To circumvent the prohibitive computational cost of resolving grain-scale heterogeneities in particulate composites, each doped sand configuration was modeled as a homogeneous continuum within an EMT framework implemented in COMSOL Multiphysics 6.4. The macroscopic transport properties of each sand-dopant mixture were derived from volumetric mixing rules. The effective density, mass-averaged isobaric heat capacity, and parallel-weighted thermal conductivity of the composite bed are expressed as a coupled property set:

(3) $$ \rho_{\text {eff }}=\phi_{s} \rho_{s}+\phi_{d} \rho_{d}, C_{p, \text { eff }}=\frac{\phi_{s} \rho_{s} C_{p, s}+\phi_{d} \rho_{d} C_{p, d}}{\rho_{\text {eff }}}, k_{\text {eff }}=\phi_{s} k_{s}+\phi_{d} k_{d} $$

where subscripts s and d denote the quartz sand matrix and dopant phase, respectively, and ϕ_i is the volumetric fraction of constituent i subject to ϕ_s + ϕ_d = 1. The volumetric thermal energy storage density of the composite, accounting for both sensible and latent contributions, is:

(4) $$ e_{\text {vol }}=\rho_{\text {eff }} C_{p, \text { eff }} \Delta T+\phi_{d} \rho_{d} L_{d} \xi(T), \xi(T) \in[0,1] $$

where L_d is the mass-specific latent heat of the dopant (non-zero only for phase-change constituents) and ξ(T) is the liquid fraction evolving according to the melting front progression. The Biot number characterizing the relative magnitude of internal conduction resistance to external convective resistance at the TEG interface is given by:

(5) $$ \mathrm{Bi}=\frac{h_{\mathrm{TEG}} L_{c}}{k_{\text {eff }}}, L_{c}=R_{o}-R_{i} $$

where h_TEG is the effective thermal contact conductance at the TEG module surface and L_c is the characteristic radial length of the annular storage domain.

Five candidate dopants were selected to span the principal thermophysical mechanisms by which solid-state additives modify packed-bed thermal performance: latent heat absorption, thermal conduction enhancement, and volumetric heat capacity augmentation. The selection drew on an initial screening of twelve candidate materials [Table 2], from which five were retained on the basis of thermal figure-of-merit, chemical compatibility with silica at temperatures up to 700 K, commercial availability, and cost^[30].

The composite index in Figure 3B is evaluated as a weighted sum of six normalized criteria. Thermal conductivity received the highest weight (0.25) because radial heat delivery to the TEG surface is governed by the effective conductivity of the annular composite layer, making it the rate-limiting property in this cylindrical geometry. Volumetric heat capacity was assigned a weight of 0.20 to reflect the sensible thermal energy accumulated over the 5 h charging window. Latent heat content, volumetric energy storage density, and chemical compatibility with the silica matrix were each weighted at 0.15. These three criteria are of comparable importance but operate through distinct mechanisms: latent heat governs isothermal buffering capacity at the phase-change transition, volumetric energy density integrates sensible and latent contributions into a single delivery metric relevant to the finite charging window, and chemical compatibility imposes a stability ceiling that no performance advantage can offset. Thermal diffusivity was assigned the lowest weight (0.10); as the ratio of conductivity to volumetric heat capacity, it carries information partly captured by the two highest-weighted terms, but its inclusion retains sensitivity to fast-transient thermal response that would otherwise be suppressed in a score dominated by steady-state properties.

Each criterion was normalized to [0, 1] by min-max scaling over the twelve candidates: $$ \tilde{x} $$ = (x - x_min)\/(x_max - x_min), with chemical compatibility treated as binary. A one-at-a-time weight perturbation of ±50%, with remaining weights rescaled to preserve unity, left Solar Salt, flake graphite, copper turnings, and SiC among the top five in all twelve variants, with Kendall rank correlations against the baseline exceeding 0.85 throughout, confirming that the selection is not sensitive to the precise weight assignments within physically reasonable bounds.

