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A new creep fatigue model for solder joints
Microelectronics Reliability 98 (2019) 153–160
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A new creep fatigue model for solder joints E.H. Wong
T
⁎
Sino-Singapore International Joint Research Institute, China Energy Research Institute at NTU, Singapore
A R T I C LE I N FO
A B S T R A C T
Keywords: Creep fatigue Analytical equation Temperature cycling Accelerated testing Acceleration factor
The simplicity of the Engelmaier equation is highly attractive; however, it suffers from two critical flaws: (i) it is incapable of modeling the full range of creep fatigue, from pure fatigue to pure creep; and (ii) the embedded creep function is inconsistent with the constitutive equation of creep. As such, the fitting constants in the Engelmaier equation are not material constants but have limited window of applicability. In this manuscript, three novel creep-integrated fatigue equations that are capable of modeling the full range of creep fatigue and that are embedded with the creep constitutive equations of Dorn, Larson-Miller, and Manson-Haferd are established. The equations are validated and benchmarked against the Engelmaier equation using the experimental creep fatigue data of Sn37Pb and An3.5Ag solders. The fitting constants in these creep-integrated fatigue equations are material constants and therefore can be used with confidence to accelerate creep fatigue testing. The acceleration factors for temperature cycle testing of solder joints in electronic assemblies have been established for these novel creep-integrated fatigue equations.
1. Introduction
adoption by other industries. Taira [2] too recognised the need to treat the damages due to creep and fatigue separately. He proposed that the two damages may be treated using their respective linear damage accumulation rules; that is, Df = ∑ ni / Ni [3] for fatigue damage and Dc = ∑ ti/ tRi [4] or
The main challenge in creep fatigue modeling is in compounding the damages due to creep and fatigue to arrive at a life prediction model. The most intuitive and simplistic approach is to consider creep to be inflicting identical mechanical damage as fatigue. Thus, the life prediction models for fatigue may then be extended to creep fatigue; e.g.
εp = CN−β → εp + εc = CN−β,
(1)
wherein εp and εc are the magnitude of the incremental plastic and incremental creep strains in a single cycle; N is the cyclic life; and C and β are fitting constants. However, this simplistic approach did not survive experimental validations. Through performing extensive experiments, Manson et al. [1] has established that creep and plastic strains of identical magnitude did not return the same cyclic life. He proposed to partition the individual strain:
εjk = Cjk Njk −βjk ,
(2)
wherein the subscripts jk represents pp, cc, and cp, respectively, and correspond to pure cyclic plastic loading, pure cyclic creep loading, and cyclic creep-plastic loading, respectively. This strain range partitioning method is widely used in the aerospace industry. However, the impracticality of conducting pure cyclic creep experiments discourages its
⁎
i
i
Dc = ∑ εi/ εRi [5] for creep damage; wherein ni and Ni are the cumulated i
number of fatigue cycles and the limiting number of fatigue cycle, respectively, under cyclic plastic strain amplitude εpi; ti and tRi are the cumulative time and the limiting creep rupture time, respectively, and εi and εRi are the cumulative creep strain and the limiting creep rupture strain, respectively, both under time-steady tensile stress σsi. Most importantly, he proposed the simple linear damage summation rule for the exhaustion of creep fatigue life:
Df + Dc ≤ 1.
(3)
In practice, the microstructural damages of creep and fatigue do frequently interact and aggravate each other [6]. This is referred to as creep fatigue interaction (CFI). The effect of such interaction has been accounted for by introducing a CFI damage factor, Dcf; that is,
Df + Dc + Dcf ≤ 1.
(4)
This method has been adopted in the design codes of power plants [7,8]. The CFI damage factor is a function of Df/Dc and its definition requires extensive characterisation effort, which is not practically
Sino-Singapore International Joint Research Institute No. 8 Feng Huang San Lu, Guangzhou, China. E-mail address: [email protected].
https://doi.org/10.1016/j.microrel.2019.05.012 Received 9 January 2019; Received in revised form 14 May 2019; Accepted 15 May 2019 Available online 27 May 2019 0026-2714/ © 2019 Elsevier Ltd. All rights reserved.
Microelectronics Reliability 98 (2019) 153–160
E.H. Wong
c (T , tc ) = 1 − c1 (T − Tref ) − c2 log(tc / tref ), T > Tref , t > tref ,
viable for the considerable number of solder alloys used in electronic assemblies. Despite the well reported evidences against lumping creep damage with fatigue damage, the electronic assembly community has been stubbornly sticking with the practice of treating creep fatigue as simply a viscoplastic behavior, in which creep is treated as time-dependent plasticity. Life prediction models for fatigue are simply extended to creep fatigue; that is, εp → εp + εc [9,10] or wp → wp + wc [11], where wp and wc are the incremental work energy densities of plastic and creep, respectively, in a single cycle. Referring to Eq. (2), this is equivalent to assuming Ccc = Cpp, βcc = βpp, and Ccp → ∞. It is no surprise that the fitting constants in these life prediction models have limited window of validity and are dependent on the geometrical shape and size of the test specimen as well as on the test conditions. The electronic assembly community has also been attracted to the simplicity of the Engelmaier equation [12], which may be expressed in the more general form as:
εp = Co N−βE (T , tc ) ,
Tref and tref are the temperature and the cycle time, respectively, below which creep is dormant. The function c(T, tc) has a value that lies between 0 and 1. The extreme conditions c(T, tc) = 0 and 1 represent pure creep and pure fatigue, respectively. By conveniently treating the creep rupture time, tR, in the Manson-Haferd stress parameter [18],
c2 =
(5)
(6)
(7)
(11)
1 c2 , c1 (σ ) = − , log(t∞/ tref ) PMH (σs )
(12)
2. Derivations 2.1. The fundamentals of creep 2.1.1. The constitutive equations of creep The steady-state creep strain rate is frequently described in the form:
εIİ = f (σs ) e−H / RT , T ≥ 0, (8)
(13)
wherein f(σs) may take the form of a power law [21], a hyperbolic sine [22], or an exponential function [23]. On the other hand, Larson-Miller [24] has proposed that the energy barrier, H, is dependent on the applied stress:
where T is in °C; and k1 and k2 are the frequency exponents for 10−3 Hz < f < 1 Hz and 10−4 Hz < f < 10−3 Hz, respectively. Unless a creep-integrated fatigue equation is capable of modeling the full range of creep fatigue, from pure creep to pure fatigue, the fitting constants would have limited window of applicability. For the Engelmaier equation, this implies the creep function, βE(T,tc), must become a constant at pure fatigue and tends to infinitely large at pure creep. None of these conditions are satisfied by Eq. (6). Not surprisingly, the fitting constants in the Engelmaier equation have been known to be dependent on the designs of the electronic assemblies and the cycling temperatures [16]. For the Coffin equation, the functions C(T) and β(T) must turn to constants and the function k(T) must turn to unity at pure fatigue; and either the function C(T) turn to nil or the function β(T) tends to infinitely large at pure creep. None of these conditions are satisfied by Eq. (8). With the necessary condition of pure creep and pure fatigue modeling in mind, Wong & Mai [17] proposed the following form of creepintegrated fatigue equation:
εp = Co c (T , tc ) N−βo,
,
wherein σs is a time-steady tensile stress in a creep rupture test; and t∞ is the time at which the iso-σs lines converge in the T − log(tR) space. However, the treatments are not mathematically sound. More fundamentally, the proposed creep function, Eq. (10), is inconsistent with any known constitutive equation of creep. There are as many creep-integrated fatigue equations as there are the number of creep constitutive equations; we shall demonstrate three creep-integrated fatigue equations that are rested on the constitutive equations of Dorn, Larson-Miller, and Manson-Haferd. The methodology for deriving the embedded creep functions from the steadystress parameters of Dorn, Larson-Miller, and Manson-Haferd has been rigorously established [20]. We shall present in this manuscript a practical and simpler approach in establishing the creep function from the constitutive equations of Dorn, Larson-Miller, and Manson-Haferd. We shall validate these creep-integrated equations using the reported creep fatigue data of the Sn37Pb solder and the Sn3.5Ag solder; and we shall benchmark these creep-integrated fatigue equations against the Engelmaier equation.
where f is the cyclic frequency. Solomon [14] and Shi et al. [15] have modeled the creep fatigue life of eutectic SnPb solder using Eq. (7). Through performing extensive mechanical creep fatigue experiments on Sn37Pb solder over a wide range of temperatures from −40 °C to 150 °C and cyclic frequency from 1 Hz to 10−4 Hz, Shi et al. [15] expressed the temperature-dependent coefficients in Eq. (7) in the form of polynomial functions:
C (T ) = 2.122 − 3.57 × 10−3T + 1.329 × 10−5T 2 − 2.502 × 10−7T 3 β (T ) = 0.731 − 1.63 × 10−4T + 1.392 × 10−6T 2 − 1.151 × 10−8T 3 k1 (T ) = 0.919 − 1.765 × 10−4T − 8.634 × 10−7T 2 k2 (T ) = 0.437 − 3.753 × 10−4T − 8.04 × 10−7T 2,
log(tR/ t∞ )
as the cyclic time, tc, and performing local collocations for Eqs. (10) and (11) at T = Tref and tc = tref respectively, Liu et al. [19] suggested that c1 and c2 may be expressed in terms of PMH(σs) as
wherein Co, βo, b1, b2, and b3 are fitting constants while T and tc are the environmental temperature and the cycle time, respectively. Creep damage is represented by the creep function, βE(T,tc); there is no need for specific computation of creep strain, which is much more computationally intensive compared to the computation of plastic strain. We shall refer to this type of equation as creep-integrated fatigue equation. Indeed, Coffin [13] was the first to propose a form of the creep-integrated fatigue equation:
εp = C (T )(Nf k (T ) − 1)−β (T ) ,
T − Tref
PMH (σs ) =
where
βE (T , tc ) = βo [1 + b1 T − b2 ln(1 + b3/ tc )],
(10)
εIİ = Be−H (σs)/ RT , T ≥ 0,
(14)
where B is a constant. Manson-Haferd [18] suggested indirectly the steady-state creep strain rate to be having the form:
εIİ = De (T − Tref ) r (σs) , T ≥ Tref ,
(15)
where D is a constant. We shall refer to Eqs. (13), (14), and (15) as the Dorn equation, the Larson-Miller equation, and the Manson-Haferd equation respectively. 2.1.2. Cyclic creep damage parameters The average creep strain rate in a single stress cycle is given by εċ = εc / tc . Assuming simplistically that εċ takes the form of εIİ , we have
PcdD (σ ) =
εc = tc e−H / RT , T ≥ 0, f (σ )
(9)
PcdLM (σ ) =
where 154
H (σ ) = T ln(Bc tc ), T ≥ 0, R
(16) (17)
Microelectronics Reliability 98 (2019) 153–160
E.H. Wong
and
PcdMH (σ ) = −
Table 1 Sn37Pb: Creep fatigue coefficients extracted from Ref. [15].
