A FEM coupling model for properties prediction during the curing of

an epoxy adhesive for a novel assembly of radio telescope panel Shouwei Hu*a,b,c, Yi Chena,b

a Nanjing Institute of Astronomical Optics & Technology, Chinese Academy of Sciences, Nanjing 210042, China;b Key Laboratory of Astronomical Optics & Technology, Nanjing Institute of Astronomical Optics& Technology, Chinese Academy of Sciences, Nanjing 210042, China;

c Graduate University of Chinese Academy of Sciences, Beijing 100049, China

ABSTRACT The curing of epoxy adhesives is a complex phenomenon where the thermal, the chemistry and the mechanics are coupled. Corresponding material properties such as mechanical and physics properties are evolving with the curing. This paper focuses on their predictions by a multiphisics FEM approach of the thermal, chemical and mechanical couplings involved by the curing for a novel assembly of radio telescope panel. The first part presents the constitutive model of an epoxy adhesive that is considered for the curing. The numerical solving, performed with a specific user subroutine of Ansys, is detailed and allows the study of real three-dimensional structure parts. Residual stresses and strains of different metallic membranes and internal adhesives in the interconnect during the assembly of radio telescope panel are investigated. The mechanical response of the interconnect is analyzed with respect to the poisson’s ratio, relaxation time and adhesive thickness. It is shown that, although the overall residual stresses at the interconnect increase with the adhesive curing, the local strains have different evolving trends, indicating the possibility of damage and decohesion that might compromise mechanical integrity and interrupt the component processing precision. Keywords: material modeling, multiphysics FEM, epoxy adhesive, cure shrinkage, viscoelastic effect, residual stresses

1. INTRODUCTION Thermosetting epoxy adhesives (TEAs) are promising candidates for low temperature joining technologies in automotive, electronics and aerospace applications. This is because they maintain an excellent balance between various properties such as adhesion properties, humidity resistance, heat resistance, and mechanical properties 1,2,3. Long-term reliability of TEA interconnect in harsh environments is crucial, especially in automotive and precision machinery industries, The downside, however, is the low impact toughness and mechanical deterioration of the joint over time, resulting in an increase in the failure of the interconnect4,5 .Therefore, it is essential that reliability predictions take processing history into account in order to correctly address premature failures as well as make sound long-term predictions. A TEA interconnect is manufactured by a sandwich structure of the antenna high-precision panels. Adhesive sticks on the surface of the metallic membrane, which is then placed on a substrate and cured at elevated temperature. Curing of TEAs is a complex process having strong influence on the application range of this class of materials. At the stage, a liquid

* swhu@niaot.ac.cn; phone +08615951907642; fax 85482261; http://www.niaot.ac.cn

Advances in Optical and Mechanical Technologies for Telescopes and Instrumentation, edited byRamón Navarro, Colin R. Cunningham, Allison A. Barto, Proc. of SPIE Vol. 9151, 915132