Solar Salt (60 wt% NaNO₃, 40 wt% KNO₃) was retained as the sole PCM, chosen for its well-characterized latent heat (L_d ≈ 100 kJ kg^-1 and a solid-liquid transition temperature (T_m ≈ 493 K) falling within the anticipated TEG operating window^[23,24]. The isothermal buffering effect of this plateau is evident in Figure 4A. Flake graphite (k_d ≈ 150 W m^-1 K^-1) and copper turnings (k_d ≈ 400 W m^-1 K^-1) occupy the high-conductivity end of the design space, targeting aggressive suppression of bed thermal resistance^[22,31]. The superior thermal delivery of these high-conductivity formulations relative to the pure sand baseline is also reflected in Figure 4A. Iron filings were included as a high-volumetric-heat-capacity additive (ρ_dC_p,d ≈ 3.5 MJ m^-3 K^-1) to evaluate whether enhanced sensible storage density can offset modest conductivity improvements^[32]. Silicon carbide (SiC, k_d ≈ 120 Wm^-1 K^-1) probes an intermediate regime, offering simultaneous elevation of conductivity and superior chemical inertness at elevated temperatures relative to metallic or carbonaceous fillers^[21,33]. The contrast in radial heat distribution between the pure sand baseline and the Solar Salt composite at t = 5 h is illustrated in Figure 4B and C, respectively.

Seven additional candidate materials were evaluated during the screening phase but ultimately excluded [Table 2]. Alumina (Al₂O₃) offers excellent chemical stability but its thermal conductivity at the grain scale (k ≈ 30 W m^-1 K^-1) provides insufficient enhancement over quartz sand to justify inclusion^[6]. Magnetite (Fe₃O₄) possesses favorable volumetric heat capacity (ρC_p ≈ 3.6 MJ m^-3 K^-1) but undergoes irreversible oxidation to hematite (Fe₂O₃) above 570 K in ambient atmosphere, compromising long-term property stability^[34]. Steatite, a magnesium silicate ceramic, was excluded for its low thermal conductivity (k ≈ 3 W m^-1 K^-1) despite high volumetric heat capacity^[35]. Aluminium turnings, although highly conductive (k ≈ 237 W m^-1 K^-1), melt at 933 K and suffer progressive surface oxidation that generates insulating Al₂O₃ layers and degrades interfacial contact conductance^[36]. Boron nitride (hexagonal, k_⊥ ≈ 30 W m^-1 K^-1, k_∥ ≈ 400 W m^-1 K^-1) was excluded because of the strong anisotropy in its thermal conductivity, which renders effective medium predictions unreliable without explicit microstructural resolution^[37]. Calcium chloride hexahydrate (CaCl₂·6H₂O) melts at 302 K, far below the target operating window^[36]. Sodium hydroxide (NaOH, T_m ≈ 596 K) presents severe corrosion risk to both quartz sand and metallic containment at the temperatures of interest^[38].

The dopant volume fraction ϕ_d was parameterized at 0.10, 0.20, and 0.30 via a global parameter sweep, yielding 30 distinct configurations (5 dopants × 3 fractions × 2 architectures). Two loading architectures were investigated. In the uniform configuration, the entire sand domain was assigned spatially invariant ρ_eff, C_p,eff, k_eff corresponding to the target ϕ_d. In the layered (functionally graded) configuration, the cylindrical domain was partitioned into two concentric sub-domains: an inner core (0 ≤ r ≤ R_c) retaining pure quartz sand, and an outer annulus (R_c < r ≤ R_o) assigned the composite material properties. At the internal interface r = R_c, temperature continuity and heat flux conservation were enforced:

(6) $$ T^{-}\left(R_{c}, t\right)=T^{+}\left(R_{c}, t\right),-\left.k_{s} \frac{\partial T}{\partial r}\right|_{r=R_{c}^{-}}=-\left.k_{\text {eff }} \frac{\partial T}{\partial r}\right|_{r=R_{c}^{+}} $$

The transient temperature field within each homogeneous sub-domain is governed by the energy equation in cylindrical coordinates:

(7) $$ \rho_{\text {eff }} C_{p, \text { eff }}^{*}(T) \frac{\partial T}{\partial t}=\frac{1}{r} \frac{\partial}{\partial r}\left(r k_{\text {eff }} \frac{\partial T}{\partial r}\right)+\frac{\partial}{\partial z}\left(k_{\text {eff }} \frac{\partial T}{\partial z}\right)+\dot{Q}(r) $$

where $$ \dot{Q}(r) $$ denotes the volumetric heat generation rate localized at the central resistive heating element. For PCM-doped configurations, the apparent heat capacity method was employed to handle the phase-change nonlinearity, replacing C_p,eff with an augmented capacity that incorporates the latent heat release through a Gaussian-smoothed liquid fraction derivative:

