 in the Mooney–Rivlin model, then WnH =µ

2(I1− 3) + 1

d(J− 1)², (1.15)

where µ = 2C10is the initial shear moudlus. This equation is known as the neo–Hookean model formu- lated by Treloar 1943 [12]. This is the simplest hyperelastic model and may act as a good starting point as it only requires one model parameter. The neo–Hookean model is only suitable for small strains.

1.3.3 Ogden

In 1972 Ogden [13] formulated the strain energy function in terms of principal stretches

WO=

N

X

i=1

µi

αi(λ^α₁ⁱ+ λ^α₂ⁱ+ λ^α₃ⁱ) +

N

X

i=1

1

di(J− 1)²ⁱ, (1.16)

where µr and αr are model parameters. In [13] Ogden used the classic Treloar data [7] for vulcanised rubber using simple tension, pure shear, and equibiaxial tensions tests. Ogden proposed a six parameter model, that is N = 3, which proved to be a good fit even for large strains. This model does however depend on all three stretches and six parameters, hence it requires a large experimental database. The initial shear modulus is defined as

µ0= 1 2

N

X

i=1

µiαi. (1.17)

Usually only d1is considered as the contribution from the other di is negligible, thus d = d1.

1.3.4 Yeoh

1990 [14] Yeoh introduced a hyperelastic model only depending on the first strain invariant.

WY =

N

X

i=1

ci0(I1− 3)ⁱ+

N

X

i=1

1 di

(J− 1)²ⁱ, (1.18)

where ci0 are the model parameters. This type of model is suitable if limited test data is available, e.g.

uniaxial test. A first order Yeoh model (N = 1) is equivalent to the neo–Hookean model. The initial shear modulus is defined as

µ0= 2c10. (1.19)

Just like the Ogden model, only d1 is usually considered.

1.3.5 Arruda–Boyce

The Arruda–Boyce boyce model, also known as the eight–chain model corresponding to the eight vertices of a cube, formulated 1993 [15] is a statistical mechanics based model requiring two model parameters.

Its strain energy equation has the form

WAB= µ

N

X

i=1

Ci

λ²ⁱ⁻²_L I₁ⁱ− 3ⁱ
+1 d

 J²− 1 2 − ln J



, (1.20)

where µ is the initial shear modulus, λL is the limiting network stretch, and Ci arise from the inverse Langevin series expansion as a result from treating the network as non–Gaussian [16]. Only the first five terms are commonly used, that is N = 5 and the five constants are C1= 1/2, C2= 1/20, C3= 11/1050, C4= 19/7050, and C5 = 519/673750. The limiting network stretch λL denotes the value at which the stress starts to increase without limit. In the limit of λL→ ∞ Arruda–Boyce becomes the neo–Hookean model. The initial shear modulus is µ = nKT where K is Boltzmann’s constant, T is temperature, and n is a model parameter referring to the chain density in the model. The Arruda–Boyce model has some similarities to the Yeoh model considering the I1 dependency, meaning this model is also suitable for limited test data.

1.3.6 Gent

Similar to the Yeoh and Arruda–Boyce model is the Gent model [17]. Another hyperelastic model dependent on the first strain invariant I1 only. The strain energy function has the following form:

WG=−µJm

2 ln



1−I1− 3 Jm

 +1

d

 J²− 1 2 − ln J



, (1.21)

1.4 Linear viscoelasticity

A viscoelastic material combines the characteristics of both elastic and viscous behaviour where the stress–strain relationship is time-dependent. Stress relaxation, creep and hysteresis are three important phenomena occurring in such materials [18]. Springs and dashpots are known to exhibit purely elastic and viscous response respectively. Combination of these are used to mathematically develop the response equations of these mechanical elements, so called rheological models [19].

The elastic component is described by

σ = Eε (1.22)

or dε

dt = 1 E

dσ

dt, (1.23)

where σ is the stress, ε is the strain, E is the elastic modulus, and t is time. The viscous component is described by

σ = ηdε

dt, (1.24)

where η is the viscosity of the material.

The two most basic rheological models are the Maxwell model (figure 1.2) and the Kelvin–Voigt model (figure 1.3), combining a spring and a dashpot in series and a spring and a dashpot in parallel respectively.

η E

Figure 1.2: Maxwell model.

η E

Figure 1.3: Kelvin–Voigt model.

For the Maxwell model the total strain is the sum of elastic and viscous strain contributions such

that σ

η + 1 E

dσ dt =dε

dt. (1.25)

The total stress in the Kelvin–Voigt model is the sum of stress of the elastic and viscous stress contri- butions such that

σ = Eε + ηdε

dt. (1.26)

Another common model is the standard linear solid (SLS) model, which combines the Maxwell model and a spring in parallel as in figure 1.4.

η1 E1

E2

Figure 1.4: Standard linear solid model.

The stress-strain relationship of the SLS model is expressed as σ + η1

E1

dσ dt = E2

 ε + η1

E1+ E2

E1E2

dε dt



. (1.27)

It can be shown [18, 20] that the Maxwell predicts stress relaxation accurately but is unable to predict the creep response. The Kelvin–Voigt model on the other hand will accurately model creep response but is unable to predict the stress relaxation. The SLS model will accurately predict both creep and stress relaxation.

1.4.1 Generalised Maxwell model

A more general and commonly used model is the generalised Maxwell model, or Wiechert model, or Maxwell–Wiechert model which consists of several Maxwell models in parallel which each other and a single spring, figure 1.5.

