Méthodologie probabiliste basée sur la simulation pour l

19e Congrès de Maîtrise des Risques et Sûreté de Fonctionnement - Dijon 21-23 octobre 2014

METHODOLOGIE PROBABILISTE BASEE SUR LA SIMULATION POUR L’EVALUATION DE LA FIABILITE DES SYSTÉMES MECATRONIQUES

EMBARQUÈS

SIMULATION-BASED PROBABILISTIC METHODOLOGY FOR RELIABILITY ASSESSMENT OF EMBEDDED MECHATRONIC SYSTEMS

AOUES Y., MAKHLOUFI A. et EL-HAMI A. POUGNET P. INSA de Rouen-LOFIMS VALEO 685 Avenue de l’Université BP 08, 14, avenue des Béguines, 76800 Saint Etienne de Rouvray. 95892 Cergy-Pontoise.

Résumé La durée de vie des systèmes mécatroniques embarqués est souvent liée à la défaillance des joints de brasure des composants électroniques par fatigue thermique. Les directives européennes actuelles interdisent l’utilisation des brasures à base de plomb et en recommandant leurs remplacements par des alliages sans plomb. Toutefois, la caractérisation de ces nouveaux alliages n’est pas encore complètement effectuée. De plus, la durée de vie de ces brasures est souvent étudiée par des modèles déterministes. Ces modèles combinent la caractérisation expérimentale et une modélisation physique du phénomène de fatigue nécessitant l’utilisation des méthodes de simulation telle que la méthode des éléments finis. Cependant, ces modèles ne considèrent pas les incertitudes des paramètres du modèle et des dimensions géométriques et la fluctuation des chargements thermiques et des conditions d’opération en service. En effet, l’évaluation de la fiabilité des composants électroniques des systèmes embarqués doit s’appuyer sur des approches probabilistes. Dans cette communication une approche probabiliste est développée pour l’estimation de la durée de vie probabiliste des joints de brasure d’un système mécatronique embarqué. Cette méthodologie combine la modélisation probabiliste des paramètres d’entrée, la modélisation numérique non linéaire de la fatigue thermique du joint de brasure, l’approximation du modèle numérique par un metamodèle de krigeage et l’utilisation des simulations de Monte Carlo pour l’estimation de la distribution probabiliste de la durée de vie en fatigue. Summary Field failure of embedded mechatronic systems is often caused by the thermal fatigue failure of the solder joint of the electronic components. Current European guidelines prohibit the use of the lead solder in the electronic products. New lead-free alloys are used to substitute the SnPb Solder. However, these lead-free alloys, generally named as SAC (SnAgCu) is not yet been fully characterized. The fatigue lifetime of the solder joint is usually evaluated by deterministic models. These models combine experimental characterization, physical modeling of the fatigue failure that requires numerical simulation like as the finite element method. However, these models do not take into account the uncertainties of model and design parameters and the variability of operating conditions in service. In order to assess more accurately the reliability of critical solder joints of an automotive mechatronic device, a probabilistic approach is developed. This methodology combines nonlinear finite element simulation of the solder joint, kriging approximation of the numerical model and Monte Carlo simulations in order to evaluate the probability distribution of the fatigue lifetime.

1. Introduction

Embedded electronic systems have a role increasingly growing up in transport: electric and hybrid vehicles, trains and aircraft. For these applications, safety and reliability are a critical point. New packaging and assembly technologies and requirements to comply with environmental friendly directives create challenges in assuring the reliability of mechatronic products for higher performance and lower cost. Larger sized or more integrated components, finer pitches, and materials for reliability requirements characterize the new packages. Mechatronic packages operate in hostile environments including wide temperature ranges and vibrations in which multiple interactive damage mechanisms occur. Field returns show that most reported failures in automotive mechatronic devices are due to the fatigue failure of solder joints. In order to meet reliability requirements the robustness of solder connexions should be validated.

Due to thermal expansion and stiffness mismatch of the different materials of the device assembly, thermal cycles induce mechanical stresses in the solder joints connecting electronic components to the circuit board. When the device is operational and subjected to external environmental conditions, these stresses can be significant leading to plastic deformation due to the visco-plastic behavior of the solder. Accumulated plastic strain can cause damage leading to crack initiation and to solder joint failure. Consequently, to avoid field failures and reduce warranty costs, robust validation of critical solder joints should be done as early as possible in the development process of an automotive mechatronic device. Lead-free alloys, generally alloys of SnAgCu composition called SAC have slightly different mechanical properties than eutectic SnPb alloys. Engelmaier (2003) has reported that lead-free solder joint reliability suffers from problems due to their creep rate, which is 100 times slower than the creep rate of eutectic tin-lead joints. Experimental studies have reported that tin-lead and lead-free solder joint lifetime depend on the package type and also on the applied loading conditions (Vandevelde et al.,2007). For example, for certain packages the

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reliability of lead-free assemblies is lower thigher.