(8) $$ C_{p, \text { eff }}^{*}(T)=C_{p, \text { eff }}+\frac{\phi_{d} \rho_{d} L_{d}}{\Delta T_{m} \sqrt{2 \pi}} \mathrm{exp}\left[-\frac{\left(T-T_{m}\right)^{2}}{2\left(\Delta T_{m}\right)^{2}}\right] $$

where ΔT_m = 5 K is the half-width of the transition interval centered at T_m. A uniform initial condition T(r,z,0) = T₀ = 300 K was imposed throughout the domain, and the outer radial boundary was treated as adiabatic ($$ \partial T /\left.\partial r\right|_{r=R_{o}}=0 $$) to isolate the intrinsic storage dynamics of each composite from external dissipation.

All simulations were carried out in COMSOL Multiphysics 6.4 using the Heat Transfer in Solids module. The computational domain was constructed as a 2D axisymmetric geometry comprising a cylindrical annulus bounded by the inner heater radius R_i and the outer casing radius R_o. For the layered architecture, a concentric partition at r = R_c divided the domain into two material sub-domains, each assigned distinct effective properties through Equation (1). The geometry was discretised with a mapped quadrilateral mesh refined in the radial direction near both the heater interface and the material discontinuity at R_c, where the steepest temperature gradients were anticipated. A mesh convergence study was performed on the pure sand baseline by successively halving the characteristic element size from 2.0 mm to 0.25 mm. The TEG-side temperature at t_f changed by 0.1 K between the two finest grids, below the 0.3 K threshold adopted as the convergence criterion, and the monotonic, asymptotic approach of the monitored temperature confirms that the solution lies within the grid-independent regime. The 0.5 mm mesh was retained for the production sweep because it reproduces the converged temperature to within 0.3 K at roughly one quarter of the degrees of freedom of the finest grid, giving the most efficient accuracy-to-cost balance for the 30-configuration study. Equivalent studies confirmed grid independence for the other two tiers: the interfacial heat flux changed by less than 0.4% beyond five boundary-layer sublayers, and the peak interfacial von Mises stress changed by less than 1% between the 60,000 and 120,000 element discretization. Across the figures, each plotted curve corresponds to a single grid-independent solution of the governing equations, and where an envelope is shown it quantifies the residual cycle-to-cycle variation of the stabilized cyclic solution. Confidence in the reported quantities therefore rests on the grid-independence levels established above and on the breadth of the parametric and sensitivity sweeps, which span the thirty composite configurations and the nine material pairings.

The time-dependent solver employed the backward differentiation formula (BDF) with adaptive time stepping. The initial time step was set to 0.1 s to resolve the early transient, and the maximum step was capped at 10 s to ensure adequate temporal resolution of the phase-change dynamics in the Solar Salt configurations. For those configurations, the apparent heat capacity formulation [Equation (6)] replaced the standard heat capacity; a segregated solver with Anderson acceleration was used to stabilize convergence during the latent heat interval, where the effective heat capacity exhibits a sharp peak near T_m. Convergence was declared when the relative residual of the temperature field fell below 10^-6 at every time step.

The 30 configurations (5 dopants × 3 volume fractions × 2 architectures) were executed through the COMSOL parametric sweep functionality. Unless otherwise stated, the thermophysical properties of the solid constituents (density, specific heat, and thermal conductivity of quartz sand and the solid dopants) were treated as temperature-independent and evaluated at the representative values listed in Table 2. Across the 300 K to 660 K range spanned by the simulations, the intrinsic conductivity and heat capacity of these solids vary by less than about 15%, which is small relative to the order-of-magnitude contrasts between dopant candidates that drive the comparative ranking. The constant-property assumption therefore leaves the relative performance ordering unaffected, although it may shift absolute terminal temperatures by a few kelvins. The two physically nonlinear responses were resolved with explicit temperature dependence: the latent contribution of Solar Salt through the apparent heat capacity formulation of Equation (8), and the thermoelectric leg properties α(T), ρ_e(T) and K(T) entering the cycle-averaged efficiency of Equation (12).

