NREL is a national laboratory of the U.S. Department of Energy, Office of Energy 

Efficiency & Renewable Energy, operated by the Alliance for Sustainable Energy, LLC.

Contract No. DE-AC36-08GO28308

Modeling Thermal Fatigue in 

CPV Cell Assemblies

Preprint

Nick Bosco, Timothy J. Silverman, 

and Sarah Kurtz

Presented at the 37th IEEE Photovoltaic Specialists Conference 

(PVSC 37)

Seattle, Washington

June 19-24, 2011

Conference Paper

NREL/CP-5200-50685

July 2011
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1

MODELING THERMAL FATIGUE IN CPV CELL ASSEMBLIES

Nick Bosco, Timothy J Silverman, and Sarah Kurtz

National Renewable Energy Laboratory, Golden, Colorado, USA

ABSTRACT

A finite element model has been created to quantify the 

thermal fatigue damage of the CPV die attach. 

Simulations are used to compare to results of empirical 

thermal fatigue equations originally developed for 

accelerated chamber cycling. While the empirical 

equations show promise when extrapolated to the lower 

temperature cycles characteristic of weather-induced 

temperature changes in the CPV die attach, it is 

demonstrated that their damage does not accumulate 

linearly: the damage a particular cycle contributes 

depends on the preceding cycles. Simulations of modeled 

CPV cell temperature histories provided for direct 

comparison of the FEM and empirical methods, and for 

calculation of equivalent times provided by standard 

accelerated test sequences.

INTRODUCTION

When in service under concentrated direct sunlight, the 

cell assembly in a concentrating photovoltaic (CPV) 

module will experience a complex time history of 

temperature. This history will contain many temperature 

reversals, or thermal cycles, that will impart stress to the 

die attach due to a mismatch in coefficient of thermal 

expansion between the cell and its substrate. With every 

cycle, inelastic strain energy will be gained by the die 

attach. The accumulation of this energy will lead to die 

attach cracking and ultimately to cell failure.

Previous work introduced the rainflow counting algorithm 

method (RFC) in an attempt to quantify thermal fatigue in

the CPV cell assembly’s die attach during outdoor 

exposure [1]. This promised to be a robust method for 

quickly and simply analyzing service times of many years, 

however, concessions on the accuracy of the calculation 

were made. For instance, the thermal fatigue equation

employed was developed for large thermal cycles 

characteristic of accelerated testing, not for small, quick 

cycles typical of a CPV cell assembly during service. This 

equation was also generalized for all materials and applied 

to the CPV cell assembly geometry, two factors that

influence the rate of damage accumulation and that are a 

departure from the mechanical test samples used for the 

equation’s development. The rate of damage 

accumulation was also assumed to be linear, meaning that 

a given cycle does a particular amount of damage 

independent of how much damage has already been 

done, thereby giving no consideration to the temperature 

history of the assembly.

A finite element model (FEM) has been created to quantify 

the thermal fatigue damage of the CPV die attach. Results 

are compared against popular thermal fatigue equations 

and the previously reported RFC method. The CPV cell 

assembly temperature history during one month in 

Golden, Colorado and Phoenix, Arizona are also 

simulated with the FEM. Comparison of the thermal 

fatigue damage accumulated during each day of the 

simulation and through accelerated thermal cycling 

provides for a direct calculation of the equivalent time of 

the test sequence.

MODELING AND APPROACH

Rainflow Counting

The method of distilling time histories of temperature into 

discrete thermal cycles via the rainflow counting algorithm 

has been previously presented and is therefore only briefly 

reproduced here for clarity [1]. First, a time history of CPV 

cell temperature is derived from the meteorological data of 

direct normal irradiance, ambient temperature and wind 

speed taken at one-minute intervals [2]. The cell 

temperature history is then distilled into segments of 

temperature change that are relevant to thermal fatigue 

using the three-parameter rainflow counting algorithm [3].