T − Tref
1 = , T ≥ Tref . r (σ ) ln(tc / tc ∞ )
Creep fatigue coefficients
(18)
wherein σ is the amplitude of the cyclic stress, Bc = B / εc, and tc∞ = εc / D. We shall refer to Pcdk(σ) as the cyclic creep damage parameter, where “k” represents “D”, “LM”, and “MF”, respectively. The cyclic creep damage parameter can be most conveniently expressed as a power-law function of stress amplitude:
Pcdk (σ ) = aσ b,
233
298
348
398
423
Cb βb
2.76 0.775
2.22 0.755
1.70 0.745
1.43 0.75
0.97 0.715
Creep fatigue coefficients
Period, tc (s), T = 298 K
(19)
where a and b are fitting constants. Applying the Ramberg–Osgood ′ relation for cyclic stress-strain, σ = K ′εpn wherein K′ and n′ are material constants, onto Eq. (19) transforms it to Pcdk (εp) = aK′bεpbn′, or simply
Pcdk (εp) = pεpq.
Cb βb
(20)
Temperature (K), tc = 1 s
100
101
102
103
104
2.22 0.755
1.92 0.735
1.67 0.715
1.23 0.715
0.60 0.670
3. Demonstrations and validations
2.2.1. The form of the creep-integrated fatigue equation The strain-life relation for a material experiencing pure fatigue may be expressed into the form:
The creep-integrated fatigue equation, εp = Coc(εp, T, tc)N−βo, with the creep function c(εp,T,tc) as defined in Section 2.2.2 shall be demonstrated and validated using the eutectic tin-lead solder Sn37Pb and the lead-free solder Sn3.5Ag, and shall be benchmarked against the Engelmaier equation.
εp, ref = Co N−βo,
3.1. Eutectic tin-lead solder Sn37Pb
2.2. The creep-integrated fatigue equation
(21)
where εp,ref is the amplitude of the alternating plastic strain. The subscript “ref” emphasizes the condition of pure fatigue. A generic creepintegrated fatigue equation may take the form:
εp = Co c (εp , T , tc ) N−βo b (εp , T , tc ) ,
3.1.1. The experimental creep fatigue data The creep fatigue coefficients (Cb, βb)j of Sn37Pb solder at five temperatures and five cycle time were extracted from Figs. 6 and 12, respectively, of Shi et al. [15] and these are tabulated in Table 1. The coefficients (Cb, βb)j were used to generate pseudo experimental creep fatigue data, (εpi, Nexp,i)j, where i = 1 to 5. These are shown in Fig. 1.
(22)
where εp is the residual capacity of plastic strain amplitude in the presence of creep. The ratio
εp/ εp, ref = c (εp , T , tc ) N−βo [b (εp , T , tc ) − 1]
3.1.2. Evaluating the fitting constants in the creep-integrated fatigue equations Let the creep fatigue coefficient, Cij, corresponding to a particular set of temperature-cycle time, (T, tc)j, and a particular plastic strain amplitude, εpi, be
(23)
gives the residual fractional fatigue capacity. The fractional creep damage is simply 1 − εp / εp,ref. Eq. (23) suggests that the fractional creep damage, and hence the state of material damage, evolves with the number of cycle. Ignoring the evolution of material damage, the creepintegrated fatigue equation shall take the form:
εp = Co c (εp , T , tc )
N−βo.
Cij = Co c (εpi , Tj , tc, j ) = Co [1 − χ (εpi ) η (T , tc ) j],
wherein χ(εp) is given by Eq. (27) and η(T,tc) by Eq. (26). The corresponding creep fatigue life, Ncom,ij, was computed as
(24)
Ncom, ij = (εpi/ Cij )−1/ βo .
2.2.2. The definition of c(εp,T,tc) The creep-integrated fatigue equation, Eq. (24), ought to possess two necessary characteristics: (i) capable of modeling the full range of creep fatigue, from pure fatigue to pure creep; and (ii) the embedded creep function is consistent with the constitutive equation of creep. Noting that the function c(εp,T,tc) is the residual fatigue capacity, let
c (εp , T , tc ) = 1 − χ (εp ) η (T , tc ),
1
(25)
1
1 1 = . Pcdk (εp ) pεpq
10
100
strainp
p
0.1
1000
N
= 1.04N-0.659 R² = 0.906
0.01
(26) 423 K
satisfy this condition. At pure creep, the fractional creep damage becomes unity. Again, referring to Eqs. (16) to (18) and applying the σ − εp transformation gives
χ (εp) =
(29)
These were regressed against Nexp,ij by minimizing the sum of the difference-square and yielded the fatigue coefficients, Co and βo, the power-law constants for cyclic damage parameters, p and q, and the fitting constants H/R, Bc and tc∞ for the respective creep rate equations.
wherein χ(εp)η(T,tc) is then the fractional creep damage. At pure fatigue, there is nil fractional creep damage, which suggests η(T,tc) = 0 at pure fatigue. Referring to the constitutive equations for cyclic creep, Eqs. (16) to (18), the functions:
ηD (T , tc ) = tc e−H / RT = 0 for T ≤ 0 ηLH (T , tc ) = T ln(Bc tc ) = 0 for T ≤ 0 , T − Tref ηMF (T , tc ) = = 0 for T ≤ Tref ln(tc / tc ∞ )
(28)
104 s
Cycle time = 1 s 398 K 348 K 298 K Temperature = 298 K 103 s 102 s 101 s
233 K
Fig. 1. Sn37Pb: Creep fatigue data, εp − Nexp, regenerated using Cb, βb of Table 1.
(27) 155
Microelectronics Reliability 98 (2019) 153–160
E.H. Wong
Table 2 Sn37Pb: Creep fatigue coefficients of creep-integrated fatigue equations embedded with the steady-state creep rate equations and benchmark with the Engelmaier equation. Creep rate equations
Fatigue coefficients βo
Co Dorn Larson-Miller Manson-Haferd, Tref = 150 K Engelmaierb Eqs. (5) & (6)
1.78 6.97a 3.75 1.49
Cyclic damage parameters
Fitting constant in the creep rate equations
Diff-square sum
p
H/R
Σ
q 1.20 × 10−9 8.39 × 103 −21.8 –
0.733 0.814 0.773 0.717
−0.039 −0.005 −0.005 –
Bc
tc∞
– 4.95 × 107 – b2 0.018
9000 – – b1 9.29 × 10−4
– – 2.34 × 107 b3 6.26 × 107
1.30 1.11 0.41 3.33
a The projection of Co to the significantly lower creep-dormant temperature gives rise to a higher magnitude of Co for the Larson-Miller equation compared to the Manson-Haferd equation. b Not a creep rate equation.
• In consistent with the smaller magnitudes of the evaluated differ-
These coefficients and the difference-square sum corresponding to the three rate equations are tabulated in Table 2. For the purpose of benchmarking, the fitting coefficients and the difference-square sum for the Engelmaier equation, Eqs. (5) & (6), were also evaluated and tabulated in Table 2. 3.1.3. Comparisons and benchmarking Among the three embedded creep constitutive equations, the Manson-Haferd equation gave a distinctively superior fitting while the Dorn equation gave the worst fitting. But even the latter is far superior than the Engelmaier equation in modeling the creep fatigue of the Sn37Pb solder. With the creep functions c(εp,T,tc) and b(T,tc) fully defined, the regenerated creep fatigue data εp − Nexp of Fig. 1 was transformed to the pure fatigue data εp,ref − Nexp using the transformation equation:
εp, ref =
εp c (εp , T , tc ) N−βo [b (T , tc ) − 1]
.
•
(30)
3.2. Lead-free solder Sn3.5Ag
The transformed pure fatigue data for the creep-integrated fatigue equations embedded with the Dorn equation, the Larson-Miller equation, the Manson-Haferd equation, and that for the Engelmaier equation are shown in Fig. 2. These have provided interesting insights into the characteristics of various creep-integrated fatigue equations:
(a)
1 1
10
ence-square sum obtained for the creep-integrated fatigue equations embedded with the constitutive equations of Manson-Haferd and Larson-Miller, the transformed εp,ref − Nexp data by these two equations collapsed nicely into coherent sets of power-law data. The creep-integrated fatigue equation embedded with the Dorn equation tends to target-transform those data associated with the longer cycle time and the highest temperature while leaving the rest of the data un-transformed. The Engelmaier equation transformed the εp,ref − Nexp data at increased effectiveness with increasing creep fatigue life. This is the inherent characteristic of the equation: it inherently and erroneously assigns increasing fractional creep damage, 1 − N−βo[b(T,tc)−1], to increasing creep fatigue life – there is nil creep damage at N = 1 and 100% creep damage at infinitely large N. In other words, the Engelmaier equation tends to under-model creep damage at low number of cycles while over-modeling creep damage at high number of cycles.
