The integration of lithium-ion batteries as structural components within electric vehicle chassis, particularly in Cell-to-Pack (CTP) and Cell-to-Chassis (CTC) architectures, has fundamentally altered the mechanical interaction between the energy storage system and the vehicle platform. In these configurations, the lithium-ion battery contributes directly to the overall structural rigidity and load-bearing capacity of the chassis. Consequently, its mechanical properties are no longer merely a matter of internal performance but become critical parameters influencing vehicle dynamics, durability, and safety. A key mechanical characteristic that undergoes significant and dynamic change during operation is the battery’s stiffness. This variation stems primarily from the electrochemically induced volume changes of the active materials during lithium (de)intercalation, a phenomenon often termed “breathing.” A precise understanding and quantification of this dynamic equivalent stiffness is therefore paramount for optimizing battery pack design, ensuring reliable interfacial contact under cyclic loading, and accurately predicting the long-term structural behavior of the integrated system.
The mechanical state of a lithium-ion battery is intrinsically linked to its electrochemical state. During charge and discharge, lithium ions shuttle between the cathode and anode, causing reversible lattice expansion and contraction in the host materials. For instance, graphite anodes expand upon lithiation, while certain cathode materials like LiFePO₄ exhibit non-monotonic volume changes. This results in a net change in the thickness of the cell. When such a cell is constrained within a rigid or semi-rigid module, these volume changes manifest as significant swelling forces. The relationship between this constrained force and the corresponding thickness change defines the cell’s equivalent stiffness. This stiffness is not a constant material property but a dynamic parameter influenced by several operational and state variables. It can be expressed functionally as:
$$ k = f(C, SOC, F, SOH) $$
where \(k\) is the equivalent stiffness, \(C\) is the charge/discharge rate, \(SOC\) is the state of charge, \(F\) is the external compressive load or preload, and \(SOH\) is the state of health. This work focuses on developing and demonstrating a robust methodology for dynamically measuring this equivalent stiffness \(k\) as a function of \(SOC\) and \(F\) for a fresh (Beginning-of-Life, BOL) lithium-ion battery, providing a foundational framework that can later be extended to study the effects of \(C\) and \(SOH\).
Experimental Methodology for Dynamic Stiffness Measurement
The core principle of the proposed method involves the simultaneous or correlated measurement of a lithium-ion battery’s free expansion and its mechanical response under a known constraint. By comparing the free geometric change with the constrained stress-strain response, the intrinsic force-displacement relationship of the cell can be isolated. Two custom experimental fixtures were designed and employed for this purpose.
Apparatus Design and Instrumentation
1. Free Expansion Measurement Fixture: This apparatus is designed to measure the thickness change of a lithium-ion battery with minimal external constraint. It consists of an aluminum alloy frame with a central sliding rail. The pouch cell is gently held on its two largest faces by adjustable clamps that allow in-plane movement, thereby minimizing shear forces and primarily measuring expansion in the thickness direction. Two high-precision Linear Variable Differential Transformer (LVDT) displacement sensors are positioned on opposite sides to measure the thickness change dynamically during cycling. The average of the two readings compensates for any minor bending. Key specifications of the displacement sensor are summarized in Table 1.
| Parameter | Value |
|---|---|
| Output Voltage | 0 – 5 V |
| Displacement Range | 0 – 5 mm |
| Repeatability | ≤ 0.15 μm |
| Operating Temperature | 0 – 40 °C |
2. Constrained Pressure Measurement Fixture: This fixture is designed to apply a controlled preload to the lithium-ion battery and measure the resulting swelling force during electrochemical cycling. As shown in the schematic, it comprises upper and lower fixed support plates, a central movable plate, a load cell, and a displacement sensor. The test cell is placed between the upper fixed plate and the central movable plate. A high-accuracy load cell (LVDT type) is positioned between the central movable plate and the lower fixed plate to measure the force. A displacement sensor is fixed above the upper plate, with its probe contacting the central movable plate to measure its displacement, which corresponds to the thickness change of the constrained cell. The stiffness of the load cell itself, \(k_0\), is a known constant critical for later calculations. Specifications are detailed in Table 2.
| Parameter | Value |
|---|---|
| Output Voltage | 0 – 5 V |
| Force Range | 0 – 20 kN |
| Resolution | 20 N |
| Accuracy | ≤ 0.1 % |
| Sensor Stiffness, \(k_0\) | 21 N/mm |
The experimental object was a commercial 10 Ah lithium iron phosphate (LiFePO₄/Graphite) pouch lithium-ion battery, with nominal dimensions of 145 mm × 91 mm × 7.7 mm.
Testing Protocol and Derivation of Equivalent Stiffness
The measurement procedure is conducted in two sequential stages on an identical cell (or cells from the same batch at identical initial states).
Stage 1: Free Expansion Characterization. The lithium-ion battery is cycled within the free expansion fixture. The thickness change relative to a reference state (e.g., 0% SOC) is recorded as a function of SOC. Let \(\Delta x_1(SOC)\) denote this free expansion displacement, where positive values indicate thickness increase.