Outputs of the rainflow count are temperature change

(ΔT), mean temperature (Tm) and cycle time (tc) for each 

cycle. Next, an empirical thermal fatigue equation is used 

to calculate the relative damage imparted to the die attach

with each cycle. In the previous work, the Engelmaier 

thermal fatigue model was employed [1]. The Engelmaier 

equation calculates a fatigue life (Nf , number of cycles to 

failure) for a specific set of stress conditions [4]. Here, the 

Engelmaier equation is presented in two forms, one for 

application on compliant systems (Ec) where the 

maximum stress in the die attach remains below its yield 

stress throughout cycling:

 

Nf = 1

2

Kc

∆T

 

  

 

  

−1 c

(1)

and one for application on stiff systems (Es) where the 

yield stress of the die attach is exceeded through 

temperature cycling:

 

Nf = 1

2

Ks

∆T2

 

  

 

  

−1 c

(2)

where the fitting constants Ks and Kc are functions of 

material properties and specimen geometry and the 

exponent c depends on the cycle time and mean 

temperature. Inputs for the Engelmaier equation are 

therefore the three outputs of the rainflow algorithm. In 

the current paper, the Coffin-Manson model is also 

presented [5]:
2

 

Nf = KCM ∆T −2

f

1 3 exp

Ea

kT

 

   

  (3)

This equation, demonstrated by experiment to be valid for 

ceramic and plastic packages attached woth SnPb 

solders, computes Nf for a unique thermal cycle 

characterized by its ΔT, cycle frequency (f) and an 

Arrhenius dependence on the maximum cycle temperature

(Tmax) where an activation energy of Ea= 0.12 eV is 

typically used for SnPb solder, and k is Boltzmann’s 

constant [6, 7]. This empirical relationship is included for 

comparison to the results of the Engelmaier equations and 

FEM simulations.

Following similar methods of quantifying thermal fatigue 

damage during the service life of electronic components, 

the Palmgren-Miner hypothesis is used in this analysis [3]. 

The Palmgren-Miner hypothesis states that die attach

damage, D, is considered to be linearly proportional to the 

ratio of the number of cycles completed at a specific set of 

stress conditions (i), ni, to the total number of cycles that 

would result in failure for those same conditions, Nf,i:

 

Di = ni

Nf,i

(4)

Because every cycle computed using the rainflow count 

represents a unique stress condition (n=1 for all i), the 

damage accumulated through the temperature history of 

interest is calculated by summing the inverse of the 

relevant fatigue equation over every cycle in the time 

history.

 

D = 1

i Nf,i

∑ (5)

Finite Element Analysis

A 3D model of a generic CPV cell assembly was 

generated using COMSOL. The assembly consists of a 

1 cm2 x 100 μm Ge die attached to a direct-bond copper 

(DBC) substrate via 25 μm of eutectic SnPb solder. The 

DBC is modeled as a 2.3 x 2.7 cm layered structure 

composed of a 250 μm thick layer of Al2O3 sandwiched by 

two 200 μm layers of Cu. The rate-dependent creep 

behavior of the solder is characterized by Anand’s 

model [8]. This constitutive model represents the two 

different creep mechanisms of Sn-containing solders: 

grain boundary sliding or dislocation glide (low stress and 

high temperature) and dislocation climb (high stress and 

low temperature).

Darveaux’s approach is taken to evaluate the solder’s 

lifetime. Derived from the Paris power law of fatigue crack 

growth, the Darveaux approach considers the inelastic 

strain energy density as a damage indicator [9]. 

Accordingly, both crack initiation and growth are functions 

of the average inelastic strain energy density (plastic work) 

accumulated per loading cycle. In the current study, the 

two phases of failure are not differentiated, therefore only 

the total plastic work is considered as the metric for 

damage.

SIMULATIONS

Three types of thermal cycle, corresponding to the thermal

cycles in the CPV design qualification test (IEC 62108) 

and shown in Table 1 (including the dwell time td), were 

simulated. To determine the applicability of the various 

methods to temperature cycles typical of service 

conditions, the sensitivity of the FEM and the empirical 

equations to the cycle parameters (ΔT, Tm, and tc) was 

explored using the ranges in Table 2.