3.2.1. The experimental creep fatigue data The raw experimental creep fatigue data of Sn3.5Ag at three temperatures and four cycle time were extracted from Fig. 8 of Ref. [25]
(b)
N 100
1000
N
1
strainp,ref
strainp,ref
1
0.1
= 1.69N-0.722 R² = 0.986
p,ref
0.01
N
(c) 10
100
1000
100
1000
= 6.63N-0.804 R² = 0.988
0.1
p,ref
(d)
0.5
1000
1
strainp,ref
1
strainp,ref
100
0.01
1
0.1
10
= 3.66N-0.768 R² = 0.996
p,ref
0.01
0.05
10
N
= 1.28N-0.686 R² = 0.963
p,ref
0.005
423 K 104 s
Cycle time = 1 s 398 K 348 K 298 K Temperature = 298 K 103 s 102 s 101 s
233 K
Fig. 2. Sn37Pb: Transformed εp,ref − Nexp data using the creep constitutive equations of (a) Dorn, (b) Larson-Miller and (c) Manson-Haferd as well as (d) the Engelmaier equation based on Eqs. (5) and (6). 156
Microelectronics Reliability 98 (2019) 153–160
E.H. Wong
N 200
2000
4. Discussions
20000
strianp
0.01
4.1. Applying to solder joints in electronic assemblies experiencing temperature cycling While the fitting constants for the creep-integrated fatigue equations presented in this manuscript are material constants, these constants are dependent on the microstructural state of the materials and the stress state in the test specimen. For this reason, the fitting constants, Co, βo, p, q, H/R, Bc and tc∞ evaluated for the Sn37Pb and the Sn3.7Ag solder alloys experiencing uniaxial stress in Section 3 will require calibration when applying to temperature cycling of solder joints in electronic assemblies. The experimental creep fatigue life of solder joints, Nexp,ij, may be generated by temperature cycling of electronic assemblies at varied cycling temperatures and cycle time; while the corresponding plastic strains in the solder joints, εp,ij, may be evaluated using finite element analysis (FEA). The fitting constants can be extracted from these data following the method described in Section 3 except that the temperature function in Eq. (26) ought to be modified to account for the cycling temperature:
= 0.392N-0.573 R² = 0.585
p,ref
0.001
Temperature (K)-Cycle time (s) 293 -1 293-100 293-1000 293-10 358-10 393-10 Fig. 3. Sn3.5Ag: Creep fatigue data, εp-Nexp, extracted from Refs. [25,26].
and Fig. 4 of Ref. [26] respectively, and these are shown in Fig. 3. It is noted that the number of data point are relatively small and are not of particularly good quality; and this shall serve to demonstrate the robustness of the creep-integrated fatigue equations. 3.2.2. Evaluating the fitting constants transforming to pure fatigue data Following the procedures described in Section 3.1.2 for the Sn37Pb solder, the fatigue coefficients, Co and βo, the power-law constants for cyclic damage parameters, p and q, and the fitting constants H/R, Bc and tc∞ for the creep-integrated fatigue equation embedded with the respective creep rate equations and the Engelmaier equation were evaluated and are tabulated in Table 3. The transformed pure fatigue data, εp,ref − Nexp, for the four creep-integrated fatigue equations are shown in Fig. 4.
t
f (T )|eq =
tc
.
(31)
Referring to Eq. (26), f(T) = e−H/RT for the Dorn equation and f (T) = T for both the Larson-Miller and the Manson-Haferd equations. In other words, Teq is simply the average cycling temperature for the last two equations. Besides the ease of use, the creep-integrated fatigue equations presented in this manuscript have significant advantages over the Darveaux equation in modeling the temperature cycling of solder joints in electronic assemblies: Every single fitting constant in the creep-integrated fatigue equation is associated with the material characteristics of the solder joints: the coefficients Co and βo describe the pure fatigue characteristics of solder joints while the coefficients p, q, H/R, Bc and tc∞ describe the creep characteristics of solder joints. These material constants are independent of the design of the electronic assemblies and thus enable reliable life prediction for new designs of electronic assemblies. To a large extend, these material constants are also independent of the cycling temperature and the cycle time, which enables the establishment of a reliable acceleration model for temperature cycling test as shall be elaborated in Section 4.3.
3.2.3. Comparisons and benchmarking Among the three embedded creep constitutive equations, the Larson-Miller equation gave marginally superior fitting than the Manson-Haferd equation while the Dorn equation a distinct last. More interestingly, the Engelmaier equation has fitted comparably well the creep fatigue data of the Sn3.5Ag solder. The transformed pure fatigue data, εp,ref − Nexp, in Fig. 4 provided some insights:
• Similar to the case of the Sn37Pb solder, the creep-integrated fatigue •
∫0 c f (T (t )) dt
equation embedded with the Dorn equation tends to target-transform only data associated with the highest temperature and the longest cycle time while leaving the rest of the data un-transformed. Unlike the case of the Sn37Pb solder, the Engelmaier equation appeared to have transformed the data almost uniformly over the range of the creep fatigue life. A possible explanation is that the Engelmaier equation assigns increasing fractional creep damage to increasing creep fatigue life; while the error is acute for the set of creep fatigue data of the Sn37Pb solder that has two orders of spread in creep fatigue life, it is much less acute for the set of creep fatigue data of the Sn3.5Ag solder that has only one order of spread in creep fatigue life.
4.2. Extending to analytical evaluation of strain The evaluation of strain in the solder joints of an electronic assembly using finite element analysis suffers from inconsistency due to the singularity of strain that arise from discontinuity of materials and geometries. The inconsistency of the computed strain by finite element analysis has in some way contributed to the popularity of the Engelmaier equation that offers a simple equation for shear strain, γ, in the solder joints of an electronic assembly:
Table 3 Sn3.5Ag: Creep fatigue coefficients of creep-integrated fatigue equations embedded with the steady-state creep rate equations and benchmark with the Engelmaier equation. Rate equations
Fatigue coefficients βo
Co Dorn Larson-Miller Manson-Haferd Tref = 150 K Engelmaier Eqs. (5) & (6)
11.3 29.1 18.5 10.1
0.961 0.930 0.925 0.801
Cyclic damage parameters
Fitting constant in the constitutive equations
Diff-square sum
p
H/R
Σ
q −5
1.70 × 10 8.06 × 103 −25.0 –
0.051 0.013 0.022 –
157
5500 – – b1 1.75 × 10−3
Bc – 2.70 × 106 – b2 0.025
tc∞ – – 4.51 × 106 b3 8.90 × 106
2.12 0.53 0.77 0.67
Microelectronics Reliability 98 (2019) 153–160
E.H. Wong
200
(a)
N
2000
20000
(b)
0.1 200
N
2000
20000
strainp,ref
strainp,ref
0.01
= 4.25N-0.835 R² = 0.869
p,ref
0.001
= 22.9N-0.900 R² = 0.967
p,ref
0.001
200
2000
N
(d)
0.1
20000
200
strainp,ref
(c)
0.1
strainp,ref
0.01
0.01 -0.881 p,ref = 13.2N R² = 0.952
0.001
N
2000
20000
0.01
= 7.56N-0.764 R² = 0.958
p,ref
0.001
Temperature (K)-Cycle time (s) 293 -1 293-100 293-1000 293-10 358-10 393-10
Fig. 4. Sn3.5Ag: Transformed εp,ref-Nexp data using the creep constitutive equations of (a) Dorn, (b) Larson-Miller and (c) Manson-Haferd as well as (d) the Engelmaier equation based on Eqs. (5) and (6).
γ=
klεT , h3
l
(32)
px/2
where εT = α21ΔT is the mismatch thermal strain between the two substrates of the electronic assembly; l is the maximum distance from the neutral point, h3 is the height of the solder joints, and k is a “nonideal” constant. However, the equation is overly simplicity - it has conveniently assumed the substrates to be perfectly stiff and the solder joints to be perfectly compliant. A more fundamental analytical equation that takes into account the moduli of the constituting materials, the thicknesses of the substrates, as well as the diameter and pitches of the solder joints is needed, and this is presented below. Consider an electronic assembly as a tri-layer assembly in which the solder joints are sandwiched between two substrates. In case of the assembly experiencing a temperature excursion ΔT, the distribution of shear stress in the solder joints, assuming it is smeared over the area of the substrates, may be modeled approximately as [27–30]:
M M
joint
h3 A3 r3
M
joint
M
Fig. 5. Schematics of the critical solder joint in equilibrium.