Each configuration was defined by a single global parameter vector [ϕ_d’ k_d’ ρ_d’ C_p,d’ L_d’] that propagated through the material property expressions and the apparent heat capacity definition. A 5 h charging window (t_f = 18,000 s) was simulated for each case, and the TEG-side boundary temperature T_TEG(t)≡T(R_o,t) was recorded at 1 s intervals as the primary output. Full-domain temperature field snapshots were exported at hourly intervals for the pure sand baseline and the optimal 30 vol% layered Solar Salt configuration, providing the full-field spatial record used for the radial heat-transport comparison reported in the Results.

To facilitate cross-dopant comparison, a dimensionless thermal enhancement ratio was defined:

(9) $$ \eta_{T}(t)=\frac{T_{\mathrm{TEG}}(t)-T_{0}}{T_{\mathrm{TEG}, \text { base }}(t)-T_{0}} $$

where η_T > 1 indicates enhanced thermal delivery to the TEG interface relative to the pure sand baseline. The temporal stability of the TEG hot-side temperature during plateau intervals was quantified by:

(10) $$ \sigma_{T}=\sqrt{\frac{1}{t_{f}-t_{p}} \int_{t_{p}}^{t_{f}}\left[T_{\mathrm{TEG}}(t)-\bar{T}_{\text {plateau }}\right]^{2} d t} $$

where t_p is the onset time of the plateau and $$ \bar{T}_{\text {plateau }} $$ is the time-averaged temperature over the plateau interval. For dopants whose loading admits a percolation-mediated conductivity transition, the practical upper bound on volume fraction was identified through the stationary point of the effective thermal diffusivity with respect to dopant fraction:

(11) $$ \frac{\partial}{\partial \phi_{d}}\left(\frac{k_{\text {eff }}}{\rho_{\text {eff }} C_{p, \text { eff }}}\right)=0 $$

The cycle-averaged TEG conversion efficiency, which accounts for the nonlinear dependence of thermoelectric performance on the instantaneous temperature differential, was evaluated as:

(12) $$ \bar{\eta}_{\mathrm{TEG}}=\frac{1}{t_{f}} \int_{0}^{t_{f}} \eta_{\mathrm{TEG}}\left[T_{\mathrm{TEG}}(t), T_{c}, \alpha_{\mathrm{TE}}(T), \rho_{e}(T), K_{\mathrm{TE}}(T)\right] d t $$

where α_TE(T), ρ_e(T), and K_TE(T) are the temperature-dependent Seebeck coefficient, electrical resistivity, and thermal conductance of the thermoelectric legs, respectively.

The interfacial thermal resistance between the composite storage medium and the TEG substrate was quantified through a dedicated 2D axisymmetric transient simulation incorporating phase-change dynamics, implemented in COMSOL Multiphysics 6.4. The simulation workflow is summarized in Figure 5.

The computational domain comprised four vertically stacked layers: the composite sand bed (incorporating the 30 vol% layered Solar Salt formulation identified in the preceding section^[39]), a compliant TIM layer of thickness d_TIM = 0.1 mm, an alumina (Al₂O₃) TEG substrate, and the TEG module body [Figure 5A]. The TIM layer was assigned a thermal conductivity k_TIM = 3.5 W m^-1 K^-1, representative of commercially available silicone-based gap pads used in power electronics packaging^[40]. The interfacial thermal resistance across this layer is expressed as:

(13) $$ R_{\mathrm{TIM}}=\frac{d_{\mathrm{TIM}}}{k_{\mathrm{TIM}}}+R_{\mathrm{c}, \text { upper }}+R_{\mathrm{c}, \text { lower }} $$

where R_c,upper and R_c,lower are the contact resistances at the upper (sand bed-TIM) and lower (TIM-substrate) interfaces, respectively. For the baseline simulation, these contact resistances were set to 5 × 10^-5 m² K W^-1 each, yielding a total interfacial resistance R_TIM ≈ 1.29 × 10^-4 m² K W^-1. The corresponding interfacial heat flux delivered to the TEG substrate under steady-state conditions is:

(14) $$ q_{\mathrm{TEG}}=\frac{T_{\mathrm{bed}}-T_{\mathrm{sub}}}{R_{\mathrm{TIM}}} $$

The contact resistances R_c,upper and R_c,lower in Equation (13) were assigned constant values of 5 × 10^-5 m² K W^-1, consistent with values reported for lightly clamped silicone-based gap pads against ceramic surfaces^[41]. In practice, contact resistance decreases with increasing contact pressure as interfacial asperities deform, with experimental correlations indicating a reduction of approximately 30%-50% as pressure increases from 0.5 MPa to 2.0 MPa for soft polymer-ceramic interfaces. The fixed-resistance assumption is therefore most representative of the reference condition at P_contact = 1.0 MPa. Across the parametric sweep, this introduces a bounded conservatism whose net effect on the identified optimum is modest, given that the sensitivity of η_del to R_c is second-order relative to its sensitivity to k_TIM in this regime. Incorporating a pressure-dependent contact resistance correlation into Equation (13) is recommended as a refinement prior to prototype testing.

