Safety performance and failure prediction model of cylindrical lithium-ion battery

Journal of Power Sources 451 (2020) 227755

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Safety performance and failure prediction model of cylindrical lithium-ion battery Wenwei Wang, Yiding Li *, Lin Cheng, Fenghao Zuo, Sheng Yang National Engineering Laboratory for Electric Vehicles, Beijing Institute of Technology, Beijing, China

H I G H L I G H T S

G R A P H I C A L A B S T R A C T

� A safety performance model based on second-order oscillation feature is proposed. � The fracture angle of the jellyroll is only related to the load property. � The soft short-circuit prediction model based on mechanical penetration is proposed. � Failure warning verification system has a certain safety margin for battery failure.

A R T I C L E I N F O

A B S T R A C T

Keywords: Cylindrical lithium-ion battery Second-order oscillation feature Safety performance model Failure prediction model Failure warning verification system

In this paper, the safety performance model of cylindrical lithium-ion batteries, which is based on a second-order oscillation feature that is subjected to mechanical abuse is proposed via a discrete Fourier transformation of experimental data. Combined with the safety performance model and Crushable-Foam material model, the simulation results show that the average predictive error of the force-displacement and failure displacement is less than 8.8% and 4.0%, respectively. A battery fracture angle model is proposed, the fracture angle is only related to the load property, and the state-of-charge has minimal influence under the same load conditions. Furthermore, the failure prediction model of the soft short-circuit and the failure warning verification system are proposed. The experimental results show that the failure prediction model can warn the soft short-circuit in time with a certain safety margin. With further research, the safety performance model and the failure prediction model, which is based on mechanical penetration displacement, can be combined with other failure signals to provide more reliable information about battery safety. The failure warning device has the advantages of low cost and simple structure.

1. Introduction In recent years, battery safety has become one of the main obstacles to the development of electric vehicles [1,2]. To improve the safety of

pure electric vehicles, China has launched a new version of the New Car Assessment Program (C-NCAP) in 2018, and the security testing pro­ gram has been added for electric vehicles. Because mechanical abuse can destroy the integrity of lithium-ion

* Corresponding author. National Engineering Laboratory for Electric Vehicles, Beijing Institute of Technology, Beijing, 100081, China. E-mail addresses: [email protected] (W. Wang), [email protected] (Y. Li), [email protected] (L. Cheng), [email protected] (F. Zuo), 1034012081@qq. com (S. Yang). https://doi.org/10.1016/j.jpowsour.2020.227755 Received 6 December 2019; Received in revised form 6 January 2020; Accepted 13 January 2020 Available online 21 January 2020 0378-7753/© 2020 Elsevier B.V. All rights reserved.

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batteries, mechanical constitutive properties should be accurately described. The homogenized mechanical properties of lithium-ion bat­ teries were proposed by Wierzbicki et al. [3], and the average stress-strain equation was deduced, thus, the mechanical behavior of the battery was revealed. By the analysis of experiment and theory, the clay-like characteristics of the batteries were proposed by Wang et al. [4], and the influence of different properties of the battery components on the mechanical characteristics were explored. Avdeev et al. [5] proposed a uniform mechanical constitutive model for lateral impact using the virtual work principle. Xu et al. [6] brought forward a model by the unified strength theory, and the mechanical integrity of the lithium-ion batteries under high strain rate and quasi-static condition was summarized. Sheidaei et al. [7] proposed the mechanical feature of the separator according to the material creep theory. A phenomenal physical model of the battery subject to mechanical abuse, which was proposed by Oh et al. [8,9], provided a new idea for the research. Most of these models explain the mechanical behavior of lithium-ion batteries under mechanical abuse conditions, but a few researches have been done on how to warn and protect batteries of thermal runaway from mechanical abuse. Three commonly employed simulation modeling methods address the mechanical properties of lithium-ion batteries. The first and most prevalent model comprises homogenized mechanical properties for the battery jellyroll, and the Crushable Foam material model is a kind of representative model that belongs to homogenized model. The homog­ enized model has the advantages of convenient modeling, fewer pa­ rameters. Based on these reasons, numerous studies are based on the homogenized mechanical properties [3–5,10,11]. However, the ho­ mogenized model can only express the battery mechanical behavior from a macroscopic perspective, the battery details cannot be clearly displayed. The second method is a hierarchical model which also called representative volume element (RVE) method. By assigning different battery structures to the corresponding parameters, the RVE model can accurately simulate the mechanical behavior and better balance the local failure characteristics and calculation cost [12–15]. The third modeling method is a detailed modeling method that models the active material, the current collectors and the separator according to the actual geometry and material properties. This model can adequately reproduce the mechanical and failure characteristics of the battery. The defects are distinct because the size of the electrodes and separator in the thickness direction are substantially smaller than that in the other two directions. Therefore, the size of the elements is extremely small and the element number is extremely large, which causes a significant increase in computational difficulty [16,17]. In summary, the homogenized modeling method has the highest computational efficiency but unclear details, the detailed modeling method can reflect the actual mechanical features of the battery but has an immense calculation cost, and the computational cost and the ability to reflect the mechanical behavior for the RVE method falls between the other two methods [2]. In this paper, we use a homogenized model with a smaller mesh size to reduce computational costs and express the characteristics of battery failure. When the batteries are subjected to the mechanical abuse, the Stateof-Charge (SOC) can affect the mechanical features of different types of batteries. Xu et al. [18,19] discovered that the SOC has a substantial influence on the cylindrical lithium-ion battery. The maximum stress with 100% SOC is 10%–25% higher than that with 0% SOC. Luo et al. [20] observed a different phenomenon in the study of pouch batteries that they can adapt to the change in battery volume during charging and discharging. Thus, the SOC has a minimal effect on the mechanical properties of pouch batteries under mechanical abuse. The battery used in this paper is a kind of hard-shell lithium-ion battery and SOC has a great influence on the safety performance, so we should pay attention to the influence of SOC on battery safety performance. In mechanical abuse experiments, several bases are employed to analyze the failure of lithium-ion batteries, such as the sudden decrease