The reliability prediction of the mechatronic devices is usually estimated through the mean time between failures (MTBF). Thicalculation is based on the handbook methods (FIDES, MILcomponent constituting the mechatronic device which are statistically obtained from field data. It is well documented that thestimation can lead to poor prediction (Lu et al., 2009)failure mechanisms should be well understood, modelling the failure mechanisms by combining mathematical modelling with accelerated life testing to predict the reliability of mechatronic devices (Wymysłowski et al., 2007failure mechanisms in solder joints subjected to temperature cycling to predict the solder joint fatigue life. These simulation tools are based on a deterministic approach which does not take into account the variability of input parameters and the uncertaintrelated to design, loading, and operational conditions. A more rigorous approach for predicting the solder joint fatigue lifetime athe reliability of mechatronic devices should take into account the uncertainties arising from the random nature of the temperature fluctuations caused by power transients and thermal environment changes, the thermal expansion mismatch of the different materials of the assembly, the material properties and the fabrication process.

The objective of this study is to develop a statistunder cyclic thermal loading environments. This paper presents a study package in an Engine Control Unit (ECU)are analyzed. In the present work, a stochastic finite reliability of SnPb, SAC305 and SnAg3.5 solder temperature cycling loading. Monte Carlo Simulations predicted lifetime whereby illustrating the rischaracteristics. The variability of the therm

In order to predict reliability in service it is a good practice inphysics of failure approach and have a deeper understanding of the failure mechanisms. Experience shows that faautomotive electronics assemblies are often caused by thermal fatigue dasolder joint reaches a certain critical level, this leads to an electrical or a mechanical failure. The physics of failure aprequires models that predict the stress and strains in the packaging materisubjected to external environmental conditions. Finite Element Method (FEM) is often used to predict the mechadegradation. This degradation evolves with repeated thermal and power cyfissure propagation and interconnect failurewith 0.5 mm pitch gull-wing leads, which is assembled on a board fixed in a tmechatronic device is mounted in a vehicle under the hood. Figure 1model. The model is based on the geometry of the Printed Circuit Board (PCB), the mouldinterconnects and gull-wing leads.

Figure 1. a) Global finite element model of the 256

2.1 Solder joint finite element model

Calculus of the global model for only one thermal cycle time, a local finite element model of the worst of a single solder joint and different layer of materials, respectively the FR4, the copper lead andshown in figure 1 b). The 3D finite element solder joint model allows us to model precisely theconsists in describing the worst case solder jointsolder material is assumed to have a rate dependecycles, where the first thermal cycle is illustrated in figure 2

ee assemblies is lower than for tin-lead, and for other packages the reliability of the lead

The reliability prediction of the mechatronic devices is usually estimated through the mean time between failures (MTBF). Thicalculation is based on the handbook methods (FIDES, MIL-HDBK-217F…etc.) and on individual failure rates for each component constituting the mechatronic device which are statistically obtained from field data. It is well documented that th

to poor prediction (Lu et al., 2009). However, in order to predict the reliability of mechatronic devices, the failure mechanisms should be well understood, in order to predict correctly the lifetime. The physical failure approach aims to

he failure mechanisms by combining mathematical modelling with accelerated life testing to predict the reliability of Wymysłowski et al., 2007). The finite element method (FEM) is generally used for

nisms in solder joints subjected to temperature cycling to predict the solder joint fatigue life. These simulation tools are based on a deterministic approach which does not take into account the variability of input parameters and the uncertaint

d to design, loading, and operational conditions. A more rigorous approach for predicting the solder joint fatigue lifetime athe reliability of mechatronic devices should take into account the uncertainties arising from the random nature of the

ure fluctuations caused by power transients and thermal environment changes, the thermal expansion mismatch of the different materials of the assembly, the material properties and the fabrication process.

y is to develop a statistically based simulation model that can predict the fatigue failure ic thermal loading environments. This paper presents a study of the reliability of critical solder joints in an electronic

trol Unit (ECU). Several solder materials eutectic Sn63Pb37, SAC305 (Sn96.5Ag3Cu0.5) and esent work, a stochastic finite element approach is employed for a comparative

SnAg3.5 solder alloy compositions for gull-wing leads in the PQFP package under harMonte Carlo Simulations (MCS) enable the calculation of the probability distribution de

whereby illustrating the risks to product lifetime of using materials and geometries and loThe variability of the thermal loading impacts strongly the lifetime prediction.

