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Journal of Power Sources 263 (2014) 22e28

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Journal of Power Sources

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Reducing diffusion induced stress in planar electrodes by plasticshakedown and cyclic plasticity of current collector

Yicheng Song a,b,*, Zongzan Li c, Junqian Zhang a,b

aDepartment of Mechanics, Shanghai University, Shanghai 200444, Chinab Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai 200072, Chinac Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, Shanghai 200072, China

h i g h l i g h t s

� A strategy to reduce stress and enhance capacity by plastic yield of current collector.� Identify 3 elastoplastic types: pure elastic, plastic shakedown, cyclic plasticity.� Plastic shakedown reduces stress and enhances capacity with good safety.� Cyclic plasticity further reduces stress and enhances capacity.� Design schemes are provided for plastic shakedown and cyclic plasticity.

a r t i c l e i n f o

Article history:Received 21 October 2013Received in revised form25 March 2014Accepted 3 April 2014Available online 13 April 2014

Keywords:Lithium ion batteryDiffusion induced stressPlastic shakedownCyclic plasticityCurrent collector

* Corresponding author. Department of MechShanghai 200444, China.

E-mail addresses: yicheng_s@hotmail.com, ycsong

http://dx.doi.org/10.1016/j.jpowsour.2014.04.0070378-7753/� 2014 Elsevier B.V. All rights reserved.

a b s t r a c t

This paper proposes a strategy to reduce the diffusion induced stress and enhance the capacity of alayered electrode by allowing the plastic deformation of current collector. Based on analytical formu-lations of the stress in whole electrode, three types of elastoplastic behaviors of current collector, i.e. pureelastic deformation, plastic shakedown and cyclic plasticity, are identified. Criterions separating the threecases are proposed. It is found applying a thin current collector and allowing it to plastically yield in thecharge/discharge cycles is beneficial not only to capacity as more space can be provided for active ma-terials but also to electrochemical stability because the stress in active layer is significantly reduced.Structural design corresponding to plastic shakedown shows good balance between the said improve-ments and structural safety, whereas the case of cyclic plasticity further enhances the improvements.Therefore, structural designing scheme is provided for the former case according to the criterion ofplastic shakedown but for the latter one based on the CoffineManson relation with expected cycle life.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction

Lithium ion batteries have been widely used in the applicationsfrom portable electronics to electric vehicles. The electrode oflithium ion battery usually employs a multilayer structure which iscomposed of active materials such as graphite, silicon or LiCoO2,and current collector made of copper or aluminum. In the cycles ofcharge and discharge, stress field is developed in the whole elec-trode partially because the lithiation induced deformation of activelayer is restricted by the current collector, leading to mechanicaland further electrochemical degradation of the electrodes [1].

anics, Shanghai University,

@shu.edu.cn (Y. Song).

Among the intensive studies devoted to the diffusion inducedstress (DIS), the current collector was customarily considered as anelastic material [2e5]. Mechanical plastic yield is usually avoidedperhaps because of the traditional believe in mechanical designthat plastic deformation is harmful to structural safety. However,the main function of current collector in batteries is to provide apassageway for electron transportation, not for load bearing. Inaddition, if the current collector plastically yields in charge anddischarge, the relaxation brought by the plastic deformation mayreduce the stress in active layer from which the electrochemicalperformance may benefit. Therefore, this article attempts toexplore the possibility of allowing the plastic deformation of cur-rent collector in designing and taking advantage of the elastoplasticbehavior of current collector to improve the performance of abattery. In literature, although the plastic yield of active material

Fig. 2. The symmetric layered electrode investigated in this work.

Y. Song et al. / Journal of Power Sources 263 (2014) 22e28 23

has been widely discussed [6e14], the impacts of plastic yield ofcurrent collector have been rarely investigated.