in force and the voltage and the sudden increase in temperature [13,16, 21–23]. For cylindrical lithium-ion batteries, the failure circumstances are distinct and can be immediately discriminated. Due to the high ca­ pacity and large size of high-capacity batteries, energy cannot be immediately released when subjected to mechanical abuse, thus the SOC cannot significantly change. Large-scale lithium-ion batteries have the problem of unclear changes in voltage and temperature [24]; thus, the early warning of the battery is difficult [25]. When batteries suffer from structural damage, such as high tem­ perature shock or other abuse conditions, thermal runaway accidents such as temperature diffusion, fire and even explosion will be triggered. Thus, the mechanism of thermal runaway and the corresponding sequence are effective ways to control disasters. Wang et al. [26,27] proposed a temperature and thermal abuse model to control the occurrence of fire according to the relationship among heat generation, heat dissipation and the theory of the combustion triangle, which pro­ vided the basis for theoretically controlling fire with lithium-ion batte­ ries. The early warning method based on battery surface temperature detection cannot alert the safety in time. The battery may have already entered an irreversible thermal runaway state when the sensor detects that the surface temperature has reached the warning threshold. Thus, many scholars have investigated the failure warning of batteries. Ouyang et al. [28] proposed a method to predict the short-circuit failure of the battery by an equivalent circuit model that is based on the change in resistance during the short circuit. Sazhin et al. [29] proposed a warning method for the short-circuit based on the change in the self-discharge current during the internal short circuit. Zhang et al. [25] advanced a short-circuit failure detection method for battery packs based on the topology of the circuit. Nascimento et al. [30] creatively combined both the FBG (Fiber Bragg grating sensor) and FP (Fabry-Perot sensor) according to the characteristics of the FBG-FP fiber sensor for battery strain and temperature detection, which has a high practical price. Nascimento et al. [31] also integrated the FBG sensor with a thermocouple, which detected the temperature by a thermocouple. Thus, the influence of temperature on the FBG sensor was eliminated, which not only realized the detection of battery strain and temperature but also reduced the cost of the sensor. Most researches have studied the failure from the outer-battery failure feature such as circuit failure and resistance changing, but few have warned about battery failure from the level of the structure failure. In this paper, a safety performance model (equivalent mechanical model) of cylindrical lithium-ion batteries based on the second-order oscillation feature is proposed by discrete Fourier transformation of the experimental data. After parameters consistency processing, the model is combined with the Crushable-Foam material model for simu­ lation. The results show that the model can adequately predict the me­ chanical behavior of the battery, and the average prediction error of the failure displacement is less than 4%. Based on the analysis of the fracture characteristics of the battery, the fracture angle model is proposed and the conclusion that the fracture angle of the battery is only related to the load property is derived. Subsequently, the fd characteristic of the bat­ tery for soft short-circuit failure prediction is proposed, and the battery failure warning verification system prototype is designed and manu­ factured. The experimental results show that the warning system can be used under the mechanical abuse for soft short-circuit prediction and has a certain safety margin that can earn time for emergency treatment of the battery. With further research, the warning method of mechanical failure based on penetration displacement can be integrated with voltage, temperature and other failure signals. Thus, battery failure warning can be more accurate and reliable. 2. Experimental equipment and process The experimental lithium-ion batteries are SONY VTC4 2100mAh 18650 cylindrical lithium-ion batteries; the universal tensile testing 2

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shown in Table 2. M is the mass, k is the elastic coefficient, D is the damping coefficient. f(t) and F(s) are the equivalent units’ forces for time domain and frequency domain, respectively. x(t) and X(s) are the displacements of the equivalent units for time domain and frequency domain, respectively. s¼ωj, where ω is the angular frequency, j is the imaginary part and the dimension of s is the reciprocal of time. The frequency response characteristics of different equivalent me­ chanical units are shown in Fig. S2 in supplementary material. The horizontal axis (Re) is the real part of the imaginary number, and the vertical axis (Im) is the imaginary part. In the frequency domain, the feature of the spring unit is a point that is located on the real part with the value of k. The frequency response characteristic of damping unit is a vertical line located on the imaginary axis, and that for the mass unit is a reciprocating line on the real axis. Thus, the results concluded that the frequency response characteristic of the spring-damping series unit is similar to an elliptical shape in the frequency domain.

Table 1 Initial displacement of infrared thermal imager data recording. Abuse Condition Rigid Rod Test Hemispherical Punch Test

SOC 0

0.2

0.4

0.5

0.6

6.5 (mm) 5.5 (mm)

5 (mm) 4 (mm)

5.5 (mm) 4.5 (mm)

5 (mm)

3.5 (mm) 3.5 (mm)

4.5 (mm)

Table 2 Equivalent mechanical unit and corresponding expression. Component

Structure Model

TimeDomain Equation

FrequencyDomain Equation

Equivalent Spring Stiffness

Mass

fðtÞ ¼ Mx ðtÞ

FðsÞ ¼ Ms2 XðsÞ

Ms2

Spring

fðtÞ ¼ kxðtÞ

FðsÞ ¼ kXðsÞ

k

Damping

fðtÞ ¼ Dx ðtÞ

FðsÞ ¼ DsXðsÞ

Ds

��

3.1. Mathematical model

�

Based on the previously described physical expression, the force and displacement of cylindrical lithium-ion batteries are analyzed in fre­ quency domain under mechanical abuse. As shown in Fig. 1(a1)- (a2), the lithium-ion battery is composed of positive electrodes, negative electrodes, separators and electrolyte. In the condition of mechanical abuse, the performance of the battery components is shown in Fig. 1(b), the electrodes and separators are given mass-spring series properties and the electrolyte is imparted with damping properties. As shown in Eq. (1), discrete Fourier transformations for the force-displacement obtained from flat tests, rigid rod tests and hemispherical punch tests are performed,