TCA-1 TCA-2 TCA-3

Tmax (°C) 85 110 65

Tmin (°C) -40 -40 -40

tc (min) 100 100 100

td (min) 10 10 10

Table 1. Simulated thermal cycles corresponding to 

options in IEC 62108

parameter ΔT Tm tc

ΔT (°C) 5-50 25 30

Tm (°C) 35 5-75 35

tc (min) 16 16 4-32

Table 2. Range of parameters explored using FEM

To address the Palmgren-Miner hypothesis of linear 

damage accumulation, the effect of temperature history 

was studied. This study was conducted by simulating a 

repetition of 50 cycles (ΔT= 25°C, Tm= 35oC, tc= 16 min),

then changing the mean temperature and continuing the

simulation for an additional 50 cycles. Fourteen trials were 

simulated, representing changes in mean temperature,

ΔTm, from –35oC to 35oC in 5oC increments.

Finally, the CPV cell temperatures for one month

experienced in Golden, Colorado and Phoenix, Arizona 

were derived from one-minute weather data and used in

FEM simulations.

Due in part to the presence of strain hardening in the 

FEM, repeated cycles can each contribute different 

amounts of work. For all temperature cycle simulations, a 

minimum of 50 cycles was repeated and the damage 

reported is the inelastic strain energy accumulated from 

the three out-of-plane stress components in the die attach 

for the last cycle. For the weather simulations, each 

month was run five consecutive times and the results of 

the fifth repetition reported.
3

RESULTS AND ANALYSIS

Thermal Cycling

The normalized inelastic strain energy (damage) 

accumulated during a single cycle of each of the three 

TCA types is presented in Fig. 1. Also shown in the figure 

are the fits of the two Engelmaier equations (Ec and Es) 

along with the result of the Coffin-Manson (C-M) model for 

the same cycles. The Engelmaier fits are denoted TCA to 

represent that their constants were adjusted to match the 

ratio of damage calculated from the FEM simulations of 

these TCA cycles (Kc,TCA= 35 and Ks,TCA= 2000). The 

Coffin-Mason model fit the FEM well using the reported 

activation energy, as did the two fitted Engelmaier 

equations.

The FEM simulations of the twenty-nine weather-type 

cycles are presented in three plots according to the varied 

parameter, Figs. 2–4. These results were used to provide 

a corresponding weather fit of the Engelmaier and CoffinManson equations, which are also included in the figures,

and normalized to the most damaging cycle as determined 

by the FEM simulation. In this case, adjustment of the Ec

constant (Kc,weather= 5.35e6) was required to achieve a 

satisfactory fit to the FEM simulations and no fit of the Es

equation was possible, so it is omitted. Similarly, 

adjustment of the Coffin-Manson’s activation energy was 

also required (Ea,weather= 0.9 eV) which corresponds 

closely to the activation energy, determined by 

experiment, for the constitutive model used in the finite 

element simulation (Ea,FEM= 0.93 eV). While the Ec

equation does appear capable of fitting cycles in both the 

TCA and weather regimes, the fitting constant is not 

intended to be adjusted according to cycle characteristics.

The results of the series of simulations to explore the 

hypothesis of linear damage accumulation are presented 

in Fig. 5. Results are presented as a ratio (closed 

symbols) of damage during the first cycle to damage 

during the last cycle after the mean temperature shift, and 

as an absolute change in damage (open symbols). A ratio 

of unity would signify a linear relationship and is only 

present at zero change in mean temperature. The 

asymmetric response of damage ratio around mean 

temperature change is an artifact of the results being 

compounded with the effects of mean temperature: at 

higher temperatures the kinetics for stress relaxation 

become more favorable. The symmetrical response of 

absolute damage illustrates that the higher temperature 

cycles are more damaging. Despite the previous results 

that show promise in reconciling the FEM simulations and 

empirical thermal fatigue models, this clear nonlinearity 

will prevent the application of an empirical model that does 

not also include some consideration of temperature 

history.