E_∞

ηN EN

η2 E2

η1 E1

Figure 1.5: Generalised Maxwell model.

If one considers a relaxation test during uniaxial tension at a constant strain ε0, the relaxation modulus can be determined by

E(t) = σ(t) ε0

, (1.28)

where σ(t) is the stress in figure 1.6. The relaxation modulus takes the following form at the limits:

E(t = 0) = σ0

ε0

= E0, (1.29)

E(t =∞) = σ_∞ ε0

= E_∞. (1.30)

σ_∞ σ0

t σ

Figure 1.6: Stress relaxation at constant strainε0.

For the generalised Maxwell model the relaxation modulus can be represented by the Prony series

N

where τi is the relaxation time for each Prony component, and N is the number of Maxwell elements in figure 1.5, [20]. The relaxation time is defined such that at t = τi the modulus will be at 1/e of its original value. A normalised relative modulus is defined as

αi= Ei

E0

. (1.32)

The relative modulus is used in ANSYS Mechanical to characterise the viscoelastic properties by rewriting equation 1.31 to

E(t) = E0 α_∞+

N

X

i=1

αie^−t/τi

!

. (1.33)

From this one can conclude that

N

X

i=1

αi≤ 1 (1.34)

is always true. If the sum of αiis equal to 1, that means α_∞= 0 leading to G(t→ ∞) = 0. Equivalently one can write

α_∞+

N

X

i=1

αi= 1. (1.35)

Extending the relaxation modulus to three dimensions for a homogeneous and isotropic material the stress tensor σij is dependent on the shear and bulk moduli, G and K respectively. It is also necessary to consider the previous history of the material, therefore a hereditary integral is used to describe the stress tensor,

σij(t) = 2 ˆt

−∞

G(t− t⁰)∂eij

∂t⁰ dt⁰+ 3δij

ˆt

−∞

K(t− t⁰)∂εkk

∂t⁰ dt⁰. (1.36)

Here G(t) and K(t) is the shear and bulk relaxation moduli respectively, εkk is the volumetric part of the strain, and eij is the deviatoric part of the strain,

eij = εij−1

3εkkδij. (1.37)

For an isotropic material the shear and bulk relaxation moduli can be written as G(t) = E(t)

2(1 + ν), (1.38)

K(t) = E(t)

3(1− 2ν), (1.39)

where ν is Poisson’s ratio. For an incompressible material ν = 0.5. It is assumed that the silicone material is fully or nearly incompressible such that the Poisson ratio ν . 0.5 and constant.

1.5 Mullins effect

Rubber-like materials exhibit significant change in their stress–strain curve after their first deformation [21]. This phenomenon has been studied for the past six decades and is still recognised as a major difficulty. This stress softening effect is not accounted for in regular hyperelastic- or viscoelastic models, but must be taken care of separately.

To account for this loss of stiffness in the material Simo [22] proposed to model the strain energy function through a virgin strain energy function W0 with damage parameter d∈ [0, 1] such that

W (Fij) = (1− d)W⁰(Fij). (1.40)

Thus every load cycle is proportional to the next.

Ogden and Roxburgh [23] presented a pseudo–elastic model describing the second loading curves in an additive way, hence no more proportionality between the curves. The Ogden–Roxburgh pseudo elastic model is a modification of the classical approach and is given by:

W (Fij, η) = ηW0(Fij) + φ(η), (1.41)

where η∈ [0, 1] is an evolving damage variable, and φ is a damage function defined by

φ(1) = 0, (1.42)

dφ(η)

dη =−W⁰(Fij), (1.43)

which implicitly defines η in terms of the deformation. The stress tensor can then be written as

Sij = ηS_ij⁰. (1.44)

In ANSYS Mechanical a modified Ogden–Roxburgh damage function [24] is used with a damage variable of the following form:

η = 1−1

rerf Wm− W⁰ m + βWm



, (1.45)

where r, m, and β are material parameters and Wm is the maximum virgin potential over the time interval t∈ [0, t⁰]. The damage variable should not be confused with η in section 1.4.

1.6 Printed circuit boards

Printed circuit boards (PCBs) are a relatively simple entity but is more complex to actually model. A PCB generally consists of several copper layers sandwiched by composite materials, often glass-epoxy.

Additionally several components of varying sizes are soldered onto the PCB and thereby increasing the stiffness of the board.

The glass epoxy laminates act as the base for the circuit board due to its preferred mechanical, thermal, and electrical properties. The most commonly used base material is the woven epoxy glass laminate FR-4, which due to its woven nature is an orthotropic material [25].

Upon modelling PCBs, Pitarresi et al. [26, 27] introduced smearing techniques removing all the com- ponents by adding their stiffness directly in the board, thereby significantly reducing the computational time while still maintaining good accuracy. In [28] large components are modelled as blocks rigidly con- nected to a PCB and in [29] a comparison between a block model and a smeared model is performed, all with good accuracy when compared to three-point bend tests, shock loading, and modal analysis. Smear- ing techniques are further investigated in [30, 31] for shock and vibrational response. Knowledge of the components elasticity modulus is necessary to achieve accurate results when smearing the components.