machine is INSTRON with the maximum load of 250 kN; the data recorder is HOKI MR8880 with 4 channels; the infrared thermal camera is FLUKE Ti400 and the Digital Signal Processing (DSP) controller is TMS320F28335 with the clock speed of 150 MHz. Flat plate tests, rigid rod tests and hemispherical punch tests are performed with the universal tensile tester to create mechanical abuse conditions. The corresponding abuse tests are shown in Fig. S1 in sup­ plementary material, Fig. S1(a) shows the flat plate test, Fig. S1(b) shows the battery shape after the experiment, Fig. S1(c) shows the rigid rod test and Fig. S1(d) shows the hemispherical punch test. The di­ ameters of the rigid rod punch and hemispherical punch are 24 mm and 14 mm, respectively, as shown in Fig. S1(e) and Fig. S1(f). During the experiment, the force-displacement, voltage and temperature are simultaneously recorded to detect the failure behavior and failure signal. The batteries are charged and discharged by Constant CurrentConstant Voltage (CC-CV) strategy, the charging current is 0.2C and the discharging current is 1 C. To eliminate the polarization of the battery, a 10-min pause has been acted during the charging and dis­ charging process, and the SOC samples of the batteries for experiments are 0%, 20%,40%, 50%, 60%, 80% and 100%. Ichimura [32] has reported that the lower penetration speed can produce more adverse results. Thus, the mechanical abuse experiments in this paper were performed under the quasi-static condition with the loading speed of 0.5 mm/min. The data recorder was directly connected to the tabs of the batteries to record the change in battery voltage. Due to the memory settings of the thermal camera, the data recording started from a certain displacement as shown in Table 1. The initial displace­ ment in Table 1 was set randomly to ensure that both the thermal camera storage settings and the recording of the battery failure process can be met. The initial displacement was independent of the battery failure displacement. However, it ensured that the initial displacement is much earlier than the actual failure displacement. The setup of the failure warning verification system is shown in Fig. 8 and will be described in section 4.3.

N X

FðqÞ ¼

f ðnÞe

j2π ðn 1Þðq 1ÞN

q ¼ 1; 2; …; N

(1)

n¼1

where f(n) is the load force, n and q are the sequence of the discrete signals. By Eq. (1), the force-displacement in time domain and the spectra features in frequency domain of the batteries under mechanical abuse are obtained, as shown in Fig. 1(c1)- (e2). In the limited frequency, the spectra features are similar to the symbol ε, as shown in Fig. 1(c2), (d2) and (e2). Although in Fig. 1 (c2) and (d2) the frequency responses of the batteries are similar to the shape of semielliptical from macroscopic viewpoint, while the characteristic remains concave near the frequency, where the Im value is 0. It means the force-displacement spectra char­ acteristics of the batteries under mechanical abuse are similar to that of the second-order oscillation units in the limited frequency. The smaller are the punch diameters, the more distinct are the characteristics of the second-order oscillation, as shown in Fig. 1 (f). Therefore, the cylindrical lithium-ion battery has the characteristics that are equivalent to the second-order oscillation system, in which the mass unit, spring unit and damping unit are connected in series under mechanical abuse. In other words, the mechanical characteristics of cylindrical lithium-ion batteries subject to mechanical abuse have the equivalent physical meaning of the second-order oscillation part. From the above description, the safety performance model with the equivalent physics significance of cylindrical lithium-ion battery under mechanical abuse is proposed. The general form of the second-order oscillation unit is expressed as Eq. (2), FðsÞ ¼

a Ts2 þ 2ςTs þ b

(2)

where T is the oscillation period coefficient, ζ is the damping ratio co­ efficient, and a and b are constant values. In Fig. 1(g), fi(t) is the input force that will be converted to the load force, xo(t) is the output displacement that will be converted to the compression displacement, fk(t) is the equivalent spring unit force and

3. Mathematical model and parameter consistency processing The expressions in different equivalent mechanical units for time domain, frequency domain and their equivalent spring stiffness are 3

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Fig. 1. Equivalent mechanical model composition diagram of lithium-ion battery with second-order oscillation feature. (a1)(a2) Internal structure of cylindrical lithiumion battery. (b) Local equivalent mechanical structure of the lithium-ion battery. (c1)- (e1) Time-domain response under flat plate test, rigid rod test and hemispherical punch test, respectively. (c2)- (e2) Frequency-domain response under flat plate test, rigid rod test and hemispherical punch test, respectively. (f) Qualitative characteristics of second-order oscillation structure. (g) Mass-springdamping system with second-order oscilla­ tion feature.

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Table 3 Equivalent mechanical model parameters under various SOC. SOC

d

0 0.2 0.4 0.5 0.6 0.8 1

h 10860 23360 27870 27550 38700 47550 54410

669.7 593.4 905.3 1052 1428 1595 2205

xo ðtÞ ¼L fi ðtÞ

m

h th 2 sin e 2m > > pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > > : h ¼ 4km d2

Dx0 ðsÞs ¼ Mx0 ðsÞs2

(4)

With GðtÞ ¼

Therefore, the transfer function of the equivalent mechanical model can be expressed as Eq. (5), xo ðsÞ 1 ¼ fi ðsÞ Ms2 þ Ds þ k

dt 2m

(6)

xo ðtÞ (7)

th We set M¼m, in Eq. (7) sin 2m is a periodic signal, and the discrete Fourier transformation in this paper is performed in a finite frequency range. Therefore, we do not consider the role of the periodic signal in the th , and the equivalent me­ equation. Thus, we disregard the term sin 2m chanical equation is reduced to Eq. (8), 8 > h > > < fi ðtÞ ¼ dt xo ðtÞ 2e 2m (8) ffi ffi ffi ffi ffi ffi ffi ffi ffiffiffiffiffiffiffiffiffiffi p > > > : h ¼ 4km d2