In this manuscript, wewill demonstrate a strategy to reduce theDIS and enhance the capacity of a layered electrode by applying athin current collector whose plastic deformation is utilized. Thereare three types of possible elastoplastic behaviors for the currentcollector, i.e. pure elastic deformation (PED), plastic shakedown(PS) and cyclic plasticity (CP). See Fig. 1, taking a cyclic uniaxialtension/compression test as example, pure elastic deformation isone in which plastic yield does not occur. Plastic shakedown is onein which plastic yield takes place only during the first loading,while the subsequent unloading and reloading are perfectly elastic.Cyclic plasticity is one in which alternating plastic yield of thematerial takes place in each loadingeunloading cycle, leading to aclosed elasticeplastic loop.

In order to characterize the impacts of the elastoplastic behaviorof current collector on battery performance and designing insights,an analytical model will be established for a planar layered elec-trode. Yield criterion of current collector will be proposed, takinginto account the impacts of material properties, electrode config-uration and charging states. Evolutions of the stresses in bothcurrent collector and active layer in the cycles of charge anddischarge will be simulated. Finally, designing insights will beprovided based on the discussions.

2. Methodology

Consider a layered electrode in which two active layers of equalthickness h1 are symmetrically bonded to a current collector ofthickness hc, Fig. 2. Let the thickness direction be alignedwith the z-axis and the in-plane of plate with x- and y-axes. Lithium ions areinserted into and extracted out of the electrode from both sidesurfaces.

The Li-ion diffusion in the active layer is assumed to be governedby Fick’s law. Other factors affecting the diffusion such as saturationcap [15], stress coupling [16], and concentration dependent elasticmodulus [2,17], are neglected because the focus here is to investi-gate the impacts of plastic yield of current collector. Therefore, thediffusion is described by

vcvt

¼ Dv2cvz2

(1)

where c is the molar concentration of Li-ions in the plate, D is thediffusivity of lithium ions.

Consider a full cycle, see Fig. 3. Both charge and discharge startwith galvanostatic operation followed by potentiostatic operation.

Fig. 1. Explanation of the terms “pure elastic deformation”, “plastic shakedown” and “cycompression test. sY is the yield strength.

In the charge phase of the cycle the lithiation of anode starts from alithium free state. Hence, the initial condition and the boundaryconditions for galvanostatic charging are

c ¼ 0 for t ¼ 0 (2a)

Dvcvz

¼ inF

for z ¼ h1 (2b)

Dvcvz

¼ 0 for z ¼ 0 (2c)

where F ¼ 96485.3 C mol�1 is Faraday’s constant and in is thesurface current density which is positive for lithiation but negativefor delithiation. Due to the symmetry, only the equations of theupper active layer are provided. The distribution of Li-ion concen-tration is provided by Crank [18]:

c z; tð Þ ¼ inh1FD

(Dth21

þ 3z2 � h216h21

� 2p2

XNn¼1

�1ð Þnn2

cosnpzh1

exp �n2p2Dth21

!)(3)

It is assumed that the charge operation switches from the gal-vanostatic to the potentiostatic at the time moment t0 with Li-ionconcentration c(z,t0). Hence, the initial and boundary conditionsfor potentiostatic charging are

c ¼ cðz; t0Þ for t ¼ t0 (4a)

c ¼ c0 for z ¼ h1 (4b)

clic plasticity” with the stressestrain curve of a strain controlled uniaxial tension/

Fig. 4. Illustration of the plastic constitutive behavior of the current collector. Dεpcycle isthe plastic strain range.

Fig. 3. Illustration of one cycle of charge and discharge.

Y. Song et al. / Journal of Power Sources 263 (2014) 22e2824

Dvcvz

¼ 0 for z ¼ 0 (4c)

where c0 is the constant concentration at the charging surface. Theexpression for c(z,t0) is obtained by Equation (3) with insertiont ¼ t0. The solution can be obtained by using the method of sepa-ration of variables and is written as

cðz; tÞ ¼ c0 �XNn¼1

8><>:

2h1

Zh1

0

½c0 � cðz; t0Þ�cos�ð2n� 1Þp

2h1z�dz

$exp

"ð2n� 1Þ2p2Dðt � t0Þ

4h21

#$cos

�ð2n� 1Þp2h1

z�9>=>;

