Simple Estimation of Mechanical Fatigue Life of Negative Electrode for Lithium-Ion Battery

Abstract

The macroscopic mechanical fatigue properties of negative electrodes in lithium-ion batteries and their estimation methods have been investigated based on a simple mechanical model. Tensile and bending fatigue tests were conducted on a negative electrode made of carbon powder and polyvinylidene fluoride (PVDF) binder. The simple model proposed in this study was used to estimate the stress and strain in the PVDF binder supporting the structure of the negative electrode. This model approximates the orientation of the carbon particles as the body-centered cubic (bcc) or the face-centered cubic (fcc), referring to the crystal lattice of metallic materials. The carbon particles in the model are bonded by the PVDF binder. The tensile fatigue test results showed that the negative electrode dissipated energy under the repeated loading, and the stress–strain curve showed hysteresis loops. The total dissipated energy of the binder obtained from the tensile fatigue test and the proposed simple model were used to estimate the mechanical fatigue life of the negative electrode with different binder ratios. The estimated S–N curve agreed with the mechanical fatigue life of the negative electrode with low binder ratios in the bending fatigue test.

 

This Paper was Originally Published in Japanese in J. Soc. Mater. Sci., Japan 73 (2024) 610–617.

Fig. 16 Results of plane bending fatigue test.

1. Introduction

Among the secondary batteries that can be used repeatedly, lithium-ion batteries have a high energy density per unit volume, making them ideal for reducing product size and weight. Therefore, lithium-ion batteries are widely used as power sources for portable electronic devices and electric vehicles. However, if a lithium-ion battery is damaged, the stored energy is instantly released, which increases the risk of battery smoke and fire [1]. Hao et al. investigated the mechanical properties of a battery as a bulk material under static loading with acoustic emission and digital image correlation [2], while Voyiadjis et al. used the finite element method [3]. Wang et al. investigated mechanical properties under impact loads using a compressive impact test and the finite element method [4].

The interior of a lithium-ion battery is composed of alternating layers of positive electrode, separator, and negative electrode. The separator is a porous material that is immersed in an electrolyte. Positive and negative electrodes are metal foils coated with electrode materials that conduct electrolytic reactions on their surfaces, producing electricity outside the battery [5]. The electrode material is composed of a bridged structure of a powdered active material held together by a polymeric binder. Reducing the amount of binder increases the number of pores within the electrode material as well as the contact area between the active material and electrolyte, thereby increasing the power generation efficiency of the electrode material while reducing its mechanical strength. Mechanical damage to the electrode material reduces the electrical performance of the lithium-ion battery [6]. In recent years, numerous studies have been conducted to link the mechanical properties of the electrode material to a decrease in the electrical performance of the lithium-ion battery [7–12]. These studies focused on the fact that the mechanical properties of electrode materials depend on the mechanical properties of the binder that supports the bridged structure.

On the other hand, when batteries are installed in mechanical structures, such as automobiles, that are constantly exposed to external loads, the vibration generated in the structure acts as a cyclic load on the electrode material [13]. However, there have been very few reports on the fatigue properties of electrode materials under cyclic loading because there is no established method for evaluating the cyclic stress generated in a randomly oriented binder inside the electrode material, and electrode materials with low binder concentrations are too low in strength to be fixed directly to the fatigue tester.

Previous studies have proposed a mechanical model to easily evaluate the stress generated in the binder and confirmed its validity by conducting a plane bending fatigue test, which can be performed with the electrode material coated on a metal foil [14, 15]. This mechanical model approximates the location of the active material particles using body-centered cubic (bcc) or face-centered cubic (fcc) lattices in relation to the crystal lattice of metallic materials. The active material particles are connected by a binder approximated as straight bars. The stress generated per binder inside the electrode material during the plane-bending fatigue test was estimated using this model. The estimated stress was equal to the tensile strength of the binder when the electrode material cracked in one cycle and one-half of the tensile strength of the binder when 10⁶ cycles were performed. The stress amplitude estimated by the model and the number of cycles at which the electrode material cracked showed a linear relationship in a single logarithmic plot.