Eliminating fk and fD, we can get Eq. (4), kx0 ðsÞ

� � Z aþj∞ xo ðsÞ 1 xo ðsÞ st ¼ e ds fi ðsÞ 2πj a j∞ fi ðsÞ

8 > > > > > < fi ðtÞ ¼

142.2 286.9 389 397.4 606.1 750.5 863

fD(t) is the equivalent damping unit force. In the frequency domain, we can obtain the equivalent forces of the different model units as shown in Eq. (3), 8 < fk ðsÞ ¼ kxo ðsÞ f ðsÞ ¼ Dxo ðsÞs (3) : D fi ðsÞ fk ðsÞ fD ðsÞ ¼ Mxo ðsÞs2

fi ðsÞ

1

h 2e

dt 2m

,t ¼

xo ðtÞ , v

and v as the load velocity, the Laplace

transformation of G(t) is performed to verify whether the reduced equation has a second-order oscillation feature, as shown in Eq. (9), � � Z ∞ fi ðtÞ ^ h st GðsÞ ¼ L dt (9) ¼ dt e xo ðtÞ 2e 2m 0

(5)

After inverse Laplace transformation of Eq. (5) by Eq. (6), the forcedisplacement of the equivalent mechanical model in time domain is obtained as shown in Eq. (7),

The expression of the reduced function Eq. (8) in the frequency domain is shown in Eq. (10). As shown in Eq. (10), we observe that the

Fig. 2. Consistency processing of the equivalent mechanical model parameters under various SOC. (a) d vs. SOC. (b) h vs. SOC. (c) m vs. SOC. 5

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Journal of Power Sources 451 (2020) 227755

Fig. 3. Simulation settings. (a) Flat plate test parameters settings. (b) Real stress-strain of the cylindrical lithium-ion battery derived by Eq. (8), Eq. (11) and Eq. (19). (c1) Flat plate test simulation. (c2) Rigid rod test simulation. (c3) Hemispherical punch test simulation.

4. Results and discussion

Table 4 Material parameters of jellyroll. Component

Young’s Modulus

Poisson’s Ratio

Density

Jellyroll

1.5 GPa

0.15

3.22 g cm

4.1. Simulation results

TSC 3

In the simulation, the real stress-real strain that is subjected to me­ chanical abuse should be acquired. To simplify the calculation, we use the average stress-average strain model proposed by Wierzbicki et al. [3] shown in Fig. 3(a), the following three conclusions are utilized in the model:

10 MPa

equivalent mechanical model, which disregards the periodic signal th , has the characteristics of second-order oscillation, a ¼ hv, T ¼ 2, sin 2m

ς ¼ md and b ¼ d 2m , 2

fi ðsÞ ¼

2s2

2

hv 2 2 2dms þ d 2m

� The total length of the battery remains constant under mechanical abuse, the length of the battery is L ¼ 58 mm; � The area in which the battery contacts the flat plate remains flat; � The circumference of the battery section area remains constant.

(10)

The response nature of Eq. (8) in the frequency domain is shown in Fig. 1(f) which infers that the equivalent mechanical model of Eq. (8) with the second-order oscillation feature is equivalent to the experiment.

In Fig. 3(a), f(x) is the force applied on the battery, x is the displacement of the flat plate, R is the original radius of the battery, R ¼ 8.8 mm, r is the radius for which the battery is not directly compressed, 2b is the contact width of the flat area, and b is expressed as Eq. (12),

3.2. Parameter consistency processing

π

(12)

b¼ x 4

The parameters of the safety performance model Eq. (8) for different SOCs are shown in Table 3. The parameters d, h and m have monotonic properties with SOC, and the fitting results of d, h and m are shown in Fig. 2. We note that d and m have a linear relationship to SOC, and h has a quadratic relationship with SOC. Thus, we can conclude that the SOC can influence the mechanical behaviors of cylindrical lithium-ion batteries. By the relationship between the SOC and parameters, the consistency processing is helpful in predicting the mechanical characteristics of batteries with a continuous SOC, and the fitting process is based on this process. The fitting results are shown in Fig. 2 and Eq. (11). 8 < d ¼ 40310⋅SOC 12200 h ¼ 1310⋅SOC2 þ 287:9⋅SOC þ 604 (11) : m ¼ 660⋅SOC 143:8

The average stress-average strain is shown in Eqs. (13) and (14),

σ av ¼

f ðxÞ 2bL

(13)

εav ¼

x 2R

(14)

Combined with Eq. (8) and Eq. (11), the relationships between the average stress and average strain are Eq. (15),

σ av ¼

h

πLe

dRεav mv

(15)

The engineering stress se, and the engineering strain ee are defined in Eqs. (16) and (17), where P is the load and A0 is the section area, se ¼

6

P A0

(16)

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Journal of Power Sources 451 (2020) 227755

Fig. 4. Simulation and mechanical test results. (a)–(d) Flat plate tests under SOC ¼ 0, 0.4, 0.8 and 1. (e)–(f) Rigid rod tests under SOC ¼ 0 and 0.6. (g–h) Hemi­ spherical punch tests under SOC ¼ 0 and 0.4.