(5)

In the discharging phase of the full cycle the delithiation willstart with galvanostatic operation from the saturation state havinghomogenous lithium-ion distribution, and continue with poten-tiostatic operation from the time moment t ¼ t2. The delithiated Li-ions of galvanostatic discharging can be obtained by simply usingEquation (3) with the replacement of t by discharging time (t � t1).And the delithiated Li-ions during the follow-up potentiostaticdischarging can be obtained by Equation (5) with replacement of tand t0 by discharging time (t � t1) and (t2 � t1).

Considering that the layered electrode is large and the Li-ionsare inserted symmetrically, the electrode does not bend. Amongthe totally six stress components, only the two in-plane biaxialstresses sx and sy are non-zero. The out-of-plane stress sz, whicharises from the compression by adjacent electrodes, is neglected asit is small and considered incapable of significantly impacting theelastoplastic behavior of current collector. Because sx ¼ sy, we uses1 and sc to denote the biaxial stress in the active layer and currentcollector, respectively. According to the equilibrium condition, s1and sc must satisfy

2Zh1

0

s1dzþZ0�hc

scdz ¼ 0 (6)

Focusing on the activematerials like graphite and LiFePO4 whichdo not show significant plastic behaviors, the active material layersare assumed to be isotropic and always elastic. In the active layerthe stress s1 is

s1 ¼ E0ðε0 � Uc=3Þ (7)

where ε0 is the in-plane strain of the plate electrode, U is the partialmolar volume, E

0 ¼ E/(1 � n) is the biaxial modulus and n is thePoisson’s ratio.

The current collector is considered as an elastoplastic materialwith linear kinematic hardening behavior, see Fig. 4. Its elasticmodulus is Ec, plastic modulus Ep, and yield strength sY. In chargeprocess the current collector is stretched due to the expansion ofactive materials, while in discharging the current collector isunloaded due to the shrinkage of active materials.

When the current collector is elastic, the biaxial stress sc de-velops linearly with the in-plane strain ε0. In the linear hardeningstage after the current collector yields, the equivalent stress

se ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3sijsij=2

qchanges linearly with the equivalent plastic strain

εp ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2εpijε

pij=3

q. In the elastic unloading stage, the incremental

stress Dsc is linear to the incremental strain Dε0. In the reversinghardening stage the incremental equivalent stress is proportionalto the incremental equivalent plastic strain. The cyclic plasticconstitutive behavior is depicted in Fig. 4 and described as

sc ¼ E0cε0 for εp ¼ 0 (8a)

Dse ¼ EpDεp for εps0; dεp > 0 (8b)

Dsc ¼ E0cDε0 for εps0; dεp ¼ 0 (8c)

where E0c ¼ Ec=ð1� nÞ is the biaxial modulus of the current col-lector. Note that sij ¼ sij � skkdij/3 is the deviatoric stress, and theplastic strain ε

pz ¼ �2εpx ¼ �2εpy due to the assumption of plastic

incompressibility, we have

se ¼ sc (9a)

εp ¼ 2εpx ¼ 2εpy (9b)

Substitute Equations (7) and (8a) into Equation (6), the biaxialstresses in the current collector and active layer in elastic stage are

sc ¼ 13E0cUcsG1

Q (10a)

s1 ¼ 13E0UcsG1

Q � 13E0Uc (10b)

where G1 ¼ 1 þ 1/2Ec/E1hc/h1 is a dimensionless variable deter-mined by the modulus ratio and thickness ratio,

Q ¼ 1=csh1

Z h1

0c dz is the SOC (state of charge) and cs is the

saturation concentration at stoichiometric limit.

Fig. 5. Yield map showing the separation of three types of elastoplastic behaviors ofcurrent collector, i.e. pure elastic deformation, plastic shakedown and cyclic plasticity,with respect to the elastic modulus ratio Ec/E1 and thickness ratio hc/h1. This map is forfully charged graphite electrodes.