In previous studies [14, 15], the effects of different binder concentrations on mechanical fatigue life were further investigated. In the tensile fatigue test for binder materials, it is difficult to obtain the fatigue property of the binder materials because the binder is significantly elongated before fracture, even if the load is applied repeatedly. Previous reports have not been able to predict the fatigue life of electrode materials based on the fatigue properties of the binder.

Therefore, in this study, a tensile fatigue test was conducted on electrode materials with a sufficiently high binder concentration for direct fixation to a testing machine, and the fatigue properties of the binder in the electrode materials were estimated using a mechanical model. Furthermore, we examined whether the fatigue life of the electrode materials with low binder concentrations could be predicted using the estimated fatigue properties of the binder. This paper describes the validity of the predicted results together with the results of the plane bending fatigue test. Although there are various types of electrode materials [16, 17], carbon-based negative electrodes, which are widely used as negative electrodes, were used as the specimens in this study.

2. Mechanical Model

Figure 1 shows a mechanical model that approximates the microscopic structure of the electrode material. The spheres in the figure represent the active material particles inside the electrode, and the straight bars represent the binders connecting the particles. In reality, active material particles are randomly located, and there are numerous methods to connect them. Previous studies [14, 15] proposed two structures to approximate the location of the active material particles: bcc and fcc lattices. In general, carbon-based negative electrodes for lithium-ion batteries consist of carbon powder as the active material and polymer binders such as polyvinylidene fluoride (PVDF) as the binder. Because the Young’s modulus of carbon is greater than 10 GPa and that of PVDF, the base material of the binder, is approximately 1 GPa, most of the deformation of the negative electrode is due to the binder. Therefore, all spheres in the figure are rigid.

Fig. 1

Microscopic mechanical models of electrode material in case of tensile direction ⟨110⟩.

As shown in Fig. 1, given the volume ratio α_a of the active material and the volume ratio α_b of the binder constituting the electrode material, the relationship between the normal stress σ of the electrode material as a bulk and the normal stress σ_t generated per binder can be expressed by eq. (1), and the relationship between the normal strain ε in the bulk of the electrode material and normal strain ε_t per binder can be expressed by eq. (2) as follows:

  

\begin{equation} \sigma = \beta_{\sigma} \cdot \frac{8\alpha_{b}}{\zeta N}\sigma_{t} \end{equation} (1)

  

\begin{equation} \varepsilon = \beta_{\varepsilon} \cdot \zeta \varepsilon_{t} \end{equation} (2)

  

\begin{equation} \zeta = \eta \left(\eta - 4\root 3 \of{\frac{3\alpha_{a}}{4\pi n}} \right) \end{equation} (3)

where η denotes a constant determined by the geometric relationship of the lattice, n represents the number of active material particles in the lattice, N denotes the number of binders inside the lattice, and β_σ and β_ε represent constants determined by the loading direction to the lattice. According to the results of the static tensile and creep tests conducted separately by the authors [18, 19], the upper limit of the stress–strain curves agreed with the bcc model subjected to tensile loading in the ⟨110⟩ direction (hereinafter referred to as the bcc⟨110⟩ model), and the lower limit of the stress–strain curves agreed with the fcc model subjected to tensile loading in the ⟨110⟩ direction (hereinafter referred to as the fcc⟨110⟩ model). Therefore, the bcc⟨110⟩ and fcc⟨110⟩ models shown in Fig. 1 were also used in this study. Table 1 lists the values of the constants η, n, N, β_σ, and β_ε for each model.

Table 1 List of constant values.

3. Experimental Procedure

3.1 Specimen

Table 2 lists the constituents of each specimen. Carbon powder with an average particle diameter of 5 µm and PVDF were used as the active material and binder, respectively. The powdered PVDF was dissolved in N-methyl-2-pyrrolidone and mixed with carbon powder to form an electrode slurry. The amount of PVDF was specified by the mass ratio of the carbon powder and was set to 20 and 30 wt% in the tensile fatigue test to increase the strength sufficiently to be mounted on a testing machine and to 6 wt% in the plane bending fatigue test, which is the same as the actual negative electrode.

Table 2 Constituent materials of specimen.