© 2014 SPIE · CCC code: 0277-786X/14/$18 doi: 10.1117/12.2055397

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polymer gradually transforms into a viscoelastic solid which is caused by cross-linking of the initially short polymer chains. As a result of cross-linking, cured epoxy matrix shrinks, creating cure shrinkage stresses and strains. Next, the assembly is cooled down to room temperature, creating additional stresses due to the mismatch in the coefficients of thermal expansion (CTE). The residual stresses from both sources have a significant effect on the overall performance and lifetime of interconnect and are especially important for understanding premature failure. The studies on reliability evaluation of TEA joints in mechanical applications treat the adhesives as a continuum and focus on the dynamic mechanical analysis6. A comprehensive and fair review on the small and large strain constitutive models for simulating the curing process of resins has been discussed in Hossain et.al 7,8,9. The time evolution of poisson’s ratio has been investigated on the basis of static tests by Bogetti and O’Brien et.al 10,11,12. The incorporation of damage of thermosets is explained by Mergheim.J.13. Kinlge 14 gives an overview of the model for the purely elastic curing in which the free energy density is assumed as a convolution integral consisting of a volumetric and a deviatoric part. An optimal temperature cycle that can reduce residual stresses during cure of composite structures is developed by White and Hahn 15,16. Residual stresses that develop in a nano-Ag ICA interconnect during the assembly of a flip-chip pin grid array are investigated, in which a multiscale modeling framework is adopted to link the nano-sized particles to the interconnect level by Kouznetsova et.al.17. Klinge and Bartels 18 deal with the modeling of viscoelastic and shrinkage effects accompanying the curing of polymers at multiple length scales. For the modeling of viscous effects, the deformation at the microlevel is decomposed into an elastic and a viscoelastic part, and a corresponding energy density consisting of equilibrium and non-equilibrium parts is proposed. Simulations of evolution of cure-induced stresses in a viscoelastic thermoset resin are presented by Bhaskar Patham19. For the simulations, the detailed kinetic and chemo-thermo-rheological models for an epoxy-amine thermoset resin system are employed. Several studies exist on modeling the cure process of commercial epoxies 20,21,22. Previous efforts in modeling TEAs have mainly focused on mechanical analysis, underfill properties and cyclic mechanical degradation by a continuum approach based on viscoelastic characterization of the adhesive mix 23,24. However curing of TEAs addressed at the multiphysics simulations in thin epoxy adhesives, and extending this information to reliability prediction have not received much attention. Additionally, as opposed to thick epoxy adhesives, experimental verification is hard for the thin epoxy adhesives for interconnect applications currently. Hence, the absence of quantitative experimental data hinders the quantitative verification of models for thin epoxy adhesive systems. Accurate predictions of residual stresses in thermosetting epoxy, adhesives, and composites require detailed accounting of multiple phenomena: mold-part interaction, curing shrinkage strains, thermal strains, the kinetics of resin cure, and the evolution of the resin properties with temperature and degree of cure. To describe accurately the evolution of stresses and strains within a thermosetting epoxy resin adhesive during cure, it is imperative to capture in detail the time-cure, and temperature dependent evolution of the resin modulus. Complex kinetic and thermo-chemo-rheological models that account for diffusion effects in kinetics and rheological complexity of the curing resin have traditionally been difficult to implement in finite element schemes, necessitating simplifying assumption. With the development of advanced computational methods involving multiphysics capabilities this complexity may be addressed in a logical fashion. In this study, multiphysics based finite element simulations of evolution of cure-induced residual stresses that develop in a thin TEA interconnect during bonding a metallic membrane on the substrate system (Fig.1) are presented. The results obtained from this study provide input for the industrial development of these novel systems and materials design for specific applications.

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Metallic membrane

Adhesive:

Chctrstp:

Metallic membrane

Adhesive.- '

1

Fig.1. Schematic of an isolated interconnect from a larger component.

The outline of the paper is as follows. First, the proposed multiphysics framework is outlined. Next, structure of the model is explained. The model geometry and constitutive behavior of epoxy are given. Finally, simulation results on curing stresses and deformations are presented in a comparative manner. Finally concluding remarks are given.

2. SIMULATIONS 2.1. The finite element model A 3D finite element model is created by considering a single interconnect from a commercial metallic membrane on a substrate, as shown in Fig.2. Table 1 gives an overview of the materials and dimensions. In order to reduce computational costs, the epoxy resin adhesive is meshed rather coarse.

Fig.2. Finite element model of a single interconnect from a metallic membrane on a substrate.

The interconnect is assumed to be taken from the adhesive of the package and thus the periodic boundary condition are applied in the rear surface of the substrate. Additionally, the model is free to expand and contract and fixed for rotations. The generated model is meshed using the auto-mesh function of the ANSYS12.0 FEM software, where all the simulations are performed.

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Table 1.Materials and dimensions in the model shown in fig.1.

Component Material Height(mm) Length(mm)

Metallic membrane Aluminum 0.18 200

Adhesive Epoxy resin 0.1 200

Substrate Structural steel 3 200

2.2. Constitutive behavior The materials used in the simulations are aluminum (Al), epoxy and structural steel. All materials, except for epoxy, are modeled linear elastically and the parameters are tabulated in Table 2; all other materials are assumed isotropic. Interfaces between materials are assumed perfect. Table 2.Material parameters used in the simulations

Material Modulus (GPa) Poisson’s ratio CTE(at300k) (10-6K-1)

Strength (MPa)

Aluminum(Al) 71 0.33 23.1 280

Structural steel 200 0.3 12 250

Adhesive Calculated Calculated 55 135 Constitutive model of the epoxy incorporates the dependences of its material behavior on the cure degree as will be described in the following. During the curing, a rubber-like state epoxy gradually transforms into a viscoelastic solid which is caused by cross-linking of the initially short epoxy chains, resulting in a net volumetric shrinkage, known as the cure shrinkage. The effect is a result of chemical reactions, binding forces and decreasing mean distances between the epoxy molecules. Additionally, a high curing compression or tension can also counteract or enhance this effect. However, the last influence is negligible under stress magnitudes which occur during technical epoxy curing such that shrinkage in this context can be considered as an autogenetic process. This transformation is weighed by the cure α，where the degree of cure α∈ [0,1] is a time-dependent parameter controlling the influence of curing on the shrinkage. The check of the rate of curing yields the following relationships 25.