Y. Song et al. / Journal of Power Sources 263 (2014) 22e28 25

In the plastic hardening stage of charge

sc ¼ sYG2

þ 2EpUcs3G2

Q (11a)

s1 ¼ �13E01Uc�

hch1

sY2G2

þ�1þ 2Ep

E0c

�E01Ucs3G2

Q (11b)

where G2 ¼ hc=h1Ep=E01 þ 1þ 2Ep=E0c. In the elastic unloadingstage, taking the end of charging as the reference state, the incre-mental stresses are

Dsc ¼ E0cUcs3G1

DQ (12a)

Ds1 ¼ E01Ucs3G1

DQ � 13E01UDc (12b)

In this stage, reversing yield would be induced if Dsc exceeds2sY. Whereas if Dsc is smaller than 2sY, there would be no reversingyield and the current collector is unloaded and reloaded elasticallyin the following cycles.

In the reversing hardening stage after the reversing yield, theincremental equivalent stress Dse is again linear to the incrementalequivalent plastic strain Dεp. Similar to Equations (11a) and (11b),we have

Dsc ¼ 2EpUcs3G2

DQ (13a)

Ds1 ¼�1þ 2Ep

E0c

�E01Ucs3G2

DQ � 13E01UDc (13b)

Finally in the elastic reloading stage, the stresses develop againlinearly with the in-plane strain as described by Equations (12a)and (12b).

In the cycles, the plastic strain range Dεpcycle is very important asit directly impacts the low-cycle fatigue life (Here Dεpcycle is used todistinguish from the incremental plastic strain Dεp). According toFig. 4, the plastic strain range Dεpcycle is equal to the incrementalplastic strain in the reversing hardening stage. Therefore, accordingto Equations (13a) and (8b)

Dεpcycle ¼ 2Ucs3G2

ðQs � 2QY Þ (14)

where Qs is the saturation SOC when delithiation starts, QY is theSOC corresponding to the initial yield and can be determined byEquation (10a), thus Qs � 2QY corresponds to the change of SOC inreversing hardening stage.

3. Results and discussions

3.1. Yield criterion, plastic shakedown and cyclic plasticity

Firstly we discuss the yield criterion. The current collector yieldswhen the equivalent stress reaches yield strength, i.e. se ¼ sY.Introducing Equations (9a) and (10a), we have the yield criterion

1E1Ecþ 1

2hch1

Q ¼ sY (15)

where sY ¼ 3sY=E01Ucs is the dimensionless yield stress. This cri-terion, depicted in Fig. 5 using the red line (in the web version),indicates that active materials with lower elastic modulus E1 and

smaller lithiation induced strain Ucs are favored to avoid the plasticyield of current collector. In addition, the elastic modulus ratio andthickness ratio of current collector to active material also impactthe yield.

Once yields, the current collector may exhibit two types ofplastic behaviors in the following cycles, i.e. plastic shakedown andcyclic plasticity. In the former case the unloading and reloading ispurely elastic, see Fig. 1(b). While in the latter case alternatingplastic yield takes place in each loadingeunloading cycle, seeFig. 1(c). Consider a discharge operation from saturation state tolithium-free state in which DQ ¼ Qs, reversing yield takes placeonce the incremental stress Dsc exceeds 2sY. Introducing thiscondition into Equation (12a), the critical condition distinguishingbetween plastic shakedown and cyclic plasticity is obtained as

1E1Ecþ 1

2hch1

Qs

2¼ sY (16)

This critical condition is plotted in Fig. 5 using the blue line.Comparing Equations (16) with Equation (15) and noting QY is theSOC corresponding to the first yield, it is clear that plastic shake-down takes place if the initial yield of current collector occurs afterthe electrode has been charged half full, i.e. QY � 0.5Qs, whereascyclic plasticity will be induced when QY < 0.5Qs.