The dumbbell-shaped specimen shown in Fig. 2 was used for the tensile fatigue test. The electrode slurry was applied to aluminum foil coated with the mold release agent to the specified thickness using the film applicator and then heated and dried at 120°C for 30 min using the electric furnace. The dried slurry was then peeled off from the aluminum foil and cut into the dumbbell shape, as shown in Fig. 2.

Fig. 2

Dumbbell specimen for tensile fatigue test.

The rectangular specimen shown in Fig. 3 was used for the plane bending fatigue test. The electrode slurry was applied to a 0.05 mm thick C1220 copper foil to the specified thickness using the film applicator and then heated and dried at 120°C for 30 min using the electric furnace. After removing the rough heat from the specimen at room temperature in air, strain gauges (Kyowa Electronic Instruments Co., Ltd. KFGS-2-120-C1-16, gauge length 2 mm) were attached to points A to E on the copper foil surface shown in the figure to measure the normal strain in the longitudinal direction of the specimen that occurred during plane bending fatigue testing.

Fig. 3

Rectangular specimen for bending fatigue test. (online color)

To measure the volume ratio of the constituent materials, the mass and dimensions of the completed specimens were measured. The thickness t of the specimen shown in Fig. 2 and the thickness t_a of the electrode material shown in Fig. 3 are the actual dimensions of the completed specimens. The average values of the volume ratio of active material (carbon) and binder (PVDF) to the PVDF content of the specimen obtained from the mass and dimension measurements are listed in Table 3. The values in the table were used for the following calculations.

Table 3 Volume ratios of constituent materials.

3.2 Experimental procedure

Tensile fatigue tests were conducted using a Shimadzu EZ-LXHS compact tabletop testing machine with a load capacity of 1 kN. The test speed was set to 0.5 Hz, and the test was performed using the triangular wave with the stress ratio of zero under load control. It is more appropriate to use displacement control for comparison with the plane bending fatigue tests. However, in the preliminary displacement-controlled tensile fatigue tests, the permanent deformation of the specimens exceeded the minimum displacement in most cases, and the specimens buckled macroscopically. Because the specimens did not buckle in the plane bending fatigue test as no load was applied during unloading, it was deemed reasonable to conduct a tensile fatigue test with load control that did not cause the specimens to buckle. The load P applied to the specimen and the displacement u of the gripper were measured during the test. The maximum load was adjusted for each specimen such that data from 10 to 10⁵ fracture cycles could be evenly obtained. The number of fracture cycles was limited to 10⁵ because when the maximum load was reduced to achieve a higher number of fracture cycles, the area of the hysteresis loop in the load–displacement curve described below became too small to be measured with the specimens and testing machine used in this study. Consequently, cyclic load–displacement curves until fracture were obtained for 19 specimens with 20 wt% PVDF and 18 specimens with 30 wt% PVDF.

The plane bending fatigue test was performed using a custom-made plane bending fatigue testing machine, as shown in Fig. 4. The specimen was repeatedly pressed against the curved plate by running the motor at the rotational speed of 5–7 Hz. During the test, the strain amplitude Δε/2 applied to the surface of the electrode material coated on the copper foil was estimated using the strain gauges attached to the copper foil, and the macroscopic aspect of the electrode material was recorded using a video camera. The radius of curvature of the curved plate was either 50 or 70 mm for each specimen to obtain data evenly across the range of 1–10⁸ cycles, at which cracks occurred on the surface of the electrode material. The strain amplitude Δε/2 applied to the surface of the electrode material was calculated using the following equation derived from the bending deformation of the specimen, as shown in Fig. 5.

