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Models for Battery Reliability 

and Lifetime

K. Smith, E. Wood, S. Santhanagopalan, 

G.-H. Kim, J. Neubauer, and A. Pesaran 

Presented at the Battery Congress 2013

Ann Arbor, Michigan

April 15-16, 2013

Conference Paper

NREL/CP-5400-57746

March 2014
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Models for Battery Reliability and Lifetime

Kandler Smith, Eric Wood, Shriram Santhanagopalan, Gi-Heon Kim, Jeremy Neubauer, Ahmad Pesaran

National Renewable Energy Laboratory

Abstract

Models describing battery degradation physics are needed to more accurately understand how battery usage 

and next-generation battery designs can be optimized for performance and lifetime. Such lifetime models may also 

reduce the cost of battery aging experiments and shorten the time required to validate battery lifetime. Models for 

chemical degradation and mechanical stress are reviewed. Experimental analysis of aging data from a commercial 

iron-phosphate lithium-ion (Li-ion) cell elucidates the relative importance of several mechanical stress-induced

degradation mechanisms.

Introduction

Lifetime predictive models are needed to better understand design and operational consequences on 

performance degradation of Li-ion batteries (LIB). Calendar-life models describing LIB degradation have shown 

reasonable promise in predicting rate transport and reactions leading to lithium (Li) loss and can be closely matched 

to data (Ploehn, 2004; Safari, 2011). Detailed elementary chemical reaction models (Christensen, 2004; Colclasure, 

2010) provide a framework for studying degradation reactions for different chemical systems. Cycle-life models of 

LIBs, however, have yet to offer a method to predict capacity fade for a wide range of cycling and environmental 

conditions. Coulombic-throughput or energy-throughput are sometimes used as proxies to describe mechanical

stress-induced fade and are regressed to experimental capacity data (Peterson, 2010). These models are difficult to 

extend to a wide range of cycling conditions (Wang, 2010).

Mechanical stress has gained increasing attention in LIB modeling literature with one motivator being to 

create physical models of capacity fade that can help guide cell design. Mechanical stress effects have been modeled 

at length scales ranging from particle-level (Christensen, 2006) to electrode sandwich level (Renganathan, 2010; 

Xiao, 2010), cell level (Sahraei, 2012) and pack level (Sahari, 2010). Particle stress investigations have shown, for 

example, possible failure during fast-rate charging where intercalation-rate of Li into a negative electrode active

particle drives a faster rate of expansion at the outer radii of the particle, generating high-tensile stress in the inner 

core that may lead to fracture. During de-intercalation, high-tensile stress occurs at the outside of the particle. 

Models predict theoretical stress levels below which no failure should occur, albeit for simplified electrode 

geometries without pre-existing flaws.

While these stress models already provide useful directional input to the design phase, further extensions 

are needed to predict capacity fade. Particle fracture models must be extended beyond crack initiation to describe 

crack propagation on a time scale relevant for life prediction. Computational models have not yet captured realistic 

geometries including dispersed flaws consequent of the manufacturing process. Fracture leading to apparent activesite isolation has been studied; however, additional factors that may lead to active-site isolation—such as binder 

failure coupled with differential expansion of various cell components—have not been studied as they relate to 

cycling fade rate. Cell and pack level models have mostly considered the impact of crush on various cell and pack 

geometries. 

In this paper, we hypothesize a fatigue model to describe mechanical-stress-induced capacity fade. The 

model is regressed to capacity fade data from the literature for a commercial iron-phosphate cell. Based on the 

model, the relative importance of various mechanical-induced degradation mechanisms on cycle life is discussed. 

Degradation Mechanism Hypotheses

Some important mechanical stress considerations leading to cell performance fade are illustrated in Figure 1. 

