thermal-stress

name	thermal-stress
description	Thermal stress analysis — constrained expansion, bi-metallic strips, composite bars, thermal fatigue, creep at temperature, thermal shock resistance, CTE mismatch in assemblies.
metadata	{"priority":7,"promptSignals":{"phrases":["thermal stress","thermal expansion","CTE","temperature stress","thermal fatigue","bimetallic","constrained expansion"],"minScore":4}}

Thermal Stress Analysis — Complete Skill

Free Thermal Expansion (Unconstrained)

ΔL = α × ΔT × L [m or in] ε_T = α × ΔT (thermal strain, no stress produced)

Coefficients of thermal expansion α [10⁻⁶/°C]:

Material	α
Carbon steel	11.7
Stainless 304	17.3
Stainless 316	16.0
Aluminum 6061	23.6
Copper	17.0
Titanium	8.9
Inconel 718	13.0
Brass	19.0
Concrete	10-12
Glass (borosilicate)	3.3
PTFE	130
CFRP (fiber direction)	-0.5 to +1

Constrained Thermal Stress

Fully constrained (both ends fixed): ε_total = ε_thermal + ε_mechanical = 0 α × ΔT + σ/E = 0 σ = -E × α × ΔT (compressive if heated, tensile if cooled)

Partially constrained: δ_allowed = some prescribed deformation δ₀ σ = E × (α × ΔT - δ₀/L)

Design check: |σ| < Sy (no yielding at operating temperature) Remember: Sy decreases with temperature — use elevated temperature Sy

Composite (Multi-Material) Bar

Example: steel rod in aluminum jacket (parallel configuration) Compatibility: δ_steel = δ_Al (same total deformation) Equilibrium: P_steel + P_Al = P_external (or = 0 if no external load)

With temperature change ΔT: P_Al × L/(A_Al × E_Al) + P_steel × L/(A_steel × E_steel) = (α_Al - α_steel) × ΔT × L

Stress in each: σ_i = P_i/A_i

For multiple materials in series: Forces equal (P₁ = P₂ = P = total force) Deformations add: δ_total = Σ P×L_i/(A_i×E_i)

Bimetallic Strip (Thermostat)

Curvature induced by CTE mismatch: κ = 6(α₂-α₁)ΔT(1+m)² / [h(3(1+m)²+(1+mn)(m²+1/mn))] Where: m = t₁/t₂ (thickness ratio), n = E₁/E₂ (modulus ratio), h = total thickness

Tip deflection (cantilever bimetallic strip, length L): δ_tip = κ × L²/2

Curvature simplification (m≈1, n≈1): κ ≈ (α₂-α₁)ΔT × 6/h² (for equal thickness strips)

Thermal Fatigue

Cyclic ΔT → cyclic σ_thermal = E × α × ΔT (if constrained) Number of cycles to failure (Coffin-Manson): Δε_p = ε'_f × (2N_f)^c Where Δε_p = plastic strain range = σ_range/E - ε_elastic_range

Design for thermal fatigue:

• Minimize CTE mismatch
• Add thermal expansion joints / bellows
• Use materials with high thermal fatigue resistance (low E×α product)
• Slow thermal cycling rate (creep relieves stress)
• Thermal barrier coatings

Creep at Elevated Temperature

Significant above T_homologous = T/T_melt > 0.4: Steel: > ~400°C, Al: > ~150°C

Steady-state creep rate (Norton's law): ε̇_ss = A × σⁿ × exp(-Q/RT) n = 3-8 (metals), Q = activation energy, R = 8.314 J/(mol·K)

Larson-Miller parameter: P = T × (C + log(t_r)) [T in Rankine or Kelvin, t_r in hours] C ≈ 20-30 for many steels Use P to find rupture time at given σ, or σ at given life from rupture curve

Stress relaxation (constant total strain, stress decreases): σ(t) = σ₀/(1+(n-1)×A×σ₀^(n-1)×E×t)^(1/(n-1)) Common in bolted joints at temperature

Thermal Shock Resistance

Rapid ΔT → σ_surface = E × α × ΔT / (1-ν) (for thick body with restrained surface)

Thermal shock resistance parameter: R = σ_f × (1-ν) / (E × α) [MPa·K or ksi·°F] — higher is better

Figure of merit for thermal shock: R' = k × σ_f × (1-ν) / (E × α) [W/m] (with thermal conductivity — higher k = faster equalization)

Material	R (MPa·K)
Si₃N₄	850
SiC	750
ZrO₂	100
Al₂O₃	70
Steel	60-90
Aluminum	500
CFRP	1500+

CTE Mismatch in Electronic/Structural Assemblies

Interface stress: σ_interface = E_avg × (α₁-α₂) × ΔT × L/(2h) Where L = bond length, h = thickness Peel stress (out-of-plane): σ_peel = 0.5×σ_shear If |σ| > σ_adhesive → delamination

Common assembly mismatches: PCB (FR4, α=17ppm) + copper trace (α=17ppm) — well matched Ceramic substrate (α=6ppm) + solder (α=25ppm) — HIGH RISK Steel housing (α=12ppm) + Invar insert (α=1.5ppm) — check joint

Output

Provide: σ_thermal [MPa/ksi], ΔL [mm/in], check vs. Sy at operating temperature, creep life if applicable, design recommendation (expansion joints, material change, coating).