Fatigue failure of solder joints is one of the major causes of failure in electronic devices. Fatigue life prediction models of solder joints were first put forward in the early 1960s, and since then, numbers of methods were used to model the fatigue mechanism of solder joints. In this article, the majority fatigue life models are summarized, with emphasis on the latest developments in the fatigue life prediction methods. All the models reviewed are grouped into four categories based on the factors affecting the fatigue life of solder joints, which are: plastic strain-based fatigue models, creep damage-based fatigue models, energy-based fatigue models, and damage accumulation-based fatigue models. The models that do not fit any of the above categories are grouped into “other models.” Applications and potential limitations for those models are also discussed.

Introduction

The electronic packaging industry is one of the largest industries in the world [1]. The role of solder in the electronic packaging has become more critical since the size of dies and chip carriers and number of interconnects have increased, while the volume of solder joints and cost have decreased [2]. Since solder joints connect components with printed circuit board (PCB), they are the weakest connection in the assembly, and failure of solder joints is one of the major reliability concerns for electronic products. Solder joints are typically exposed to various real-life conditions such as vibration, drop, thermal shock, elevated temperature, and changing temperature conditions. All have been proved as threats for the reliability of solder joints, and the thermal cycling has been known as the major threat [26]. When solder joints are at high extreme temperature, creep failure dominates; and when exposed to temperature cycling, fatigue failure dominates [715]. If under the combination of both, the failure mechanism is generally referred to as creep-fatigue [16]. The creep and fatigue of solder joints are mainly due to the mismatch of coefficient of thermal extension (CTE) between component and PCB.

Through history, development of solder joint materials can be roughly categorized into three phases: Tin-lead phase, lead-free phase, and doped solder material phase. Tin-lead solder joints were once extensively used in electronic assemblies [3] due to their relatively low cost and low melting point. However, lead was discovered to be one of the top 17 chemicals that posed great threat to human life and environment by Environment Protection Agency (EPA). After being considered and debated for years, European Union's Waste in Electrical and Electronic Equipment (WEEE) and restriction of hazardous substances announced that by July 1, 2006, lead must be replaced in electronic equipment. Over the years, the industry has tried to find a suitable replacement solder material for Tin-lead; however, till now, no new alloy has been found that could be a “drop in” replacement for eutectic Tin-lead alloy [17,18]. Among hundreds of lead-free solder alloys that have been created, Sn–Ag–Cu (SAC) solder alloys are the most prevalent and are regarded as the best alternatives to eutectic Sn–Pb solder alloys, since SAC solder joints have shown some outstanding properties over traditional Sn–Pb solder alloys [19,20]. However, their use in some areas is still in question, as compared to Sn–Pb solder alloys, including melting temperatures, unit cost, or mechanical strengths. Given the disadvantages of SAC solder alloys, research is currently focusing on developing the “next generation” of solder alloys: doped solder joints. This refers to adding rare earth elements into the traditional eutectic or noneutectic solder alloy to form another new solder alloy [21]. Test results show that adding rare earth elements has great potential to improve solder joints properties and make them better alternatives to replace SnPb solder joints [2224].

Based on information from ASTM International, fatigue of a solder joint is defined as “the process of progressive localized permanent structural change occurring in a solder joint subjected to conditions which produce fluctuating stresses and strains at some points which may culminate in cracks or complete fracture [25].” Typically, each cycle of fluctuating stress would not be large enough to cause an immediate failure. Mainly, there are two stages to describe the fatigue failure of a solder joint: crack initiation and crack propagation. For the crack initiation, as described by Schijve, Shang et al., and Miller, it could be basically described as the initiation of the microcrack, the nucleation of the microcrack, and finally, the initiation of the major physical crack [2628]. Sometimes, the initiation of the crack can be caused by overloading, where the crack is generated as a result of an accident or void on the crack region. However, the main cause of the crack initiation in a fatigue test is due to the cyclic deformation of the joint by the stress [3]. During exposure to thermal cycling, fatigue, it is mainly due to the thermomechanical stresses in the solder joints arising from the differences in the CTE between PCB and component [29,30]. Also, as the locations where the crack is initiated vary from alloy to alloy, so are the failure modes. For example, as reported by Jang et al., in the case of SAC305 solder joints with electroless nick immersion gold surface finish, the cracks initiated at the interface between (CuNi)6Sn5 and the Ni layer and the fracture behaviors appear to be that of interfacial fracture. However, in the case of SAC105, the solder joints include both interfacial and bulk fracture [31]. The direction of crack propagation is generally orthogonal to the direction of the principal stress [32]. Some papers have also found that the propagation of the crack can either be along with grain boundary of bulk solder or near intermetallic compound interfacial regions [3335].

By understanding the basic mechanism of the failure process in solder joints, and with proper assumptions, many researchers have proposed models to predict the fatigue life of solder joints since the early 1960s. Review of solder joints life prediction models has been conducted by Lee et al., which was published in 2000 [36]. Many solder joints fatigue life prediction methods have been published since then, and systematic review is seriously needed. The purpose of this paper is to carry out a state-of-the-art review about solder joints fatigue life prediction models with special emphasis on reviewing the latest development on the fatigue life prediction, including modifications on some of the classic prediction models. Models reviewed in the paper are classified into four major categories (sometimes there will be overlaps among each group): plastic strain-based fatigue models, creep damage-based fatigue models, energy-based fatigue models, and damage accumulation-based fatigue models as suggested by Lee and Jeong [37]. The models that do not fit any categories are grouped as “the other models” category. Table 1 summarized most of the fatigue life prediction models that have been reviewed in this paper. “Damage parameter” refers to the variables that the models used, for example, strain energy density or plastic strain range, represents the factors that the author thought will play key impact on the fatigue life of test sample. “Package applied” refers to the type of test vehicle (experimental sample) that the prediction models have been applied to. “Applicable condition” refers to the type of test condition/test design that the test vehicles have been exposed to collect the experimental data used for the prediction model.