2. Finite Element model

In order to predict reliability in service it is a good practice in the electronics industry to combine accelerated testing and a physics of failure approach and have a deeper understanding of the failure mechanisms. Experience shows that fa

assemblies are often caused by thermal fatigue damage. For example, when damage accumulation in a solder joint reaches a certain critical level, this leads to an electrical or a mechanical failure. The physics of failure aprequires models that predict the stress and strains in the packaging materials of the assembly when the device is operational and subjected to external environmental conditions. Finite Element Method (FEM) is often used to predict the mecha

This degradation evolves with repeated thermal and power cycling and ultimately causes solder joint cracking, fissure propagation and interconnect failure. The electronic package under investigation is a 256-pin plastic quad flat package

wing leads, which is assembled on a board fixed in a tight metallic housing. In application, this whole mechatronic device is mounted in a vehicle under the hood. Figure 1a) shows a perspective view of the meshed 3D global model. The model is based on the geometry of the Printed Circuit Board (PCB), the moulding compound and its solder

Global finite element model of the 256-PQFP b) Geometry of the local solder joint model.

odel for only one thermal cycle has a very high computational cost. In order to reduce the nite element model of the worst case solder joint is developed. This 3D geometric model is

layer of materials, respectively the FR4, the copper lead and the molding compound, as element solder joint model allows us to model precisely the inelastic strain of the solder jo

the worst case solder joint with fine mesh elements and in taking into account material nonlinearities. The is assumed to have a rate dependent plasticity. The solder joint finite element model is solved

cycle is illustrated in figure 2 a).

a)

packages the reliability of the lead-free assemblies is

The reliability prediction of the mechatronic devices is usually estimated through the mean time between failures (MTBF). This ) and on individual failure rates for each

component constituting the mechatronic device which are statistically obtained from field data. It is well documented that this However, in order to predict the reliability of mechatronic devices, the to predict correctly the lifetime. The physical failure approach aims to

he failure mechanisms by combining mathematical modelling with accelerated life testing to predict the reliability of ) is generally used for modelling the physical

nisms in solder joints subjected to temperature cycling to predict the solder joint fatigue life. These simulation tools are based on a deterministic approach which does not take into account the variability of input parameters and the uncertainties

d to design, loading, and operational conditions. A more rigorous approach for predicting the solder joint fatigue lifetime and the reliability of mechatronic devices should take into account the uncertainties arising from the random nature of the

ure fluctuations caused by power transients and thermal environment changes, the thermal expansion mismatch of the

can predict the fatigue failure of solder joints solder joints in an electronic

37, SAC305 (Sn96.5Ag3Cu0.5) and SnAg3.5 omparative study of the mechanical

the PQFP package under harsh the probability distribution density of the

materials and geometries and loads of unknown

the electronics industry to combine accelerated testing and a physics of failure approach and have a deeper understanding of the failure mechanisms. Experience shows that failures in

mage. For example, when damage accumulation in a solder joint reaches a certain critical level, this leads to an electrical or a mechanical failure. The physics of failure approach

als of the assembly when the device is operational and subjected to external environmental conditions. Finite Element Method (FEM) is often used to predict the mechanical solder joint

cling and ultimately causes solder joint cracking, pin plastic quad flat package

ight metallic housing. In application, this whole shows a perspective view of the meshed 3D global

ing compound and its solder

Geometry of the local solder joint model.

al cost. In order to reduce the computational ped. This 3D geometric model is based on the geometry

the molding compound, as inelastic strain of the solder joint. It

taking into account material nonlinearities. The finite element model is solved for five thermal

b)



Figure 2. a) Cyclic temperature profile.

The commercial finite element analysis (FEA) software ANSYS V.13 was used for the nonlinear finite element computation. Figure 2 b) shows the finite element mesh of the solder joint model. As the temperature loading is time dependent, several substeps are required to carry out the nonlinear analysis and to ensure the convergence. For instance, the local model is as well time-consuming. However, the correctness of the finite element analysis is significantly dependent on the mesh density and the substeps iterations, accuracy of material properties used in the model and suitability of constitutive material models. Some of the assumptions in the numerical model are made: all materials including the solder joint were assumed homogeneous; intergrowth was not considered; linear elastic materials except the solder joint were is assumed visco

2.2 Material properties and thermal loading The thermo-mechanical behaviour of FR4 PCB board is assumed as orthotropic, linear elastic and temperature depenproperties. In order to obtain accurate simulation, temperature dependent material properties obtained from the stress strain curves, Hence, a statistical method incorporating linear regression was employed to quantify the temperaturemechanical properties of FR4 PCB board

where T is the current temperature (°K), EAnd Ez is the Young's modulus in the z directionThe copper lead is assumed isotropic linear elasti

where T is in °K. All the solder materials SnAg3.5, Sn67Pb37, behavior, as creep plays a very important role in deThe Young’s Modulus of SnAg3.5 (Lu et al., 1997) and SnPb (Mao et a

3.5

SnPb

SnAg

E T T

E T T T

All material properties shown in Table.1 are given for the reference temperature 20of plastic strains is dependent on the rate of loading(SnPb) solder joints during creep and proposedincorporates viscoplasticity and time-dependbehavior of SnAgCu solder materials which are referred to the Anand widely used for electronic solder applications in FEA of thermomodel is expressed by a flow and three evolution equations below.

in Aε ξ=&

where �� is the inelastic strain rate, A the preabsolute temperature, σ the tensile stress, deformation resistance. In addition, �� is an internal state variable

Cyclic temperature profile. b) Local finite element model of the solder joint

The commercial finite element analysis (FEA) software ANSYS V.13 was used for the nonlinear finite element computation. shows the finite element mesh of the solder joint model. As the temperature loading is time dependent, several required to carry out the nonlinear analysis and to ensure the convergence. For instance, the local model is as well

consuming. However, the correctness of the finite element analysis is significantly dependent on the mesh density and the rations, accuracy of material properties used in the model and suitability of constitutive material models. Some of the

assumptions in the numerical model are made: all materials including the solder joint were assumed homogeneous; inters not considered; linear elastic behavior with temperature dependent properties was assumed for the package

materials except the solder joint were is assumed visco-plastic-creep material behavior.