Chapter 2

Method

2.1 Determining the material parameters

2.1.1 Hyperelasticity

The principal Cauchy stresses can be determined with equation 1.8 inserted in equation 1.11 leading to:

σi=J⁻¹λi∂W

∂λi. (2.1)

Determining the model parameters for a material is done through different tests. Common test include, but are not limited to, uniaxial tension tests, biaxial tension tests, and shear tests. Under the assumption of incompressibility λ1λ2λ3= 1 andJ = 1, the deformation tensor is easily determined for each test. Limiting the study to I1-dependent models and rewriting the principal stretches to a single variable λ such that I1(λ1, λ2, λ3) = I1(λ), the derivative in equation 2.1 can be written as

dW dλ = dW

dI1

dI1

dλ. (2.2)

The first derivative on the right hand side can then be calculated for each model:

dWnH

dI1

=µ

2, (2.3)

dWY

dI1

= c10+ 2c20(I1− 3) + 3c³⁰(I1− 3)², (2.4) dWAB

dI1

= µ

5

X

i=1

Ci

λ²ⁱ_L⁻²iI₁ⁱ⁻¹, (2.5)

dWG

dI1

=µJm

2

1

Jm− I¹− 3. (2.6)

For every dataset a least square method is used to fit the data to a specific stress function determined for each type of test. Thus the model parameters can be specified and the material characterised.

Only uniaxial data is available, however both biaxial and shear tests are described in the following sections. They are also important as they can give hints of stability in the model. A good example of instability can be seen in a higher order Yeoh model and may occur when some parameters ci0 are negative, figure 2.1.

−0.4 −0.2 0 0.2 0.4 0

εeng

σeng[Pa]

uniaxial biaxial shear

Figure 2.1: Examples of a third-order Yeoh model exhibiting a non-physical behaviour where the stress suddenly changes direction.

2.1.1.1 Uniaxial tension

In uniaxial tension (figure 2.2) with a stretch ratio of λ1= λ with a corresponding Cauchy stress of σ1, and the other principal stresses σ2= σ3= 0, since no other loads are applied. Due to incompressibility the other stretch ratios λ2= λ3= λ^−1/2. This will lead to

I1= λ²+2

λ. (2.7)

The principal Cauchy stress is then determined by σ^(uni)₁ = λdW

dλ =

 λ²−1

λ

 dW dI1

. (2.8)

Figure 2.2: Uniaxial tension test.

2.1.1.2 Equibiaxial tension

In equibiaxial tension (figure 2.3) two of the principal stresses are equal, σ1= σ2, while the third σ3= 0.

The corresponding stretches are written as λ1= λ2= λ and λ3= λ⁻² leading to I1= 2λ²+ 1

λ⁴. (2.9)

The Principal Cauchy stresses are then determined by σ₁^(bia)= σ₂^(bia)= λdW

dλ = 4

 λ²− 1

λ⁴

 dW dI1

. (2.10)

Figure 2.3: Equibiaxial tension test.

2.1.1.3 Pure shear

In pure shear (figure 2.4) one of the principal stretch ratios are fixed, λ2= 1, while the others are λ1= λ and λ3= λ⁻¹ with the corresponding principal stresses σ1, σ2= 0, and σ36= 0, thus

I1= λ²+ 1

λ² + 1. (2.11)

The principal Cauchy stress of interest is then determined by σ^(she)₁ = λdW

dλ = 2

 λ²− 1

λ²

 dW

dI1. (2.12)

Figure 2.4: Pure shear test.

2.1.2 Viscoelasticity

Only the shear relaxation modulus G(t) will be considered because the bulk relaxation modulus K(t) is usually negligible due to the near incompressible characteristic of these types of material. For an incompressible material the bulk modulus will approach infinity and the volumetric part of the strain will be zero.

To characterise the viscoelastic part in ANSYS, the shear bulk modulus

G(t) = G0 a_∞+

N

X

i=1

αie^−t/τi

!

(2.13)

is used. The modulus is determined via equation 1.38, thus it is only a scaled quantity of the uniaxial stress from the relaxation test. Using the uniaxial stress directly will provide a similar result as the relative relaxation modulus is of interest,

α(t) = α_∞+

N

X

i=1

αie^−t/τi. (2.14)

2.2 Pad test - uniaxial compression

Uniaxial tests on a 10× 10 mm pad with varying thickness of 1 − 3.5 mm. The tests were conducted by the manufacturers in room temperature with strain controlled test machines utilizing two hard plates to compress the material, a schematic of the test can be seen in figure 2.5, and an example test machine is shown in figure 2.8. The full details of the test machines are unknown. Different strain rates ˙ε was used to observe the viscoelastic effects and allow comparison in simulations. Due to the varying thickness of the pads a compression rate [mm/min] is defined instead of a strain rate [1/s].

F ˙ε

εeng

Figure 2.5: Schematic of the uniaxial compression test. The grey box represents the pad. F [N] is the reaction force from the pad and σ^eng=F [N/cm²].

The tests performed are not purely uniaxial due to friction, which is significant because of shear forces and tacky surfaces. This is a flaw in the uniaxial compression test and must be considered during the computer simulation in ANSYS Mechanical.

Compression to -0.5 strain with stress relaxation was conducted at different compression rates (10 mm/min, 1 mm/min) to characterise the material models. From the 10 mm/min test, the stress–strain loading curve acquired was used to characterise the hyperelastic model. From the 1 mm/min test, the stress–time of the relaxation was used to characterise the viscoelastic model. A sufficiently slow strain rate should be used such that inertia effects can be ignored [20]. The faster compression rate data was used for the viscoelastic parts for some materials as well due to better results.