Figure 1. FEM damage accumulated through the TCA 

thermal cycles and corresponding empirical fits.

Figure 2. Relative comparison of FEM and empirical 

fits sensitivity to magnitude of cycle temperature 

change.

Figure 3. Relative comparison of FEM and empirical 

fits sensitivity to mean cycle temperature.
4

Figure 4. Relative comparison of FEM and empirical 

fits sensitivity to cycle time.

Figure 5. Ratio of damage (closed symbols) of the 

first to last cycle following a shift in mean cycle 

temperature. The same results are reported in terms 

of absolute damage (open symbols).

Figure 6. Inelastic strain energy (FEM) accumulated

for Golden and Phoenix.

The Weather

The simulations of each month in Golden, Colorado and 

Phoenix, Arizona are first summarized by the accumulated 

inelastic strain energy (damage) by day in Fig. 6. Over the 

course of these months, the die attach of an identical 

assembly in Phoenix would accumulate roughly 45% of 

the damage accumulated in Golden. Three days in Golden 

are presented separately in Figs. 7–9. Each plot contains 

the modeled cell temperature and its associated inelastic 

strain energy (damage) accumulated throughout the day. 

The relative damage calculated through the FEM 

simulation and the RFC method using both the C-M and 

Ec weather regime fits for the thermal fatigue equation are 

also included. First, July 29, a very damaging, partly 

cloudy day is presented and used for the normalization, 

Fig. 7. On this day, several long periods of clouding

occur. Next, July 10, a sunny day with no clouds is 

presented where, according to the FEM, roughly half the 

damage is accumulated compared to July 29, Fig. 8. 

While both the C-M and Ec results agree with the FEM 

result that the sunny day causes less damage, both of the 

empirical damage equations substantially underestimate 

the magnitude of the difference.

Figure 7. Golden July 29: Modeled cell temperature, 

accumulated inelastic strain energy (FEM) and RFC 

method results.

Figure 8. Golden July 10: Modeled cell temperature, 

accumulated inelastic strain energy (FEM) and RFC 

method results.
5

Figure 9. Golden July 12: Modeled cell temperature, 

accumulated inelastic strain energy (FEM) and RFC 

method results.

Finally, another partly cloudy day is presented, July 12, 

which contains many short periods of clouding, Fig. 9. In 

this case, according to the FEM, the simulated damage is 

only slightly larger than on the sunny day. The C-M model 

overestimates the relative damage of July 12 while the Ec 

model does not significantly change with respect to July 

10, the sunny day.

Two similar figures for days in Phoenix are presented in 

Figs. 10–11. The most damaging day of the month,

according to FEM results, is August 22 and is presented in 

Fig. 10. This day causes damage very comparable to the 

similarly cloudless day in Golden, July 10. The last day 

presented, August 5, contains many short periods of 

clouding, similar to July 12 in Golden, but causing ~12% 

less damage, Fig. 11. The simulated cell temperature and 

accumulated damage for August 22 are superimposed 

(light grey and light blue curves) to illustrate the similar 

temperature profiles of these days, save the effect of short 

period clouding on Aug 5.

Figure 10. Phoenix, AZ August 22: Modeled cell 

temperature, accumulated inelastic strain energy 

(FEM) and rainflow results.

Figure 11. Phoenix August 5: Modeled cell 

temperature, accumulated inelastic strain energy 

(FEM) and RFC method results. 

Equivalent Time

The preceding simulation results and analysis provide an 

estimate of equivalent time for standard accelerated 

testing sequences, Table 3. The values in this table are

produced by relating the energy accumulated through the 

TCA cycles simulated, Fig. 1, to the energy accumulated 

during each characteristic day or month. Therefore the 

values of equivalent time represent consecutive 

characteristic days or months. As per IEC 62108, the 

number of cycles considered for each test are 1000, 500 

and 2000 for the TCA1, 2 and 3 sequences, respectively.

equivalent time (years)

Golden 

July 10

Golden

July 29

Golden

July

Phoenix

Aug 22

Phoenix

Aug

TCA1 15 9 13 20 29

TCA2 14 8.5 12 18 27

TCA3 18 11 15 23 34

Table 3. Equivalent times for the standard IEC 62108 

testing sequences.