The mesh strategy was designed to resolve the steep thermal gradients within and adjacent to the 0.1 mm TIM gap^[41]. A five-layer boundary layer mesh was deployed on both the upper and lower surfaces of the TIM domain, with a first-layer thickness of 2 μm and a geometric growth ratio of 1.5 (Figure 5A, inset). The maximum element size was constrained to 5 μm within the TIM layer itself, transitioning to a coarser mapped quadrilateral mesh in the bulk sand bed and substrate domains. A mesh independence study confirmed that further refinement of the boundary layer beyond five sub-layers altered the computed interfacial heat flux by less than 0.4%.

The thermal boundary conditions replicated the operating scenario of the sand battery during a single charging cycle [Figure 5B]. A time-dependent pulsed heat flux was applied at the bottom boundary to represent the resistive heating input:

(15) $$ q_{\text {in }}(t)=12000 \cdot\left(1-\exp \left(-\frac{t}{3600}\right)\right) \mathrm{W} \cdot \mathrm{~m}^{-2} $$

where the exponential ramp approximates the thermal inertia of the resistive heating element during start-up, reaching 95% of the steady-state flux of 12,000 Wm^-2 within the first hour. The top surface was subjected to a convective boundary condition with h = 150 W m^-2 K^-1, representing the combined natural convection and radiative exchange at the TEG cold side. The initial temperature was set uniformly to 300 K throughout all domains. The transient solver employed BDF, and the apparent heat capacity method [Equation (8)] was retained for the Solar Salt layer to capture phase-change nonlinearity^[42].

The computed 2D temperature field at t = 2.5 h is presented in Figure 5C. The thermal field reveals a pronounced temperature discontinuity across the TIM gap, with the sand bed side reaching approximately 520 K while the substrate side remains near 480 K, confirming that the 0.1 mm interfacial layer accounts for a temperature drop of approximately 40 K. This drop is comparable in magnitude to the temperature differential exploited by the TEG module itself, indicating that the interfacial resistance constitutes a non-negligible fraction of the total thermal pathway resistance. The fraction of stored thermal energy that is effectively deliverable to the TEG interface within the 5 h charging window was quantified by the thermal delivery efficiency:

(16) $$ \eta_{\mathrm{del}}=\int_{0}^{t_{f}} q_{\mathrm{TEG}}(t) d t / \int_{0}^{t_{f}} q_{\text {in }}(t) d t $$

For the baseline TIM configuration, η_del was computed to be 0.74, indicating that 26% of the input thermal energy remains trapped upstream of the interfacial barrier at the end of the charging cycle^[43]. A parametric sweep over k_TIM from 1.0 W m^-1 K^-1 to 10.0 W m^-1 K^-1 showed that increasing the TIM conductivity to 6.0 W m^-1 K^-1 raised η_del to 0.82, beyond which further gains were marginal.

The cyclic thermal loading imposed during repeated charge-discharge operation induces mechanical stresses at material interfaces where CTE values are mismatched. To evaluate the structural integrity and fatigue life of the TIM joint under service conditions, a one-way thermo-mechanical coupling analysis was performed in ANSYS Workbench 2024 R1.

A three-dimensional conformal mesh comprising approximately 500,000 hexahedral-dominant elements was generated over the full cylindrical assembly [Figure 6A]. Edge fillets of 0.5 mm radius were introduced at the perimeter corners of the TIM layer to suppress the geometric stress singularity that would otherwise arise at sharp bimaterial interfaces.

The coupling topology is shown in Figure 6B: the transient temperature history computed in the Transient Thermal module was mapped as an imported body temperature load onto the Static Structural module, enabling sequential resolution of the thermal and mechanical fields on a shared mesh. This sequential strategy avoids the convergence difficulties associated with fully coupled thermomechanical solvers while preserving the fidelity of the thermal driving force at each structural time step.