Table 1

Summary of fatigue life prediction models

Fatigue modelEquation #Damage parameterPackage appliedApplicable conditionRequired parameters
Coffin–Manson1Plastic strainN/ALCFΔεp: plastic strain range;
m: fatigue exponent;
C: ductility coefficient
Shi2Frequency-modified plastic strainN/ALCFΔεp: plastic strain range
v: frequency
k: frequency exponent
m: fatigue exponent
C: ductility coefficient
Engelmaier5Shear strainLCC, CCCLCFΔγ: cyclic shear strain range;
ϵf: fatigue ductility coefficient;
c: fatigue ductility exponent
Engelmaier6Shear strainLeadless surface mount deviceLCF and applied condition comparisonsΔγ: shear strain range;
c: fatigue exponent;
Engelmaier8Package geometry and temperatureStiff leadless SMD and leaded SMDLow-cycle fatigue and applied condition comparisonsLD: half maximal distance between component solder joints;
Δα: absolute difference in the coefficients of thermal expansion;
β: Weibull shape parameter;
F: empirical factor; ΔTe: equivalent temperature; εf : ductility coefficient;
c: fatigue ductility exponent
Solomon9Shear strain60/40 solder sheetLCFΔγp: plastic shear strain range; α and θ are constants
Norris–Landzberg11Temperature and cycle frequencyN/AOperating condition versus test conditionΔT: temperature range;
f: thermal cycling frequency; Tmax: maximum temperature;
Ea: activation energy;
k: Boltzmann's constant
Syed15Accumulated creep strainPBGAPower cyclingEcr: accumulated creep strain for the whole cycle;
C: creep ductility
Syed17, 18Accumulated creep strain and creep energy densityBGA and CSP with SAC and SnPb solderThermal cycling with differentεacc: accumulated creep strain;
wacc: accumulated creep energy density;
C: creep strain constant;
W: creep energy constant
Manson22Creep strain and plastic strainExtruded steel pipeCreep fatigue under isothermal temperatureΔϵpp : completely reversed plasticity;
Δϵpc: tensile plasticity reversed by compressive creep;
Δϵcp : tensile creep reversed by compressive plasticity; Δϵcc: completely reversed creep
Akay24Volume-weighted average creep strainLCCThermal cycling with different thermal profilesγ¯av: volume-weighed average effective creep shear strain range;
ϵf : fatigue ductility coefficient;
k: fatigue ductility exponent
Akay25Total creep strain energyLCCThermal cyclingΔWtotal : strain energy c;
W0 and k are load-independent material constants
Knecht and Fox26Creep strainSolder joints in SMDThermal cyclingΔγmc: creep strain;
C: constants
Stowell28Energy loss for cycling between two stress levelsSteelMechanical cycling in stress amplitudesσf and ϵf are true stress and strain at fracture;
E, σ0, and ϵ0 are the known constants
Pan29Strain energy density and geometry of the componentLCCCThermal cyclingC: critical strain energy density;
Ep: time-dependent plastic energy density;
Ec: time-dependent creep energy density;
a and b are weighting factors
Kujawski30Strain energy density per cycleMetalLCF
HCF
ΔWt : total strain energy density;
ΔWend: elastic strain energy density;
Wf and d are constants
Joseph31Cumulative hysteresis energy failure densityFEA simulation model of elastic-viscoplastic materialThermal cycling with different temperature profilesEintrinsic : the average of all the hysteresis energy failure density for each test; Wt: total energy density accumulated; W: energy density dissipated in one cycle
Morrow32Plastic strain energy densityMetal thin-walled tubular specimensAxial tension-compression LCFWp : plastic strain energy density;
m: fatigue exponent;
C: ductility coefficient
Solomon33Frequency modified strain energy density60/40 solderFatigue cycling at different temperaturesWp : plastic strain energy density;
v: frequency;
k and n are frequency exponents;
C: material ductility coefficient
Letcher39Cumulative “critical” strain energyAluminum rodLoad-controlled fatigue cyclingWcrit : critical energy;
Wm: monotonic tension energy;
σf: true stress at fracture;
εf: true strain at fracture;
E: modulus of elasticity;
ε0 and σ0 are material parameters for monotonic strain;
σc: material parameter for cyclic strain;
σa: amplitude of alternating stress
Jahad42Elastic and plastic energy induced by tensile or torsion loadingVarious steelAxial and torsional fatigue loadingΔEA: energy due to pure tensile loading;
ΔET: energy due to pure torsion loading;
E : total elastic-plastic energy; NA: fatigue life from purely tensile loading;
NT: fatigue life from purely torsion loading
Miner44Damage per cycle to the total damageN/AFatigue cyclingni: number of cycles accumulated at a given stress amplitude;
Ni : number of cycles to failure at the same stress amplitude;
k: number of stress levels
Grover45,46Total damage for crack initiation and propagationSteelFatigue cyclingni: number of cycles accumulated at a given stress amplitude;
Ni : number of cycles to failure at the same stress amplitude;
a: constant
Corten and Dolan47Total damage with considering stress effectSteel wireFatigue cyclingN1: fatigue life at the highest stress amplitude σ1;
ni: number of cycles at stress amplitude σi;
d: material constant for the model
Hamasha50Damage per cycle to the total damage with considering stress effectSolder joint on the packageLCF
HCF
nmi: number of cycles at low amplitude;
Nm: life at low amplitude;
nhi: number of cycles at high amplitude;
Nh: life at high amplitude; fi: hysteresis energy in the low amplitude cycles
Cheng53Total damage to form a critical crackPressure vessel steelLCFΔσ: stress amplitude; A(R) and β are temperature dependent material constants; Ψ is the ductility of the material
Darveaux59, 60Damage leads to the crack initiation and propagationAssemblies with SnPb solder jointsThermal cycling with different thermal profilesa: final crack length;
da/dN: crack growth rate; ΔWave: average viscoplastic strain energy density per cycle for the interface elements;
K1K4: material constants
Gustafsson62Initiation and propagation of primary and secondary cracksLead-free BGAThermal cyclingN0S and N0P: secondary and primary crack initiation energy-based terms;
a: total possible crack length; dasdN and dapdN: secondary and primary crack propagation rates
Huang70Fatigue modulusGlass epoxy composite materialStress-controlled fatigue cyclingF0: fatigue modulus at zeroth cycle;
Ff: fatigue modulus at fracture; B and c are material constants
Nam72Number of nucleated cavities per unit areaP containing stainless steelCreep fatigue testP: cavity nucleation factor; Δεp: plastic strain range;
C: material constant;
Qg: activation energy of grain boundary diffusion;
R and T are Boltzmann's constant and temperature;
σ(t) is the tensile peak stress relaxation term during hold time
Fatigue modelEquation #Damage parameterPackage appliedApplicable conditionRequired parameters
Coffin–Manson1Plastic strainN/ALCFΔεp: plastic strain range;
m: fatigue exponent;
C: ductility coefficient
Shi2Frequency-modified plastic strainN/ALCFΔεp: plastic strain range
v: frequency
k: frequency exponent
m: fatigue exponent
C: ductility coefficient
Engelmaier5Shear strainLCC, CCCLCFΔγ: cyclic shear strain range;
ϵf: fatigue ductility coefficient;
c: fatigue ductility exponent
Engelmaier6Shear strainLeadless surface mount deviceLCF and applied condition comparisonsΔγ: shear strain range;
c: fatigue exponent;
Engelmaier8Package geometry and temperatureStiff leadless SMD and leaded SMDLow-cycle fatigue and applied condition comparisonsLD: half maximal distance between component solder joints;
Δα: absolute difference in the coefficients of thermal expansion;
β: Weibull shape parameter;
F: empirical factor; ΔTe: equivalent temperature; εf : ductility coefficient;
c: fatigue ductility exponent
Solomon9Shear strain60/40 solder sheetLCFΔγp: plastic shear strain range; α and θ are constants
Norris–Landzberg11Temperature and cycle frequencyN/AOperating condition versus test conditionΔT: temperature range;
f: thermal cycling frequency; Tmax: maximum temperature;
Ea: activation energy;
k: Boltzmann's constant
Syed15Accumulated creep strainPBGAPower cyclingEcr: accumulated creep strain for the whole cycle;
C: creep ductility
Syed17, 18Accumulated creep strain and creep energy densityBGA and CSP with SAC and SnPb solderThermal cycling with differentεacc: accumulated creep strain;
wacc: accumulated creep energy density;
C: creep strain constant;
W: creep energy constant
Manson22Creep strain and plastic strainExtruded steel pipeCreep fatigue under isothermal temperatureΔϵpp : completely reversed plasticity;
Δϵpc: tensile plasticity reversed by compressive creep;
Δϵcp : tensile creep reversed by compressive plasticity; Δϵcc: completely reversed creep
Akay24Volume-weighted average creep strainLCCThermal cycling with different thermal profilesγ¯av: volume-weighed average effective creep shear strain range;
ϵf : fatigue ductility coefficient;
k: fatigue ductility exponent
Akay25Total creep strain energyLCCThermal cyclingΔWtotal : strain energy c;
W0 and k are load-independent material constants
Knecht and Fox26Creep strainSolder joints in SMDThermal cyclingΔγmc: creep strain;
C: constants
Stowell28Energy loss for cycling between two stress levelsSteelMechanical cycling in stress amplitudesσf and ϵf are true stress and strain at fracture;
E, σ0, and ϵ0 are the known constants
Pan29Strain energy density and geometry of the componentLCCCThermal cyclingC: critical strain energy density;
Ep: time-dependent plastic energy density;
Ec: time-dependent creep energy density;
a and b are weighting factors
Kujawski30Strain energy density per cycleMetalLCF
HCF
ΔWt : total strain energy density;
ΔWend: elastic strain energy density;
Wf and d are constants
Joseph31Cumulative hysteresis energy failure densityFEA simulation model of elastic-viscoplastic materialThermal cycling with different temperature profilesEintrinsic : the average of all the hysteresis energy failure density for each test; Wt: total energy density accumulated; W: energy density dissipated in one cycle
Morrow32Plastic strain energy densityMetal thin-walled tubular specimensAxial tension-compression LCFWp : plastic strain energy density;
m: fatigue exponent;
C: ductility coefficient
Solomon33Frequency modified strain energy density60/40 solderFatigue cycling at different temperaturesWp : plastic strain energy density;
v: frequency;
k and n are frequency exponents;
C: material ductility coefficient
Letcher39Cumulative “critical” strain energyAluminum rodLoad-controlled fatigue cyclingWcrit : critical energy;
Wm: monotonic tension energy;
σf: true stress at fracture;
εf: true strain at fracture;
E: modulus of elasticity;
ε0 and σ0 are material parameters for monotonic strain;
σc: material parameter for cyclic strain;
σa: amplitude of alternating stress
Jahad42Elastic and plastic energy induced by tensile or torsion loadingVarious steelAxial and torsional fatigue loadingΔEA: energy due to pure tensile loading;
ΔET: energy due to pure torsion loading;
E : total elastic-plastic energy; NA: fatigue life from purely tensile loading;
NT: fatigue life from purely torsion loading
Miner44Damage per cycle to the total damageN/AFatigue cyclingni: number of cycles accumulated at a given stress amplitude;
Ni : number of cycles to failure at the same stress amplitude;
k: number of stress levels
Grover45,46Total damage for crack initiation and propagationSteelFatigue cyclingni: number of cycles accumulated at a given stress amplitude;
Ni : number of cycles to failure at the same stress amplitude;
a: constant
Corten and Dolan47Total damage with considering stress effectSteel wireFatigue cyclingN1: fatigue life at the highest stress amplitude σ1;
ni: number of cycles at stress amplitude σi;
d: material constant for the model
Hamasha50Damage per cycle to the total damage with considering stress effectSolder joint on the packageLCF
HCF
nmi: number of cycles at low amplitude;
Nm: life at low amplitude;
nhi: number of cycles at high amplitude;
Nh: life at high amplitude; fi: hysteresis energy in the low amplitude cycles
Cheng53Total damage to form a critical crackPressure vessel steelLCFΔσ: stress amplitude; A(R) and β are temperature dependent material constants; Ψ is the ductility of the material
Darveaux59, 60Damage leads to the crack initiation and propagationAssemblies with SnPb solder jointsThermal cycling with different thermal profilesa: final crack length;
da/dN: crack growth rate; ΔWave: average viscoplastic strain energy density per cycle for the interface elements;
K1K4: material constants
Gustafsson62Initiation and propagation of primary and secondary cracksLead-free BGAThermal cyclingN0S and N0P: secondary and primary crack initiation energy-based terms;
a: total possible crack length; dasdN and dapdN: secondary and primary crack propagation rates
Huang70Fatigue modulusGlass epoxy composite materialStress-controlled fatigue cyclingF0: fatigue modulus at zeroth cycle;
Ff: fatigue modulus at fracture; B and c are material constants
Nam72Number of nucleated cavities per unit areaP containing stainless steelCreep fatigue testP: cavity nucleation factor; Δεp: plastic strain range;
C: material constant;
Qg: activation energy of grain boundary diffusion;
R and T are Boltzmann's constant and temperature;
σ(t) is the tensile peak stress relaxation term during hold time