  

\begin{equation} \frac{\Delta \varepsilon}{2} = \frac{1}{2}\left(\varepsilon_{s,\text{max}} + \frac{t_{a} - t_{s}}{\rho + \Delta \rho} - \varepsilon_{s,\text{min}} \times \frac{t_{a} + t_{b} - \Delta \rho}{t_{s} + t_{b} - \Delta \rho} \right) \end{equation} (4)

where ε_s,max and ε_s,min denote the strain on the copper foil surface measured by the strain gauges under loading and unloading, respectively. t_a represents the thickness of the electrode material, t_b denotes the thickness of the copper foil (= 0.05 mm), t_s is the height of the measuring section of the strain gauge, ρ denotes the radius of curvature of the curved plate (50 or 70 mm), and Δρ represents the position of the neutral axis of the specimen. The height of the strain gauge measuring section t_s and the position of the neutral axis of the specimen Δρ were determined by the preliminary test in which the specimen was pressed against round bars with a diameter of 15–140 mm (radius of curvature 7.5–70 mm). The results for the three specimens subjected to preliminary tests are shown in Fig. 6. As shown in the figure, the equation for bending strain was fitted by the least-squares method, and t_s = 0.019 [mm] and Δρ = 0.025 [mm] were obtained, and these values were used to calculate eq. (4). The relationship between the number of cycles of plane bending and strain amplitude until crack initiation in the electrode material was obtained from 27 specimens at a motor speed of 5 Hz, two specimens at 6 Hz, and three specimens at 7 Hz.

Fig. 4

Plane bending fatigue testing machine.

Fig. 5

Schematic of deformation of specimen in bending fatigue test.

Fig. 6

Results of preliminary bending test.

4. Results and Discussions

4.1 Tensile fatigue test

Figure 7 shows a photograph of the specimen after the tensile fatigue testing. The specimens were macroscopically fractured at the neck. Once a macroscopic crack was initiated, the specimen fractured within one cycle, and no crack propagation was observed.

Fig. 7

Macroscopic observation.

Figure 8 shows a SEM photograph of the fracture surface of the specimen captured using the scanning electron microscope (Hitachi High-Tech Corporation SEM, SU8230). As indicated by the arrows in the figure, several binder pieces were observed to have been pulled and fractured. The tips of the fractured portions were thin and fully elongated. This indicates that the binder was gradually elongated and its cross-section became thinner as the tensile load was repeatedly applied. Furthermore, the sufficiently thin binder broke sequentially, leading to macroscopic fracture. On the other hand, no fatigue fracture initiation points or striations, which are observed in general metallic materials, could be observed. This would be ascribable to the fact that the electrode material had a structure with many internal pores, which were supported by a large number of randomly oriented binders. The binder that generated the maximum stress was also randomly determined, and the fracture initiation point and striation could not be clearly observed.

Fig. 8

SEM observation.

Figure 9 shows an example of cyclic load–displacement curves of the specimen with the maximum load of 4.40 N and 596 fracture cycles. As shown in the figure, at the same displacement u, the load value at unloading was always lower than the load value at loading, and the load–displacement curve showed a hysteresis loop. Permanent deformation remained at the load value of 0 N and increased with repeated loading. This indicates that the specimen eventually fractured with energy dissipation under cyclic loading. In this study, the mechanical model described in Section 2 was used to estimate the dissipated strain energy in the binder supporting the specimen structure as follows. As shown in Fig. 8, the length, orientation, generated stress, and timing of fracture of the binder inside the actual specimen were extremely random. Note that the dissipated strain energy and other values estimated in the following procedure correspond to the mean value for all the specimens.

Fig. 9

Example of load–displacement curve in tensile fatigue test. (PVDF 20 wt%, Load amplitude ΔP/2 = 2.20 N, Number of cycles to fracture N_f = 596 cycles)

First, using the shape of the load–displacement curve shown in Fig. 9 as a reference, the stress–strain curve of the specimen as a bulk at each cycle was approximated by a power series as in the following equation.

  

\begin{equation} \begin{split} &\text{In loading process:}\\ &\quad \varepsilon = \sum_{k = 1}^{n} C_{k}\sigma^{\frac{1}{k}} = \sum_{k = 1}^{n} C_{k}\frac{P^{\frac{1}{k}}}{W^{\frac{1}{k}}t^{\frac{1}{k}}}\\ &\text{In unloading process:}\\ &\quad \varepsilon = \sum_{k = 1}^{n} \{D_{k}(\sigma^{\frac{1}{k}} - \sigma_{\text{rev}}{}^{\frac{1}{k}}) + C_{k}\sigma_{\text{rev}}{}^{\frac{1}{k}}\} \\ &\quad\ = \sum_{k = 1}^{n} \frac{D_{k}(P^{\frac{1}{k}} - P_{\text{max}}{}^{\frac{1}{k}}) + C_{k}P_{\text{max}}{}^{\frac{1}{k}}}{W^{\frac{1}{k}}t^{\frac{1}{k}}} \end{split} \end{equation} (5)