Material properties and thermal loading

hanical behaviour of FR4 PCB board is assumed as orthotropic, linear elastic and temperature depenIn order to obtain accurate simulation, temperature dependent material properties obtained from the stress strain

Hence, a statistical method incorporating linear regression was employed to quantify the temperatureFR4 PCB board. The constitutive relation, as given by (Mao et al., 2008):

( ) ( )( )

27924 37 (MPa)

12204 16 (MPa)X Y

Z

E T E T T

E T T

= = −= −

Ex the Young's modulus in the x direction, Ey is the Young's modulus in the direction. The molding compound material is assumed to be isotropic and linear elastic.

linear elastic and temperature dependent (Otiaba et al., 2013) given by:

( )( )

15.64 0.0041 (ppm/K)

141.92 0.0442 (GPa)

Copper

Copper

T T

E T T

α = +

= −

er materials SnAg3.5, Sn67Pb37, SAC305 (Sn96.5Ag3Cu0.5) are assumed with elastoays a very important role in deformation behavior of solder joint at homologous temperature of SnAg3.5 (Lu et al., 1997) and SnPb (Mao et al., 2008) are considered temperature dependent given by

( )( ) 2

3.5

75842 152 (MPa)

52708 67.14 0.0587 (MPa)

SnPbE T T

E T T T

= −

= − −

are given for the reference temperature 20 °C. In solder joint materials, thedent on the rate of loading. Many authors have studied the response of lead free (SnA

joints during creep and proposed constitutive equations. One of the equations developed is Anand’s dependent plasticity. (Wang et al., 2001) proposed a unified framewor

which are referred to the Anand constitutive equations. The viscoplastic Anandwidely used for electronic solder applications in FEA of thermo-mechanical behavior of solders (Chengmodel is expressed by a flow and three evolution equations below.

1

exp sinh

QmRTA

s

σε ξ − =

the pre-exponential factor, Q the activation energy, R the universal gas constant, the tensile stress, ξ the stress multiplier, m the strain rate sensitivity of the stress and

is an internal state variable whose evolution is described by:

�� 1 ��∗�

� sign �1 �

�∗��

Local finite element model of the solder joint.

The commercial finite element analysis (FEA) software ANSYS V.13 was used for the nonlinear finite element computation. shows the finite element mesh of the solder joint model. As the temperature loading is time dependent, several required to carry out the nonlinear analysis and to ensure the convergence. For instance, the local model is as well

consuming. However, the correctness of the finite element analysis is significantly dependent on the mesh density and the rations, accuracy of material properties used in the model and suitability of constitutive material models. Some of the

assumptions in the numerical model are made: all materials including the solder joint were assumed homogeneous; inter-metallic with temperature dependent properties was assumed for the package

hanical behaviour of FR4 PCB board is assumed as orthotropic, linear elastic and temperature dependent In order to obtain accurate simulation, temperature dependent material properties obtained from the stress strain

Hence, a statistical method incorporating linear regression was employed to quantify the temperature dependent

{1}

Young's modulus in the y direction The molding compound material is assumed to be isotropic and linear elastic.

given by:

{2}

SAC305 (Sn96.5Ag3Cu0.5) are assumed with elasto-plastic-creep joint at homologous temperature (Syed, 2007).

considered temperature dependent given by:

{3}

joint materials, the development response of lead free (SnAgCu) and eutectic

the equations developed is Anand’s model which proposed a unified framework for the viscoplastic

The viscoplastic Anand model is Cheng et al., 2001) the Anand

{4}

the universal gas constant, T the the strain rate sensitivity of the stress and s is the

{5}



where h0 is the hardening or softening constant, a the hardening or softening strain rate sensitivity and s* is the saturation value of associated with a given temperature and strain rate pair and is described by:

* ˆ exp

nQ

RTins sA

ε

=

& {6}

where �̂ is a coefficient for saturation and n is the strain rate sensitivity for s*.