Note: BGA: ball grid array; PBGA: plastic ball grid array; SMD: surface mount device; CSP: chip scale package; HCF: high cycle fatigue; and LCC: leadless chip carrier.

Plastic Strain-Based Fatigue Model

Failure in low-cycle fatigue (LCF) is mainly due to cyclic plastic deformation. When a solder joint is loaded, the stress–stain curve would be linear with elastic behavior until the yield point; then it would be permanently deformed (plastic deformation) with the increase in load. Electronic assemblies in actual service are often exposed to cyclic temperature changes. The cyclic change in temperature results in a cyclic plastic strain of the interconnected solder joints due to the CTE mismatch between electronic components and substrate materials. This is a low-cycle fatigue of solder joints and is the dominant factor that affects the reliability of surface mount electronic devices [3840]. Low-cycle fatigue is associated with shorter fatigue life and higher stress, where the stress level usually steps into the plastic strain range. The following prediction models use the plastic strain as a measurable factor to predict the fatigue life.

Coffin–Manson model, Eq. (1), [41] is widely used to predict the low-cycle fatigue life Nf as a function of plastic strain range Δεp for solder alloys
NfmΔεp=C
(1)

where m is the fatigue exponent and C is the ductility coefficient.

As proven by Shi et al. [42], the fatigue life varies with the cycling frequency. When the frequency is above 10−3 Hz, the fatigue life decreases with decreasing frequency at a low rate. However, when the frequency is reduced from 10−3 to 10−4 Hz, the fatigue life decreased drastically. Therefore, when applying the Coffin–Manson model to predict the fatigue life of solder alloys, large differences are seen in the fatigue life at different frequencies. As a result, the frequency-modified Coffin–Manson model was introduced by Shi et al. (Eq. (2)) [42]
Nfv(k1)mΔεp=C
(2)
where v is the frequency and k is the frequency exponent. k is calculated differently depending on the frequency range; roughly as 0.91 for 10−3 to 1 Hz range and 0.42 for 10−3 to 10−4 Hz range. Then, different k values were used to calculate the frequency-modified coefficient v(k1) using the following equations:
v(k1)=v(k11)for1Hzv103Hz
(3)
v(k1)=v103k21103k11for103Hz>v104Hz
(4)
Engelmaier modified the Coffin–Manson model by incorporating parameters like cyclic frequency, solder, and substrate temperature into the equation, known as the Engelmaier model [43] as shown in the following equation:
Nf=12Δγ2ϵf1/c
(5)

where Nf is the mean cycle to failure, Δγ is the cyclic shear strain range, ϵf is the fatigue ductility coefficient, c is the fatigue ductility exponent. c can be calculated for near-eutectic tin-lead solder alloy by: c=0.4426×104T¯S+1.74×102ln1+f, where T¯S is the mean solder joint temperature (e.g., the average of the maximal and minimal temperature in °C), and f is the cycling frequency (cycles/day).