The typical strain control used for testing is shown in figure 2.6.

−0.5

−1

t εeng

Figure 2.6: Compression and relaxation.

Cyclic loading was also applied to observe the Mullins effect in the materials. In this types of test a successively increasing strain is applied keeping a constant loading-unloading rate in order to see the stress softening effects, figure 2.7.

−0.5

−1

t εeng

Figure 2.7: Cyclic loading.

Figure 2.8: Strain controlled test machine used for uniaxial compression.

2.3 Pad modelling

The data acquired from the compression test are used to determine the parameters needed to characterise the hyperelastic models. Only uniaxial data is available hence a suitable I1-dependent model should be chosen, i.e. neo–Hookean, Yeoh, Arruda–Boyce, or Gent. A least square curve fitting tool available in ANSYS is used to determine the parameters according to section 2.1.1 using the formulation for uniaxial tension.

The viscoelastic part is also included in the material model according to section 2.1.2 making the material time-dependent.

Some care must be taken when considering how time is defined in a static structural simulation in ANSYS Mechanical using the available data. The hyperelastic part of the model is time-independent, however the measurement from which it is determined is strain rate dependent. Once the time-dependent viscoelasticity is included stress relaxation will always occur during straining of the material. Thus a relatively fast simulated loading time compared to real time can be used to accurately follow the same stress–strain curve from the 10 mm/min compression test in all cases. Hence making the viscoelastic part obsolete during the loading phase.

2.3.1 Contact surfaces

Upon modelling the contact surfaces one must consider the origin of the dataset that is used.

When conducting a uniaxial compression test it is not possible to avoid friction effects, which is an inherited flaw in this type of measurement and the effect is significant. When implementing the model a frictionless state or a friction coefficient near zero should therefore be used because of the fact that friction is included in the measurements. There will still be an error here due to the different contact surfaces in the test and in the application, none of which has been quantified in any way. In [32] friction is discussed and a more complicated equibiaxial test may complement the uniaxial test to get a more accurate material model of the silicone pads [33], however a friction coefficient will also have to be determined and applied to the contact surfaces. Determining the characteristics of the material through an equibiaxial test would be more difficult and time consuming.

A low friction coefficient was used at the contact surfaces of the pad. The reason for using this type of connection as opposed to a completely frictionless contact was to prevent rigid body motion in a simple way.

2.3.2 Compressibility

Although not absolutely necessary, compressibility is introduced in the model by using a Poisson ratio of ν = 0.49. This will have little effect on the actual end result (partly due to frictionless state) but it will allow the solution to converge with more ease. Hence this is merely a numeric application with physical repercussions where the chosen quantity of ν has no scientific basis other then to keep a nearly incompressible state. Thus the volumetric part Wb(J) in equation 1.2 may be nonzero, thereby including the compressibility term d which is determined for each hyperelastic model according to section 1.3.

2.3.3 One–element model

A one–element model, such as figure 2.9, with the same dimensions as the tests was used to verify the different models and compare to the experimental data. The reason for using this type of model is its simplicity and because modelling the actual experiment (figure 2.5) is difficult due to convergence problems and unknown friction. This one–element model ought to be sufficient to study the forces in one direction which assumes a fully uniaxial state as it is completely frictionless. This is also the way to model the material due to the inherited flaws from a uniaxial compression test in terms of friction and shear stresses which is included in the data.

D

Figure 2.9: One element model where the top surface is displaced a distance D. The bottom surface is fixed, the pad is free to expand in the other directions.

It is also possible to calculate these stresses by hand by combining the hyperelastic and viscoelastic model. For the uniaxial case the stress can be determined as

2.3.4 Single pad in product

One of the thermal pads in the radio was chosen, such as figure 2.10, to analyse how the solution would converge when the top plate was displaced 0.5 mm in a downward direction. This was useful in order to prepare for the larger time consuming complete model.

2.54 mm ξ

ξ plate

pad component

Figure 2.10: Cross-section of how the thermal pad may appear in the product. The sizes of the different parts may vary. A friction coefficient of ξ added between the surfaces.

2.3.4.1 Element technology

Meshing of the pad was done using tetrahedals as this improved convergence and computation time.

Two different element types are suitable for hyperelastic materials, namely Solid187, and Solid285.

Solid187 is a higher order element consisting of 10 nodes, each having three degrees of freedom (transla- tional).

Solid285 is a lower order element consisting of 4 nodes, each having four degrees of freedom (transla- tional and hydrostatic pressure), hence it uses a mixed u–P formulation. The mixed u–P formulation is commonly used to treat volumetric locking and is also available for Solid187.

Solid285 is used in all simulations as it significantly decreases computation time.

(a)Solid187 (b) Solid285

Figure 2.11: Tetrahedon elements.

2.3.4.2 Mesh convergence

A mesh convergence study was performed to have sufficiently accurate reaction forces with reasonable computation time. The mesh is altered according to the 2.54 mm thick pad resulting in figure 2.12. A similar test is done for the thicker pad (5.5 mm) using only one of the materials and shown in figure 2.13. Mesh improvements may be necessary as the convergence is not good. A finer mesh will however significantly increase the computation time.

0 1 2 3 4 5

·10⁵ 110

120 130 140 150 160

elements

max(F)[N]

A C D E

Figure 2.12: Mesh convergence, 2.54/N [mm] element size and friction of ξ = 0.02.