Parameter description Mean value Coefficient of variation

Distribution

FR4 Young Modulus EX= EY (GPa) Eq.1 0.05 Lognormal

FR4 Young Modulus EZ (GPa) Eq.1 0.05 Lognormal

FR4 CTE αx = αy (ppm/°K) 16 0.05 Lognormal

FR4 CTE αz (ppm/°K) 84 0.05 Lognormal

Copper lead Young Modulus (GPa) Eq.2 0.05 Lognormal

Copper lead CTE (ppm/K) Eq.2 0.05 Lognormal

SAC305 Solder Young Modulus (GPa) 38.7 0.05 Lognormal

SAC305 solder CTE (ppm/K) 21 0.05 Lognormal

SnAg3.5 Solder Young Modulus (GPa) Eq.3 0.05 Lognormal

SnAg3.5 solder CTE (ppm/K) 22.25 0.05 Lognormal

SnPb Solder Young Modulus (GPa) Eq.3 0.05 Lognormal

SnPb solder CTE (ppm/K) 23.75 0.05 Lognormal

Liquidus temp. of SnPb (°C) 183 - deterministic

Liquidus temp. of SnAg3.5 (°C) 221 - deterministic

Liquidus temp. of SAC305 (°C) 219 - deterministic

Table 1. Material properties and statistical data of random variables.

The material parameters of the Anand’s model for SAC305, SnPb and SnAg3.5 solders are obtained from experimental results and by the separated elasto-plasto-creep constitutive relations. There are nine constants to be curve-fitted to the Anand model. ANSYS has an option for viscoplastic analysis using Anand’s model to describe rate-dependent material behavior with large strain solid, solid185 element, for three-dimensional large strain.. When using Anand’s model, nine material constants from C1 to C9 are required. These material parameters are shown in Table 2 as given by many authors. The accelerated thermal profile testing recommended from JEDEC accelerated thermal cycling standards (JEDEC, 2009) is applied as the thermal loading in the finite Element analysis. These harsh temperatures vary between-40 °C and 125 °C with 30 min dwell at the peak and lowest temperature, the ramp rate is 15 °C/min are applied. Devices for automotive applications are typically tested within this temperature range. Figure 3 shows the cyclic temperature profile. Furthermore, this thermal cycling range can be suitable for testing microelectronics applications under the hood of automotive (Otiaba et al., 2013). In the FEM calculus the material solders were assumed as stress free at the liquidus temperature. The first step consists to simulate the process of reflow soldering to take into account the initial stress. This process consists to apply the temperature profile going from the melting temperature of solder joints to room temperature of 25°C in 150 seconds. In a second step, five thermal cycles between -40 °C and 125 °C are applied as loading condition in the finite element model.



Solder A (s-1) Q/R (K) ξ m �̂ (MPa) n h0 (MPa) a s0 (MPa)

SnPb (Liu et al, 2008) 4×106 9400 1.5 0.303 13.79 0.07 1378.95 1.3 12.41

SnAg3.5 (Wang et al., 2007)

2.23×104 8900 6 0.182 73.81 0.018 3321.15 1.82 39.09

SAC305 (Otiaba et al., 2013)

5.87×106 7460 2 0.0942 58.3 0.015 9350 1.5 45.9

Table 2. Material parameters of Anand’s model.

2.3 Coffin-Manson Fatigue model

Most failures in mechatronic devices are due to solder joints fatigue caused by the thermomechanical damage mechanisms during the component operation life. Solder joint fatigue life prediction involves combining finite element simulations with a thermal fatigue model. The fatigue model is generally obtained by using experimental data and accelerated testing. This model is used to determine the number of cycles that a component can resist before fatigue failure.

Several models have been proposed to predict solder joint fatigue life and are often classified into four categories: stress-based, plastic and creep strain-based, energy-based, and damage-based. Generally the stress-based approach is used in the high cycle fatigue where it is applied to vibration, where the mechanical behavior remains largely elastic. In low cycle fatigue, plastic and creep strain-based approach are used. Thermal cycles lead to permanent plastic deformation. This deformation is often due to the viscoplastic behavior of the solder, thus the total strain can be decomposed into creep strain and plastic strain. The plastic-creep strain fatigue model is based on the inelastic deformation derived from the time-independent plastic material behavior (plastic strain) and the time dependent effects (creep strain). The energy-based fatigue models are based on the calculation of the overall stress/strain hysteresis energy, where the damage-based fatigue models are based on computing the accumulated damage caused by crack propagation and fracture mechanics (Lee et al., 2000).

Since inelastic strain (creep and plastic deformation) is a dominant parameter that influences low-cycle fatigue, this study uses the Coffin-Manson type equation classified into the plastic-creep strain-based approach. The inelastic strain is computed by nonlinear finite element simulation with the unified viscoplasticity Anand’s model, which combines plastic and creep deformation. The Coffin-Manson fatigue model is the most widely used approach, where the number of cycles to failure N_f depends on the inelastic strain amplitude εin generally given as the following equation:

Δ�� {7}

Where ∆εin is the inelastic strain range, Nf is the number of cycles, α is the fatigue ductility exponent and θ is the fatigue ductility coefficient. A variety of strain life engineering data are available on most of the solders used in electronic packaging (Shang et al., 2007). In this study, the parameter coefficients of the fatigue model identified by Kanchanomai et al. (2002a) for SAC305, SnAg3.5 and SnPb (Kanchanomai et al, 2002b) are used. Table 3 gives these parameters of the studied solders. Generally the fatigue model parameters are matched to the experimental data which are highly dispersed because of their limited number. The natural variability of these parameters should be taken into account in fatigue life prediction. In this work, these parameters are considered as random variables modeled by a lognormal distribution with mean values given in Table 3 and a coefficient of variation of 2%.