τsmear ≈
εT e β (x − l) for βl ≥ 3, λ x κs
(33)
τjo int ≈
2
where
β= 2
λ x / κs ;
λ x = ∑ (4 + 3h3/ hi )/(Ei hi )
εT e−βpx /2, Ar λ x κs
(34)
and wherein Ar = A3 / pxpy. The equilibrium of the solder joint is shown in Fig. 5. The shear couple at the two ends of the solder joint must be balanced by the end moments; that is, 2M = τjointA3h3. The magnitude of the moment decreases linearly away from the ends of the solder joint along its length as shown in Fig. 5. For a typical solder joint that has a barrel shape, the maximum bending stress occurs at the two ends. Assuming the solder joint undergoes simple bending, the bending stress at the ends is simply
i=1
κs = h3/ G3∗ + ∑ hi /(8Gi ) are the in-plane stretch compliance and the i=1
shear compliance, respectively, of the assembly, wherein Ei, Gi, hi are the stretch modulus, the shear modulus, and the thickness of substrate #i; G3⁎ = G3A3 / (pxpy) is the smeared shear modulus of the solder joints – the smearing of the shear modulus, G3, of a single solder joint with an equivalent cross-sectional area, A3, over a rectangular area defined by the pitches between the solder joints along the x-coordinate, px, and the y-coordinate, py; and x is the distance from the neutral point. Eq. (33) has been validated by finite element analysis [28–30]. The solder joint nearer the edge of an electronic assembly, at x = l − px / 2 (refer to Fig. 5), experiences the largest magnitude of shear stress. The magnitude of the smeared shear stress at the center of this solder joint is given by substituting x = l − px / 2 into Eq. (33). The magnitude of the shear stress experienced by the solder joint with an equivalent cross-sectional area A3 is approximately
σb, jo int ≈
2h3 τjo int, r3
(35)
where r3 is the radius of the solder joint at the ends. It is worth highlighting that Eq. (35) is valid for solder joints of barrel or cylindrical shape. It is also worth highlighting that σb,joint > τjoint for typical solder joints. The corresponding shear strain and normal strain are approximately γjoint = τjoint / G3 and εjoint = σb,joint / E3. The equivalent Von Mises strain is given by 158
Microelectronics Reliability 98 (2019) 153–160
E.H. Wong
5. Conclusion
2 3 3 εjo int 2 + γjo int 2 3 2 4 2τjo int 6(h3/ r3 )2 + 3(1 + ν )2 = 3E3
εeq =
≈ εT e−βpx /2
2 3E3 λ x Ar
2h3 1+ν . + (1 + ν ) r32 h3
Three creep-integrated fatigue equations embedded with the creep strain rate equations of Dorn, Larson-Miller, and Manson-Haferd have been established. These novel creep-integrated fatigue equations have been validated and benchmarked against the Engelmaier equation using the experimental creep fatigue data of Sn37Pb solder and the Sn3.5Ag solder. The creep-integrated fatigue equations embedded with the Larson-Miller equation and the Manson-Haferd equation have been found to be capable of modeling very well the creep fatigue of both solder alloys. On the other hand, the Engelmaier equation tends to under-model creep damage at low number of cycles while over-model creep damage at high number of cycles. The singularity of strain has resulted in the inconsistency of the evaluated plastic strain in the solder joints of an electronic assembly by finite element analysis. An analytical equation for the elastic strain in the critical solder joint of an electronic assembly experiencing temperature excursion has been presented. The elastic strain does not suffer from singularity and the analytical equation has two essential novelties: (i) it accounts for the compliances of the electronic assembly and (ii) it accounts for the shear and the bending stresses in the critical solder joints. The fitting constants in the novel creep-integrated fatigue equations are material constants, independent of the design of the electronic assemblies and the test conditions. These creep-integrated fatigue equations offer reliable life prediction for new designs of electronic assemblies and offer reliable acceleration of temperature cycling test. The acceleration factor for temperature cycling has been established for these novel creep-integrated fatigue equations.
(36)
Thus, the magnitude of εeq can be reduced by increasing the in-plane compliance of the assembly, λx, increasing the distribution density, Ar = A3 / pxpy, of the solder joints, and increasing the radius of the critical solder joint at the ends, r3. The equivalent strain, εeq, ought to be used in place of the plastic strain amplitude, εp, in the creep-integrated fatigue equations presented in Section 2. The use of the analytically evaluated elastic strain, εeq, in place of the FEA evaluated plastic strain, εp, in the creep-integrated fatigue equation requires the recalibration of the fitting coefficients. 4.3. Accelerating the temperature cycling test With the fitting constants in the creep-integrated fatigue equation that are independent of the cycling temperature and the cycle time, the creep-integrated fatigue equation can be used with confidence to accelerate temperature cycling test, which is a currently very time consuming. The acceleration factor for temperature cycling test is defined as
AF =
N1 tc1 = AFN AFtc . N2 tc 2
(37)
The cycle time for the temperature cycling test can be reduced by raising the cycling temperature. The function η(T,tc) in Eq. (26) provides the relationship between temperature and cycle time. Let η(T1,tc1) = η(T2,tc2) and after some manipulation yields AFtc for the Larson-Miller and the Manson-Haferd equations:
Declaration of Competing Interest None. References
AFtc |Larson − Miller = (Bc tc 2 )T2/ T1− 1 AFtc |Manson − Haferd = (tc 2/ tc ∞ )(T1− T2)/(T2− Tref ) .
(38)
[1] S.S. Manson, G.R. Halford, M.H. Hirschberg, Creep-fatigue analysis by strain range partitioning, Proc. First Symposium on Design for Elevated Temperature Environment, 1971, pp. 12–28 San Francisco. [2] S. Taira, Lifetime of structures subjected to varying load and temperature, Creep in Structures, Hoff NJ Ed, Springer-Verlag, Berlin, 1962, pp. 96–124. [3] Palmgren AG: Die Lebensdauer von Kugellagern. Zeitschrift des Vereines Deutscher Ingenieure, vol. 68, (1924), pp. 339–341. [4] E.L. Robison, Effect of temperature variations on the creep strength of steels, Trans. ASME 60 (1938) 253–259. [5] R.M. Goldhoff, Uniaxial creep–rupture behaviour of low alloy steel under variable loading conditions, J Basic Engng 86 (1965). [6] R. Penny, D. Marriott, Design for Creep, Chapman & Hall, 1995. [7] ASME Boiler and Pressure Vessel Code, Section III Rules for the Construction of Nuclear Facility Components, Div. 1-Subsection NH, Class 1 Components in Elevated Temperature Service, ASME, New York, 2007. [8] RCC-MR, Design and Construction Rules for Mechanical Components of Nuclear Installations, AFCEN, Paris, 2012. [9] H.U. Akay, N.H. Paydar, A. Bilgic, Fatigue life predictions for thermally loaded solder joints using a volume-weighted averaging technique, Trans. ASME J Electron Packag 119 (1997) 228–235. [10] J. Liang, N. Golhardt, P.S. Lee, S. Heinrich, S. Schroeder, A integrated fatigue life prediction methodology for optimum design and reliability assemssment of solder integ-connections, Int Intersociety Electronic and Photonic Packaging (1997) 1583–1592. [11] R. Darveaux, Effects of simulation methodology on solder joint crack growth correlation and faituge life prediction, Trans. ASME J. Electron. Packag. 124 (2002) 147. [12] W. Engelmaier, Fatigue life of leadless chip carrier solder joint during power cycling, IEEE Trans on Components, Hybrids and Manufacturing Technology 6 (1983) 232–237. [13] L.F. Coffin Jr., Fatigue at high temperature, ASTM STP 520 Fatigue at elevated temperature (1973) 5–34. [14] H. Solomon, Fatigue of 60/40 solder, IEEE Trans Components, Hybrids, and Manufacturing Technology 9 (4) (1986) 423–432. [15] X.Q. Shi, H.L.J. Pang, W. Zhou, Z.P. Wang, Low cycle fatigue analysis of temperature and frequency effects in eutectic solder alloy, Int. J. Fatigue 22 (2000) 217–228. [16] P. Chauhan, M. Osterman, S.W.R. Ricky, M. Pecht, Critical review of the Engelmaier model for solder joint creep fatigue reliability, IEEE Trans on Comp Packag
It is worth highlighting that the temperatures, T1 and T2, in Eq. (38) are indeed the equivalent cycling temperature: Teq = ∮ T(t)dt/tc. The number of cycles for the temperature cycling test can be reduced by raising the magnitude of the equivalent strain through raising the range of the cycling temperature. From Eq. (24), 1/ βo
AFN =
εeq2 c1 (εeq , T , tc ) ⎞ N1 = ⎛⎜ ⎟ N2 ⎝ εeq1 c2 (εeq , T , tc ) ⎠
. (39)
The creep function c(εeq,T,tc) is given by 1 − χ(εeq)η(T,tc); noting that η(T,tc) has been kept a constant and that χ(εeq) is a weak function of εeq, the acceleration factor for N may be approximated as AFN ≈ (εeq2/εeq1)1/βo. Substituting εeq with Eq. (36) and noting that εT = α21ΔT yields
AFN ≈ (ΔT2/ΔT1 )1/ βo .
(40)
4.4. Limitations The methodology described in this manuscript is a simplification of a more comprehensive methodology presented in Ref. [20]. This methodology does not account for the exact profile of the cycling strain on creep damage. Consequently, the creep coefficients: p, q, H/R, Bc, and tc∞ may be dependent on the time-temperature profile of the cycling temperature. This methodology also assumes the fitting constants Bc and tc∞ to be constants, which is not strictly correct, though it did appear to be acceptable for the Sn37Pb and the Sn3.5Ag solder alloys as presented in Section 3. 159
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E.H. Wong
[24] F. Larson, J. Miller, A time-temperature relationship for rupture and creep stresses, Trans. ASME 174 (5) (1952) 765–775. [25] C. Kanchanomai, Y. Mutoh, Effect of temperature on isothermal low cycle fatigue properties of Sn–Ag eutectic solder, Mater. Sci. Eng. A 381 (2004) 113–120. [26] C. Kanchanomai, Y. Miyashita, Y. Mutoh, S.L. Mannan, Influence of frequency on low cycle fatigue behavior of Pb-free solder 96.5Sn/3.5Ag, Mater. Sci. Eng. A 345 (2003) 90–98. [27] E. Suhir, Stresses in bi-metal thermostats, ASME J Appl Mech 53 (1986) 657–660. [28] E.H. Wong, Interfacial stresses in sandwich structures subjected to temperature and mechanical loads, Comp Struct 134 (2015) 226–236. [29] Wong E.H. and Liu Johan. Interface and interconnection stresses in electronic assemblies – a critical review of analytical solutions. Microelectron. Reliab., DOI:https://doi.org/10.1016/j.microrel.2017.03.010. [30] E.H. Wong, Liu Johan, Design analysis of bonded structures, Polymers 9 (2017) 664, https://doi.org/10.3390/polym9120664.
Technologies 32 (3) (2009) 693–700. [17] E.H. Wong, Y.-W. Mai, A unified equation for creep fatigue, Int. J. Fatigue 68 (2014) 186–194. [18] S.S. Manson, A.M. Haferd, A linear time-temperature relation for extrapolation of creep and stress-rupture data, NACA TN 2890. National Advisory Committee for Aeronautics, 1953. [19] D. Liu, D.J. Pons, E.H. Wong, Creep-integrated fatigue equation for metals, Int. J. Fatigue 98 (2017) 167–175. [20] E.H. Wong, Derivation of novel creep-integrated fatigue equations, Int. J. Fatigue (2019) (under review). [21] F.H. Norton, The Creep of Steel at High Temperatures, McGraw-Hill, London, 1929. [22] P.G. McVetty, Creep of metals at elevated temperatures: the hyperbolic sine relation between stress and creep rate, Trans. ASME 65 (1943) 761–769. [23] J.E. Dorn, Some fundamental experiments on high temperature creep, J Mech Phys Solids 8 (1954) 85–116.