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Table 5 Average relative error between experiment and simulation under flat plate tests. SOC

0

0.2

0.4

0.5

0.6

0.8

1

Average relative error

8.58%

7.97%

8.11%

9.84%

6.49%

9.17%

11.03%

Table 6 Failure displacement and relative error by experiment and simulation under various mechanical abuse conditions. SOC

0

0.2

0.4

0.5

0.6

Rigid rod test

Experimental failure displacement (mm) Failure displacement by simulation (mm) Relative error of failure displacement between experiment and simulation (mm)

8.368 8.37 0.02%

8.766 8.46 3.49%

8.238 8.34 1.24%

7.95 8.28 4.15%

7.3 7.68 5.21%

Hemispherical punch test

Experimental failure displacement (mm) Failure displacement by simulation (mm) Relative error of failure displacement between experiment and simulation (mm)

8.9 8.04 10.70%

8.3 7.92 4.58%

8.15 7.83 3.93%

8.483 8.22 3.10%

8.45 8.22 2.72%

ee ¼

x 2R

The real strain ε is Eq. (18), Z 2R dx ¼ lnð1 þ ee Þ ε¼ 2R x x

fails. The material of the flat plate punch is defined as Rigid and the property is Solid, the materials of the rigid rod punch and the hemi­ spherical punch are Rigid and the properties are defined as Shell. The experimental and simulation results are shown in Fig. 4, Fig. 4 (a)–(d) display the comparison results of the flat plate tests for various SOC, and the average relative error between the experiment and the simulation is shown in Table 5. The average relative error of the forcedisplacement on the simulation level by the equivalent mechanical model is less than 8.8%, which reflects the mechanical characteristics of the battery. In the elastic part (x < 4 mm), the simulation results produce a large error compared with the experimental results due to an increase of the SOC. Thus, the strength of the battery and Young’s modulus in­ crease [35]. To simplify the simulation process, Young’s modulus is set to a constant. These results inspire us to reflect the actual mechanical properties of batteries and comprehensively consider the influence of the SOC on the material constant in the future research. As shown in Fig. 4(a)–(d), the strength of the battery increases with an increasing in the SOC because a large number of lithium ions are inserted into the anode and cause higher structure stiffness under the higher SOCs [19]. The comparison results for the rigid rod tests and hemispherical punch tests between experiment and simulation are shown in Fig. 4(e)– (f) and Fig. 4(g)-(h), respectively. In the rigid rod tests, the simulation results can better predict the experiment. For the peak force, however, the simulation value is higher than the experimental value due to a local internal short-circuit. The battery internal electrochemical reaction heat and short circuit ohmic heat will work together to heat the battery to a higher temperature, and the current collector will soften. In the simu­ lation, the Crushable-Foam model cannot consider the effect of tem­ perature on the material. Thus, the simulation peak force is greater than that in the experiment. In the hemispherical punch tests, the simulation results can also predict the experiment but the simulation peak force does not appear similar to the phenomenon under the rigid rod punch tests. The simulation results are lower than the experimental values, because the size of the hemispherical punch is small, which produce a stress concentration in the jellyroll, and part of the structure of the battery loses the ability to bear force, which causes the battery to fail in advance. The analysis of the mechanical failure displacement of the battery by the experiment and simulation model is shown in Table 6. Due to the battery inconsistency of the mechanical properties and the measurement error of the test equipment, distinct decreasing trend for the failure displacement is not observed under the rigid rod tests and the hemi­ spherical punch tests. However, the simulation results indicate that the battery failure displacement decreases with an increase in the SOC because a large number of lithium ions are inserted into the anode and increase the stiffness of the battery while reducing the impact toughness, which renders the battery more prone to failure under high SOCs. The

(17)

(18)

Thus, the real stress-real strain of cylindrical lithium-ion battery is shown in Eq. (19), 8 > > h > <σ ¼ dRεav ð1 þ εav Þ (19) πLe mv > > > : ε ¼ lnð1 þ εav Þ The stress-strain of the battery under various SOC, is shown in Fig. 3 (b). The results indicate that the amount of lithium ions that are embedded in the anode with high SOC increases; thus, the strength of the battery increases. Therefore, the higher is the SOC under the same strain, the greater is the stress of the battery [19]. The homogenized model can unify the electrode, current collector, separator and other components into a whole part. It can take into ac­ count the average failure conditions of different components under mechanical abuse, thus, the homogenized model with small element size have good performance in balancing the calculation cost and model accuracy [41]. In this paper, a homogenized modeling method and Crushable-Foam material model are used to simulate the mechanical behavior of the battery. The Crushable-Foam material model defines the yield criterion via the Self-Similar yield surface model [33]. The yield stress F is shown in Eq. (20), rffiffiffiffiffiffiffiffiffi�ffiffiffiffiffi� ffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi α 2 F ¼ q2 þ α2 p2 σc 1 þ ¼0 (20) 3 where σ c is the absolute value of the uniaxial compression yield strength, p is the compressive stress, q is the Mises equivalent stress, and α is the shape factor for the p-q stress elliptic yield surface (aspect ratio of el­ lipse). In the compression process, as long as the element stress satisfies the requirement of yield stress, the material will fail and not bear the load. Concurrently integrating Eq. (11), Eq. (19) and Crushable-Foam material model, this paper has simulated and validated the flat plate tests, rigid rod tests and hemispherical punch tests for cylindrical lithium-ion batteries. The simulation settings are shown in Fig. 3(c1)(c3). In Ls-Dyna, the jellyroll material is Crushable-Foam, the property is Solid with a mesh size of 0.3 mm, and the total number of the elements is 755 200. The material constants of the jellyroll are shown in Table 4, the Tensile Cut-Off Value (TSC) is set to calibrate the battery failure, and when the element stress reaches TSC ¼ 10 MPa [10,18,23], the element 8

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Journal of Power Sources 451 (2020) 227755

Fig. 5. Mechanical simulation results and thermal images. (a1)- (b4) Failure modes of the batteries for simulations and experiments under the hemispherical punch tests and rigid rod tests. (a5)- (b6) Thermal results under the hemispherical punch tests and rigid rod tests. (a7) and (b7) Temperature history of the battery under hemispherical punch tests and rigid rod tests.