These are:

(i) Particle and electrode isolation caused by high-concentration, gradient-driven strains leading to 

fracture; rate is likely accelerated by high intercalation C-rates and low temperatures cycling
2

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(ii) Particle and electrode isolation caused by loss of adhesive/cohesive properties leading to debonding of 

particles from one another, or delamination of active materials from current collectors; degradation rate 

acceleration factors likely include:

a. High temperature leading to binder breakdown

b. Depth-of-discharge (DOD) swings causing bulk strain of active materials 

c. Temperature swings during cycling and cool-down (∆T) leading to differential strain amongst 

various cell components

(iii) External pressure and forcing conditions on the cell packaging transmitted to the jelly-roll; pressure 

supplied by a module housing, for example, can impact the cycle life of pouch cells. 

For the present cylindrical cell tested in a laboratory environment, impact of external pressure and forcing can 

be neglected.

Fig. 1 Mechanical stress effects contributing to damage and performance fade

Current Collector (Cu)

- +Current Collector (Al)

sep δ − δ δ + Negative Electrode Separator Positive Electrode

L

Li+

er

LixC6

r

Li

y

CoO2

eElectrolyte

x

cs,e

ce

cs(r) cs

(r)

cs,e

Intercalation strains

Thermal strain

Residual 

stresses of 

manufacturing

Binder/Separator Swelling

Fracture

Delamination

External Loads 

and Packaging

Winding Tension

Negative electrode

Separator

Positive electrode

Spiral cell image 

© Panasonic
3

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Mathematical Model

Estimation of relative effects of mechanical stress on capacity fade under different aging conditions is 

performed using a sequence of local model regression, degradation-rate visualization/hypothesis of rate functional 

form, and rate-law model regression. Local models and hypothesized rate laws are described below.

Local models

Local models describe capacity fade with time t and electro-thermo-mechanical cycles N for a single fixed 

aging condition. In the present model, relative capacity, q (available capacity at a given state-of-life divided by 

beginning-of-life capacity), is interpreted as

min( , ). Li sites q = q q [1]

where qLi is capacity limited by cycleable-Li in the cell,

q b b t b N z

Li = 0 + 1 + 2 , [2]

and qsites is capacity limited by active-sites. Provided the cell is not stored in an over-discharged or over-charged 

state, we find qsites predominantly correlates with number of cycles rather than time,

qsites = c0 + c2N . [3]

Equations [1-3] reproduce several common features observed in capacity fade data: 

(i) Graceful fade regime observed for cells aged under pure storage conditions (b1t

z

) or in low-tomoderate cycling conditions (b2N) in [2]

(ii) Linear fade regime (c2N) in [3] either observed

a. immediately starting at BOL when a cell is repeatedly cycled at moderate-to-high stress levels 

such as during an accelerated cycling test, or if not then

b. nearing EOL when an initial graceful fade region experiences a sudden knee where apparent 

capacity degradation suddenly accelerates.

The min function in Eq. [1] has several consequences for interpreting capacity fade data. First, multiple 

shapes of capacity versus age are possible. Second, depending on the chosen aging condition and extent of aging, 

degradation rates b1, b2, and c2 may not all be observable from total capacity measurements for a single aging 

condition. To investigate mechanical stress degradation, the present paper only considers regression of a rate model 

for c2 to data for the commercial graphite/iron-phosphate cell. Regression of parameters b1 and b2 will be described 

in a future publication.

Rate model

Tracking damage as a function of cycles N is common in fatigue literature. The fatigue approach used in 

the present work is practical when a mechanical-stress/life-predictive model must retain close relationship with 

supporting experimental data. Thermo-mechanical fatigue literature suggests formulas to combine effects of 

multiple simultaneous degradation mechanisms occurring in parallel. For gas turbine engines for example, the 

number of cycles to end-of-life (EOL) is expressed as harmonic mean of cycle-life for separate mechanisms 

1 1 1 1 . NEOL = Ncrack + Ncreep + Noxidation [4]

In [4], crack propagation is due to pure mechanical loading, creep is the flow of materials induced when mechanical 

cycles are superimposed with thermal cycles, and oxidation is material change due to accumulated exposure at high 

temperatures.
4

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For a battery, one might hypothesize a similar model where life is limited by bulk DOD cycles, thermal cycles, and 

fracture driven by species concentration gradients

1 1 1 1 . NEOL = N DOD + N∆T + N fracture [5]