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Contents lists available at ScienceDirect
Microelectronics Reliability journal homepage: www.elsevier.com/locate/microrel
A new creep fatigue model for solder joints E.H. Wong
T
⁎
Sino-Singapore International Joint Research Institute, China Energy Research Institute at NTU, Singapore
A R T I C LE I N FO
A B S T R A C T
Keywords: Creep fatigue Analytical equation Temperature cycling Accelerated testing Acceleration factor
The simplicity of the Engelmaier equation is highly attractive; however, it suffers from two critical flaws: (i) it is incapable of modeling the full range of creep fatigue, from pure fatigue to pure creep; and (ii) the embedded creep function is inconsistent with the constitutive equation of creep. As such, the fitting constants in the Engelmaier equation are not material constants but have limited window of applicability. In this manuscript, three novel creep-integrated fatigue equations that are capable of modeling the full range of creep fatigue and that are embedded with the creep constitutive equations of Dorn, Larson-Miller, and Manson-Haferd are established. The equations are validated and benchmarked against the Engelmaier equation using the experimental creep fatigue data of Sn37Pb and An3.5Ag solders. The fitting constants in these creep-integrated fatigue equations are material constants and therefore can be used with confidence to accelerate creep fatigue testing. The acceleration factors for temperature cycle testing of solder joints in electronic assemblies have been established for these novel creep-integrated fatigue equations.
1. Introduction
adoption by other industries. Taira [2] too recognised the need to treat the damages due to creep and fatigue separately. He proposed that the two damages may be treated using their respective linear damage accumulation rules; that is, Df = ∑ ni / Ni [3] for fatigue damage and Dc = ∑ ti/ tRi [4] or
The main challenge in creep fatigue modeling is in compounding the damages due to creep and fatigue to arrive at a life prediction model. The most intuitive and simplistic approach is to consider creep to be inflicting identical mechanical damage as fatigue. Thus, the life prediction models for fatigue may then be extended to creep fatigue; e.g.
εp = CN−β → εp + εc = CN−β,
(1)
wherein εp and εc are the magnitude of the incremental plastic and incremental creep strains in a single cycle; N is the cyclic life; and C and β are fitting constants. However, this simplistic approach did not survive experimental validations. Through performing extensive experiments, Manson et al. [1] has established that creep and plastic strains of identical magnitude did not return the same cyclic life. He proposed to partition the individual strain:
εjk = Cjk Njk −βjk ,
(2)
wherein the subscripts jk represents pp, cc, and cp, respectively, and correspond to pure cyclic plastic loading, pure cyclic creep loading, and cyclic creep-plastic loading, respectively. This strain range partitioning method is widely used in the aerospace industry. However, the impracticality of conducting pure cyclic creep experiments discourages its
⁎
i
i
Dc = ∑ εi/ εRi [5] for creep damage; wherein ni and Ni are the cumulated i
number of fatigue cycles and the limiting number of fatigue cycle, respectively, under cyclic plastic strain amplitude εpi; ti and tRi are the cumulative time and the limiting creep rupture time, respectively, and εi and εRi are the cumulative creep strain and the limiting creep rupture strain, respectively, both under time-steady tensile stress σsi. Most importantly, he proposed the simple linear damage summation rule for the exhaustion of creep fatigue life:
Df + Dc ≤ 1.
(3)
In practice, the microstructural damages of creep and fatigue do frequently interact and aggravate each other [6]. This is referred to as creep fatigue interaction (CFI). The effect of such interaction has been accounted for by introducing a CFI damage factor, Dcf; that is,
Df + Dc + Dcf ≤ 1.
(4)
This method has been adopted in the design codes of power plants [7,8]. The CFI damage factor is a function of Df/Dc and its definition requires extensive characterisation effort, which is not practically
Sino-Singapore International Joint Research Institute No. 8 Feng Huang San Lu, Guangzhou, China. E-mail address: [email protected].
https://doi.org/10.1016/j.microrel.2019.05.012 Received 9 January 2019; Received in revised form 14 May 2019; Accepted 15 May 2019 Available online 27 May 2019 0026-2714/ © 2019 Elsevier Ltd. All rights reserved.
Microelectronics Reliability 98 (2019) 153–160
E.H. Wong
c (T , tc ) = 1 − c1 (T − Tref ) − c2 log(tc / tref ), T > Tref , t > tref ,
viable for the considerable number of solder alloys used in electronic assemblies. Despite the well reported evidences against lumping creep damage with fatigue damage, the electronic assembly community has been stubbornly sticking with the practice of treating creep fatigue as simply a viscoplastic behavior, in which creep is treated as time-dependent plasticity. Life prediction models for fatigue are simply extended to creep fatigue; that is, εp → εp + εc [9,10] or wp → wp + wc [11], where wp and wc are the incremental work energy densities of plastic and creep, respectively, in a single cycle. Referring to Eq. (2), this is equivalent to assuming Ccc = Cpp, βcc = βpp, and Ccp → ∞. It is no surprise that the fitting constants in these life prediction models have limited window of validity and are dependent on the geometrical shape and size of the test specimen as well as on the test conditions. The electronic assembly community has also been attracted to the simplicity of the Engelmaier equation [12], which may be expressed in the more general form as:
εp = Co N−βE (T , tc ) ,
Tref and tref are the temperature and the cycle time, respectively, below which creep is dormant. The function c(T, tc) has a value that lies between 0 and 1. The extreme conditions c(T, tc) = 0 and 1 represent pure creep and pure fatigue, respectively. By conveniently treating the creep rupture time, tR, in the Manson-Haferd stress parameter [18],
c2 =
(5)
(6)
(7)
(11)
1 c2 , c1 (σ ) = − , log(t∞/ tref ) PMH (σs )
(12)
2. Derivations 2.1. The fundamentals of creep 2.1.1. The constitutive equations of creep The steady-state creep strain rate is frequently described in the form:
εIİ = f (σs ) e−H / RT , T ≥ 0, (8)
(13)
wherein f(σs) may take the form of a power law [21], a hyperbolic sine [22], or an exponential function [23]. On the other hand, Larson-Miller [24] has proposed that the energy barrier, H, is dependent on the applied stress:
where T is in °C; and k1 and k2 are the frequency exponents for 10−3 Hz < f < 1 Hz and 10−4 Hz < f < 10−3 Hz, respectively. Unless a creep-integrated fatigue equation is capable of modeling the full range of creep fatigue, from pure creep to pure fatigue, the fitting constants would have limited window of applicability. For the Engelmaier equation, this implies the creep function, βE(T,tc), must become a constant at pure fatigue and tends to infinitely large at pure creep. None of these conditions are satisfied by Eq. (6). Not surprisingly, the fitting constants in the Engelmaier equation have been known to be dependent on the designs of the electronic assemblies and the cycling temperatures [16]. For the Coffin equation, the functions C(T) and β(T) must turn to constants and the function k(T) must turn to unity at pure fatigue; and either the function C(T) turn to nil or the function β(T) tends to infinitely large at pure creep. None of these conditions are satisfied by Eq. (8). With the necessary condition of pure creep and pure fatigue modeling in mind, Wong & Mai [17] proposed the following form of creepintegrated fatigue equation:
εp = Co c (T , tc ) N−βo,
,
wherein σs is a time-steady tensile stress in a creep rupture test; and t∞ is the time at which the iso-σs lines converge in the T − log(tR) space. However, the treatments are not mathematically sound. More fundamentally, the proposed creep function, Eq. (10), is inconsistent with any known constitutive equation of creep. There are as many creep-integrated fatigue equations as there are the number of creep constitutive equations; we shall demonstrate three creep-integrated fatigue equations that are rested on the constitutive equations of Dorn, Larson-Miller, and Manson-Haferd. The methodology for deriving the embedded creep functions from the steadystress parameters of Dorn, Larson-Miller, and Manson-Haferd has been rigorously established [20]. We shall present in this manuscript a practical and simpler approach in establishing the creep function from the constitutive equations of Dorn, Larson-Miller, and Manson-Haferd. We shall validate these creep-integrated equations using the reported creep fatigue data of the Sn37Pb solder and the Sn3.5Ag solder; and we shall benchmark these creep-integrated fatigue equations against the Engelmaier equation.
where f is the cyclic frequency. Solomon [14] and Shi et al. [15] have modeled the creep fatigue life of eutectic SnPb solder using Eq. (7). Through performing extensive mechanical creep fatigue experiments on Sn37Pb solder over a wide range of temperatures from −40 °C to 150 °C and cyclic frequency from 1 Hz to 10−4 Hz, Shi et al. [15] expressed the temperature-dependent coefficients in Eq. (7) in the form of polynomial functions:
C (T ) = 2.122 − 3.57 × 10−3T + 1.329 × 10−5T 2 − 2.502 × 10−7T 3 β (T ) = 0.731 − 1.63 × 10−4T + 1.392 × 10−6T 2 − 1.151 × 10−8T 3 k1 (T ) = 0.919 − 1.765 × 10−4T − 8.634 × 10−7T 2 k2 (T ) = 0.437 − 3.753 × 10−4T − 8.04 × 10−7T 2,
log(tR/ t∞ )
as the cyclic time, tc, and performing local collocations for Eqs. (10) and (11) at T = Tref and tc = tref respectively, Liu et al. [19] suggested that c1 and c2 may be expressed in terms of PMH(σs) as
wherein Co, βo, b1, b2, and b3 are fitting constants while T and tc are the environmental temperature and the cycle time, respectively. Creep damage is represented by the creep function, βE(T,tc); there is no need for specific computation of creep strain, which is much more computationally intensive compared to the computation of plastic strain. We shall refer to this type of equation as creep-integrated fatigue equation. Indeed, Coffin [13] was the first to propose a form of the creep-integrated fatigue equation:
εp = C (T )(Nf k (T ) − 1)−β (T ) ,
T − Tref
PMH (σs ) =
where
βE (T , tc ) = βo [1 + b1 T − b2 ln(1 + b3/ tc )],
(10)
εIİ = Be−H (σs)/ RT , T ≥ 0,
(14)
where B is a constant. Manson-Haferd [18] suggested indirectly the steady-state creep strain rate to be having the form:
εIİ = De (T − Tref ) r (σs) , T ≥ Tref ,
(15)
where D is a constant. We shall refer to Eqs. (13), (14), and (15) as the Dorn equation, the Larson-Miller equation, and the Manson-Haferd equation respectively. 2.1.2. Cyclic creep damage parameters The average creep strain rate in a single stress cycle is given by εċ = εc / tc . Assuming simplistically that εċ takes the form of εIİ , we have
PcdD (σ ) =
εc = tc e−H / RT , T ≥ 0, f (σ )
(9)
PcdLM (σ ) =
where 154
H (σ ) = T ln(Bc tc ), T ≥ 0, R
(16) (17)
Microelectronics Reliability 98 (2019) 153–160
E.H. Wong
and
PcdMH (σ ) = −
Table 1 Sn37Pb: Creep fatigue coefficients extracted from Ref. [15].