However, as Engelmaier mentioned in his paper, there are also limitations including: the joint geometry is based on a cylindrical solder joint with filets on both sides, where the structure of solder joint of ceramic chip carrier (CCC) have much more complex geometry structure; therefore, the fatigue life prediction of CCC based on Eq. (5) could be misleading.

Further, Engelmaier put forward an acceleration transformation, which can compare results from different test conditions by normalizing the data to one common comparison condition; predict the fatigue life operating condition (condition 2) reliability from the accelerated test results (condition 1) [44] based on the following equation:
N¯f2=122N¯f1c1Δγ2Δγ11/c2
(6)
where N¯f2 is the fatigue life in operating condition 2, N¯f1 is the fatigue life in accelerated test (condition 1), c1 is the fatigue exponent in condition 1, and c2 is the fatigue exponent in condition 2. For leadless surface mount solder joints, the total solder joint cyclic shear strain range Δγ in either condition is given by
Δγ=Fd2hΔαΔT
(7)
where F is an empirical factor for second-order effects, d is the longest distance on the component between solder joints, h is the solder joint height, ΔαΔT is thermal expansion differential. Later, he combined a two-parameter Weibull distribution into his model to address the performance fluctuation within a component [40]. The fatigue life Nfx%, at a given acceptable failure probability x, could be predicted by
Nfx%=12F2εfLDΔαΔTeh1/cln10.01xln0.51/β
(8)

where LD is the half maximal distance between component solder joints measured from solder joint centers, Δα is the absolute difference in the coefficients of thermal expansion between the component and the substrate materials. β is the Weibull shape parameter for Weibull distribution. ΔTe is the equivalent cycling temperature swing. εf is fatigue ductility coefficient, c is fatigue ductility exponent.

Solomon proposed a low-cycle fatigue model (Eq. (9)) that relates the plastic shear strain to the fatigue life cycles for near eutectic solder joints at four different temperatures (−50 °C, 35 °C, 125 °C, and 150 °C) [45]
ΔγpNfα=θ
(9)

where Δγp is the plastic shear strain range, α and θ are constants. For 60/40 Tin-lead solder material from −50 °C to 150 °C, α and θ are estimated to be 0.51 and 1.14, respectively.

Norris and Landzberg had proposed a fatigue life prediction method based on three factors: the maximum temperature Tmax, the frequency of the temperature cycle f, and the thermal excursion range ΔT [46] based on the Coffin–Manson model. Those factors are known to have a significant impact on the reliability of solder joints. By taking the ratio of the lifetime in field Nfield and test Ntest environment, this requires no information on the test vehicle, neither the geometry nor the material properties. By applying acceleration factor models, acceleration testing data can be quickly related to the potential solder-joint field reliability as
AF=NfieldNtest
(10)
AF=ΔTtestΔTfield1.9×ffieldftest1/3eEa/k1Tmax,field1Tmax,test
(11)

where AF is the acceleration factor. The subscripts “field” and “test” denote different conditions. The activation value Ea is estimated to be 0.122 eV for Sn–Pb solder alloy and k is Boltzmann's constant.

Pan et al. provided a new set of parameters on Sn–Ag–Cu solder materials for Norris–Landzberg (NL) equation, as follows [47]:
AF=ΔTtestΔTfield2.65×ffieldftest0.136eEa/k1Tmax,field1Tmax,test
(12)

where the activation energy is estimated to be 0.189 eV.

Dauksher [48] proposed a modification of Norris–Landzberg (NL) model (Eq. (13)) to predict the fatigue life of SAC solder alloy under accelerated thermal test. In this equation, the frequency term was replaced by Thot term which accounts for the time per cycle when the component is hot. In this way, the Thot term will only consider the ramp-to-hot and hot dwell times per cycle which will better fit the test data during the thermal cycling test
AF=N2N1=ΔT1ΔT21.75t1hott2hot1/4exp1600T2MAX1600T1MAX
(13)

The subscripts 1 and 2 indicate separate environments.

Creep Damage-Based Model

Solder joints creep refers to a time-dependent deformation while solder joints are under an applied load that is below their yield strength. The creep behavior often happens when solder material is exposed to an elevated constant temperature. The closer temperature is to its melting point, the heavier the creep. For most of the solder materials, creep develops in three stages: primary, secondary, and tertiary creep. The creep response begins with an initial instantaneous strain, which consists of the elastic deformation as soon as the constant load is applied [49]. Then creep enters the second stage, which is “steady-state creep;” where the microstructure does not evolve, and solder strength is constant. In the tertiary creep, strain rates are dramatically increased with the stress, as the accumulation of internal cracks or voids decrease the effective area. Solder strength is significantly dropped, and the shape of solder joints is permanently transformed (necking), which leads to fracture.

Most of the fatigue prediction models of solder joints only focus on the plastic deformation induced by cyclic strain, since at low temperature, fatigue damage is independent of time [50]. However, since solder joints are normally exposed to elevated temperature and thermal cycling, the elevated temperature induced creep strain in the solder joints can also be a major concern. It was found that in the creep range, fatigue endurance are significantly reduced than that predicted by fatigue theories at ambient temperature. Therefore, creep-fatigue generated indicates the mechanics of damages due to the combination of both creep effects and fatigue effects [51]. Creep-fatigue deformation is viewed as one of the major failure modes of solder joints for electronic packaging modules [2].

Syed [52] combined Monkman–Grant equation [53] for creep rupture and Miner rule [54] to have a new life prediction model, where the cyclic loading is considered as a special case of creep. In Monkman–Grant equation (Eq. (14)) [53], the time to rupture is inversely related to the steady-state creep strain rate, ɛ.cr
tr=Cεcr˙
(14)
then Eq. (15) predicts the cyclic creep-fatigue life under varying and repeated stress for one single creep mechanism
Nf=(EcrC)percycle1
(15)
where Ecr is the accumulated creep strain for the whole cycle, equals to i=1nΔti×ɛcri. Δti is the time spent at given stress level within one cycle. C is the creep ductility for the creep mechanism. For two creep mechanisms, the equation becomes:
Nf=E1C1+E2C2percycle1
(16)
the constants are determined as 50 and 15.87 for C1 and C2, respectively, for Sn–Pb eutectic solder alloy reported by the author.
Syed [55] then proposed two fatigue life equations based on creep strain and creep energy density to predict fatigue life for SnAuPb and SnPb solder alloy. In this equation, the author assumes that the damage in solder joints during thermal cycling tests is mainly from the steady-state creep strain accumulation. When considering an approach using fracture mechanics, creep strain-based model is proposed
Nf=CIεaccI+CIIεaccII1
(17)
then when switching creep strain related constants to energy density constants, we have the energy density model
Nf=WIwaccI+WIIwaccII1
(18)

where Nf is the number of cycles to failure for both equations. CI and CII represent the constants related to creep strain-based model, respectively, εacc represents accumulated creep strain per cycle for each creep mechanisms. WI and WII represent constants related to dissipated creep energy density based model, respectively. wacc represents accumulated creep energy density per cycle for each creep mechanisms.