Solder α θ

SnPb [14] 0.68 0.85

SAC305 [11] 0,73 3,7

SnAg3.5 [11] 0.93 21.9

Table 3. Fatigue model parameters.

3. Probabilistic methodology

Probabilistic approaches in structural simulation can be classified into three categories: second moment approaches, structural reliability methods and stochastic finite element methods (Sudret et al., 2000). Second moment approaches such as perturbation methods aim to evaluate the two statistical moments of the responses. The structural reliability approach consists in determining the probability of failure of structural systems. The stochastic finite element approach aims at evaluating the global probabilistic structural response quantities. All these probabilistic approaches take into account the uncertainties arising from the random nature of loading fluctuations, geometry dimension and material properties. The mechatronic device may be not reliable if these uncertainties are not considered during product process design. So far, probabilistic methods are used to improve design



robustness and to estimate the impact of parameter uncertainties on the system response. Moreover, combining probabilistic methods to the finite element simulations does not only enable one to establish the scope and the limits of usual deterministic approaches as well as providing more information and effective utilization of empirical data. Stochastic approaches are an extension of the deterministic analysis where the input parameters are considered as random variables instead of having deterministic fixed values. In this paper, a probabilistic methodology is developed to estimate the probability distribution density of the predicted lifetime. This approach is based on surrogate the mechanical model by a kriging metamodel and uses Monte Carlo simulations (MCS) to characterize the random response and the probability density function (PDF) of the response.

3.1 Monte Carlo Simulations

Monte Carlo Simulation (MCS) is a method which is widely used in the probabilistic approaches (Ditlevsen et al., 1996). MCS consists in generating a set of random samples. The mechanical model is run for each of these samples. The obtained results are then used to compute estimators of the response. However, to bring accurate results MCS needs a large set of samples. In the context of thermomechanical nonlinear finite element simulation, the computational cost makes the MCS impracticable. For this reason, the solution consists in building an explicit function which can approximate the finite element response. The kriging metamodeling approximation is adopted in order to surrogate the mechanical model because it allows one to quantify the surrogate error. The global kriging explicit function is then used in Monte-Carlo Simulations.

3.2 Kriging approximation

The stochastic approach computes the response variability of a system when the input parameters themselves vary around their means. The use of Monte Carlo simulations for estimating the probability density function may require very time-consuming computation and becomes impracticable for complex structures when complex computer analysis and simulation codes such as finite element method are involved. Approximation methods are used to build simplified approximations or metamodels providing an alternative to the original finite element model. The most popular metamodel is the response surface methodology, which typically employs second order polynomial approximation using least-square regression techniques. Several authors have used response surface methods in reliability analysis (Kaymaz et al., 2005). Few studies have treated the use of kriging approximation in structural reliability approaches and the propagation of uncertainties (Echard et al., 2013). In this paper, the kriging approximation is used as alternative to the traditional response surface method, to approximate the mechanical model. The main advantage of the kriging approximation is that it gives accurate global approximation while controlling the computational cost and accuracy.

The construction of a kriging approximation model is based on data from a computer experiment. The kriging model then replaces the original computer model. The kriging approximation assumes that the implicit function of the mechanical model is considered as the realization of a stochastic field G(x). The first step is to define the stochastic field parameters according to the computer design of experiments. The model for G(x) is given as: �(�) � !(�, #) + %(�) {8}

where x is a sample point that corresponds to a realization of the random variables X, F(x) is the deterministic part which gives an approximation of the mean response. It shows the trend of the approximation and is represented by a regression model written as: !(�, #) � #&'&(�) + #('((�) + ⋯+ #*'*(�) {9}

where, f(x)={f1(x),…,fp(x)} is the vector of the basis functions and α={α1,…,αp} the vector of regression parameters. z(x) is a stationary Gaussian process with zero mean and covariance between two points of space s1 and s2 defined by: +,-((%(�&), %(�()) � ./(01(�&, �() {10}

where, σz2 is the process variance and Rθ the correlation model with parameters θ. Several models exist to define the correlation

function. Here the anisotropic Gaussian model is used.