where C_k and D_k denote constant coefficients, P represents the tensile load, W denotes the specimen width, t represents the specimen thickness, σ_rev denotes the stress at the beginning of the unloading, and P_max represents the maximum load for the cycle. If the elongation λ of the specimen is presumed to be the difference between the displacement u and the initial displacement of the cycle, the following equation is obtained by integrating eq. (5) in the tensile direction of the specimen, that is, in the x direction, as shown in Fig. 2.

  

\begin{equation} \begin{split} &\text{In loading process:}\\ &\quad \lambda = \sum_{k = 1}^{n} C_{k}P^{\frac{1}{k}}t^{- \frac{1}{k}} \int W^{- \frac{1}{k}}dx\\ &\text{In unloading process:}\\ &\quad \lambda = \sum_{k = 1}^{n} \{D_{k}(P^{\frac{1}{k}} - P_{\text{max}}{}^{\frac{1}{k}}) + C_{k}P_{\text{max}}{}^{\frac{1}{k}}\}t^{-\frac{1}{k}} \int W^{-\frac{1}{k}}dx \end{split} \end{equation} (6)

The specimen width W is a function of x because the specimen is dumbbell shaped, as shown in Fig. 2. Given the dimensions of the specimen, the integral of $W^{ - \frac{1}{k}}$ in eq. (6) can be obtained numerically: Using the least-squares method, the values of the coefficients C_k and D_k were obtained by fitting eq. (6) to the load–displacement curve. The fit was considered adequate up to n = 5. Once the coefficients C_k and D_k were obtained, the stress–strain curve for an arbitrary stress was drawn using eq. (5).

As an example, Fig. 10 shows the stress–strain curve of the 298th cycle in Fig. 9 obtained by the above calculation. The figure shows the stress–strain curve when the maximum load P_max (4.40 N) divided by the cross-section area of the fractured part (1.11 mm²) was presumed to be the maximum stress σ_max (3.96 MPa) and the stress σ_rev at the start of unloading was set between 60 and 100% of the maximum stress σ_max (2.38 to 3.96 MPa). Half of the maximum strain in this curve was the strain amplitude Δε/2, and the area enclosed by the hysteresis loop was the dissipated strain energy ΔU. Figure 11 shows the time history of the strain amplitude Δε/2 and dissipated strain energy ΔU for the specimen shown in Figs. 9 and 10 from the start of the test to the number of cycles of fracture. The stress σ_rev at the start of unloading was set to the maximum stress σ_max (3.96 MPa) at the fractured part of the specimen. This corresponds to the strain amplitude and dissipated strain energy in the fractured part of the specimen. The figure shows that the strain amplitude and dissipated strain energy remained approximately constant until the specimen fractured, except for the final cycle when there was no unloading process, although the tensile fatigue test was conducted under load control.

Fig. 10

Calculated stress–strain curve at 298th cycle. (PVDF 20 wt%, Maximum stress σ_max = 3.96 MPa, Number of cycles to fracture N_f = 596 cycles)

Fig. 11

Strain amplitude and dissipated energy at fracture site. (PVDF 20 wt%, Stress at start of unloading σ_rev = σ_max = 3.96 MPa, Number of cycles to fracture N_f = 596 cycles)

Because the strain amplitude Δε/2 and dissipated strain energy ΔU obtained in the above procedure are values for the bulk of the specimen, using eqs. (1) and (2), the strain amplitude Δε_t/2 and dissipated strain energy ΔU_t of the binder can be estimated as follows:

  

\begin{equation} \frac{\Delta \varepsilon_{t}}{2} = \frac{1}{\beta_{\varepsilon}\zeta} \cdot \frac{\Delta \varepsilon}{2} \end{equation} (7)

  

\begin{equation} \Delta U_{t} = \frac{N}{8\alpha_{b}\beta_{\sigma}\beta_{\varepsilon}} \cdot \Delta U \end{equation} (8)