Manson et al. [56] have provided a creep-based fatigue prediction method, which partitioned any one cycle of a completely reversed inelastic strain into three strain range components: Δϵpp–completely reversed plasticity, Δϵpc–tensile plasticity reversed by compressive creep or Δϵcp–tensile creep reversed by compressive plasticity, and Δϵcc–completely reversed creep. Based on the authors, “the first letter of the subscript (c for creep and p for plastic strain) refers to the type of strain imposed in the tensile portion of the cycle, and the second letter refers to the type of strain imposed during the compressive portion of the cycle.” For each component of inelastic strain, based on the experimental results, they all follow a power law equation similar to “Coffin–Manson Equation”:
Δϵij=αNijβ
(19)

where subscripts i and j are corresponding to p and c, respectively.

Then damage (Dij) is associated with Δϵij by a cycle ratio N /Nij, where N is the total predicted of cycles to failure for the combination effects and Nij is the predicted number of cycles to failure containing only ij strain range. Failure occurs when
Dpp+Dcc+DpcorDcp=1
(20)
By applying Manson's strain range partitioning method, Yoshiharu et al. [57] partitioned the entire reversed inelastic strain range into four generic components with creep and plasticity: ɛpp: time-independent plasticity reversed by time-independent plasticity; ɛpc: time-independent plasticity reversed by creep; ɛcp: creep reversed by time-independent plasticity; ɛcc: creep reversed by creep. The partitioning equation is giving by
Δεij=Aij×Nijmij
(21)
where subscripts i and j are corresponding to p and c, respectively. Nij is the cyclic life corresponding to the specific partitioned strain range Δεij and m is the slope of Coffin–Manson plot. Δεcp, Δεpc, Δεcc are determined from the hysteresis loops in a series of tests that feature the three generic inelastic components. With all the partitioned cyclic fatigue life known, Miner's rule was used to predict creep-fatigue life for each solder alloy
1Nf=1Npp+1Ncc+1Ncp+1Npc
(22)
For Sn-3.5Ag solder alloy, the equation is provided below:
1Nf=Δεpp1.061.92+Δεcp1.921.67+Δεpc4.760.98+Δεcc0.241.56
(23)
Akay et al. [16] put forward a fatigue life estimation of solder alloys based on the volume-weighed averaging method that involves a creep–fatigue interaction based on Engelmaier model and energy-partitioning approach. Volume-weighed averaging method considers the stress–strain information at the point of highest stress and strain values within the solder joints. This method assumes that the fatigue behaviors of the whole solder join will lead to the failure of this joint, and the damage effect in each finite element will be proportional to the element's geometrical contribution to the solder joint. The inelastic shear strain range and total inelastic strain energy calculated from each finite element were used to calculate the fatigue life of the solder joint of three different assemblies (68-pin LCC, 20-pin LCC, 20-pin LCC) as:
Nf=12×Δγ¯avϵf1/k
(24)

where γ¯av is volume-weighed average effective creep shear strain range, ϵf is fatigue ductility coefficient, and k is fatigue ductility exponent. From three different counts of lead-free assemblies, fatigue ductility exponent k and fatigue ductility coefficient ϵf were calculated to be −0.564 and 0.4655 mm/mm, respectively, using a least-squares curve-fitting analysis.

After that, the volume-weighted average total strain energy was also correlated to the experimental data through:
Nf=ΔWtotalW01/k
(25)

where ΔWtotal (N·mm) is the total strain energy calculated from the volume-weighed average over a cycle in a steady-state region. W0 and k are load-independent material constants that equals to 0.358 N·mm and −0.8165, respectively, when fitted using a least square method.

Knecht and Fox [58] predicted the fatigue life of solder joint by correlating the median fatigue life with creep strain component due to matrix creep Δγmc, which is based on the creep deformation
Nf=CΔγmc
(26)

where value C depends on factors like solder microstructure and failure criteria used (increasement of electrical resistance or percentage of reduction in load) and is calculated from the least square fit between cycles to failure (Nf) and matrix creep strain (Δγmc). For 60Sn–40Pb solder joint, C was given as 890% by the authors.

Energy-Based Model

The idea that using hysteresis energy as a damage indicator relies on the assumption that, during cyclic stress–strain test, energy was absorbed cycle by cycle because of the plastic deformation [59]. The energy is a measure of the damage that is accumulated in the solder joint until a point at which it is enough to cause a fracture or crack in the solder joint. Many studies tended to correlate hysteresis energy to fatigue life as energy reflects both stress and strain that solder joints experienced in the fatigue testing, and is, therefore, a better parameter to analyze the fatigue behaviors with more accuracy than creep or plastic strain [60].

Stowell [59] came up with one of the earliest fatigue life prediction models by assuming that the total energy to the fracture of the material (Wf) is a known quantity and can be calculated from the static stress–strain hysteresis loop. Then the correlation between the energy required to fracture (Wf) and the (Wm), energy losses for cycling between positive stress level (+σ) to negative stress level (−σ) is used to predict the number of cycles to failure (N). Given the criterion for fatigue life is
Nf×Wm=Wf
(27)
The number of cycles to failure (Nf) is then calculated for cyclic stress amplitudes by
Nf=2σfσ0×ϵfσf/2Eϵ0cosσfσ0+1σσ0sinσσ02cosσσ01
(28)

where σf and ϵf are true stress and strain at fracture. E, σ0, and ϵ0 are the known constants.

Pan [61] provided a strain energy-based fatigue life prediction model (Eq. (29)) called critical accumulated strain energy failure criterion based on the assumption that solder joint fails when a critical value (C) is reached because of the accumulation of strain energy during thermal cycling.

In this study, leadless ceramic chip carrier (LCCC) consists of 84 I/O and 0.64 mm pitch size was soldered on printed wiring board with near-eutectic 60Sn–40Pb solder alloy. The size of solder square pattern is around 16.51 mm with an average distance from joint to LCCC center 9.32 mm. The equation is provided below:
C=Nf*(aEp˙+bEc)˙
(29)

where C is the critical strain energy density and geometry dependent. In this study, C was reported to be 4.55 MPa/mm3 from thermal cycling test with various thermal profiles. Nf* is the number of cycles to failure. (aEp˙+bEc)˙ is the strain energy density accumulation rate per cycle contributing to the fatigue life. Two strain energies were involved in this study: Ep and Ec are the time-dependent plastic and time-dependent creep energy generated within the solder joints and can be calculated using finite element analysis. a and b are the weighting factors for the corresponding energies.

Kujawski [62] proposed a method to predict the fatigue life (Eq. (30)) in low and high cycle fatigue, using strain energy density. In their theory, the strain energy density is viewed as a constant damage parameter in low and high life region.
ΔWt=Wf2Nfd+ΔWend
(30)

In this equation, ΔWt is the total cyclic strain energy and is the combination of the cyclic plastic strain energy enclosed within the stress–strain hysteresis loop ΔW and the linear elastic strain energy, which can be calculated by ΔWt=(1/2)ΔW+(1/2)ΔσΔε. Wf and d can be determined through the best fit of the experiment. ΔWend is the elastic strain energy density associated with the material endurance level.

Joseph and Jeries [63] proposed a similar creep-fatigue life prediction method by assuming a constant fatigue damage parameter
Nf=EintrinsicW
(31)

It assumes that there exists an “intrinsic material property”—Eintrinsic, the average of all the hysteresis energy failure density for each test in one experiment, and was viewed as a constant value of hysteresis damage energy at failure.