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Journal of Power Sources 451 (2020) 227755

where γ is the fracture angle, α is the half angle between the jellyroll and the punch at the starting failure point, β is the angle between the fracture line and the vertical direction, Rp is the radius of the punch, and dB is the battery diameter. Before the battery is compressed, AO1 ¼ RP þ dB , and O1 O2 ¼ x is the displacement of the punch. After the battery is com­ pressed, AO2 ¼ AO1 x. By geometric relation, we obtain AC2 ¼ AO22 þ CO22 2⋅AO2 ⋅CO2 ⋅cos α,

Table 7 Fracture angle under different abuse conditions and SOC by simulation (the maximum fracture angle). SOC

0

0.2

0.4

0.5

0.6

Hemispherical Punch Rigid Rod

74.1� 69.6�

74.3� 70.4�

74.7� 69.7�

73.9� 69.7�

74.2� 69.7�

average prediction error of the failure displacement under the rigid rod test is less than 2.83%, and that for the hemispherical punch test is less than 5.10%. Under the hemispherical punch test, when a soft short-circuit occurs, in which the blue elements inside the red box fail in Fig. 5(a1), the local structure does not carry the load force, and the soft short-circuit failure occurs in the upper and middle parts of the battery. Fig. 5 (a5) shows the infrared image, where a temperature increase is observed in the upper part of the battery. Fig. 5 (a3) shows the battery cross-sectional, where an electrode layer failure occurs in the upper part of the section due to the external load. The stress concentration is generated near the punch, where the in-plane fracture [42] is the main failure mode, the temper­ ature slowly increases and the voltage slightly drops. When the load increases further, as shown in Fig. 5(a2) and (a6), interlayers fracture [42] occurs, the temperature sharply rises and the voltage rapidly drops. The motion of the electrode increases the contact between the anode and cathode and exacerbates the interlayers failure, which causes the battery to enter an irreversible hard short-circuit [14]. As shown in Fig. 5(a2), the fracture areas in the simulations are marked with red lines, and the fracture angle of the jellyroll under different SOCs is approximately 74� . In Fig. 5(a4), the fracture areas in the experiments are denoted with red lines, and the two main fracture angles are 73.4� and 72.5� , respectively. However, total layer failure does not occur in contrast with the simu­ lation results in Fig. 5(a2), because the impact of the battery mandrel is not taken into account in the simulation, and the mandrel limits the force to the jellyroll. Thus, interface fractures occur only in the upper part of the battery. Similarly, under the rigid rod test as shown in Fig. 5(b1), (b3) and (b5), the local in-plane failure is observed at the center and bottom of the battery. When the force further increases, the fracture spreads throughout the battery, and a hard short-circuit occurs. As shown in Fig. 5(b2), (b4) and (b6), which differs from the hemispherical punch test, the contact surface between the rigid rod punch and the battery is larger, and the mandrel does not have sufficient capacity to stop the load. At the bottom of the battery, the structure is damaged due to tensile stress, and the fracture angle under different SOCs is 69� . As shown in Fig. 5(a7) and (b7), consistent with the simulation, the first heating point and the maximum temperature point under the two abuse conditions are located near point P2. Under the hemispherical punch test, P2 is located below the punch, as shown in Fig. 5(a1), at the first failure point, the temperature increase firstly occurs here. Under rigid rod test, P2 is located on the bottom of the battery. As shown in Fig. 5(b7), although P2 has the maximum temperature, the temperature difference with the surrounding point is unclear because the metal shell has excellent conductivity. Thus, the thermal on the interplane fracture layers can quickly transmit to the surrounding, and the temperature field distribution is more uniform. With the internal electrochemical reaction, the reaction at point P2 simultaneously decreases. After that, because the bottom of the battery contacts the metal base of the uni­ versal tensile tester, the heat dissipation conditions are good, and the temperature of P2 is lower than the surrounding temperature.

β ¼ arccos

π

γ¼

2

AC2 þ AO22 CO22 2⋅AC⋅AO2

(21) (22)

β

Using Eq. (21)- (22), the battery fracture angles under different abuse conditions and SOCs compared with the simulation results are shown in Fig. 6(b). Under the same abuse conditions, the fracture angles in different SOCs tend to be consistent. Thus, the fracture angle is related to the load property and is not closely related to the SOC, the similar phenomenon has also been observed by Chung et al. [39] and Wang et al. [40] for the pouch cells and the prismatic cells. In the experiment, the battery is in the elastic stage when the compression displacement is less than 4 mm, and almost no failure oc­ curs before 4 mm. Therefore, we only consider the failure properties of the batteries when the load displacement is greater than or equal to 4 mm. In our previous research [34], the fd feature of the battery for failure prediction subject to mechanical abuse was discovered, as shown in Eq. (23). The force will change the nature at the corresponding displacement when the fd attains the extreme value, that is, the soft short-circuit caused by the mechanical abuse appears inside the battery. The fd model can be used for failure prediction and early warning application of the battery. 8 fd ðxÞ ¼ > > > > > > D1 þ D2 e > > > > > > > > > <

D1 D2 kx D1 v

þ 0

D22 kxe

ðx > 4mmÞ

kx D1 v

B v@D1 þ D2 e

12 kx D1 v

> > > > > D1 ¼ 33:89⋅SOC þ 57:45 > > > > > > > D2 ¼ 91:06⋅SOC þ 9807 > > > : k ¼ 1355⋅SOC þ 4258

C A (23)

In this paper, the extreme value of the x-SOC-fd surface is solved by the discrete numerical method, and the extreme value curve is solved by the maximum of the matrix row, as shown in Eq. (24)- (25). The dis­ cretization matrix is shown in Table 8, and the data are shown in Fig. 6 (c), � m 1 m ¼ 0; x 2 ½4; 10� (24) lim xSOC xSOC n 1 n

n→∞

lim ðSOCm

n→∞

SOCm 1 Þ ¼ 0; SOC 2 ½0; 0:8�

(25)

m m where xSOC is the displacement at SOCm . We set am;n ¼ fd ðxSOC Þ ¼ n n D1 D2 . SOCm kx