Stage 2: Constrained Cycling under Preload. The same type of lithium-ion battery, initially at the same reference state (0% SOC), is placed in the pressure fixture. A known initial preload \(F_0\) is applied via the tightening mechanism. This compresses the cell, and the resulting thickness reduction from its free-state reference thickness is measured by the displacement sensor as \(\Delta x_2\). Subsequently, the battery is electrochemically cycled while under this constraint. Throughout the cycle, the total force \(F(SOC)\) on the load cell and the displacement of the movable plate are recorded.
The displacement sensor reading in Stage 2, however, does not directly give the cell’s thickness change. It measures the movement of the central plate, which is affected by the compression of both the lithium-ion battery and the load cell. Since the load cell has a known linear stiffness \(k_0\), its compression \(\Delta x_3\) under a force change can be calculated. From force equilibrium, the change in force relative to the preload is borne by the load cell:
$$ F(SOC) – F_0 = k_0 \cdot \Delta x_3(SOC) $$
Thus, \(\Delta x_3(SOC) = (F(SOC) – F_0) / k_0\).
The actual compression of the lithium-ion battery under constraint, denoted \(\Delta x_{SOC, F}\), is then derived from a kinematic compatibility condition. The total movement of the plate accounts for the battery’s compression from its free-state thickness. The relationship is:
$$ \Delta x_1(SOC) + \Delta x_2 – \Delta x_3(SOC) = \Delta x_{SOC, F} $$
Here:
- \(\Delta x_1(SOC)\): How much the free cell would have expanded at this SOC.
- \(\Delta x_2\): The initial static compression from the preload \(F_0\).
- \(\Delta x_3(SOC)\): The additional compression of the load cell due to swelling force change.
- \(\Delta x_{SOC, F}\): The net compression of the battery cell relative to its free state at SOC under the current total force \(F(SOC)\).
Finally, the equivalent dynamic stiffness \(k\) of the lithium-ion battery at a specific SOC and force \(F\) is defined as the ratio of the total force exerted by the cell to its net compression:
$$ k(SOC, F) = \frac{F(SOC)}{\Delta x_{SOC, F}} $$
This value represents the instantaneous stiffness of the battery pack element under the given electrochemical and mechanical state.
For this study, the focus was on characterizing the BOL stiffness. The lithium-ion battery was tested under three different initial preloads \(F_0\): 800 N, 1200 N, and 1800 N. To isolate the effects of SOC and force from significant thermal artifacts, a low constant current discharge rate of 0.3C was primarily used for stiffness derivation. The detailed test matrix is shown in Table 3.
| Test Group | Initial Preload \(F_0\) (N) | Primary Discharge Rate for Stiffness Analysis |
|---|---|---|
| Free Expansion | 0 | 0.3C |
| 1 | 800 | 0.3C |
| 2 | 1200 | 0.3C |
| 3 | 1800 | 0.3C |
Results and Discussion
Free Expansion Behavior of the Lithium-ion Battery
The thickness change of the unrestrained lithium-ion battery during a 0.3C discharge is plotted against Depth of Discharge (DOD = 1 – SOC) in Figure 1. The curve reveals a complex, non-monotonic behavior. The thickness initially decreases in the early stage of discharge (high SOC), then increases to a local maximum around mid-DOD, before decreasing again as the cell approaches full discharge. This “bell-shaped” profile is characteristic of lithium-ion batteries using LiFePO₄ cathode and graphite anode. It results from the competing volume changes of the two electrodes: graphite contracts continuously as lithium deintercalates during discharge, while LiFePO₄ expands as lithium intercalates. The non-linear expansion profile of the LiFePO₄ cathode, particularly its more pronounced expansion in the mid-SOC range, dominates the net cell thickness change in that region, leading to the observed peak. This free expansion data, \(\Delta x_1(SOC)\), serves as the essential baseline for all subsequent constrained stiffness calculations.
Swelling Force Evolution under Constraint
The swelling force measured during 0.3C discharge for the three different initial preloads is presented in Figure 2. A critical observation is that the *trend* of force evolution with DOD is remarkably consistent across all preload levels. All curves follow a similar trajectory: an initial force drop, followed by a rise to a peak in the mid-DOD range, and a final decrease towards the end of discharge. This trend is the direct mechanical consequence of the free expansion profile being resisted by the fixture. When the cell tries to expand (mid-DOD), the constraint generates a higher swelling force; when it tries to contract (high and low DOD), the force decreases. The absolute force level is naturally shifted vertically according to the initial preload \(F_0\).
This consistent pattern confirms that the source of the force variation is the intrinsic electrochemical volume change of the lithium-ion battery materials, not an artifact of the preload magnitude. The preload essentially sets the operating point on the cell’s non-linear force-compression curve.