Rationalizing this model with [3] implies that degradation rates, expressed as capacity fade per cycle, of the various 

mechanisms are additive

. 2 2,DOD 2, T 2, fracture c = c + c + c ∆ [6]

The fatigue model must express the functional dependence of degradation rate on operating parameters, c2

(DOD,∆T,T,Crate,tpulse). Consistent with the degradation mechanism hypotheses discussed above, we expect fracture 

degradation in [6] to be accelerated by sluggish intercalation at low temperatures (Safari, 2011), and high C-rates 

together with long (dis)charge times (Zhao, 2010). We propose a non-dimensional acceleration factor θ for 

intercalation-induced fracture,

exp( ( ))( ) . , ,

intercal. 1 1

intercal. R 



 



 = − −

pulse ref

pulse

rate ref

rate

ref

a

t

t

C

C

T T

E θ [7]

Binder breakdown is likely to lead to electrode site isolation, particularly for high temperature cycling. The

capacity fade impact of binder failure is likely convoluted with occurrence of differential strain of particles, 

electrodes, and other cell components. A proposed capacity fade rate model describing binder degradation, 

differential expansion of cell components, and particle fracture effects is

{exp( ( ))[ ] }. intercal.

4

intercal.

1 2 3

1 1

2 2,

binder

c c m DOD m T m θ m θ

ref

a

R T T

E = ref − + ∆ + + − [8]

Several considerations are noted for regression of this rate model. As written, [8] is over-determined. 

Parameter m1 is removed from the parameter search, given the constraint m1+m2+m3+m4=1. Next, it is difficult to 

regress two conflicting activation energies simultaneously. In [8], Ea

binder > 0 describes high temperature 

acceleration and Ea

intercal. < 0 describes low temperature acceleration. We fix activation energy for intercalationinduced fracture to Ea

intercal. = 43321 J/mol representing transport limitations in the graphite electrode (Safari, 2011) 

as studies have shown that particle damage is negligible for the iron-phosphate electrode with nano-sized particles.

The following section describes regression of parameters p = [c2,ref, m2, m3, m4, Ea

binder] in [8] and 

interpretation of mechanical stress-induced capacity fade created by this hypothesized model. 

Data Regression Analysis

We compiled capacity fade data from multiple sources including Delacourt et al. (Safari 2011); HRL Labs 

consortium (Wang, 2010); and NASA Jet Propulsion Laboratory (Smart 2008). The total set of data encompasses 50 

different aging conditions for a 2.2 Ah A123 26650 cell with graphite/iron phosphate chemistry.

In this work, we restrict our analysis to the knee regime of the capacity fade curves. Figure 2 shows the 

subset of aging data within the apparent knee regime. Each tested condition is labeled according to the test 

laboratory, cycling profile and temperature, in the format “Lab, DOD, C-rate,dis/C-rate,chg, test chamber 

temperature.” The knee regime is identified mathematically using thresholds for first and second derivatives of 

relative capacity, dq/dt < 0 and d2

q/dt2 < ε where ε is a small negative number.
5

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Fig. 2 Aging data for 2.2 Ah graphite/iron-phosphate cell; entire dataset (green dotted lines); subset of data in 

knee regime (black solid lines)

Local model regression

A value of parameter p = [c2] in [3] is regressed to fit the knee region in each of the 13 datasets, creating 13 

rate values that can be visualized for dependence on temperature and cycling condition. In [3] we assume c0 = 1 

which, for the present cell, is supported by the data. Local model fits to capacity fade trajectories are shown in 

Figure 3. The combined models capture the knee-region of capacity fade with 1.37% root-mean-square error 

(RMSE). . The rate of capacity fade, c2, for each aging condition is shown in Figure 4 versus the inverse of 

temperature. Confidence intervals are narrow in most cases. High C-rate cycling at 0o

C is not estimated as well. This 

could be partly attributable to error in the thermal model we employed to estimate T(t) for each aging condition, 

together with the assumption that associated duty-cycle stress variables T and ∆T stay constant throughout life.
6