When compared to Wt (energy density dissipated in one cycle (w) multiplied by the total number of cycles to failure, Nf), WtEintrinsic means a failure happens. When the intrinsic energy Eintrinsic is divided by the energy density dissipated per cycle (w), we have the fatigue life for this material. Similar work is shown in Ref. [64] for validation.

Morrow's energy model [65] has been widely used to estimate the fatigue life of solder joints with this equation
NfmWp=C
(32)

where m is fatigue exponent and C is material ductility coefficient. Wp represents the plastic strain energy density for the steady-state loop.

Recently, researchers have found that the Morrow's model had ignored the effect of frequency and temperature on fatigue life, therefore causing errors when predicting fatigue life at different frequencies and temperatures [39,66].

Solomon and Tolksdorf [66] introduced a frequency modified energy model (Eq. (33)) by incorporating frequency modified strain energy density Wp/vn, and frequency modified fatigue life Nfv(k1)
Nfv(k1)mWpvn=C
(33)

where v is frequency and k is frequency exponent determined from the relationship between fatigue life and frequency. n is another frequency exponent determined from the relationship between strain energy density and frequency.

Shi et al. [39] proposed a temperature modified Morrow energy model (Eq. (34)) by applying a temperature-dependent material parameter into the frequency-modified energy model.
Nfv(k1)mWp2σflow=C
(34)

In order to yield the effect of temperature to constant C, the temperature parameter should decrease with the increase in temperature. Since σflow represents the average value of flow stress, which is the yield point and the highest point in the stress–strain curve within a hysteresis loop, it serves as the same function as the “temperature parameter,” σflow is applied to limit the influence of temperature on constant C.

Lee et al. [37] used a new physical quantity named cracking energy density (CED) to replace strain energy density in the current Morrow model and other modified Morrow models. The authors stated that strain energy density does not provide direction such as the direction of a “soon-to-be-crack” plane. However, the cracking energy density did provide the direction, which improves the accuracy when calculating the fatigue life under biaxial loading conditions. So, with new physical quantity in the Morrow's model, the fatigue life is expressed as
NfmΔCED=C
(35)
Tchankov and Vesselinov [67] reported a fatigue life prediction approach under random loading which does not need any cycle counting procedure. Normally, we only consider fatigue failure under constant amplitude situation, during which the inelastic loop is closed and the area enclosed is the dissipated energy per cycle. However, during a random loading situation, it is not possible to observe a closed-loop [67]. Under random loading situation, the total hysteresis energy is estimated by adding up all the increments of the energy during the loading process:
Wf=0failuredW
(36)
The fatigue failure under random loading will occur after hitting a boundary value of the total dissipated energy. Therefore, it is assumed that the random loading history can be divided into N* equal representative parts, each consisting of an equal number of n cycles. Therefore, the intensity of the hysteresis energy dissipation ΔW can be defined as
ΔW=WfN*
(37)
The intersection between the boundary curve and fatigue life versus hysteresis energy under constant loading line is then predicted as the number of cycles to failure, Npr, as proposed
DNprdΔWpNpr+ΔWpn2Wp1=0
(38)

where Wp1=n1ΔW1+n2n1ΔW2++np1np2ΔWp1. ΔW1, …. ΔWp is the intensity of dissipation of hysteresis energy for all the regimes. The length of each regime is n1, … np.

Letcher et al. [68] argued that the basic assumption of the energy-based fatigue model is not accurate enough to predict the fatigue life of the test sample since the energy accumulated during cycling is not equal to the energy required for the fracture. Therefore, they proposed a new definition called “critical lifetime,” which refers to the point in the specimen's lifetime when the 5% threshold is crossed. The cumulative strain energy up to the “critical lifetime” point was found to be almost constant for all the stress levels tested. Thus, the authors claimed that the “critical energy” is a proper estimation for the failure energy of the specimen. A new equation to estimate the fatigue life Nc is put forward with consideration of both curve fit approximation and critical energy
Nc=WcritWmσfεfσf22Eε0σ0cosσfσ0+12σcCσaσcsin2σaσccos2σaσc1
(39)

where σf is true stress at fracture, εf is true strain at fracture, E is modulus of elasticity, ε0 and σ0 are defined material parameter for monotonic strain, σc is material parameter for cyclic strain, σa is amplitude of alternating stress. The fatigue prediction equation is multiplied by the ratio of critical energy Wcrit to monotonic tension energy Wm.

Jahad et al. [69] provided a method to predict the fatigue life between upper and lower fatigue life limits. In this equation, the authors assumed that the total energy in the engineering area is purely axial and torsional loading.
ΔEA=EeNAB+EfNAC
(40)
ΔET=WeNTB+WfNTC
(41)
where ΔEA and ΔET are energies due to either pure tensile loading or pure torsion loading, respectively. NA and NT are the fatigue life from purely tensile or torsion loading conditions. Ee and We are the shear fatigue strength coefficients and Ef and Wf are the fatigue toughness for each loading conditions. Then the authors provided the equation to calculate the fatigue life by incorporating both limits of axial and torsional loading conditions:
Nf=ΔEAΔENA+ΔETΔENT
(42)

where Nf is the fatigue life for the material, ΔE is the total elastic-plastic energy calculated from the cyclic hysteresis loop.

Zhang et al. [70] proposed a damage evolution model to predict the fatigue life of solder joints under thermal cycling for double-beam samples. The fatigue life prediction for 63Sn–37Pb solder alloy with two beams is proposed as
D=0.00001×N2+0.0009NΔWcurrentΔW2beam
(43)

where D is the damage evolution function: D=1(S/S*), where S* is the area of the largest stabilized loop and S is the area of the current cycle. D = 0 means the undamaged state and D = 1 for the final failure of the solder material. ΔW2beam is the creep strain energy density range of the first stabilized cycle from the double-beam sample and ΔWcurrent is the current case under investigation. The two variables are collected from the center of solder joints based on the single solder joint finite element models to account for the differences in geometry and thermal loading. N is fatigue life of the solder joint. When using this equation, set D equal to 1, find ΔW2beam and ΔWcurrent from the center of the finite model of this solder, and then solve the fatigue life N.

Damage Accumulation-Based Fatigue Models

The increase of the fatigue damage with applied loads in a cumulative manner will lead to the failure of materials [54]. With the amount of damage accumulated, the lifetime under certain loads becomes limited. Therefore, it is important to study the mechanism of damage accumulation in the material that results in the failure. Comprehensive reviews about the damage accumulation theories have been conducted earlier by Fatemi and Yang [71]. Generally, the damage models can be further classified into two categories: (1) linear damage cumulative theory and (2) nonlinear damage cumulative theory.

Linear damage cumulative theory predicts that the damage caused by a stress cycle that is independent of its load sequences and assumes the ratio of damage accumulation is also independent of the stress level. Linear damage models are used due to their simplicity and close approximation to reality [72].

Among many of those models, the Miner's rule has been extensively utilized to estimate the fatigue life exposed to cyclic loading conditions with varying stress amplitudes [54]
i=1kniNi=1
(44)

where ni is the number of cycles accumulated at a given stress amplitude and Ni is the number of cycles to failure at the same stress amplitude. k indicates the number of stress levels.

Moreover, several modifications of Miner's rule have also been provided by Gatts and Grover [73,74].