( )( )1 2 1 nmd K Kdta

a a= + - , 1,2

1,2 1,2 exp AEK A

RT-æ ö

= ç ÷ç ÷è ø

(1)

Where m and n denote experimentally determined constants, A is the reaction rate. R is the universal gas constant. EA is the activation energy and T is the Kelvin temperature. The cure parameters are tabulated in Table3. And the cure curve obtained with these parameters is plotted in Fig3.

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100 s_p___90

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Fig.3. Evolution of the cure degree (a ) at different temperature

Table3. Cure kinetics parameters for epoxy 17.

m n A1(s-1) A2(s-1) EA1(J mol) EA2(J mol)

0.75 1.25 1.411x103 2.359x106 4.920x104 6.638x104

For this modeling, a sum decomposition of the total strain tensor tote into the cure shrinkage strain ce , the thermal strain

the and the elastic strain ele .

tot c th ele e e e= + + (2)

In this proposal, the maximum magnitude of the shrinkage volume used in this study is 6%. The thermal dynamical description of the viscoelastic curing including shrinkage require the following assumption: the magnitude of the shrinkage volume is related to the cure degree α, the cure shrinkage strain tensor can be expressed as follows:

( ) ( )( )13

max, 1 , 1c ct T t T Ie a eæ ö

= - -ç ÷è ø

(3)

For the sake of deriving the model used to describe the thermal strain, the following assumption that the CTE of the epoxy is independent of the cure degree is considered. The thermal strain in this case haves the following form:

thTd dTIe b= (4)

Where βT represents the CTE and dT is the change in temperature. The derivation of the model for the stress evolution equation is based on Hooke’s law for linear elasticity. Since the small strains are is considered. The incremental form of Hooke’s law is given as:

( )( ) ( )( ) ( )( ), . : ,eld t T C t T d t Ts a a e a= (5)

Where the fourth-order tensor C(α(t, T)) can be decomposed into the cure degree dependent shear (G) and bulk (K)

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7

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The experimental results for the bulk modulus in the non-cured(α=0) and fully cured(α=1) states are explained as K0

=3.0Gpa and K ¥ =6.3Gpa respectively26. The measured evolution modulus as a function of the cure degree is obtained by a perfect mixture assumption.

( ) ( ) 01 ( , )K K t T Ka a a ¥= - + (7)

The evolution of the shear modulus has been calculated according to the following recurrence formula 26.

( )( )

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for ap and bp . gela denotes the so-called gel point taken as gela =0.55. Fig4 displays the bulk and shear modulus as

function of time during epoxy curing at T1=90℃, T2=120℃ and T3=180℃. The evaluation of another material parameter (i.e. Young’s modulus) can be obtained using the numercal interconversion or if the data on another combination of the parameters is known. For example, a common situation is that the data on shear and bulk modulus is available and used for the evaluation of Young’s modulus and Poission’s number.

(a) Bulk modulus (b) Shear modulus

Fig.4. Evolution of the (a) bulk and (b) shear modulus of the epoxy as a function of time during curing at different temperature.

2.3. The assembly process The process of bonding a metallic membrane on a substrate is simulated. Bonding is achieved by curing the epoxy resin in the TEA. The temperature function includes the heating phase with increasing load, at which the assembly is assumed to be stress free, heating the uncured assembly from room temperature T=22℃ to the three different cure temperature

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.111M8 ti 8

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T1=120℃, T2=150℃, and T3=180℃. The holding phase with constant load for 1000s at which the resin will cure into a solid material and fix the connection is applied. And the last phase with the temperature cooling down to T=20℃(fig.5). The process is applied to the samples consisting of different material, and the calculated stresses are used to qualify the influence of individual material parameters. At each processing step, the transient effects are neglected and the specified temperature is assumed to be reached instantaneously.