Where N is an integer. Pad B is excluded because it is much larger in magnitude but identical to the others.

0 1 2 3 4 5 6 7 8

·10⁴ 38

40 42 44 46 48 50 52

elements

max(F)[N]

Figure 2.13: Mesh convergence thick pad, C only. 5.5/N [mm] element size and friction of ξ = 0.02. Where N is an integer.

2.3.4.3 Friction study

A short friction study was conducted to observe the effect of altering the friction coefficient ξ in the model. Figure 2.14a may indicate that the true stress, σtrue, converges for lower values of ξ. Studying the true stress, it will significantly increase for increasing values of ξ. Because the pads are compressed approximately 20%, corresponding to a true strain of εtrue≈ −0.22, the stresses observed are similar in magnitude therefore a frictionless state or a friction coefficient near zero should be used. As previously mentioned in section 2.3.1, the material of the contact surfaces will introduce some errors.

2.4 PCB - three–point bend test

0 0.1 0.2 0.3 0.4 0.5 2

4 6 8

·10⁵

ξ max(σtrue)[Pa]

(a)Friction study.

−0.6

−0.4

−0.2 0

−4

−3

−2

−1

0

·10⁵

εtrue

σtrue[Pa]

(b) True stress–strain, further presented in the fol- lowing sections.

Figure 2.14: Friction effects compared with stress–strain data from the uniaxial com- pression test to determine a valid friction coefficient ξ to be used at the contact surfaces, for pad A only. The other pad materials will use the same coefficient.

weight by hand on top of the PCB acting as a point load and measuring the deflection, figure 2.15. This test was conducted on a PCB without any components. Figure 2.16 shows the setup in the workshop.

F

D

Figure 2.15: Cross-section of the circuit board when applying a point load F at the center of the board, measuring the deflectionD at a suitable point nearby.

The circuit board is supported by two beams along the edges of the board on both the left and right side.

Figure 2.16: Bend test performed in the workshop. The red wiring to the left was used to determine when the displacement measuring tool was in contact.

When analysing PCBs in ANSYS it is possible to import a circuit trace (figure 2.17) from an electronic computer-aided design (ECAD) tool, a useful method to investigate thermal stresses when the circuits are operating. This will however require a very fine mesh significantly increasing computational time.

Therefore a comparison using pure copper layers between the FR-4 with a coarser mesh is carried out.

The PCB consists of 31 layers of varying thickness, 14 of which are copper layers and the remaining layers are FR-4.

The three-point bend test was simulated using both an imported trace and pure copper layers.

Figure 2.17: PCB with an imported trace in ANSYS Mechanical from an ECAD tool.

Colour scale indicate copper alloy, that is red shows is copper.

2.5 Complete model - the setup

The complete model aim to simulate the radio when attaching the lid. No thermal considerations were done hence no electronic trace was imported. Thus allowing a coarse mesh on the PCB. Most of the frame was cut out, leaving only the screw pillars and cooling plates as these were the only interesting parts in this purely mechanical model. Figure 2.18 shows the schematic of the radio without the lid and thermal pads.

(a)Lidless frame (b) Frame and PCB

Figure 2.18: The encircled pillar in (a) is an added support because of a structural weakness found at a later stage. It is not included in simulation models.

Because the highest compression rate data available was 10 mm/min, this is the fastest compression rate which can be modelled. In order to follow this curve closely the loading time in ANSYS Mechanical was set to 0.1 s avoiding any relaxation during this phase. As a reminder it should also be noted that time ought to be considered as an arbitrary unit based on the experimental data.

A small value of friction (ξ = 0.02) was applied at the pad surface connected to the components and the PCB making uniaxial compression the only substantial part of the system. Bonded connections were used between the PCB and the components, and the PCB and the screw pillars.

The simplified model in figure 2.19 now includes the pads and the plates cut out from the lid.

1 2

3 4

5

Figure 2.19: The complete setup where every top plate will be displaced −0.5 mm in negative z direction. The large piece of the bottom frame is not included in the simulation. Only the top part of the screw towers are included and fixed.

Chapter 3

Results & Discussion

3.1 Pad data

The engineering stress σeng and engineering strain εeng from the uniaxial compression test is shown in figure 3.1. The remaining pads are presented in the appendix, section A. From this test data the hyperelastic and viscoelastic models are characterised easily using the built in tools in ANSYS and are presented in the following section. An interesting note is that both tests converge in the stress relaxation part in figure 3.1b.

−0.5

−0.4

−0.3

−0.2

−0.1 0

−8

−7

−6

−5

−4

−3

−2

−1 0

·10⁵

εeng

σeng[Pa]

10 mm/min 1 mm/min (a)stress–strain

0 20 40 60 80 100 120

−8

−7

−6

−5

−4

−3

−2

−1 0

·10⁵

t [s]

σeng[Pa]

10 mm/min 1 mm/min (b)stress–time

Figure 3.1: Uniaxial compression test for pad A.

3.2 Material parameters

3.2.1 Hyperelasticity

The third order Yeoh model was observed to be unstable all of the pads. Second order Yeoh is stable for some of the pads, due to the low values of c20, relative to c10, and sometimes being negative deems it unfit to characterise the pads. The first order Yeoh is already equal to the neo–Hookean model. The limiting values λL and Jm is mostly very large and therefore the Arruda–Boyce and Gent models are equivalent to the neo–Hookean model. Hence the neo–Hookean model was chosen for all the pads. Even though λLhas a suitable value for pads B and D, this is most certainly coincidental due to the similarities between all material data. Due to low strains in the product the neo–Hookean model should suffice.