D1 þD2 e

SOC kxn m D1 v

þ

me D2 kxSOC n

02

B v@D1 þD2 e

n D1 v

SOC kxn m D1 v

12 C A

i The maximum value of fd under different SOC is max½f SOC � ¼ ai;maxi . d

4.2. Failure prediction model

i The corresponding displacement is x½maxðf SOC Þ� ¼ xmax . Thus, the xmax i i d under various SOC is shown in Table 9 and the a1 curve of Fig. 6(d). The corresponding displacement value of xmax is the predicted soft shorti circuit failure value. The projection of the extreme value curve a1 on the Displacement-

The simulation results under different abuse conditions and SOCs are analyzed, as shown in Table 7. The fracture angle under hemispherical punch tests is around 74� and that in the rigid rod tests is about 69� . The fracture angle model is shown in Fig. 6(a) and Eq. (21)- (22), 10

W. Wang et al.

Journal of Power Sources 451 (2020) 227755

Fig. 6. Fracture angle model and failure prediction model. (a) Fracture angle model diagram. (b) Fracture angle model result. (c) Failure surface. (d) Extreme value curve a1 in the failure surface. (e) Extreme value curve a2 in the displacement-SOC surface.

and that for the mechanical experiment, the results are shown in Table 10. The prediction and experimental soft short-circuit failure displacement with an increase in SOC shows a decreasing trend, which suggests that the performance of the battery that bears the external load decreases as the SOC increases. The battery becomes more brittle under the high SOCs, and the batteries are more prone to fail. The larger the

SOC surface is shown in curve a2 on Fig. 6(e). The curve a2 has the analytical form of xfailure ¼ ηSOC3 þ λSOC2 þ μSOC þ ϕ. By the param­ eter fitting, a2 is expressed as Eq. (26) for the Displacement-SOC surface, xfailure ¼

0:5276⋅SOC3

0:4568⋅SOC2

1:503⋅SOC þ 7:899

(26)

Compared with the soft short-circuit failure prediction displacement 11

W. Wang et al.

Journal of Power Sources 451 (2020) 227755

verification system will be designed according to the fd failure prediction method in chapter 4.3, and the actual failure of the battery will be tested.

Table 8 Discrete matrix of fd. SOC1 SOC2 SOCi

SOCm

x1

x2

x3

…

xmax i

…

xn

a1;1

a1;2

a1;3

…

a1;max1

…

a1;n

a2;1

a2;2

ai;1

ai;2

am;1

am;2

a2;3

…

a2;max2

…

4.3. Application of the failure prediction model

a2;n

ai;3

…

ai;maxi

…

ai;n

am;3

…

am;maxm

…

am;n

Fig. 7 shows the principle prototype and the installation effect of the failure warning verification system. In Figs. 7 and 1 is the universal tensile tester, 2 is the compressed battery sample, 3 is the infrared thermal cameral, 4 is the control computer, 5 is the warning system control computer DSP with the warning model, 6 is the data recorder to record the voltage and failure signal of the battery, 7 is the hardware of the warning system and 8 is the displacement sensor. The working principle of the failure warning verification system is that the Putter is connected to universal tensile tester. When the tester compresses the battery, the Putter pushes the sensor and moves with it, the Barrier Plate contacts the bottom of the sensor, which produces a relative motion and outputs the voltage signal. The DSP transforms the received voltage to a displacement signal and compares it with the xfailure prediction displacement. When the displacement reaches and exceeds the warning displacement, the DSP outputs a warning signal, and the voltage of the warning signal changes from low (0 V) to high (3 V), which indicates that the battery has attained a critical failure state. The results of the mechanical abuse experiments with the failure warning verification system are shown in Fig. 8. The pink dotted line

Table 9 xmax under various SOC. i SOC

SOC1

SOC2

…

SOCm

x

xmax 1

xmax 2

…

xmax m

SOC, the higher is the battery energy; as a result, the consequences of thermal runaway are more critical. The predicted failure displacement of soft short-circuit characterized by fd can better predict the actual soft short-circuit, and the average relative error is less than 5%. Compared with the failure prediction method that is based on voltage and temperature, the failure analysis based on mechanical penetration has the advantages of being instanta­ neous and no lag; thus, it is more suitable for the failure warning of a large capacity battery under mechanical abuse. The failure warning

Table 10 Failure prediction displacement and corresponding relative error by experiment and fd. SOC

0

0.2

0.4

0.5

0.6

Rigid rod test

Soft short-circuit prediction displacement by fd (mm) Soft short-circuit displacement by experiment (mm) Relative error of soft short-circuit failure displacement between fd and experiment

7.899 7.401 6.73%

7.576 7.515 0.81%

7.191 7.265 1.02%

6.967 6.773 2.86%

6.719 6.433 4.44%

Hemispherical punch test

Soft short-circuit prediction displacement by fd (mm) Soft short-circuit displacement by experiment (mm) Relative error of soft short-circuit failure displacement between fd and experiment

7.899 7.233 9.21%

7.576 7.183 5.47%

7.191 7.15 0.57%

6.967 6.916 0.74%

6.719 7.433 9.91%

Fig. 7. Principle prototype of the failure warning verification system (left) and experimental installation effect (middle and right). 12

W. Wang et al.

Journal of Power Sources 451 (2020) 227755

Fig. 8. Experimental results of the failure warning verification system. (a) Rigid rod test with SOC ¼ 0.2. (b) Rigid rod test with SOC ¼ 0.6. (c) Hemispherical punch test with SOC ¼ 0.2. (d) Hemispherical punch test with SOC ¼ 0.6. (*1) Infrared photos of the battery under thermal runaway. (*2) Infrared photos of the battery at the start of the soft short-circuit. (* is the symble of Fig. a, b, c and d).