The mesh was locally refined at the TIM boundaries to capture the steep stress gradients associated with the coefficient of thermal expansion (CTE) mismatch between the composite sand bed (α_TES ≈ 11.5 × 10^-6 K^-1) and the alumina substrate ($$ \alpha_{A l_{2} O_{3}} $$ ≈ 7.2 × 10^-6 K^-1) [Figure 6C]. The thermally induced strain at the interface scales with the product of the CTE mismatch and the applied temperature differential:

(17) $$ \varepsilon_{\mathrm{th}}=\Delta \alpha \cdot \Delta T=\left(\alpha_{\mathrm{TES}}-\alpha_{\mathrm{Al}_{2} \mathrm{O}_{3}}\right) \cdot\left(T(t)-T_{\mathrm{ref}}\right) $$

where T_ref = 300 K is the stress-free reference temperature. For a peak interfacial temperature of 520 K, this expression yields ε_th ≈ 9.46 × 10^-4, corresponding to a thermal mismatch stress that concentrates at the perimeter edge of the TIM joint where the radial constraint is most severe.

The transient temperature field T(t) was imported as a nodal body temperature load onto the structural model [Figure 6C], and a static structural solution was obtained at each time step corresponding to the peak and valley temperatures of a representative charge-discharge cycle (300 K to 520 K to 300 K). The equivalent (von Mises) stress field exhibited pronounced concentration at the perimeter edge of the TIM layer, where the displacement magnification factor reached approximately 100 under the applied thermal loading [Figure 6D]. The maximum von Mises stress at this location was 48.3 MPa, which remains below the tensile strength of the silicone-based TIM (typically 55-65 MPa) but exceeds 74% of the yield threshold, placing the joint in the high-cycle fatigue regime^[44].

Fatigue life estimation was performed using the ANSYS Fatigue Tool with the Goodman mean-stress correction. The alternating stress amplitude σ_a and mean stress σ_m were extracted from the computed stress cycle at the critical perimeter location. The fatigue life N_f was evaluated from the modified Basquin relation:

(18) $$ \sigma_{a}=\sigma_{f}^{\prime} \cdot\left(1-\frac{\sigma_{m}}{\sigma_{\mathrm{UTS}}}\right) \cdot\left(2 N_{f}\right)^{b} $$

where $$ \sigma_{f}^{\prime} $$ is the fatigue strength coefficient, σ_UTS is the ultimate tensile strength, and b is the fatigue strength exponent of the TIM material. The computed fatigue life at the most critically stressed location was N_f ≈ 8,400 cycles. Assuming one full charge-discharge cycle per day, this corresponds to a projected service life of approximately 23 years, which exceeds the 20-year design horizon specified for the HEV infrastructure.

The 20-year horizon corresponds to N = 7,300 cycles at one cycle per day and reflects the typical asset depreciation period for community-scale off-grid infrastructure, it serves as the structural design criterion for the containment vessel. The 23-year figure (N_f ≈ 8,400 cycles) is a computed output of the Goodman fatigue model at the baseline interface condition, not an independently specified target, and its role is to confirm that a sufficient margin exists before the interface is driven toward higher conductivity in the optimization study of Section 3.2.

The thermal ratcheting deformation visible at the outer perimeter in Figure 6D remained below 0.12 mm per cycle, confirming that progressive plastic accumulation does not compromise the geometric stability of the TIM joint within the anticipated service window^[45].

To evaluate the global thermomechanical response of the storage vessel under sustained cyclic operation, a 2D axisymmetric simulation was constructed in COMSOL Multiphysics 6.4 using a one-way coupling between the Heat Transfer in Solids (ht) and Solid Mechanics (solid) modules. The computational domain comprised four concentric annular regions: the central heating element, the composite storage layer (inner pure-sand core and outer 30 vol% layered Solar Salt-sand composite), a ceramic insulation shell (alumina, 5 mm), and an outer metallic containment shell (stainless steel, 3 mm), as depicted in Figure 7A. All inter-domain boundaries were constructed with shared nodes to enforce displacement continuity^[46,47].

