Solid-State Battery Stack Pressure, Swelling and Ceramic Fracture

The compression path has total compliance

The plate input is the combined stiffness of both plates. Bolts and springs are internally parallel banks; these banks, plates, pads and stack act in series.

Lithium thickness is $h_{plated}=q_A/Q_V$, with $Q_V=2061.24$ mAh/cm³ (Wu et al., 2023, supplementary Note 1). Graphite and silicon use full-lithiation active-material volume increases 0.10 and 2.80 (Otero et al., 2018). Their volume-weighted blend, active fraction and pore accommodation determine external volume growth. Fixed footprint converts it directly to thickness; isotropic expansion takes the cube root. Cathode thickness change is linear in state of charge. This is a composite kinematic model inspired by Koerver et al., not a fit to their particular cells.

Let $s(z)$ be free stack swelling relative to assembly at the lower state of charge and $x$ the irreversible lithium closure:

Dormand–Prince integration uses relative tolerance $10^{-6}$ and restarts at each charge, hold, discharge and rest boundary. The effective constrained Norton law requires calibration; Duan et al., Eq. 1, supports the power-law form and exponent, but its bulk deviatoric-stress rate is not inserted as a stack-pressure rate. The initial closure is zero. Sampling includes every protocol corner, with 80 points per ramp and 40 per nonzero hold.

For each unsupported patch, $D=E t^3/[12(1-\nu^2)]$ and net load $q=f_{load}p$. The Navier solution includes 21 odd modes in each direction (indices 1…41), including twisting moments, on a 49 × 37 grid. Surface stress tensors follow $6M/t^2$. At depth $u\in[-1,1]$, add residual equibiaxial tension and retain the positive maximum principal tensile stress. Its Weibull power is integrated analytically through thickness and by trapezoidal quadrature in the plane. Supported regions retain residual tension only:

The fracture-qualified pressure is searched up to the user pressure ceiling. The effective ceiling also includes spring travel. The preload search first caps assembly pressure by remaining lithium inventory, then solves the lower and upper cycle-pressure constraints on that valid domain. Every candidate checks the complete sampled inventory history. Because closure is nondecreasing and lithium thickness is smallest at the final low-SOC endpoint, that endpoint is also the continuous history's global inventory minimum. Safeguarded secant steps use a bisection every third iteration; a boundary is returned only after its residual is checked on the feasible side. A failed search reports nonconvergence and returns no recommended interval. The reported preload interval rounds inward to 0.001 MPa; a narrower interval is not reported.

Uniform mean stack pressure and constant loaded footprint; quasistatic force equilibrium; linear incremental compliance of stack, plates, bolts, springs and pads. Assembly preload is applied at the lower state of charge before any modeled creep.

Identical layers and identical non-overlapping unsupported patches with simply supported edges and retained corner support. Uniform differential pressure over each patch; an isotropic homogeneous electrolyte; constant residual equibiaxial tension. Normal compression is not subtracted from in-plane tensile bending stress.

Volume-controlled Weibull flaws with a maximum-principal tensile-stress criterion and independent flaw populations across the stack. Identical cycling loads do not create independent new specimens. The probability is for the maximum load reached, not a cumulative cycle-life prediction.

Illustrative porous-LLZO defaults are E = 67 GPa, m = 6.4, characteristic strength 55 MPa at effective reference volume 7.5 mm³ (Yan, 2020, printed pp. 100–101). Fixture stiffnesses, differential load fraction, flaw coverage and effective creep rate are design examples, requiring measured replacements.

Use the plate result when the shorter unsupported span is at least ten electrolyte thicknesses, maximum deflection is below about 0.2 thickness, and patch aspect ratio is between 0.25 and 4. Transverse shear, large-deflection membrane action, partially clamped edges and elastic-foundation support need a different local model. The field is evaluated on 49 × 37 nodes; smooth colour shading blends between these solved values. Layer thicknesses and displacement in the first schematic are illustrative; swelling minus creep is magnified 20-fold and limited for layout.

Cycle extrema use dense sampling with exact protocol endpoints; extrema inside a ramp are located to the sampling resolution, not by a separate continuous optimization. The adaptive ODE retains its own smaller integration steps. Preload searches require a final pressure residual no larger than 0.00001 MPa on the feasible side, within 96 iterations per boundary. A target halfway inside this tolerance prevents numerical sign ambiguity. Inventory qualification retains a reserve of 0.1% of assembly lithium thickness, with a minimum total-stack reserve of 0.0000001 mm; its boundary residual must be within 5% of that reserve, again on the feasible side. This reserve protects the numerical search near complete depletion; it is not a material safety factor. The interval is conditional on the schedule, cycle count, stiffnesses and calibration. A single pressure limit is named only when it is the minimum; coincident ceilings within 0.000001 MPa are named together. The fracture ceiling is searched only up to the user pressure ceiling, so a clipped but nonbinding fracture value does not create a spurious tie.

The effective creep law represents constrained thickness relaxation at the calibration temperature and confinement. It does not resolve lithium’s local multiaxial stress, diffusion, hardening, void healing, extrusion paths, or loss of electrochemically available lithium. Do not extrapolate a bulk uniaxial creep constant directly to mean stack pressure. Exhausted local lithium inventory invalidates subsequent pressure predictions. Reaching the inventory reserve blocks all result cards for the selected case. The pressure history and instantaneous pressure are explicitly blank after the first detected exhaustion. An assembly stack thinner than its known ceramic plus selected-anode thickness is rejected; the entered total thickness must additionally allow for the actual cathodes, collectors and other internal layers. Host swelling is a reversible volume mixture with pore accommodation, not a particle stress or hysteresis model.

Weibull output is fast fracture for the entered flaw population and effective volume convention. Surface-flaw statistics, subcritical crack growth, fatigue, environmental attack and dendrite-driven fracture are excluded. The published porous-LLZO parameters do not characterize every LLZO or sulfide. No universal lower or upper operating pressure follows from Doux’s experiments. Bolts need separate proof-strength and fatigue checks; springs need separate stress, buckling and solid-height checks.