If the uniaxial compression tests would be complemented with uniaxial tension tests (positive strains) the other models may prove more useful.

It is also clearly visible that pad B is stiffer than the other materials.

Table 3.1: Hyperelastic material parameters using least square curve fitting tool in AN- SYS. From the table it is apparent that for the third-order Yeoh model every c30 value is negative which may cause a non-converging solution. The com- pressibility parameter d is determined using a Poisson ratio of ν = 0.49.

Model & parameters Pads

A B C D E

neo–Hookean µ^[Pa] 2.20× 10⁵ 1.22× 10⁶ 2.73× 10⁵ 2.06× 10⁵ 2.33× 10⁵ d^[Pa−1] 3.66× 10⁻⁷ 6.58× 10⁻⁸ 2.95× 10⁻⁷ 3.91× 10⁻⁷ 3.46× 10⁻⁷ 1st Yeoh c10 ^[Pa] 1.10× 10⁵ 6.12× 10⁵ 1.36× 10⁵ 1.03× 10⁵ 1.16× 10⁵

d^[Pa−1] 3.66× 10⁻⁷ 6.58× 10⁻⁸ 2.95× 10⁻⁷ 3.91× 10⁻⁷ 3.46× 10⁻⁷

2nd Yeoh

c10 ^[Pa] 1.12× 10⁵ 5.82× 10⁵ 1.49× 10⁵ 9.98× 10⁵ 1.18× 10⁵

c20 ^[Pa] −1090 2.16× 10⁴ −7860 2000 −759

d1 ^[Pa−1] 3.60× 10⁻⁷ 6.92× 10⁻⁸5 2.70× 10⁻⁷ 4.03× 10⁻⁷ 3.43× 10⁻⁷

3rd Yeoh

c10 ^[Pa] 9.38× 10⁴ 4.04× 10⁵ 1.44× 10⁵ 8.18× 10⁴ 9.63× 10⁴ c20 ^[Pa] 3.03× 10⁴ 3.88× 10⁵ 645 3.43× 10⁴ 3.69× 10⁴ c30 ^[Pa] −1.45 × 10⁴ −1.99 × 10⁵ −3870 −1.52 × 10⁴ −1.76 × 10⁴ d1 ^[Pa−1] 4.29× 10⁻⁷ 9.97× 10⁻⁸ 2.79× 10⁻⁷ 4.93× 10⁻⁷ 4.18× 10⁻⁷

Arruda–Boyce

µ^[Pa] 2.20× 10⁵ 1.02× 10⁶ 2.73× 10⁵ 1.83× 10⁵ 2.33× 10⁵

λL 2.97× 10⁷ 2.19 1.30× 10¹¹ 2.71 2.96× 10⁷

d^[Pa−1] 3.66× 10⁻⁷ 7.93× 10⁻⁸ 2.95× 10⁻⁷ 4.40× 10⁻⁷ 3.46× 10⁻⁷

Gent

µ^[Pa] - 1.22× 10⁶ 2.73× 10⁵ 2.06× 10⁵ -

Jm - −2.62 × 10¹⁴ 2.79× 10¹⁴ −9.25 × 10¹³ -

d^[Pa−1] - 6.58× 10⁻⁸ 2.95× 10⁻⁷ 3.91× 10⁻⁷ -

−1 −0.5 0 0.5 1

−1

−0.5 0

·10⁷

εeng

σeng[Pa]

A B C D E

Figure 3.2: Comparison of the neo–Hookean hyperelastic model.

3.2.2 Viscoelasticity

In table 3.2 the viscoelastic parameters are presented. In figure 3.3 the relative relaxation modulus is shown, that is equation 2.14 for every pad. The most interesting parameter here is α_∞ which can interpreted as the ratio of which the stress will converge to. That is the maximum force achieved during loading can easily be predicted by

σ(t→ ∞) = max(σ)α∞ (3.1)

to approximate the final stress after a long time has passed. However accurate values of α_∞is required for this simple approximation which is very limited.

Table 3.2: Viscoelastic material parameters using least square curve fitting tool in AN- SYS. Using further terms would only result in similar repeated parameters or quantities ofαi→ 0.

Parameters Pads

A B C D E

α_∞ 0.295 0.611 0.342 0.475 0.0368 α1 0.344 0.0871 0.267 0.278 0.221 τ1^[s] 0.283 1.57 0.515 1.05 0.699 α2 0.164 0.0629 0.162 0.126 0.130

τ2^[s] 1.36 4.21 3.27 11.2 5.40

α3 0.110 0.0619 0.109 0.121 0.108

τ3^[s] 6.56 23.4 16.3 93.8 40.4

α4 0.0867 0.177 0.120 - 0.505

τ4^[s] 43.8 1770 114 - 2550

0 100 200 300 400 500 600 0.3

0.4 0.5 0.6 0.7 0.8 0.9 1

t [s]

α(t)

A B C D E

Figure 3.3: Relative moduli.