Figure 12 shows the relationship between the strain amplitude Δε_t/2 and dissipated strain energy ΔU_t of binder for all the 37 specimens calculated using the above equations, separately for the PVDF content (20 and 30 wt%) and the mechanical model used in the calculation (bcc⟨110⟩ and fcc⟨110⟩). The plots in the figure show the average values for each specimen, excluding the final cycle. The stress σ_rev at the start of unloading was set between 50% of the maximum stress σ_max (= stress at the maximum cross-section area of the specimen) and 100% (= stress at the fractured part of the specimen) in 10% increments. This means that a total of 12 different calculations were obtained from a single specimen: 6 different stresses σ_rev at the start of unloading × 2 different mechanical models (bcc⟨110⟩ and fcc⟨110⟩). The figure shows that the dissipated strain energy of the binder increased monotonically with strain amplitude. The lower boundary of the dissipated strain energy was observed in the strain amplitude range of 0.0045 ± 0.002. Differences due to the mechanical model used in the calculations were not clearly observed because of the relatively large scatter in the test results. Therefore, the regression curve Δε_t/2 = f(log₁₀ ΔU_t) was obtained for all plots by the least-squares method, as shown in the figure. The regression curve fitted well with the power series up to the fifth order.

Fig. 12

Dissipated strain energy of PVDF binder.

$\left( \begin{array}{l} f(s) = 8.13 \times 10^{ - 5}s^{5} + 1.08 \times 10^{ - 3}s^{4} + 5.71 \times 10^{ - 3}s^{3} \\ \quad \quad \quad {}+ 1.61 \times 10^{ - 2}s^{2} + 2.64 \times 10^{ - 2}s + 2.62 \times 10^{ - 2} \\ \quad\ \delta = 0.002 \\ \end{array} \right)$

The sum of the dissipated strain energy of the binder until the specimen fractures is defined as the total dissipated strain energy. Figure 13 shows the relationship between the number of fracture cycles N_f and cumulative dissipated strain energy U_t of the binder for all 37 specimens. For calculating the dissipated strain energy, the stress σ_rev at the start of unloading was set to the maximum stress σ_max for each specimen. Thus, the total dissipated strain energy in the figure corresponds to the strain energy required to fracture the binder. The figure shows that the total dissipated energy linearly increased as the number of fracture cycles increased on double logarithmic plots. This tendency agreed with that observed for general materials [20, 21]. In addition, the scatter in the test results became relatively small by summing the results. This fact clarified the difference in the mechanical model used to calculate the dissipated strain energy, and the specimens with PVDF 20 wt% and those with PVDF 30 wt% were located on the same straight line. Therefore, the fatigue life of electrode materials with different binder concentrations can be predicted using the total dissipated strain energy as a criterion for fatigue failure of the electrode material. For example, in the plane bending fatigue test that provides constant strain amplitude as described in the next section, the following equation can be applied if the dissipated strain energy of the binder in each cycle is constant.

  

\begin{equation} U_{t} = N_{f} \times \Delta U_{t} \end{equation} (9)

Using eq. (9), the regression curves Δε_t/2 = f(log₁₀ ΔU_t) and U_t = cN_f^m (where c and m are constants) shown in Figs. 12 and 13 and eq. (2), the S–N curve of the bulk of electrode material can be predicted as follows:

  

\begin{equation} \frac{\Delta \varepsilon}{2} = \beta_{\varepsilon}\zeta \cdot f((m - 1)\log_{10}N_{f} + \log_{10}c) \end{equation} (10)

where function f(s) and constants c and m are set to the function in Fig. 12 and the values in Fig. 13, respectively.

Fig. 13

Total dissipated strain energy of PVDF binder.

4.2 Plane bending fatigue test

Figure 14 shows a photograph of the specimen after the plane bending fatigue testing. As indicated by the arrow in the figure, a macroscopic crack was observed vertically on the surface of the electrode material in the longitudinal direction of the specimen. This crack exhibited a brittle fracture, similar to that observed in the tensile fatigue test. The crack initiated during the loading process and propagated until the maximum bending deformation was reached, that is, until the arm and rod of the plane bending fatigue testing machine reached their lowest positions, as shown in Fig. 4. However, the cracks did not propagate during the subsequent unloading process or after the next cycle. Therefore, the number of cycles required for macroscopic cracks to appear on the surface of the electrode material was defined as the fatigue life N_f in the plane bending fatigue test.