01((�&, �() � 2exp7 ��(�& �()(8�

�9& {11}

For a set S of m design of experiments S = [x1, x2,…, xm] where nix R∈ is the ith sample point, where n is the number of random

variables X. Y is the vector of mechanical response for all the set of sample points, as: : � 7G(�<), G(�=),… , G(�?)8 {12}

So, A is the expanded m×p design matrix with Ai,j = fj (xi), as: @ � 7'(�<), '(�=),… , '(�?)8 {13}

The matrix R of stochastic-process correlation between the design experiments is:

0�,A � 01(�B, �C)

{14}

Now, the parameters of the kriging approximation (θ,α,σ_z^2) should be determined. The Maximum Likelihood Estimation (MLE) technique is used as detailed by Lophaven et al. (2002). The likelihood of the data is maximized with respect to the parameters (θ, α, σz

2). The parameters α and σz2can be derived analytically using the Karush-Kuhn-Tucker (KKT) necessary optimality

conditions (called also the first order optimality conditions), where it depends on the auto-covariance parameters θ. Thus, these parameters are estimated by solving the following algebraic equation:

#⋆ � (@EFG&@)G&@EFG&: {15}

./( � &H (: @#⋆)EFG&(: @#⋆)

The best unbiased prediction of the response is:



DACE toolbox (Lophaven et al., 2002) anisotropic Gaussian model for the correlation function are selected. An anisotropic correlation function is preferred in relanalysis studies [18]. The Latin hypercube sampling is used as a strategy for generating random sample points ensuring that all portions of the vector space is represented. For each random sample point the mechanical model is performed. All these randomsample points and their response quantities represent the design of experiments to be used in the building of the kriging metamodel. The mean square error (MSE) of Kriging approximation is equal to zero at the training point. However, at the testipoints which are away from these training points, the MSEs increase highly.respect to the random variables, composed of the material parameters selected in Table 1, the solder thickness, the PCB thickness, the peak and the lowest temperature loading, given in Table 4. A 100 sample points are generated and thus 100 nonlinear finite element runs are performed to estimate of the inelastic strain for each trial point. a representation of the kriging approximation where the modulus and the Copper CTE, all the other parametersError of this approximation.

Figure 3. a) Kriging approximation of the inelastic

Description

Peak temperature (

Lowest temperature

PCB thickness (mm)

Solder thickness

Table 4. Geometric dimension and thermal profil

4. Probability distribution

The probability density function of the fatigue lifetime of points by using Monte Carlo Simulations. of the fatigue lifetime. Table 5 shows that the generalized scale parameter(λ) and location parameterdensity function of the number of cycles before failure initiation is shown for SnPb inSnAg3.5 in figure 6a). The mean value of the fatigue lifetime for SnPb Solder is 212SnAg3.5 1981 cycles. Figure 6b) shows the cumulative density function of the fatigueconfirm that lead-free solder has a better thermal fatigue resistance compared to SnPbwhere the probability of failure of 0.9 can be reached at 3000 cycles for SnAg3.5,less than 300cycles for SnPb.

For a nominal value of the random variables, as hysteresis loop. It is used to study the stabilization and detcycle in solder joints subjected to multiple temperature cycle (SAC305, SnAg3.5, SnPb). is stabilized after five thethat is stored in the solder joints. High area value the hysteresis loop of the tin-lead (SnPb)SnAg3.5). The stress and plastic strain ranges of SAC305 are similar by Kanchanomai et al. (2002a).

�I(�) � J(�). #⋆ $ LE�� FG&�: @#⋆

is used to construct the kriging metamodel. A quadratic regression model and the anisotropic Gaussian model for the correlation function are selected. An anisotropic correlation function is preferred in rel

ysis studies [18]. The Latin hypercube sampling is used as a strategy for generating random sample points ensuring that all portions of the vector space is represented. For each random sample point the mechanical model is performed. All these random

points and their response quantities represent the design of experiments to be used in the building of the kriging metamodel. The mean square error (MSE) of Kriging approximation is equal to zero at the training point. However, at the testi

are away from these training points, the MSEs increase highly. In this study, the inelastic strain is approximated with respect to the random variables, composed of the material parameters selected in Table 1, the solder thickness, the PCB

the lowest temperature loading, given in Table 4. A 100 sample points are generated 100 nonlinear finite element runs are performed to estimate of the inelastic strain for each trial point.

kriging approximation where the approximated plastic strain is plotted according to the solder modulus and the Copper CTE, all the other parameters being fixed to their nominal value. Figure 3 b)

Kriging approximation of the inelastic strain. b) Mean Squared Error of the kriging approximation.

Description Mean value Coefficient of variation

Peak temperature (°C) 125 0.03

Lowest temperature (°C) -40 0.03

(mm) 1.6 0.05

(mm) 0.15 0.05

Geometric dimension and thermal profil random variables.