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Fig. 3 Local model [3] fit to 13 separate data sets; data (·), model (-), 95% confidence intervals (--)

Fig. 4 Capacity fade rate determined by local model fits [3] for 13 tested aging conditions; error bars are 95%

confidence intervals for parameter c2 (they relate to confidence intervals in Fig 3)
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Fig. 5 Capacity fade rate; rates from local models (blue “o”) and global rate-law model [7-8] (purple “x”, 

together with confidence intervals of global model)

Rate model regression

Although not shown here, the proposed global rate-law model [7-8] cannot represent the NASA-JPL 

2C/2C, -20o

C test condition well with reasonable parameter values. It is possible that this cell’s steep fade trajectory 

is controlled by Li plating rather than active site loss. Li plating is a likely side reaction to occur at low temperatures 

and high charge C-rates, such as for this test condition. We do not include this test condition when fitting the global 

rate-law model. Also not discussed further, over the course of our investigation we compared numerous alternate 

model forms to that described here. This paper only describes the most plausible model [7-8]. 

Simultaneous regression of parameters p = [c2,ref, m2, m3, m4, Ea

binder] is challenging for traditional nonlinear regression tools, such as the Levenberg-Marquardt algorithm used here. Though sub-optimal, by separately 

fitting different regions of the data (e.g., low C-rates and limited temperature ranges) we find Ea

binder has a value of 

around 36,000 to 53,000 J/mol. A typical median value that represents the entire dataset well is 49000 J/mol. 

Parameters m2, m3, and m4 are determined by direct search, stepping through the parameter space between 0 and 1 

for each mi while regressing c2,ref, sometimes in combination with other parameters. The best model fit identified 

uses m1 = 0.83, m2 = 0.04, m3 = 0, m4 = 0.13 with quality of fit R2 = 0.96.

Figure 5 compares the global rate-law prediction with rates estimated from regression of local models. 

While the data may be less than optimally fit—a subject of future effort—the hypothesized model and parameters 

offer interesting physical interpretation. Reference conditions used in [7-8] are Tref= 23o

C, Crate,ref = 1C, DODref = 

100%. The parameters mi thus imply that, under typical cycling conditions of 100% DOD, 1C charge and discharge 

at room temperature:

• 83% of active site loss is attributed to bulk volumetric expansion/contraction of the active material

• 4% of active site loss is attributed to temperature swings encountered by the cell

• 13% of active site loss is attributed to particle fracture owing to intercalation stress at high C-rates.

These ratios will change depending on temperature and cycling condition.
8 

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Conclusions

While conclusions are tentative pending validation with further data, the hypothesized fatigue model and 

analysis of graphite/iron-phosphate aging data suggests that some 83% of mechanical-induced active-site loss is 

attributable to bulk DOD swings during cycling. As the iron-phosphate electrode is made with mechanically stable 

nano-sized particles, we attribute the expansion/contraction-driven site loss to the graphite electrode with micronsized particles. The present model partially supports the use of Amp-hour throughput as a proxy for modeling 

capacity fade due to site loss, since the product of DOD with number of cycles N relates directly to Amp-hour 

throughput. This work extends Amp-hour throughput models to include the effects of C-rate, temperature, and—also 

importantly—calendar fade.

While DOD swings apparently cause the most damage, we estimate that C-rate-driven particle fracture 

contributes on the order of 13% to active site loss. The model and data further estimate that temperature swings, 

causing differential thermal expansion of cell components, contributes weakly on the order of 4% to active site loss. 

The presented fatigue model, while simple, provides a relatively robust description of the 2.2 Ah cylindrical 

graphite/iron-phosphate cell life data. It will be applied in future work within a more sophisticated multidimensional physical degradation model framework.

Acknowledgements 

The authors gratefully acknowledge funding by U.S. Department of Energy Office of Vehicle Technologies 

- Energy Storage, program managers David Howell and Brian Cunningham, and U.S. Army Tank Automotive 

Research, Development and Engineering Center, program manager Yi Ding.

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