Fig.5. Prescribed temperature as a function of time during bonding process

2.4. Effect of temperature on the equivalent von Mises stresses of TEA The first group of results obtained here deals with stress distribution over the epoxy resin three time steps in which the individual loading phases start or finish (Fig.6, Fig7 and Fig 8). The comparison of Fig.6a and b(or alternatively Fig.6c) indicates that the viscoelastic curing, in contrast to elastic curing27, is characterized by the stress change during the holding phases. The interfaces between dissimilar materials suffer most from the CTE mismatch and from the stresses caused by the temperature changes. In fig.6a these stresses are plotted by displaying the equivalent von Mises stress during the temperature is increased to the cure temperature. In fig.6b, the stress distribution resulting from the fast time-dependent material parameters during holding temperature is also observed. After the assembly is cooled down, these stresses decrease by about 70% in Fig.6c. The stresses in the TEA part are very low at the beginning of isothermal curing, since the epoxy resin is in a rubber-like state. When the gel point is reached, the stresses start to increase significantly. At the end of curing the maximum stresses in the TEA material are 93Mpa when the temperature is 120℃. After cooling, the metallic membrane tends to return to its original state but is restricted by the now solid interconnect, resulting in residual stresses. For comparison, the results obtained using the two other temperatures (150℃ or 180℃) are also shown in fig.7.and fig.8. However, it is worth noticing these high stress regions during curing, as they can lead to cracking or delamination. This observation is even more distinct if the stress state at an integration point is studied. For this purpose, point A in the middle of the epoxy resin (indicated in fig2.) is chosen and the stress state with respect to the time and with respect to the applied temperatures is monitored (Fig.9). Since the fist stress phase coincides with the beginning and the second loading phase with the middle of the curing process, the time parameters evolve fast and the stress response in both loading phases is nonlinear. At the end of this phase the stress state tends to the stationary value coinciding with the value typical of the curing process without considering viscous effects. The comparison of diagrams corresponding to different temperature indicates that lower temperature is related to less distinct stress.

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(t=1025s, T=22℃).

(a) Heating up (b) Curing (c) Cooling down

Fig7. different stresses (Mpa) in the model: (a) heating up (t=1s, T=73℃). (b) isothermal curing (t=600s ,T=150℃). (c) cooling down

(t=1025s, T=22℃).

(a) Heating up (b) Curing (c) Cooling down

Fig.8. different stresses (Mpa) in the model: (a) heating up (t=1s, T=85℃). (b) isothermal curing (t=600s ,T=180℃). (c) cooling down

(t=1025s, T=22℃).

(a) Heat up and Curing (b) Cool down

Fig.9. Equivalent von Mises stresses (Mpa) at node A obtained from the model

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2.5 Effect of temperature on the x-direction displacement of TEA The x-direction displacements in the TEA at different moments during the processing time line are monitored in Fig.10 and 11 (or alternatively Fig.12). At the beginning of curing, the x-direction displacement is increased as the scale of endless belt in the vertical direction increase. During curing, the inner parts along with the endless belt are affected by the shrinkage TEA. After cooling, ductile shear zone is observed near the two side surface edges of TEA. It is worth noticing these obvious phenomena, as they can lead to the surface figure error or the contraction of the metallic membrane by the now stiff TEA interconnect, resulting in a permanent deformation at the end of the assembly. These deformations are, however, very small, due to the small dimensions. The x-direction displacement as a function of time at point A is shown in Fig.13. The x-direction displacement decreases linearly as environment temperature rises in a special range by the shrinkage. And the value shows a linear increase up to the maximum, followed by a linear decrease until it closes to the plateau. For comparison, the results obtained using the two other temperature are simulated and both resulting diagrams show that although the time nodes have different activated time, they have similar x-displacement variation history. The comparison of diagrams corresponding to different temperature indicates that the lower temperature is related to less distinct shrinking effects.

(a) 550s (b) 1000s (c) 1025s

Fig.10. Distribution of x-direction displacement for time steps: (a) t=550s, T=120℃; (b) t=1000s, T=120℃; (c) t=1025s, T=22℃

(a) 220s (b) 1000s (c) 1025s

Fig.11. Distribution of x-direction displacement for time steps: (a) t=220s, T=150℃; (b) t=1000s, T=150℃; (c) t=1025s, T=22℃

(a) 220s (b) 1000s (c) 1025s

Fig.12. Distribution of x-direction displacement for time steps: (a) t=220s, T=180℃; (b) t=1000s, T=180℃; (c) t=1025s, T=22℃

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2.6 Effect of poisson’s ratio The influence of poisson’s ratio is checked in the analogous way: the test shown in Fig.6 is repeated for materials with different values of this parameter lying in the range 0.2-0.499. The von Mises stress distributions for the different poisson’s ratio are shown at the end of the cure holding in Fig.14. In Fig.15a, the time-dependent evolution of poisson’s ratio is depicted. All time step sizes provide similar results. The considerable increase of poisson’s ratio at the beginning of curing process is correctly reflected and in the later phase of the curing process the poisson’s ratio tends to stationary value. Although the time-dependent poisson’s ratio evolve fast, the stress response as a function of time at point A in all time phases is nonlinear and gradually increases which is known as the relaxation phenomenon(Fig.15b). The subsequent comparison of corresponding diagrams leads to the conclusion that higher poission’s ratio corresponds to higher stresses. However, this conclusion remains valid for better surface figure error which is the most favorable from a high precision surface point of view.