DISCUSSION

The FEM simulations are easily fit by the three popular 

empirical equations considered in this study. Both forms of 

the Engelmaier equation required fitting in the TCA regime

because they include a material- and specimen-dependent 

fitting constant. The Coffin-Manson equation required no 

fitting in the TCA regime when used with the activation 

energy reported from the literature for SnPb solder. In the 

weather regime, only the compliant form of the Engelmaier 

equation provides a satisfactory fit to the FEM simulations. 

This result is consistent with the CPV assembly acting as

a compliant part (σmax<σyield) for small cycles, however a 

large adjustment of the fitting constant was required to 

achieve this fit. Considering that the form of the constant 

is not dependent on cycle character, this may be a 

fortuitous result. The Coffin-Manson equation is also 

capable of providing a fit within the weather regime. In 

this case, the best fit is achieved with an activation energy 

very similar to the one employed in the constitutive 
6

equations of the FEM, suggesting a more physical 

meaning.

The period and duration of clouding and the mean 

temperature appear to have the largest influence on the 

amount of damage accumulated during a day. The above 

results suggest that fewer, longer-period clouds are more 

damaging than many short-period clouds. This effect may 

be compounded by the lower temperature achieved during 

a longer shading period, and by the length of time the 

solder is given to relieve the stress of the associated 

temperature change by inelastic strain, thereby 

accumulating energy. The short shading events do not 

provide as large a driving force (temperature change) nor 

do they allow adequate time for stress relief through 

inelastic strain. These effects are clearly illustrated in 

Fig. 11, where short periods of clouding are shown not to 

contribute much additional damage over a similar 

temperature day without clouds. The inaccuracy of the 

empirical models may primarily be attributed to the 

nonlinear accumulation of energy as illustrated by the 

FEM analysis, Fig. 5.

CONCLUSIONS

A FEM was created to quantify the thermal fatigue 

damage accumulated in the CPV cell assembly die attach

through thermal cycling and outdoor exposure. First,

results are compared to three popular thermal fatigue 

equations and indicate positive correlations when 

evaluating 29 types of small thermal cycles motivated by 

weather-induced temperature changes. The accumulation 

of damage, however, is shown to be nonlinear for adding 

adjacent cycles of different character. This observation 

disproves the Palmgren-Miner hypothesis, which is 

extensively used in the literature for similar methods of 

quantifying thermal fatigue. Considering the trends in 

damage with cycle parameter elucidated from the FEM, 

characteristic days from Golden, Colorado and Phoenix, 

Arizona were analyzed. It was observed that periods of 

long clouding contribute more damage than shorter ones. 

For the months analyzed, these types of days were 

prevalent in Golden and absent in Phoenix. The analysis 

of the TCA cycles and weather data provided for an 

estimate on the equivalent time for the IEC 62108 thermal 

cycling sequences. The range in equivalent times varied 

between 8.5 and 18 years for the most damaging day 

considered in Golden and Phoenix, respectively.

While the accuracy of the FEM is desirable, software and 

hardware costs and computing times are prohibitive. 

Alternative approaches based on the direct computation of 

damage from a given sequence of temperatures are much 

simpler to implement. But the empirical damage equations 

investigated here are inappropriate for computing damage 

due to weather. A new model is needed that properly 

computes damage done by small temperature cycles and 

that accounts for the history dependence of damage 

accumulation without the cost and complexity of the FEM. 

The result will be an easily implemented method to 

calculate both the relative thermal fatigue damage 

accumulated between deployment locations and the 

equivalent times of exposure for those locations through 

thermal cycling. Methods to explore the consequence in 

reliability on cell assembly materials and geometry are 

also required.

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