Probability distribution s of the fatigue lifetime

tion of the fatigue lifetime of the solder joint presented in the figure 5 is estimated with 5000 nte Carlo Simulations. The Chi-square goodness-of-fit test is used to find the best fitting probability distribution

shows that the generalized extreme value distribution with three parameters (shape paramelocation parameter µ) matches to the probabilistic distribution of the fatigue lifetime. The probability

number of cycles before failure initiation is shown for SnPb in figure 5a), for SAC305 in figure 5b) and for The mean value of the fatigue lifetime for SnPb Solder is 212 cycles, for SAC305 1261 cycles and for

shows the cumulative density function of the fatigue lifetime of the three solders. The results solder has a better thermal fatigue resistance compared to SnPb Solder, particularly the SnAg3.5 solder,

of failure of 0.9 can be reached at 3000 cycles for SnAg3.5, less than 2000 cycles for SAC305 solder and

For a nominal value of the random variables, Figure 4 shows the stress magnitude as a function of strain magnitude, also known study the stabilization and determine the magnitude of induced accumulated plastic work per to multiple temperature cycle loads. The hysteresis loops of the

stabilized after five thermal cycles. The area under the hysteresis loop represents the strasolder joints. High area value induces a reduction of the fatigue strength. Figure 4

lead (SnPb) solder is larger than the hysteresis loop area of the lead-ranges of SAC305 are similar to those of SnAg3.5, which confirms the

{16}

is used to construct the kriging metamodel. A quadratic regression model and the anisotropic Gaussian model for the correlation function are selected. An anisotropic correlation function is preferred in reliability

ysis studies [18]. The Latin hypercube sampling is used as a strategy for generating random sample points ensuring that all portions of the vector space is represented. For each random sample point the mechanical model is performed. All these random

points and their response quantities represent the design of experiments to be used in the building of the kriging metamodel. The mean square error (MSE) of Kriging approximation is equal to zero at the training point. However, at the testing

In this study, the inelastic strain is approximated with respect to the random variables, composed of the material parameters selected in Table 1, the solder thickness, the PCB

the lowest temperature loading, given in Table 4. A 100 sample points are generated by LHS method 100 nonlinear finite element runs are performed to estimate of the inelastic strain for each trial point. Figure 3 a) shows

plotted according to the solder Young’s 3 b) shows the Mean squared

Mean Squared Error of the kriging approximation.

Distribution

Normal

Normal

Uniform

Uniform

is estimated with 5000 sample fitting probability distribution

three parameters (shape parameter (κ), of the fatigue lifetime. The probability

5a), for SAC305 in figure 5b) and for cycles, for SAC305 1261 cycles and for lifetime of the three solders. The results Solder, particularly the SnAg3.5 solder,

less than 2000 cycles for SAC305 solder and

strain magnitude, also known accumulated plastic work per

he hysteresis loops of these three tin based solder hysteresis loop represents the strain energy

4 indicates that the area of -free solders (SAC305 and

confirms the behavior reported



Solder Mean Value

SnPb 212

SnAg3.5 1981

SAC305 1261

Table 5. Probability distribution parameters of the fatigue lifetime.

Figure 4

Figure 5. Probability density function of the fatigue lifetime of the

Figure 6. a) Probability density function of the fatigue lifetime of the

In this study a nonlinear thermomechanical finite element of an electronic package. This approach reveals material properties such as Young’s modulus, have a strong effect on the lifetime of the robustness. The above probabilistic methodthe fatigue lifetime of three solders. Furthermore, the fatigue

a)

a)

Mean Value STDV κ λ

50 0.0119 41.63

576 0.0503 418.54

432 0.1051 287.13

Probability distribution parameters of the fatigue lifetime.

4. Hysteresis loop of each solder during thermal cycling.

Probability density function of the fatigue lifetime of the a) SnPb b) SnAg3.5

function of the fatigue lifetime of the SAC305 b) Cumulative distribution func

5. Conclusion

rmomechanical finite element analysis is used to predict the reliability performance of the package. This approach reveals that the uncertainties related to the variability of the geometry

s Young’s modulus, thermal expansion coefficient mismatch as well as thermal ct on the lifetime of the electronic package. This efficient probabilistic approach en

methodology predicts the probability distribution and the cumulative Furthermore, the fatigue lifetime can be modeled by the generalized extreme value

b)

b)

µ

188.37

1717.93

1062

SnAg3.5.

Cumulative distribution functions of all solders.

reliability performance of the solder joint the variability of the geometry and of the

ent mismatch as well as thermal loading fluctuations ic approach ensures more design

distribution and the cumulative density function of the generalized extreme value



distribution with three parameters (shape parameter, scale parameter, and location parameter) as the Weibull distribution. This probabilistic approach helps to compare the reliability of solder joints of three different compositions (SnPb, SnAg3.5 and SAC305). The lead-free solders have a better thermal fatigue resistance compared to SnPb Solder and the SnAg3.5 solder is more reliable than SAC305. A possible extension of this study is to investigate the effect of solder voids and residual stresses induced during processing due to the mismatch in thermal expansion/contraction of the constituents, which can have more impact on the lifetime of the electronic packaging. A spectral method also known as stochastic finite element method can be used to represent the randomness of the fatigue life in a more efficient way.

Acknowledgment

This project is co-financed by the European Union with European regional development fund (ERDF).

References

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