T − Tref
1 = , T ≥ Tref . r (σ ) ln(tc / tc ∞ )
Creep fatigue coefficients
(18)
wherein σ is the amplitude of the cyclic stress, Bc = B / εc, and tc∞ = εc / D. We shall refer to Pcdk(σ) as the cyclic creep damage parameter, where “k” represents “D”, “LM”, and “MF”, respectively. The cyclic creep damage parameter can be most conveniently expressed as a power-law function of stress amplitude:
Pcdk (σ ) = aσ b,
233
298
348
398
423
Cb βb
2.76 0.775
2.22 0.755
1.70 0.745
1.43 0.75
0.97 0.715
Creep fatigue coefficients
Period, tc (s), T = 298 K
(19)
where a and b are fitting constants. Applying the Ramberg–Osgood ′ relation for cyclic stress-strain, σ = K ′εpn wherein K′ and n′ are material constants, onto Eq. (19) transforms it to Pcdk (εp) = aK′bεpbn′, or simply
Pcdk (εp) = pεpq.
Cb βb
(20)
Temperature (K), tc = 1 s
100
101
102
103
104
2.22 0.755
1.92 0.735
1.67 0.715
1.23 0.715
0.60 0.670
3. Demonstrations and validations
2.2.1. The form of the creep-integrated fatigue equation The strain-life relation for a material experiencing pure fatigue may be expressed into the form:
The creep-integrated fatigue equation, εp = Coc(εp, T, tc)N−βo, with the creep function c(εp,T,tc) as defined in Section 2.2.2 shall be demonstrated and validated using the eutectic tin-lead solder Sn37Pb and the lead-free solder Sn3.5Ag, and shall be benchmarked against the Engelmaier equation.
εp, ref = Co N−βo,
3.1. Eutectic tin-lead solder Sn37Pb
2.2. The creep-integrated fatigue equation
(21)
where εp,ref is the amplitude of the alternating plastic strain. The subscript “ref” emphasizes the condition of pure fatigue. A generic creepintegrated fatigue equation may take the form:
εp = Co c (εp , T , tc ) N−βo b (εp , T , tc ) ,
3.1.1. The experimental creep fatigue data The creep fatigue coefficients (Cb, βb)j of Sn37Pb solder at five temperatures and five cycle time were extracted from Figs. 6 and 12, respectively, of Shi et al. [15] and these are tabulated in Table 1. The coefficients (Cb, βb)j were used to generate pseudo experimental creep fatigue data, (εpi, Nexp,i)j, where i = 1 to 5. These are shown in Fig. 1.
(22)
where εp is the residual capacity of plastic strain amplitude in the presence of creep. The ratio
εp/ εp, ref = c (εp , T , tc ) N−βo [b (εp , T , tc ) − 1]
3.1.2. Evaluating the fitting constants in the creep-integrated fatigue equations Let the creep fatigue coefficient, Cij, corresponding to a particular set of temperature-cycle time, (T, tc)j, and a particular plastic strain amplitude, εpi, be
(23)
gives the residual fractional fatigue capacity. The fractional creep damage is simply 1 − εp / εp,ref. Eq. (23) suggests that the fractional creep damage, and hence the state of material damage, evolves with the number of cycle. Ignoring the evolution of material damage, the creepintegrated fatigue equation shall take the form:
εp = Co c (εp , T , tc )
N−βo.
Cij = Co c (εpi , Tj , tc, j ) = Co [1 − χ (εpi ) η (T , tc ) j],
wherein χ(εp) is given by Eq. (27) and η(T,tc) by Eq. (26). The corresponding creep fatigue life, Ncom,ij, was computed as
(24)
Ncom, ij = (εpi/ Cij )−1/ βo .
2.2.2. The definition of c(εp,T,tc) The creep-integrated fatigue equation, Eq. (24), ought to possess two necessary characteristics: (i) capable of modeling the full range of creep fatigue, from pure fatigue to pure creep; and (ii) the embedded creep function is consistent with the constitutive equation of creep. Noting that the function c(εp,T,tc) is the residual fatigue capacity, let
c (εp , T , tc ) = 1 − χ (εp ) η (T , tc ),
1
(25)
1
1 1 = . Pcdk (εp ) pεpq
10
100
strainp
p
0.1
1000
N
= 1.04N-0.659 R² = 0.906
0.01
(26) 423 K
satisfy this condition. At pure creep, the fractional creep damage becomes unity. Again, referring to Eqs. (16) to (18) and applying the σ − εp transformation gives
χ (εp) =
(29)
These were regressed against Nexp,ij by minimizing the sum of the difference-square and yielded the fatigue coefficients, Co and βo, the power-law constants for cyclic damage parameters, p and q, and the fitting constants H/R, Bc and tc∞ for the respective creep rate equations.
wherein χ(εp)η(T,tc) is then the fractional creep damage. At pure fatigue, there is nil fractional creep damage, which suggests η(T,tc) = 0 at pure fatigue. Referring to the constitutive equations for cyclic creep, Eqs. (16) to (18), the functions:
ηD (T , tc ) = tc e−H / RT = 0 for T ≤ 0 ηLH (T , tc ) = T ln(Bc tc ) = 0 for T ≤ 0 , T − Tref ηMF (T , tc ) = = 0 for T ≤ Tref ln(tc / tc ∞ )
(28)
104 s
Cycle time = 1 s 398 K 348 K 298 K Temperature = 298 K 103 s 102 s 101 s
233 K
Fig. 1. Sn37Pb: Creep fatigue data, εp − Nexp, regenerated using Cb, βb of Table 1.
(27) 155
Microelectronics Reliability 98 (2019) 153–160
E.H. Wong
Table 2 Sn37Pb: Creep fatigue coefficients of creep-integrated fatigue equations embedded with the steady-state creep rate equations and benchmark with the Engelmaier equation. Creep rate equations
Fatigue coefficients βo
Co Dorn Larson-Miller Manson-Haferd, Tref = 150 K Engelmaierb Eqs. (5) & (6)
1.78 6.97a 3.75 1.49
Cyclic damage parameters
Fitting constant in the creep rate equations
Diff-square sum
p
H/R
Σ
q 1.20 × 10−9 8.39 × 103 −21.8 –
0.733 0.814 0.773 0.717
−0.039 −0.005 −0.005 –
Bc
tc∞
– 4.95 × 107 – b2 0.018
9000 – – b1 9.29 × 10−4
– – 2.34 × 107 b3 6.26 × 107
1.30 1.11 0.41 3.33
a The projection of Co to the significantly lower creep-dormant temperature gives rise to a higher magnitude of Co for the Larson-Miller equation compared to the Manson-Haferd equation. b Not a creep rate equation.
• In consistent with the smaller magnitudes of the evaluated differ-
These coefficients and the difference-square sum corresponding to the three rate equations are tabulated in Table 2. For the purpose of benchmarking, the fitting coefficients and the difference-square sum for the Engelmaier equation, Eqs. (5) & (6), were also evaluated and tabulated in Table 2. 3.1.3. Comparisons and benchmarking Among the three embedded creep constitutive equations, the Manson-Haferd equation gave a distinctively superior fitting while the Dorn equation gave the worst fitting. But even the latter is far superior than the Engelmaier equation in modeling the creep fatigue of the Sn37Pb solder. With the creep functions c(εp,T,tc) and b(T,tc) fully defined, the regenerated creep fatigue data εp − Nexp of Fig. 1 was transformed to the pure fatigue data εp,ref − Nexp using the transformation equation:
εp, ref =
εp c (εp , T , tc ) N−βo [b (T , tc ) − 1]
.
•
(30)
3.2. Lead-free solder Sn3.5Ag
The transformed pure fatigue data for the creep-integrated fatigue equations embedded with the Dorn equation, the Larson-Miller equation, the Manson-Haferd equation, and that for the Engelmaier equation are shown in Fig. 2. These have provided interesting insights into the characteristics of various creep-integrated fatigue equations:
(a)
1 1
10
ence-square sum obtained for the creep-integrated fatigue equations embedded with the constitutive equations of Manson-Haferd and Larson-Miller, the transformed εp,ref − Nexp data by these two equations collapsed nicely into coherent sets of power-law data. The creep-integrated fatigue equation embedded with the Dorn equation tends to target-transform those data associated with the longer cycle time and the highest temperature while leaving the rest of the data un-transformed. The Engelmaier equation transformed the εp,ref − Nexp data at increased effectiveness with increasing creep fatigue life. This is the inherent characteristic of the equation: it inherently and erroneously assigns increasing fractional creep damage, 1 − N−βo[b(T,tc)−1], to increasing creep fatigue life – there is nil creep damage at N = 1 and 100% creep damage at infinitely large N. In other words, the Engelmaier equation tends to under-model creep damage at low number of cycles while over-modeling creep damage at high number of cycles.