A yield map is provided based on the above critical conditions,see Fig. 5. This figure shows separation of the three possibleelastoplastic behaviors of current collector in a fully chargedgraphite electrode. It shows that the elastoplastic type changesfrom pure elastic deformation to plastic shakedown and furtherto cyclic plasticity with increasing elastic modulus ratio Ec/E1 aswell as decreasing thickness ratio hc/h1. The reason is that thecurrent collector with higher Ec applies stronger restriction to theactive layer, and smaller thickness hc results in smaller cross-sectional area of the current collector, both of which lead tohigher stress and larger plastic strain in the current collectorduring the cycles.

According to Fig. 5, the thickness ratio hc/h1 is preferably higherthan 2.4 to avoid the plastic yield of current collector if we choose acopperegraphite electrode whose modulus ratio Ec/E1 is about 10.This thickness ratio is higher than that employed in real electrodes[19]. It may be an error induced by assigning the maximum lith-iation induced volumetric strain Ucs ¼ 8% of pure graphite to the

Y. Song et al. / Journal of Power Sources 263 (2014) 22e2826

graphite composite whose Ucs should be smaller due to the exis-tence of pores and other non-active compositions in the composite.Therefore, the structural designs provided by this paper areconsidered conservative.

3.2. Diffusion induced stresses

In the following sections, impacts of the plastic yield of currentcollector on DISes will be discussed. The current collector is chosenas copper and the active material as graphite. The employed ma-terial constants of copper are yield strength sY ¼ 300 MPa, elasticmodulus Ec ¼ 150 GPa and plastic modulus Ep ¼ 3 GPa. The elasticmodulus of graphite is E1 ¼ 15 GPa. The operation of charge anddischarge follows that provided by Fig. 3.

Fig. 6(a) and (b) shows the stresses along plate thickness in atime moment after the current collector has yielded in charge anddischarge, respectively. The thickness ratio hc/h1 is set to 0.25 ac-cording to Fig. 5 to ensure the plastic yield of current collector. It isfound that the estimations of stresses, in both current collector andactive layer, are significantly higher once the plastic yield of currentcollector is neglected. The reason is that a substantial amount ofdiffusion induced stress in the layered electrode arises from theconstraint of active layer by current collector. Once the currentcollector yields, the plastic relaxation enables the layered electrodeoccur relatively large deformation with respect to small incrementof stress. Therefore, ignoring the plastic yield of current collector

Fig. 6. The dimensionless biaxial stress s ¼ 3s=E01Ucs in the electrode: (a) in chargeprocess, and (b) in discharge process.

may lead to overestimations of the stresses and underestimationsof the strength of electrodes.

As seen, the diffusion induced stresses in active layers can bereduced by the plastic relaxation of current collector. Therefore,Fig. 7 is further prepared to illustrate the impacts of the plastic yieldby plotting the evolutions of the DISes in certain positions of theelectrode in cycles. According to the yield map provided by Fig. 5,

Fig. 7. Evolutions of the biaxial stress against SOC in cycles of charge and discharge: (a)in the current collector, (b) at the interface between current collector and active layerwhere z/h1 ¼ 0, and (c) at the inlet surface where z/h1 ¼ 1. Three thickness ratios areemployed according to Fig. 4 so that the blue, red and dark lines describe the cases ofpure elastic deformation (PED), plastic shakedown (PS) and cyclic plasticity (CP),respectively. The thin lines represent the stresses in the first cycle. (For interpretationof the references to color in this figure legend, the reader is referred to the web versionof this article.)

Fig. 8. The in-plane strain ε0 of graphiteecopper electrode and the plastic strain rangeDεpcycle at saturation state vs. thickness ratio hc/h1. The peak value of in-plane strain isabout 2.6% when hc ¼ 0, consistent with the lithiation induced volumetric strain 8% offree expansion graphite.

Y. Song et al. / Journal of Power Sources 263 (2014) 22e28 27

the thickness ratio hc/h1 is set to 3, 1.5 and 0.25, respectively, cor-responding to the case of pure elastic deformation, plastic shake-down and cyclic plasticity.