Grover [74] provided a two-phase damage model, which counts damage evolution as the combination of crack initiation and crack propagation
niαNi=1forcrackinitiation
(45)
ni(1α)Ni=1forcrackpropagation
(46)

It has been proved recently by Hamasha et al. [75,76] that Miner's rule is not valid for predicting the fatigue life of lead-free solder joints, giving these reasons: (1) Lead-free solder material is stress dependent, which means that the accumulated damage after a specified fraction of fatigue life is different for different stress amplitudes. However, the Miner's rule assumes the material is stress independent and (2) Lead-free solder material is damage interactive, which means that the way damage is accumulated could be changed by previously applied stress. However, the Miner's rule assumes that the material is free of damage interaction.

Corten and Dolan provided a nonlinear accumulative damage model to account for the effects of load interaction during fatigue cycling [77]. It assumes that the damage accumulation is nonlinear and can be expressed in terms of a number of cycles in a power law equation, and the damage can propagate at both high and low stress amplitudes that are applied to the material. The damage accumulation according to the Corten–Dolan model is expressed by
iniN1σiσ1d=1
(47)

where N1 is the fatigue life at the highest stress amplitude σ1, ni is the number of cycles at stress amplitude σi, and d is the material constant for the model.

According to Corten–Dolan's model, material constant d should be constant for all the materials. However, results indicate that the exponent d is smaller at a higher maximal stress than at a lower stress [78]. Many researchers have focused on determining the value of d in order to make Corten–Dolan's model more accurate and effective [76,78].

Zhu et al. [78] proposed a “dynamic” Corten–Dolan's equation, where exponent d is defined as a function that decreases with the increasing loading stress amplitude
dσi=μσiλδf1λσi
(48)
where λ is a load–interaction factor, μ is material constant, σi is the i th level stress amplitude, δf is initial static strength. Therefore, the fatigue life of Corten–Dolan's model can be rewritten as:
Ng=N1i=1kαiσi/σ1d(σi)
(49)

where Ng is the number of cycles to failure and N1 is the number of cycles to failure at σ1.

By combining the concept of Miner's rule (linear damage theory) with Corten–Dolan theory (stress interactive theory), Hamasha et al. [76] proposed a modification of Miner's rule for combination of two alternating amplitudes
1=i=0sfinmiNm+nhiNh
(50)

where nmi is the number of cycles at low amplitude, Nm is the life at low amplitude, nhi is the number of cycles at high amplitude, and Nh is the life at high amplitude. The amplitude factor fi is the hysteresis energy in the low amplitude cycles after exposure to “i” sets of nhi high amplitude cycles divided by the hysteresis energy in the low amplitude cycles before it gets exposed to high amplitude cycles.

Cheng and Plumtree [79] came up with a model based on continuum damage mechanics and ductility exhaustion using a pressure vessel steel (16MnR) alloy for verification. In the first part, the specimens were subjected to single level constant stress amplitude. In the second part, the specimens were subjected to two levels of stress: a stress amplitude for number of cycles and then lower stress amplitude for a given number of cycles. Damage variable related to ductility exhaustion has been successfully used to predict cyclic damage evolution and this variable is expressed by
D=1ϵ̃fϵf
(51)
where ϵf is the fracture strain of the uncycled material and ϵ̃f is the residual strain after n loading cycles. ϵf=ln(1/1Ψ) where Ψ is an original reduction of section area and Ψ̃f is the residual reduction of area after n cycles. In their work, the fatigue damage evolution equation is expressed as
D=Dc11niNi1/1Ψ1/1+β
(52)
where Dc is the critical damage value that corresponds to the development of 1 mm long crack in the material. Fatigue failure will occur when damage (D) reaches or exceeds Dc. Ni is the fatigue life defined by the number of cycles to form critical crack
Ni=(1+β)(1Ψ)(ΔσA(R))β1
(53)
There are two methods to forecast the structure fatigue life: one is fatigue life prediction based on fatigue cumulative damage theories, and the other is damage tolerance method based on fracture mechanics. According to Wang et al. [80], the brittle fatigue damage model is modified and combined with constitutive damage model to predict the fatigue life of titanium alloy components. The degeneration of material mechanical properties results from nucleation, growth, and integration of microcracks and voids within metal material. Thus, for one-dimensional structure, damage variable can be defined as:
D=SDS
(54)

where S is the total area of cross section, and SD is the total area of microcracks and cavities.

By applying strain equivalence theory after damage initiation, the constitutive relation of material can be defined under a state of one-dimensional stress as
ε=σ̃E0=σE0(1D)
(55)
where σ̃, E0, and ϵ are the true stress of damaged material, the initial elasticity modulus of the material, and the elastic strain, respectively. D = 0 means the material is undamaged and D = 1 means material is damaged. And the damage evolution model is defined by
dDdN=B¯q1D2q(σeqM2qσeqm2q)
(56)

where B¯ and q are material constants, σeqM is maximum equivalent stress of loading, and σeqm is the minimum equivalent stress. This model was applied using finite element method to smooth and notched specimens and predicted the fatigue life at given stress level.

As stated by Makkonen, total fatigue life of a component can be divided into three phases: crack initiation, stable crack growth, and unstable crack growth. The last phase is rapid and can be ignored in practical work when estimating the total fatigue life [81]. Stable crack growth can be reliably modeled using linear elastic fracture mechanics. In the work by Makkonen [81], a new statistical method for predicting the fatigue crack initiation life is presented. In the initiation phase, the short crack growth is very irregular. Crack growth stops at grain boundaries and other barriers for a long time. Most of the cracks stop completely and never reach the critical size. Some of them may grow large enough and change to another mode, stable crack growth.

The crack initiation phase has long been compared to the crack growth time, especially when small test specimens are in question. For instance, in the test results referred to in this paper [81], crack initiation takes approximately 40%–90% of the total fatigue life. Since initiation is a slow process, the following working hypothesis is adopted in this work: an initiated fatigue crack will change to stable crack growth as soon as it reaches the critical size that makes this possible. This means that the initiated crack is defined as the smallest crack depth where linear elastic fracture mechanics can be applied. The depth of the initiated crack (ai) is given by
αi=1π(ΔKl,thβΔσ)2
(57)

where K is the stress intensity range threshold, and β is a geometry factor.

Darveaux [32] had put forward a method to predict the thermal cycling of solder joint fatigue life from crack growth data based on the finite element model of solder joints. Given the assumption that the crack growth rate is constant during thermal cycling, the prediction of fatigue life can be approximately calculated by adding the number of cycles for crack initiation and the number of cycles to grow cross the interface as it is defined by
Nw=N0+ada/dN
(58)
Nw is the characteristic life for the solder joint and a is the final crack length. N0 is the fatigue life at the initiation of the crack. The equation for crack initiation is given by
N0=K1×ΔWaveK2
(59)
And the equation for crack growth is given by
dadN=K3×ΔWaveK4
(60)
where ΔWavg is the average viscoplastic strain energy density accumulated per cycle for the interface elements from the finite element model built based on the solder joint. Where K1K4 are life correlation constants.
ΔWavg=i=1#ofelementsΔWi×Vi/i=1#ofelementsVi
(61)

where ΔWi is the plastic work density in the ith element and Vi is the volume of that element [32]. However, some have argued that the fatigue life of solder joint is only caused by either crack initiation or crack propagation, rather than the effect of both [82].