13

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Journal of Power Sources 451 (2020) 227755

divides the results into three areas. STAGE I is the linear growth zone of the force; in this area, the properties of the battery will not change. However, in this area, we find that the increase of the force is not linear. This is because in the actual experiment to prevent the volatilization of the electrolyte we do not remove the battery metal case, the outer case of the cell first bares the compression force, the buckling of the battery case causes fluctuation of the force [18,19,35]. STAGE II is the soft short-circuit judgement zone; in this area, the resistance to the external force will produce a gradual softening phenomenon, a slight decrease in voltage and a slight increase in temperature. Therefore, failure warning in this area is suitable. STAGE III is the thermal runaway zone, the battery fails completely, the force decreases, the voltage sharply drops and the temperature quickly rises. In the experimental results, the blue solid line is the forcedisplacement curve. The blue dotted line is the first-order derivative of the force to displacement, which indicates the change of the carrying capacity of the battery to the external force. The green solid line rep­ resents the temperature curve, and the green dotted line represents the voltage curve. The red solid curve is the warning signal, the low point indicates safe and the high point is dangerous. On the Y axis, the value in the parentheses is the derivative value and voltage, and the values outside the parentheses are the force and temperature. As shown in Fig. 8, the displacement of the first-order derivative extreme value of the force to displacement is defined as a soft shortcircuit. Under the four working conditions, when the displacement reaches xfailure, a slight decrease in the voltage is expressed by the green dotted line. Due to the low thermal conductivity of the battery, a small increase in temperature is detected after a period of time and donate as a green line. The values of the warning displacement represented by xfailure are 7.576 mm and 6.719 mm in SOC ¼ 0.2 and SOC ¼ 0.6, respectively. As shown in Fig. 8, the warning signal is defined by a red solid line, and at the specified displacement where the soft short circuit occurs, the signal is converted from low to high, and the failure warning is trigged. The displacements of the soft short-circuit from mechanical force and voltage under rigid rod tests and hemispherical punch tests are 7.33 mm and 6.75 mm,7.55 mm and 6.542 mm, respectively. The errors of the actual soft short-circuit and prediction displacement does not exceed 3.5%, and the warning by xfailure is always earlier than that by the change of the temperature. Under dynamic load, the operating time of the single battery from contact external load to complete failure is generally less than 1 s, for high-speed collision, the entire failure process may be only a few mil­ liseconds, the battery safety warning in dynamic collision is still hard [6, 11,36]. Most of the studies are based on quasi-static conditions, the action time of the single cell can last from tens of seconds to hundreds of sec­ onds from contact external load to complete failure [4,34,37,38]. However, due to the physical characteristics of the battery, the lower thermal conductivity and during the soft short circuit the smaller voltage drop make the battery safety warning from the temperature and voltage has a certain lag. The principle prototype of the failure warning verifi­ cation system uses a DSP controller with the clock frequency of 150 MHz, the sampling frequency is 50 MHz, it can monitor and alert failure displacements in real time. The battery safety warning model based on mechanical penetration displacement can provide a warning time of no less than 20 s under quasi-static mechanical abuse conditions, and the warning signal is always ahead of the temperature and voltage signals. Thus, it has a certain safety margin which can gain time for the emer­ gency operation of the battery. When the prediction model is applied to the battery system, the displacement sensors can be fixed around the outermost batteries, the sensor probes detect the battery penetration displacements and output penetration displacements. The output signal line of the displacement sensor is connected to the bus of the Battery Management System (BMS), the BMS compares the received displacement signal with the warning value, when the displacements reach the warning value and the BMS

initiates fire protection measures. The type of displacement sensor is KTRC-20, the total length is 121 mm, the weight is less than 100 g and the price of each sensor is 104 yuan (CNY). Therefore, the large-scale use of this type of sensor does not significantly increase the cost of the battery system, and does not affect the energy density of the battery systems and the driving range of the vehicles. At present, the safety warning strategy of the battery by mechanical penetration displacement is based on the single cell level. In the next study, the inter-battery kinematic relationship and failure displacement signal of the battery module under mechanical abuse conditions will be studied, and propose the warning strategy to the battery module based on mechanical penetration displacement. 5. Conclusions In this paper, a safety performance and failure prediction model of the cylindrical lithium-ion battery is proposed, and the mechanical behavior is simulated accordingly. The safety performance model is based on the second-order oscillation feature. The average predictive error of the force-displacement and failure displacement between ex­ periments and simulations is less than 8.8% and 4.0%, respectively. Based on the analysis of the simulation results, the battery fracture angle model and the failure prediction model are deduced. A failure warning verification system is designed and manufactured. The prediction error of the warning system for the failure of the soft short circuit is less than 3.5%, and has a certain safety margin. In future studies, the research will focus on the battery pack and battery system level and discuss the relationship between the kinematics and the mechanics of the batteries. Combined with the mechanicalelectrochemical-thermal multi-physical failure characteristics, the fail­ ure warning model for the battery system based on mechanical pene­ tration should be proposed finally. The BMS of the pure electric vehicle with lithium-ion batteries will analyze the safety condition of the battery system according to param­ eters such as temperature, voltage, State of Health (SOH) and circuit condition of the battery. In the future, the failure warning model based on mechanical penetration displacement can be integrated into the BMS and perform safety warnings of the batteries under mechanical abuse to protect the safety of both drivers and passengers. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This work was supported by National Key R&D Program of China (No. 2017YFB0103801). Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi. org/10.1016/j.jpowsour.2020.227755. References [1] J.Q. Tian, Y.J. Wang, D. Yang, X. Zhang, Z.H. Chen, A real-time insulation detection method for battery packs used in electric vehicles, J. Power Sources 385 (2018) 1–9. [2] J.E. Zhu, T. Wierzbicki, W. Li, A review of safety-focused mechanical modeling of commercial lithium-ion batteries, J. Power Sources 378 (2018) 153–168. [3] T. Wierzbicki, E. Sahraei, Homogenized mechanical properties for the jellyroll of cylindrical Lithium-ion cells, J. Power Sources 241 (2013) 467–476. [4] W.W. Wang, S. Yang, C. Lin, Clay-like mechanical properties for the jellyroll of cylindrical Lithium-ion cells, Appl. Energy 196 (2017) 249–258.

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