3.2.1. The experimental creep fatigue data The raw experimental creep fatigue data of Sn3.5Ag at three temperatures and four cycle time were extracted from Fig. 8 of Ref. [25]
(b)
N 100
1000
N
1
strainp,ref
strainp,ref
1
0.1
= 1.69N-0.722 R² = 0.986
p,ref
0.01
N
(c) 10
100
1000
100
1000
= 6.63N-0.804 R² = 0.988
0.1
p,ref
(d)
0.5
1000
1
strainp,ref
1
strainp,ref
100
0.01
1
0.1
10
= 3.66N-0.768 R² = 0.996
p,ref
0.01
0.05
10
N
= 1.28N-0.686 R² = 0.963
p,ref
0.005
423 K 104 s
Cycle time = 1 s 398 K 348 K 298 K Temperature = 298 K 103 s 102 s 101 s
233 K
Fig. 2. Sn37Pb: Transformed εp,ref − Nexp data using the creep constitutive equations of (a) Dorn, (b) Larson-Miller and (c) Manson-Haferd as well as (d) the Engelmaier equation based on Eqs. (5) and (6). 156
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N 200
2000
4. Discussions
20000
strianp
0.01
4.1. Applying to solder joints in electronic assemblies experiencing temperature cycling While the fitting constants for the creep-integrated fatigue equations presented in this manuscript are material constants, these constants are dependent on the microstructural state of the materials and the stress state in the test specimen. For this reason, the fitting constants, Co, βo, p, q, H/R, Bc and tc∞ evaluated for the Sn37Pb and the Sn3.7Ag solder alloys experiencing uniaxial stress in Section 3 will require calibration when applying to temperature cycling of solder joints in electronic assemblies. The experimental creep fatigue life of solder joints, Nexp,ij, may be generated by temperature cycling of electronic assemblies at varied cycling temperatures and cycle time; while the corresponding plastic strains in the solder joints, εp,ij, may be evaluated using finite element analysis (FEA). The fitting constants can be extracted from these data following the method described in Section 3 except that the temperature function in Eq. (26) ought to be modified to account for the cycling temperature:
= 0.392N-0.573 R² = 0.585
p,ref
0.001
Temperature (K)-Cycle time (s) 293 -1 293-100 293-1000 293-10 358-10 393-10 Fig. 3. Sn3.5Ag: Creep fatigue data, εp-Nexp, extracted from Refs. [25,26].
and Fig. 4 of Ref. [26] respectively, and these are shown in Fig. 3. It is noted that the number of data point are relatively small and are not of particularly good quality; and this shall serve to demonstrate the robustness of the creep-integrated fatigue equations. 3.2.2. Evaluating the fitting constants transforming to pure fatigue data Following the procedures described in Section 3.1.2 for the Sn37Pb solder, the fatigue coefficients, Co and βo, the power-law constants for cyclic damage parameters, p and q, and the fitting constants H/R, Bc and tc∞ for the creep-integrated fatigue equation embedded with the respective creep rate equations and the Engelmaier equation were evaluated and are tabulated in Table 3. The transformed pure fatigue data, εp,ref − Nexp, for the four creep-integrated fatigue equations are shown in Fig. 4.
t
f (T )|eq =
tc
.
(31)
Referring to Eq. (26), f(T) = e−H/RT for the Dorn equation and f (T) = T for both the Larson-Miller and the Manson-Haferd equations. In other words, Teq is simply the average cycling temperature for the last two equations. Besides the ease of use, the creep-integrated fatigue equations presented in this manuscript have significant advantages over the Darveaux equation in modeling the temperature cycling of solder joints in electronic assemblies: Every single fitting constant in the creep-integrated fatigue equation is associated with the material characteristics of the solder joints: the coefficients Co and βo describe the pure fatigue characteristics of solder joints while the coefficients p, q, H/R, Bc and tc∞ describe the creep characteristics of solder joints. These material constants are independent of the design of the electronic assemblies and thus enable reliable life prediction for new designs of electronic assemblies. To a large extend, these material constants are also independent of the cycling temperature and the cycle time, which enables the establishment of a reliable acceleration model for temperature cycling test as shall be elaborated in Section 4.3.
3.2.3. Comparisons and benchmarking Among the three embedded creep constitutive equations, the Larson-Miller equation gave marginally superior fitting than the Manson-Haferd equation while the Dorn equation a distinct last. More interestingly, the Engelmaier equation has fitted comparably well the creep fatigue data of the Sn3.5Ag solder. The transformed pure fatigue data, εp,ref − Nexp, in Fig. 4 provided some insights:
• Similar to the case of the Sn37Pb solder, the creep-integrated fatigue •
∫0 c f (T (t )) dt
equation embedded with the Dorn equation tends to target-transform only data associated with the highest temperature and the longest cycle time while leaving the rest of the data un-transformed. Unlike the case of the Sn37Pb solder, the Engelmaier equation appeared to have transformed the data almost uniformly over the range of the creep fatigue life. A possible explanation is that the Engelmaier equation assigns increasing fractional creep damage to increasing creep fatigue life; while the error is acute for the set of creep fatigue data of the Sn37Pb solder that has two orders of spread in creep fatigue life, it is much less acute for the set of creep fatigue data of the Sn3.5Ag solder that has only one order of spread in creep fatigue life.
4.2. Extending to analytical evaluation of strain The evaluation of strain in the solder joints of an electronic assembly using finite element analysis suffers from inconsistency due to the singularity of strain that arise from discontinuity of materials and geometries. The inconsistency of the computed strain by finite element analysis has in some way contributed to the popularity of the Engelmaier equation that offers a simple equation for shear strain, γ, in the solder joints of an electronic assembly:
Table 3 Sn3.5Ag: Creep fatigue coefficients of creep-integrated fatigue equations embedded with the steady-state creep rate equations and benchmark with the Engelmaier equation. Rate equations
Fatigue coefficients βo
Co Dorn Larson-Miller Manson-Haferd Tref = 150 K Engelmaier Eqs. (5) & (6)
11.3 29.1 18.5 10.1
0.961 0.930 0.925 0.801
Cyclic damage parameters
Fitting constant in the constitutive equations
Diff-square sum
p
H/R
Σ
q −5
1.70 × 10 8.06 × 103 −25.0 –
0.051 0.013 0.022 –
157
5500 – – b1 1.75 × 10−3
Bc – 2.70 × 106 – b2 0.025
tc∞ – – 4.51 × 106 b3 8.90 × 106
2.12 0.53 0.77 0.67
Microelectronics Reliability 98 (2019) 153–160
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200
(a)
N
2000
20000
(b)
0.1 200
N
2000
20000
strainp,ref
strainp,ref
0.01
= 4.25N-0.835 R² = 0.869
p,ref
0.001
= 22.9N-0.900 R² = 0.967
p,ref
0.001
200
2000
N
(d)
0.1
20000
200
strainp,ref
(c)
0.1
strainp,ref
0.01
0.01 -0.881 p,ref = 13.2N R² = 0.952
0.001
N
2000
20000
0.01
= 7.56N-0.764 R² = 0.958
p,ref
0.001
Temperature (K)-Cycle time (s) 293 -1 293-100 293-1000 293-10 358-10 393-10
Fig. 4. Sn3.5Ag: Transformed εp,ref-Nexp data using the creep constitutive equations of (a) Dorn, (b) Larson-Miller and (c) Manson-Haferd as well as (d) the Engelmaier equation based on Eqs. (5) and (6).
γ=
klεT , h3
l
(32)
px/2
where εT = α21ΔT is the mismatch thermal strain between the two substrates of the electronic assembly; l is the maximum distance from the neutral point, h3 is the height of the solder joints, and k is a “nonideal” constant. However, the equation is overly simplicity - it has conveniently assumed the substrates to be perfectly stiff and the solder joints to be perfectly compliant. A more fundamental analytical equation that takes into account the moduli of the constituting materials, the thicknesses of the substrates, as well as the diameter and pitches of the solder joints is needed, and this is presented below. Consider an electronic assembly as a tri-layer assembly in which the solder joints are sandwiched between two substrates. In case of the assembly experiencing a temperature excursion ΔT, the distribution of shear stress in the solder joints, assuming it is smeared over the area of the substrates, may be modeled approximately as [27–30]:
M M
joint
h3 A3 r3
M
joint
M
Fig. 5. Schematics of the critical solder joint in equilibrium.
τsmear ≈
εT e β (x − l) for βl ≥ 3, λ x κs
(33)
τjo int ≈
2
where
β= 2
λ x / κs ;
λ x = ∑ (4 + 3h3/ hi )/(Ei hi )
εT e−βpx /2, Ar λ x κs
(34)
and wherein Ar = A3 / pxpy. The equilibrium of the solder joint is shown in Fig. 5. The shear couple at the two ends of the solder joint must be balanced by the end moments; that is, 2M = τjointA3h3. The magnitude of the moment decreases linearly away from the ends of the solder joint along its length as shown in Fig. 5. For a typical solder joint that has a barrel shape, the maximum bending stress occurs at the two ends. Assuming the solder joint undergoes simple bending, the bending stress at the ends is simply
i=1
κs = h3/ G3∗ + ∑ hi /(8Gi ) are the in-plane stretch compliance and the i=1
shear compliance, respectively, of the assembly, wherein Ei, Gi, hi are the stretch modulus, the shear modulus, and the thickness of substrate #i; G3⁎ = G3A3 / (pxpy) is the smeared shear modulus of the solder joints – the smearing of the shear modulus, G3, of a single solder joint with an equivalent cross-sectional area, A3, over a rectangular area defined by the pitches between the solder joints along the x-coordinate, px, and the y-coordinate, py; and x is the distance from the neutral point. Eq. (33) has been validated by finite element analysis [28–30]. The solder joint nearer the edge of an electronic assembly, at x = l − px / 2 (refer to Fig. 5), experiences the largest magnitude of shear stress. The magnitude of the smeared shear stress at the center of this solder joint is given by substituting x = l − px / 2 into Eq. (33). The magnitude of the shear stress experienced by the solder joint with an equivalent cross-sectional area A3 is approximately
σb, jo int ≈
2h3 τjo int, r3
(35)
where r3 is the radius of the solder joint at the ends. It is worth highlighting that Eq. (35) is valid for solder joints of barrel or cylindrical shape. It is also worth highlighting that σb,joint > τjoint for typical solder joints. The corresponding shear strain and normal strain are approximately γjoint = τjoint / G3 and εjoint = σb,joint / E3. The equivalent Von Mises strain is given by 158
Microelectronics Reliability 98 (2019) 153–160
E.H. Wong
5. Conclusion
2 3 3 εjo int 2 + γjo int 2 3 2 4 2τjo int 6(h3/ r3 )2 + 3(1 + ν )2 = 3E3
εeq =
≈ εT e−βpx /2
2 3E3 λ x Ar
2h3 1+ν . + (1 + ν ) r32 h3
Three creep-integrated fatigue equations embedded with the creep strain rate equations of Dorn, Larson-Miller, and Manson-Haferd have been established. These novel creep-integrated fatigue equations have been validated and benchmarked against the Engelmaier equation using the experimental creep fatigue data of Sn37Pb solder and the Sn3.5Ag solder. The creep-integrated fatigue equations embedded with the Larson-Miller equation and the Manson-Haferd equation have been found to be capable of modeling very well the creep fatigue of both solder alloys. On the other hand, the Engelmaier equation tends to under-model creep damage at low number of cycles while over-model creep damage at high number of cycles. The singularity of strain has resulted in the inconsistency of the evaluated plastic strain in the solder joints of an electronic assembly by finite element analysis. An analytical equation for the elastic strain in the critical solder joint of an electronic assembly experiencing temperature excursion has been presented. The elastic strain does not suffer from singularity and the analytical equation has two essential novelties: (i) it accounts for the compliances of the electronic assembly and (ii) it accounts for the shear and the bending stresses in the critical solder joints. The fitting constants in the novel creep-integrated fatigue equations are material constants, independent of the design of the electronic assemblies and the test conditions. These creep-integrated fatigue equations offer reliable life prediction for new designs of electronic assemblies and offer reliable acceleration of temperature cycling test. The acceleration factor for temperature cycling has been established for these novel creep-integrated fatigue equations.