3.3 One–element pad

By using a high compression rate when simulating the pads one will force the stress–strain curve to follow the hyperelastic model. This is a simple way to avoid any relaxation during loading and get an accurate maximum value once the target strain is reached, in this case for the 10 mm/min compression rate. To model a slower compression rate one must consider how time is defined in ANSYS Mechanical using different datasets. During a static structural simulation the time unit is arbitrary depending on how the different models are defined. This is due to the fact that the hyperelastic constitutive laws are time-independent but in reality time has passed during the compression in the test. This is further complicated when combining the hyperelastic model with a time-dependent viscoelastic model and thereby introducing a time scale to the problem.

It is apparent that once the material is relaxing, the stress tend to converge towards the same value no matter the initial thickness or strain rate as seen in figure 3.4. This may not always be true as seen in figure 3.5 for this particular material, where the data is significantly different depending on the thickness (see appendix A, figure A.2). In simulations the relaxation curves will always converge.

0 50 100 150 200 250 300 20

30 40 50 60

εeng

F[N]

2.5 mm 3.5 mm

Figure 3.4: Relaxation curve for pad C with different thickness at a compression rate of 1 mm/min.

0 20 40 60 80 100 120

100 150 200 250 300 350 400 450

εeng

F[N]

1 mm 2 mm

Figure 3.5: Relaxation curve for pad B with different thickness at a compression rate of 10 mm/min.

3.4 Mullins effect during cyclic loading

It is apparent that stress softening occurs in the pads when exposed to cyclic loading, figure 3.6. The unloading curve is very steep and non-physical, likely due to the pads exhibiting permanent deformation and will not rebound to their original configuration, it should not be confused with a linear elastic behaviour. Thus the reaction force during the unloading measurement show a strange behaviour, either due to the tacky surfaces of the pads, or lack of contact because the pad does not rebound. Further stress softening data is presented in the appendix, section C.

Because of the plastic behaviour, Mullins effect has not been modelled but merely observed in mea- surements. Including the softening effects in ANSYS requires the Ogden–Roxburgh model from section 1.5 further complicating the material models. To simulate these effects the computation time would

significantly increase due to adding an extra unloading and loading phase. The reason for wanting to model this would be to investigate the effects of opening the product for maintenance.

It is also difficult to extract parameters to characterise this effect completely as there is currently no tool for curve fitting Mullins effect in ANSYS 17.1. A third-party tool, or a user made method is required. The only question being how necessary it is to model stress softening effects.

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1 0

−7

−6

−5

−4

−3

−2

−1 0 1

·10⁵

εeng

σeng[Pa]

loading softening (a)stress–strain

0 200 400 600 800

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1 0

t [s]

εeng

loading softening (b)Strain control

Figure 3.6: Softening effects for pad A.

3.5 PCB - comparing results

From figure 3.7 it is evident that there is no reason to use an imported ECAD trace for purely mechanical problems, of course this may not always be true but may at least act as a good starting point. Importing a trace may also be of great interest when looking at thermal induced stress in circuit boards. Furthermore the FR-4 material already existing in ANSYS Mechanical has proven to be a suitable choice of material in this case.

In figure 3.7 a comparison of the three-point bend test and simulation is shown, where it is apparent that the slope of the three curves are similar which indicates a good match between the materials. It is also clear that the experimental linear regression line does not cross through the origin meaning the point of reference is slightly off 0.3 mm. Zero applied force should per definition result in zero displacement for any structure. The linear behaviour also indicates that bending occurs within the elastic region.

0 10 20 30 40 50

−0.5 0 0.5 1 1.5 2 2.5

F [N]

D[mm]

experimental no trace trace

0.0530· F − 0.3457 0.0523· F + 0.0112 0.0515· F + 0.0000

Figure 3.7: Simulated bend test with and without imported trace compared to the exper- imental three–point bend test. A linear regression line is plotted for each dataset.

3.6 Complete model - stress and deformation analysis

In figure 3.9 the deformation of the complete model is shown after displacing the top plates 0.5 mm, before and after letting the pads relax for 120 s. The printed forces in figure 3.9a is the maximum force occurring after 0.1 s. In figure 3.9b the printed forces is after the relaxation time of 120 s. The remaining pads are presented in the appendix, section D.

1 2

3

4 5

Figure 3.8: Figure 2.19 is reiterated here for conveniance.

An interesting note here is the low force occurring at pad 3 relative to the others, (see pad numbering in figure 3.8). This is the region furthest away from any screw pillar making it susceptible to applied loads. The low force for the pad in this region indicate a problem area where the pad may experience bad contact and thereby cause the components to overheat. It is also apparent that most of the load is applied at pad 2 , a region encircled by several screw pillars. At pad 5 the forces is relatively low.

This is where the thicker pad is used meaning it is exerted to small strains compared to the other pads of the same dimensions.

47.7 N 83.8 N

6.73 N

42.9 N 29.8 N

(a)Loading

25.7 N 34.0 N

7.12 N

24.6 N 11.6 N

(b) Relaxation

Figure 3.9: Pad A, deformation in z direction with the force at each pad. Relaxation occurs for two minutes. Computation time: 15h 7m

The force required to compress the pads at the different locations is shown in figure 3.10, the remaining material types are presented in the appendix, section D. The total force needed to compress all pads is shown in figure 3.11. In figure 3.12 the maximum displacement of the circuit board is shown, where the board will generally bend around 0.4 mm.

The area susceptible to applied loads around pad 3 should in some way be altered to maintain the crucial contact between pad and component to avoid overheating. At some point during the design of this particular product a supporting pillar was added to overcome this problem. Simulations show this flaw at en early stage which can lead to smart solutions at an early stage.