Fig. 7(a) shows the evolutions of the stress in the current col-lector. It is clearly seen the change from pure elastic deformation toplastic shakedown and finally to cyclic plasticity with respect todecreasing thickness ratio. In the case of hc/h1 ¼ 3, the currentcollector is thick so that it cannot be stretched to yield, the stress incurrent collector changes linearly with SOC. In the case of shake-downwhen hc/h1 ¼1.5, the plastic strain is limited and the stress incurrent collector changes linearly with SOC since the second cycle.Nonlinear behavior occurs only in the first charge. In the case ofcyclic plasticity when hc/h1 ¼ 0.25, the current collector is too thinso that the current collector yields in every charge and dischargeand the stress evolves in the form of a closed loop. With the changeof plastic type, the stress in current collector evolves in largerranges with higher peak values and higher residual stresses.

Fig. 7(b) and (c) illustrates the stresses in the position of z/h1¼0and z/h1 ¼1 in the active layer, respectively. It is seen that both thepeak compressive stress and cyclic stress range in the charge/discharge cycle are significantly decreased when the elastoplastictype changes from pure elastic deformation to cyclic plasticity,although tensile stress is increased due to the residual deformationof current collector. This figure for one hand reveals one mecha-nism how tensile stress is generated in anode, for the other hand itis interesting to see that applying thinner current collectors andallowing them to yield is beneficial to the integrity of active layeras the peak stress and the cyclic stress range are significantlydecreased.

The designs allowing plastic yield show two advantagescompared with that free of plastic yield. Firstly, the thickness ofcurrent collector is decreased significantly, with which electrodecapacity can be enhanced as more space can be provided for theactive materials. It is found in Fig. 7 that the thickness of currentcollector in the case of plastic shakedown is only about 50% of thatin the case of pure elastic deformation. This ratio is even lower inthe case of cyclic plasticity. Secondly, the cycle life of electrode maybenefit from the plastic yield because both the peak stress and thecyclic stress range in active layers, which is critical to the me-chanical fade and electrochemical stability, are significantlydecreased. Therefore, applying a thinner current collector andallowing it to yield is beneficial not only to the capacity but also tothe structural integrity and cycle life of electrode.

It should be noted that the stress in active layer is decreasedwith the sacrifice of higher stress and cyclic stress range in the thincurrent collector. However, this increase of stress may be not soimportant to the battery performance because current collector,which functions as a passageway of electrons, does not involve inthe electrochemical reaction and storage of Li-ions. Therefore, thisincrease of stress in current collector may be acceptable on thecondition that the plastic deformation does not lead to crack.

3.3. On the plastic yield

In this section we discuss whether the plastic yield of currentcollector is allowable. If yes, which type is better between plasticshakedown or cyclic plasticity? How to determine the thickness ofcurrent collector in designing?

For the first question, it is known plastic yield should be avoidedfor ordinary load-bearing structures because cyclic fatiguemay leadto structural failure. However, the electrode of a Li-ion battery maybe not so sensitive to fatigue for two reasons. Firstly, the cycle life ofa real battery may be less than one thousand of times and one cycleof charge and discharge may take several hours to several days.They are considered not adequate to induce fracture in the case of

finite plastic strain because experiments showed that poly-crystalline copper took more than 105 cycles to fracture with con-stant and low plastic strain amplitude [20]. Secondly, the materialsof current collector like copper and aluminum are knownwith goodductility. The reported ultimate fracture strain of copper film is 6%[21], whereas the extreme in-plane strain ε0 of the copperegraphite electrode is calculated as about 2.6%, which is limited bythe expansion of graphite and far below the reported value.Therefore, finite plastic deformation of current collector may beallowable.

The second question is which type is better between plasticshakedown and cyclic plasticity. Generally, the case of plasticshakedown shows good balance among all aspects, e.g. capacityenhancing due to thinner current collector, better electrochemicalstability due to lower stress in active layer, and good fatigueresistance due to the zero plastic strain range since the secondcycle. Electrode structure can be designed according to the criticalcriterion of plastic shakedown provided by Equation (16). Accord-ing to calculation, changing the thickness ratio hc/h1 from 2.4 to 1.1,corresponding to the transition from pure elastic deformation toplastic shakedown, may lead to a reduction of the volume of cur-rent collector by more than 50% as well as the decrease of stress inactive layer by 40%.