Dynamic Equivalent Stiffness of the Lithium-ion Battery
Using the methodology outlined, the equivalent compression \(\Delta x_{SOC, F}\) was calculated for each data point. The results for the 0.3C discharge are plotted in Figure 3, showing how the net compression of the cell varies with DOD for the different preloads. Notably, the compression is lowest (meaning the cell is “thickest” relative to its free state) in the mid-DOD region where the swelling force peaks, indicating a strong coupling between force and displacement.
Applying the stiffness formula \(k = F / \Delta x_{SOC, F}\), the dynamic equivalent stiffness was computed. To visualize the continuous relationship between stiffness \(k\), SOC, and force \(F\), the discrete data points were interpolated to generate a stiffness contour map, shown in Figure 4. The colormap represents the magnitude of the equivalent stiffness in N/mm.
The analysis yields several significant findings regarding the mechanical behavior of this lithium-ion battery:
- Stiffness Magnitude and Range: Under the investigated preload range of 800 N to 1800 N, the equivalent stiffness of the fresh lithium-ion battery varied approximately between 13,550 N/mm and 27,550 N/mm. This order of magnitude is critical for mechanical engineers integrating cells into structural systems.
- Dependence on Preload/Force: A clear positive correlation exists between the applied force (or initial preload) and the measured stiffness. At higher compressive forces, the lithium-ion battery exhibits greater resistance to further deformation. This is characteristic of many porous, multi-layer materials where internal components (electrodes, separators) are progressively compacted.
- Dependence on State of Charge (SOC): The most striking observation is the strong dependence of stiffness on SOC. The lithium-ion battery is significantly stiffer at low SOC (high DOD) and softer at high SOC. This can be explained by the location of lithium ions. At low SOC (discharged state), most lithium resides in the cathode (LiFePO₄). The cathode material, especially in its lithiated state, is inherently denser and mechanically stiffer than the graphite anode. When the cell is mechanically loaded in this state, the stiffer cathode bears more of the stress, leading to a higher overall cell stiffness. Conversely, at high SOC, lithium is stored in the graphite anode, which is a less stiff material, resulting in a more compliant cell response.
- Engineering Implication for Constant Stiffness Design: The contour plot reveals that following a constant-stiffness trajectory during operation would require a specific force-SOC path. For example, maintaining a stiffness of ~20,000 N/mm would require starting with a high preload at 100% SOC and then systematically *reducing* the applied force as the cell discharges. This counter-intuitive strategy highlights the complex interplay and is vital for designing adaptive battery housings or compliant interfaces that can maintain optimal pressure and potentially extend cell life.
The relationship can be qualitatively summarized by extending our initial function for the BOL, low-rate case:
$$ k(SOC, F) \approx g(SOC) \cdot h(F) $$
where \(g(SOC)\) is a decreasing function and \(h(F)\) is an increasing function.
Conclusion and Future Perspectives
This work has established and demonstrated a practical methodology for the dynamic measurement of the equivalent stiffness of a lithium-ion battery. By independently characterizing free expansion and constrained swelling force, the method successfully decouples the electrochemical strain from the mechanical stress-strain response, enabling the calculation of stiffness as a dynamic function of SOC and applied force. The results unequivocally show that the equivalent stiffness of a commercial LiFePO₄ lithium-ion battery is not a fixed property but varies substantially—by over a factor of two within the typical operational range—depending on its state of charge and the mechanical preload it experiences. The cell is stiffer when discharged (lithium in the cathode) and under higher pressure.
This knowledge has direct and profound implications for the safety and design of advanced battery systems. In structural battery applications (CTP, CTC), ignoring this dynamic stiffness can lead to inaccurate predictions of pack rigidity, fatigue life of connections, and load distribution during events like impacts. Furthermore, for prismatic and pouch cells that require a controlled preload for optimal performance, understanding the \(k(SOC, F)\) relationship is key to designing spring-loaded or compliant housings that can maintain an ideal contact pressure window throughout the entire SOC range, potentially mitigating degradation mechanisms like particle cracking and solid electrolyte interphase (SEI) growth.
The presented framework is foundational. Future work should systematically expand upon it to create a comprehensive stiffness model for the lithium-ion battery lifecycle:
- Rate Dependency (\(C\)): High C-rate tests will introduce thermal effects and possible lithiation gradients. Measuring stiffness under these conditions is essential for understanding performance during acceleration, regenerative braking, and fast charging.
- State of Health Dependency (\(SOH\)): The most critical extension is to track how stiffness evolves with aging. Irreversible processes like gas generation, SEI growth, and particle disconnection will likely alter the force-compression relationship. Mapping \(k(SOC, F, SOH)\) could provide a novel mechanical health indicator for battery management systems.
- Geometry and Chemistry: Applying this method to different form factors (prismatic, cylindrical) and cathode chemistries (NMC, NCA) will build a valuable database for mechanical design across the spectrum of lithium-ion battery technologies.
In conclusion, dynamically assessing the equivalent stiffness of lithium-ion batteries is a crucial step towards designing safer, more reliable, and longer-lasting battery packs for electric vehicles and energy storage systems. The method outlined here provides a clear path to obtaining this critical mechanical property, bridging the gap between electrochemistry and structural engineering.