0 20 40 60 80 100 120

0 20 40 60 80

t [s]

F[N]

1 2 3 4 5

(a)force–time

0 0.1 0.2 0.3 0.4 0.5

0 20 40 60 80

D [mm]

F[N]

1 2 3 4 5

(b)force–displacement Figure 3.10: Force required to compress each A pad0.5 mm.

0 20 40 60 80 100 120 0

100 200 300 400 500

t [s]

F[N]

A B C D E

Figure 3.11: Total force required to compress the different pads0.5 mm.

0 20 40 60 80 100 120

0.1 0.2 0.3 0.4 0.5 0.6 0.7

t [s]

D[mm]

A B C D E

Figure 3.12: Maximum displacement of the circuit board during the compression.

3.7 Future work

3.7.1 Temperature dependence

As the pads are used to cool different units on a circuit board they will operate in a high temperature region. The structural properties of the pad will undoubtedly be altered at high temperatures of say 90

◦C. The current model is based on experimental data carried out in room temperature. The same tests could be performed in an oven, implemented in ANSYS, and interpolated over the temperature region of interest.

In [4] the influence of temperature on filled and unfilled silicone rubbers are investigated and may give some insight on how these types of silicone materials are affected by temperature changes.

The effects of temperature cycling should also be considered and is discussed in [3] for different types of thermal interface materials. This is something that may have to be considered over a longer time

perspective as well.

3.7.2 Long term effects

Viscoelasticity is the only time-dependent part which has been modelled through the use of relaxation tests, the longest being only five minutes, a lot of work still remains to discover what happens over a longer period of time. A short discussion of using Arrhenius plots to estimate the lifetime of rubber materials through relaxation tests at different temperatures is found in [34].

Environmental effects may also effect the pads performance. Dust, dirt, drying, and humidity could all be a cause for loss of contact thereby cause overheating.

3.7.3 Plasticity consideration - permanent deformation

Plastic deformation is not accounted for in hyperelastic constitutive laws where zero stress will by defini- tion result in zero strain when unloading the material. It is however apparent that permanent deformation occurs after compressing the thermal pads. If the deformation of the pads are of interest, some form of plastic consideration should be performed.

An advanced model that may be of interest is the three network model by J¨orgen Bergstr¨om [35]

combining hyperelastic and viscoplastic elements.

3.7.4 Mullins effect

Combining stress softening with a hyperelastic model is an easy task under the assumption that the material in question is relatively elastic and will nearly return to its original configuration. However due to the plastic behaviour of the pad complications arise as the Ogden–Roxburgh model is based on a strain energy function W . As previously mentioned this sort of modelling may be of interest when opening the product (e.g. maintenance) and sealing it again. If the pad is not replaced the stress–strain will no longer follow the virgin curve and therefore loss of contact may occur.

3.7.5 Additional tests - biaxial, shear, tension

As previously discussed in section 2.3.1 a uniaxial compression test is simple but also flawed in the way that that friction is not accounted for, but included in the measured data. The more complicated equibiaxial test can complement the compression test to more accurately predict the material behaviour and at the same time allow for other hyperelastic model to be used, e.g. Mooney–Rivlin and Ogden.

The major differences between the I1-dependent models occurs at positive strain, hence a test covering this region is recommended which may help in moving away from the very simple neo–Hookean model.

An argument for performing these additional complicated tests is that it may allow for better pre- dictions of the deformed state of the pads. As of now the frictionless state used is non-physical allowing the pad to freely expand.

One should also be mindful of the applications intended and stay within the expected strain regions.

One should not expect data for 50% strain correlate well for strain of 200%. Because the pads investigated are compressed in a product a suitable test is also a uniaxial compression test if used in the correct way.

Through this work none of the conducted test has been performed according to any standard test.

3.8 Conclusions

A radio product has been mechanically investigated by determining the material properties of thermal gap pads commonly employed to cool the various components on a printed circuit board. It was found that simple material models from simple uniaxial tests is sufficient to acquire enough knowledge in the design of the product. In the example product investigated a weak point was found. Similar methods can be used to determine the mechanical properties of other products, where the available pad data may serve as a starting point. Collecting additional data according to standardised test should be considered.

The basics of strain energy function was introduced along with some common hyperelastic models, all available for use in ANSYS compatible with a built in curve fitting tool. An introduction to linear

Mullins effect was clearly observed in the pads but not modelled. This is a more difficult subject, especially when the material exhibit plastic behaviour which is not captured by the hyperelastic and viscoelastic models. Stress softening should be considered in some form if the product is to be maintained, unless the pad is replaced. Modelling stress softening effects may overcomplicate matters and should probably be left out.

A primitive three–point bend test conducted on the circuit board showing surprisingly good results with an expected linear behaviour. Measured and simulated data showed good agreement except for the reference point in the experiment being slightly off. This offset is clearly visible in all the data points.

With this work the design process can be shortened with less trial-and-error and instead approach the problem at hand with simulation as guidance. Problematic areas can be found and eliminated at earlier stages before manufacturing the product.

Suggestions for improvements includes temperature dependent effects, long term effects, and poten- tially additional test data in the form of a biaxial test. Plasticity should also be considered due to the permanent deformation observed and may be necessary when considering cyclic loading where stress softening effect also plays an important part. Due to the shortcomings of the uniaxial compression test the results should be interpreted with care.