(36)
Thus, the magnitude of εeq can be reduced by increasing the in-plane compliance of the assembly, λx, increasing the distribution density, Ar = A3 / pxpy, of the solder joints, and increasing the radius of the critical solder joint at the ends, r3. The equivalent strain, εeq, ought to be used in place of the plastic strain amplitude, εp, in the creep-integrated fatigue equations presented in Section 2. The use of the analytically evaluated elastic strain, εeq, in place of the FEA evaluated plastic strain, εp, in the creep-integrated fatigue equation requires the recalibration of the fitting coefficients. 4.3. Accelerating the temperature cycling test With the fitting constants in the creep-integrated fatigue equation that are independent of the cycling temperature and the cycle time, the creep-integrated fatigue equation can be used with confidence to accelerate temperature cycling test, which is a currently very time consuming. The acceleration factor for temperature cycling test is defined as
AF =
N1 tc1 = AFN AFtc . N2 tc 2
(37)
The cycle time for the temperature cycling test can be reduced by raising the cycling temperature. The function η(T,tc) in Eq. (26) provides the relationship between temperature and cycle time. Let η(T1,tc1) = η(T2,tc2) and after some manipulation yields AFtc for the Larson-Miller and the Manson-Haferd equations:
Declaration of Competing Interest None. References
AFtc |Larson − Miller = (Bc tc 2 )T2/ T1− 1 AFtc |Manson − Haferd = (tc 2/ tc ∞ )(T1− T2)/(T2− Tref ) .
(38)
[1] S.S. Manson, G.R. Halford, M.H. Hirschberg, Creep-fatigue analysis by strain range partitioning, Proc. First Symposium on Design for Elevated Temperature Environment, 1971, pp. 12–28 San Francisco. [2] S. Taira, Lifetime of structures subjected to varying load and temperature, Creep in Structures, Hoff NJ Ed, Springer-Verlag, Berlin, 1962, pp. 96–124. [3] Palmgren AG: Die Lebensdauer von Kugellagern. Zeitschrift des Vereines Deutscher Ingenieure, vol. 68, (1924), pp. 339–341. [4] E.L. Robison, Effect of temperature variations on the creep strength of steels, Trans. ASME 60 (1938) 253–259. [5] R.M. Goldhoff, Uniaxial creep–rupture behaviour of low alloy steel under variable loading conditions, J Basic Engng 86 (1965). [6] R. Penny, D. Marriott, Design for Creep, Chapman & Hall, 1995. [7] ASME Boiler and Pressure Vessel Code, Section III Rules for the Construction of Nuclear Facility Components, Div. 1-Subsection NH, Class 1 Components in Elevated Temperature Service, ASME, New York, 2007. [8] RCC-MR, Design and Construction Rules for Mechanical Components of Nuclear Installations, AFCEN, Paris, 2012. [9] H.U. Akay, N.H. Paydar, A. Bilgic, Fatigue life predictions for thermally loaded solder joints using a volume-weighted averaging technique, Trans. ASME J Electron Packag 119 (1997) 228–235. [10] J. Liang, N. Golhardt, P.S. Lee, S. Heinrich, S. Schroeder, A integrated fatigue life prediction methodology for optimum design and reliability assemssment of solder integ-connections, Int Intersociety Electronic and Photonic Packaging (1997) 1583–1592. [11] R. Darveaux, Effects of simulation methodology on solder joint crack growth correlation and faituge life prediction, Trans. ASME J. Electron. Packag. 124 (2002) 147. [12] W. Engelmaier, Fatigue life of leadless chip carrier solder joint during power cycling, IEEE Trans on Components, Hybrids and Manufacturing Technology 6 (1983) 232–237. [13] L.F. Coffin Jr., Fatigue at high temperature, ASTM STP 520 Fatigue at elevated temperature (1973) 5–34. [14] H. Solomon, Fatigue of 60/40 solder, IEEE Trans Components, Hybrids, and Manufacturing Technology 9 (4) (1986) 423–432. [15] X.Q. Shi, H.L.J. Pang, W. Zhou, Z.P. Wang, Low cycle fatigue analysis of temperature and frequency effects in eutectic solder alloy, Int. J. Fatigue 22 (2000) 217–228. [16] P. Chauhan, M. Osterman, S.W.R. Ricky, M. Pecht, Critical review of the Engelmaier model for solder joint creep fatigue reliability, IEEE Trans on Comp Packag
It is worth highlighting that the temperatures, T1 and T2, in Eq. (38) are indeed the equivalent cycling temperature: Teq = ∮ T(t)dt/tc. The number of cycles for the temperature cycling test can be reduced by raising the magnitude of the equivalent strain through raising the range of the cycling temperature. From Eq. (24), 1/ βo
AFN =
εeq2 c1 (εeq , T , tc ) ⎞ N1 = ⎛⎜ ⎟ N2 ⎝ εeq1 c2 (εeq , T , tc ) ⎠
. (39)
The creep function c(εeq,T,tc) is given by 1 − χ(εeq)η(T,tc); noting that η(T,tc) has been kept a constant and that χ(εeq) is a weak function of εeq, the acceleration factor for N may be approximated as AFN ≈ (εeq2/εeq1)1/βo. Substituting εeq with Eq. (36) and noting that εT = α21ΔT yields
AFN ≈ (ΔT2/ΔT1 )1/ βo .
(40)
4.4. Limitations The methodology described in this manuscript is a simplification of a more comprehensive methodology presented in Ref. [20]. This methodology does not account for the exact profile of the cycling strain on creep damage. Consequently, the creep coefficients: p, q, H/R, Bc, and tc∞ may be dependent on the time-temperature profile of the cycling temperature. This methodology also assumes the fitting constants Bc and tc∞ to be constants, which is not strictly correct, though it did appear to be acceptable for the Sn37Pb and the Sn3.5Ag solder alloys as presented in Section 3. 159
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[24] F. Larson, J. Miller, A time-temperature relationship for rupture and creep stresses, Trans. ASME 174 (5) (1952) 765–775. [25] C. Kanchanomai, Y. Mutoh, Effect of temperature on isothermal low cycle fatigue properties of Sn–Ag eutectic solder, Mater. Sci. Eng. A 381 (2004) 113–120. [26] C. Kanchanomai, Y. Miyashita, Y. Mutoh, S.L. Mannan, Influence of frequency on low cycle fatigue behavior of Pb-free solder 96.5Sn/3.5Ag, Mater. Sci. Eng. A 345 (2003) 90–98. [27] E. Suhir, Stresses in bi-metal thermostats, ASME J Appl Mech 53 (1986) 657–660. [28] E.H. Wong, Interfacial stresses in sandwich structures subjected to temperature and mechanical loads, Comp Struct 134 (2015) 226–236. [29] Wong E.H. and Liu Johan. Interface and interconnection stresses in electronic assemblies – a critical review of analytical solutions. Microelectron. Reliab., DOI:https://doi.org/10.1016/j.microrel.2017.03.010. [30] E.H. Wong, Liu Johan, Design analysis of bonded structures, Polymers 9 (2017) 664, https://doi.org/10.3390/polym9120664.
Technologies 32 (3) (2009) 693–700. [17] E.H. Wong, Y.-W. Mai, A unified equation for creep fatigue, Int. J. Fatigue 68 (2014) 186–194. [18] S.S. Manson, A.M. Haferd, A linear time-temperature relation for extrapolation of creep and stress-rupture data, NACA TN 2890. National Advisory Committee for Aeronautics, 1953. [19] D. Liu, D.J. Pons, E.H. Wong, Creep-integrated fatigue equation for metals, Int. J. Fatigue 98 (2017) 167–175. [20] E.H. Wong, Derivation of novel creep-integrated fatigue equations, Int. J. Fatigue (2019) (under review). [21] F.H. Norton, The Creep of Steel at High Temperatures, McGraw-Hill, London, 1929. [22] P.G. McVetty, Creep of metals at elevated temperatures: the hyperbolic sine relation between stress and creep rate, Trans. ASME 65 (1943) 761–769. [23] J.E. Dorn, Some fundamental experiments on high temperature creep, J Mech Phys Solids 8 (1954) 85–116.
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