Each domain was assigned a linear elastic constitutive model with isotropic thermal expansion referenced to T_ref = 300 K. The CTE values span nearly two orders of magnitude: α_sand ≈ 0.5 × 10^-6 K^-1, α_salt ≈ 40 × 10^-6 K^-1 (liquid phase), α_ceramic ≈ 8.0 × 10^-6 K^-1, and α_steel ≈ 17.3 × 10^-6 K^-1. This mismatch, identified as the principal driver of interfacial degradation in multi-material TES structures^[48,49], governs the radial thermal stress at any interface between adjacent layers i and j:

(19) $$ \sigma_{r}^{i j}(t)=\frac{E_{i} E_{j}}{E_{i}\left(1-v_{j}\right)+E_{j}\left(1-v_{i}\right)}\left(\alpha_{i}-\alpha_{j}\right)\left[T\left(r_{i j}, t\right)-T_{r e f}\right] $$

where i and j denote the indices of the two adjacent material layers; E_i(E_j), ν_i(ν_j), and α_i(α_j) denote the Young’s modulus, Poisson’s ratio, and coefficient of thermal expansion of layer i (layer j), respectively; and r_ij is the radial coordinate of their shared boundary. The finite-element mesh incorporated five-layer boundary layer elements at each material interface with a first-layer thickness of 0.05 mm [Figure 7B], and the total element count was maintained between 60,000 and 80,000, which a convergence study confirmed to be adequate for the 2D axisymmetric domain^[50,51]. This boundary-layer resolution at each bi-material junction is necessary to capture the steep stress gradients arising from CTE mismatches that span nearly two orders of magnitude across adjacent layers. The temperature field T(r,t) was obtained from the transient heat transfer solution and mapped onto the structural domain at each time step through the built-in variable ht.T, completing the one-way coupling illustrated in Figure 7C. The cyclic thermal loading followed the boundary condition profile shown in Figure 7D: each cycle comprised a 5 h charging phase during which the pulsed heat flux [Equation 15)] was applied at the inner boundary, followed by a 5 h cooling phase with zero heat input and natural convective dissipation (h = 5 - 10 W m^-2 K^-1, T_amb = 300 K) at the outer shell surface.

The multiaxial stress state at each interface was quantified through the von Mises equivalent stress:

(20) $$ \sigma_{v M}=\sqrt{\sigma_{r}^{2}+\sigma_{\theta}^{2}+\sigma_{z}^{2}-\sigma_{r} \sigma_{\theta}-\sigma_{\theta} \sigma_{z}-\sigma_{z} \sigma_{r}+3 \tau_{r z}^{2}} $$

where σ_r, σ_θ, σ_z, and τ_rz are the radial, circumferential, axial, and shear stress components in the axisymmetric coordinate system. Ten complete charge-discharge cycles were computed explicitly, and the incremental plastic strain per cycle $$ \Delta \varepsilon_{p}^{(n)} $$ at the most critically stressed interface was monitored for convergence. The rate of strain energy density dissipation per cycle was estimated as^[52,53]:

(21) $$ \frac{d W}{d N}=\oint \sigma_{i j} d \varepsilon_{i j}^{p} \approx \sigma_{v M}^{{peak }} \cdot \Delta \varepsilon_{p} $$

where the integral is evaluated over one complete loading-unloading path at the critical material point. Extrapolation to the target design life of N = 7,300 cycles (20 years at one cycle per day) was performed by fitting the accumulated plastic strain data from the first ten cycles to a power-law ratcheting model:

(22) $$ \varepsilon_{p}^{a c c}(N)=\varepsilon_{p}^{(1)}+A \cdot N^{\gamma} $$

where $$ \varepsilon_{p}^{(1)} $$ is the plastic strain after the first cycle, and A and γ are regression constants determined by least-squares fitting in MATLAB, following the methodology validated by Hrifech et al.^[47] for packed-bed TES systems. It should be noted that the present continuum model does not resolve discrete particle-level rearrangement within the granular storage medium; such phenomena have been shown by discrete element analysis to contribute additional hoop stress on the containment wall^[29]. The continuum-scale results therefore represent a lower bound on the total mechanical demand.

A parametric sweep over nine candidate insulation-shell material combinations (alumina, mullite, and zirconia paired with AISI 304 stainless steel (304SS), carbon steel, and Inconel 718) was conducted to identify the pairing that minimized the peak interfacial von Mises stress and long-term ratcheting strain. The results are presented in the Results and Discussion section. The key parameters governing each of these three computational tiers are consolidated in Table 4, providing a unified reference for the simulation framework described above.