Gustafsson [83] then reported a modified Darveaux model to predict the fatigue life of lead-free BGA under thermal-mechanical test as expressed by
Naw=N0s+aN0sN0pdapdNdasdN+dapdN
(62)

where Naw is the characteristic life of the solder joint and α is the total possible crack length. The modification model views both primary and secondary crack initiations grow toward each other, indicated by the subscripts “p” for primary crack and “s” for secondary crack. N0S and N0P are the secondary and primary crack initiation energy-based terms, (das/dN) and (dap/dN) are the secondary and primary crack propagations.

Similarly to Darveaux, Lau et al. [84] put forward a thermal fatigue life prediction model for solder bumped flip chip. It combines the measured thermal fatigue crack growth rate of the corner solder joint and simulates nonlinear fracture characteristics (average strain energy density per cycle) around the crack tip of the corner solder joint with different crack lengths. By fitting the crack lengths in the corner solder joints with average strain energy density range, the fatigue life prediction model can be expressed by
dadN=0.06ΔW0.25
(63)

where da/dN is the fatigue crack growth rate of the corner solder joint. ΔW is the average strain energy density range in terms of the crack length (a) for given flip chip assembly. N is the number of cycles to failure. Once the ΔW is determined for a given condition, the numbers of cycles to failure can be calculated based on the proposed equation.

Paris and Erdogan [85] indicated that the crack propagation is related to the “stress-intensity factor range” ΔK, at which ΔK is considered as a measure of the effect of loading and geometry of the body at a given stress intensity near the root of the crack. The equation is given by
dadN=CΔKn
(64)

where da/dN is the rate of crack extension. C and n are material constants.

However, Forman et al. [86] proposed that two effects need to be considered when applying Paris's model, which are: (1) various growth rates due to the load ratio R and (2) the crack may grow differently when near fracture stress-intensity factor Kc. Therefore, edited Paris equation is put forward as
dadN=CΔKn1RKcΔK
(65)

where R is load ratio and Kc is fracture stress-intensity factor.

Other Models

Tee et al. [87] had provided a Weibull equation from thermal cycling tests to predict the fatigue life of solder joints under corresponding failure rate (F) with the availability of two Weibull parameters, η and β
N=ηln1F1β
(66)

where N is the solder ball fatigue life at corresponding failure rate (F), β is slope of Weibull plot, and η is the characteristic life at 63.2% failure rate, which correlated directly from finite element analysis (FEA) predicted fatigue life based on the strain energy density per cycle.

Luan et al. [88] proposed a fatigue life prediction model for the drop test as
F=βlnN+α
(67)
where N is the impact life at corresponding failure rate (F, %). β is the rate of change, α is a constant. Fatigue-free life (Nf) is the number of drops without any failure, which is viewed at a safe zone for the product and F equals to 0 at fatigue-free life. By setting F = 0, the equation becomes
Nf=eαβ
(68)
Hwang and Han [89] predicted the fatigue life using “fatigue modulus,” which is reported that the fatigue modulus degradation rate at an arbitrary fatigue cycle can be expressed in a power function of fatigue cycle:
Fn,r=σaε(n)
(69)
where F(n, r) is the fatigue modulus at nth loading cycle; ɛ(n) is the resultant strain at the nth loading cycle, and σα is the stress applied. Then they provided the fatigue life equation as the fatigue modulus at the zeroth and final cycles to failure as
N=B1Ff/E01/c
(70)

where F0 is the fatigue modulus at zeroth cycle, and Ff is the fatigue modulus at fracture (the number of cycles to failure). B and c are the material constants.

Nam et al. [90] reported that for well-defined grain boundaries and thermodynamically stable phase testing samples, the main damage under high-temperature, creep-fatigue interaction is nucleation and growth of grain boundary cavities. The number of cavities increased with the number of cycles to fatigue. Therefore, the number of nucleated cavities during cyclic loading per unit area of the grain boundary, n is related to
n=PΔεpN
(71)
where P is the cavity nucleation factor, Δɛp is the plastic strain range, and N is the number of cycles to failure. The failure of creep-fatigue interaction is controlled by the creep cavitation damage, and it is assumed that the load carrying capacity is dramatically decreased by the coalescence of grain boundary cavities and unstable crack growth. Thus, the number of cycles to failure is formulated as
Ncr=CPΔεp3/5exp(Qg/RT)T0tσ(t)dt2/5
(72)

where C is a material-related constant, Qg is the activation energy of grain boundary diffusion, σ(t) is the tensile peak stress relaxation term during hold time, R and T are the Boltzmann's constant and temperature in absolute scale.

Summary

In this paper, fatigue models that apply to solder joints are systematically reviewed. Models are categorized into four groups based on which factor is hypothesized to be more likely to contribute to fatigue damage and can predict fatigue life with better accuracy. However, some overlaps were observed among the groups:

  • Plastic strain-based models: Plastic strain was associated with fatigue life since plastic deformation was proved as the main driving force for solder joints exposed to temperature variation. Experimental work or FEA was normally required to calculate the plastic strain range in terms of solder joints geometry [36]. Acceleration factor (AF) approach was introduced. AF model makes it possible to compare between two different scenarios, normally between a “field” scenario to a “test” scenario.

  • Creep damage-based models: The aim of creep damage model is to identify the damage caused by the mixture of creep damage due to high-temperature creep and fatigue damage. Damage could be further categorized into three approaches based on the hypotheses of how creep and fatigue are compounded [91]: (1) pure creep damage, (2) partial creep and partial fatigue effecting separately, and (3) partial creep, partial fatigue, and creep–fatigue interaction damage affecting the solder joints by linear summation.

  • Energy-based models: Energy was calculated as the area within the stress–strain hysteresis loop. Solder joints fatigue life was associated with energy with various forms based on how to better calculate the area within the stress–strain loop and how to accurately link energy with the fatigue life. However, Lee et al. pointed out that one limitation for energy models is that they can only calculate the crack initiation and cannot predict the crack propagation inside the solder joint [36].

  • Damage accumulation-based models: Damage accumulation models can be further categorized as (1) linear damage accumulation approach, (2) nonlinear damage accumulation approach, which has the ability to unify damage triggered by various types of load, and (3) crack initiation and propagation approaches [71].

From the fatigue test result published by Albert in 1837 [92] till now, within about 180 years, many predictions methods have been published trying to predict fatigue life with various aspects. However, these theories still cannot predict fatigue life with a satisfactory accuracy since fatigue mechanism is much more unpredictable and complicated, and many limitations existed in those models based on Refs. [36], [92], and [93]:

  • Lack of clear definition about fatigue failure of solder joints. Some models defined the fatigue failure of solder joints based on IPC JSTD (standard lists requirements for the manufacture of electrical and electronic assemblies) standard such as changes in resistance. Others are based on the mechanical behaviors, such as maximal stress amplitude drops to half. However, no clear definition has been provided to fit all the cases of fatigue testing. Without a universal fatigue failure standard, it can be misleading to compare fatigue life among various testing conditions.

  • The prediction of fatigue life under variable amplitudes is not well predictable. Since the basic fatigue mechanism for load interaction among various amplitudes has not been studied thoroughly, no prediction approaches related to this field provide sufficient accuracy.

  • The transferability of fatigue data from testing specimens to realistic samples is poor. Some models take transferability for granted between each case. However, researchers have found that damage of realistic specimens is lower than that in test samples due to the difference in physical size.

  • There is no explanation for mechanism and root cause of corrosion fatigue.

  • Microstructure effects have been ignored for the majority of the fatigue prediction models.

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