(a) (b) (c)

(d) (e) (f)

Fig.14. Comparison of the stress state for materials with different time-dependent poisson’s ratio: (a) 0 0.01u = ； 0.2u¥ = ; （b）

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0 0.01u = ； 0.499u¥ = ;

(a) time-dependent poisson’s ratio (b) von Mises stresses

Fig.15. (a) time-dependent poisson’s ratiou VS. time from 0.2-0.499. (b) von Mises stresses at point A for materials with different

poisson’s ratio in the range 0.2-0.499.

2.7 Effect of adhesive thickness The final example studying the behavior of homogeneous TEA deals with the effect of its thickness on peel stress. The peel stress behavior of the TEA interconnect with the influence of an external load on the metallic membrane is observed in Fig.16. It is also observed that local stress concentration occurs in the beginning and middle part of adhesive thickness. These high stress regions are located at the corner of interfaces. These local high stresses will prossibly forebode damage at the interfaces and thereby interrupt the strength of interconnect in the use of actual situations. The investigation encompasses viscoelastic curing epoxy with different thickness (Fig.17). The obtained values show that the stress rate (at point A) in the first phase (0.1-0.2) is low. Different from this, the later stage with increasing thickness is characterized by a high increase of stress which now tends to a maximum value (0.2-0.3), whereas the situation is opposite in the final phase. This endorses the conclusion that a neglecting of the evolution of adhesive thickness might lead to a significant underestimation of the peel stress values after the curing process.

(a) 0.1mm (b) 0.2mm (c) 0.3mm

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the metallic membrane.

Fig.17. Equivalent von Mises stresses (MPa) at node A for varying adhesive thickness

3. Conclusions In this paper a mutiphysics FEM approach to predict the residual stresses that appear in an isotropically thermosetting epoxy resin adhesive during the assembly process has been presented. Boding of a metallic membrane on a substrate by TEA interconnect with different situations in the epoxy matrix was simulated. The development of residual stresses at different processing steps has been investigated for varying the temperature, possion’s ratio and cure pressure. And the influence of adhesive thickness on peel stress is modeled simultaneously. The simulation results can be summarized as follows: The residual stresses after the assembly are a combination of stresses caused by the cure shrinkage of the epoxy in the interconnect and the large CTE mismatch between metallic membrane and substrate. The stresses corresponding to loading phases show the same feature as in the case of viscoelastic curing, namely the nonlinear growth due to the time evolving material parameters. The temperature arrangement in the TRA results in stress bands across the two vertical sides of interconnect. Decohesion of metallic membrane and epoxy cracking impair the mechanical integrity and threaten the functioning of the interconnect. The new effect observed here is the x-direction displacement decrease during the holding phase, which is known as the shrinkage effects. Furthermore, the simulation of the shrinkage together with the mechanical influences indicates that this phenomenon can eliminate the shrinkage effects and cause a later increase of x-direction displacement during the holding phase. Increasing the poisson’s ratio of adhesive from 0.2 to 0.499 increases

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the residual stresses at the interconnect by approximately 65%. However, the better surface figure error, the most desired case, is observed to increase with increasing poisson’s ratio. The corresponding peel stresses depending on the thickness of adhesive is observed. Lowest stresses occur in the beginning and end of the adhesive thickness procedure. The achieved results certainly indicate that the model can be applied successively for multiphysics simulation. In the future, the approach presented in this paper will be extended to incorporate damage and decohesion. Additionally, it is necessary to devise experimental methods to estimate residual stresses in these materials that can address the adhesive interconnect. In this way, the calculation can be fully validated.

Acknowledgements This research was supported by the Natural Science Foundation of China under Grant Number 11173039 and by the Key Laboratory of Astronomical Optics & Technology, Nanjing Institute of Astronomical Optics& Technology, Chinese Academy of Sciences.

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