Fig. 14

Macroscopic observation in plane bending fatigue test.

Figure 15 shows an example of the measured maximum strain ε_s,max and minimum strain ε_s,min on the copper foil surface and the strain amplitude Δε/2 on the electrode surface calculated by eq. (4) for each cycle. As shown in the figure, the maximum strain ε_s,max remained constant until the crack initiated, while the minimum strain ε_s,min gradually increased with plastic deformation of the copper foil in the initial stage of repetition (N < 10), resulting in the gradual decrease in the strain amplitude Δε/2. However, in the specimens used in this study, the minimum strain ε_s,min stopped increasing after approximately 10 cycles, and thereafter, the minimum strain ε_s,min and strain amplitude Δε/2 remained almost constant. Therefore, it is presumed that the surface of the electrode material was subjected to the constant strain amplitude in the plane bending fatigue test, and the strain amplitude Δε/2 is hereinafter arranged as the average value of all cycles.

Fig. 15

Example of measured strain in plane bending fatigue test. (Number of cycles to fracture N_f = 8.34 × 10⁴ cycles)

Figure 16 shows the S–N plot obtained from the plane bending fatigue test and the S–N curve predicted using eq. (10). The figure also indicates the predicted curve corresponding to the deviation δ = 0.002 of the binder strain amplitude Δε_t/2, as shown in Fig. 12. The figure shows that the fatigue life tends to increase with a decrease in the strain amplitude applied to the electrode material in the plane bending fatigue test. It was also confirmed that the test results were not affected by the rotational speed of the motor in the testing machine within the scope of this study.

Fig. 16

Results of plane bending fatigue test.

Comparing the predicted S–N curves, the fcc⟨110⟩ model predicted shorter fatigue life than the bcc⟨110⟩ model. This is attributable to the fact that the value of c in eq. (10) was larger for the fcc⟨110⟩ model, as shown in Fig. 13; however, the value of ζ was smaller for the fcc⟨110⟩ model when the values listed in Tables 1 and 3 were substituted. Most of the S–N plots obtained from the plane bending fatigue test indicated longer fatigue lives than the predicted S–N curves, and half of them fell within the range up to the prediction curve of the bcc⟨110⟩ model corresponding to the binder strain amplitude Δε_t/2 + δ side, which is the longest fatigue life prediction. Because the S–N curve prediction using the results of the tensile fatigue test was slightly shorter and safer than the fatigue life in the plane bending fatigue test, the prediction using eq. (10) was valid for estimating the fatigue life of electrode material. There are two possible reasons why the prediction evaluates the safe side. First, the tensile test tends to fracture materials from their structurally weak part because the stress is uniform across the cross-section, while the tensile test tends to show lower strength than the bending test because cracks tend to initiate in the part where the maximum bending stress occurs. Second, the strain amplitude Δε/2 used to organize the plane bending fatigue test does not take into account the permanent strain in the binder, and the experimental strain amplitude Δε/2 is an overestimate. In particular, the latter is an issue to be addressed in the future, because detailed investigation of the factors affecting the amount of permanent strain in the binder can be expected to improve the prediction accuracy.

5. Conclusion

Tensile and plane bending fatigue tests were conducted using carbon-based negative electrode materials that are widely used as negative electrodes in lithium-ion batteries. The tensile fatigue test results indicated that the binder, which was superimposed on the electrode material, caused energy dissipation and resulted in macroscopic fracture of the electrode material. Based on these results, the S–N curve of the specimen with a low binder concentration was predicted to be on the safe side with respect to the plane bending fatigue test results.

Acknowledgments

This study was supported by JSPS KAKENHI (Grant Number JP19K04078) and Tokyo City University Prioritized Studies. The use of the SEM was supported by the Tokyo City University Interdisciplinary Research Center for Nano Science and Technology.

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