Compared with plastic shakedown, the case of cyclic plasticity iseven superior in the aspects of capacity enhancing and electro-chemical stability, though the plastic yield in every reversal maylead to low-cycle fatigue and significant shortening of cycle life. Thenumber of cycle to failure in low-cycle fatigue is usually charac-terized by the CoffineManson relation [22,23]:

Dεpcycle ¼ ε0f ðNÞc (17)

where N is the expected cycle number to failure, ε0f is the fatigueductility coefficient and c is the fatigue ductility exponent. Beingaware that both ε

0f and c are known empirical constants for a given

material, the allowed plastic strain range Dεpcycle thus can bedetermined according to the designed cycle numbers N. Finally, thethickness ratio hc/h1 can be looked up from Fig. 8. This designingmethod enhances electrode capacity and decreases the stress inactive layer as much as possible with the aim of an expected cyclelife.

Y. Song et al. / Journal of Power Sources 263 (2014) 22e2828

4. Conclusions

This work studies the impacts of plastic yield of current collectoron the diffusion induced stress in a symmetric layered electrode.Based on analytical formulations, three types of elastoplastic be-haviors of current collector, i.e. pure elastic deformation, plasticshakedown and cyclic plasticity, are identified. Criterions sepa-rating the three cases are proposed. It is found the plastic behaviorof current collector would change from pure elasticity to plasticshakedown and finally to cyclic plasticity with increasing elasticmodulus ratio Ec/E1 and decreasing thickness ratio hc/h1 of currentcollector to active layer.

As for the diffusion induced stress, it is found that neglecting theplastic yield of current collector would result in overestimations ofthe stress in whole electrode and underestimation of electrodestrength. By simulating the evolution of stress in cycles of chargeand discharge, it is suggested that applying a thin current collectorand allowing finite plastic deformation is beneficial not only to thebattery capacity as more space can be provided for active materials,but also to the electrochemical stability and cycle life of electrodebecause the stress in active layer is significantly decreased.

Regarding the plasticity of current collector, firstly finite plasticdeformation is considered acceptable due to the few cycle numbers,long cycle period and limited plastic strain of current collector.Secondly, electrode structures responsible for plastic shakedownand cyclic plasticity are discussed and compared. The case of plasticshakedown shows good balance among all aspects, e.g. capacityenhancing due to thinner current collector, better electrochemicalstability due to lower stress in active layer, and good fatigueresistance due to the zero plastic strain range. In contrast, the caseof cyclic plasticity is even superior in the former two aspects butwith the sacrifice that the cycle life is significantly shortened due tolow-cycle fatigue. Therefore, structural design is provided by thecritical criterion of plastic shakedown in the former case while bythe CoffineManson relation according to an expected cycle life inthe latter case.

It should be noted this paper discusses the impact of plasticyield by taking a graphiteecopper anode for example. However, themain conclusion of this work, i.e. applying a thin current collectorand allowing it to yield is beneficial to both the capacity and cyclelife, is also applicable for cathode. The only difference is the sign ofstress. For a cathode which is initially lithium-saturated anddeformation free, the current collector would be compressed toyield in charge (delithiation) operation. In addition, applying a thin

current collector in cathode and allowing it to yield results indecrease of tensile stress in active layer.

Acknowledgments

The authors gratefully acknowledge the financial support by theNational Natural Science Foundation of China under grant numbers11332005, 11102103 and 11172159, the Shanghai Municipal Edu-cation Commission, China, under grant number 13ZZ070, ResearchFund for the Doctoral Program of Higher Education of China, undergrant number 20113108120006, and the Science and TechnologyCommission of Shanghai Municipality, China, under grant number12ZR1410200.

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