Thermal Expansion & Mismatch

Bi-Material Strip — Timoshenko Curvature

Two bonded layers bend under ΔT. κ from the 1925 Timoshenko solution; strip drawn to its true (exaggerated) arc.

Layer 1 (top) material

α₁ (ppm/K)

E₁ (GPa)

h₁ (mm)

Layer 2 (bottom) material

α₂ (ppm/K)

E₂ (GPa)

h₂ (mm)

Length L (mm)

ΔT (K): 100

Solder Joint Shear Strain & Coffin-Manson Life

γ = (α_sub−α_die)·ΔT·D / (2h); N_f = C·(Δγ)^−k. Curve shows life vs distance-to-neutral-point (DNP).

Die material

Substrate material

α die (ppm/K)

α sub (ppm/K)

ΔT cycle (K)

DNP D (mm)

h solder (mm)

C (fit const.)

k (exponent)

Free Thermal Expansion

ΔL = α·L₀·ΔT

Material

α (ppm/K)

L₀ (mm)

ΔT (K)

Constrained Thermal Stress

σ = −E·α·ΔT (heating → compression)

Material

α (ppm/K)

E (GPa)

ΔT (K)

Yield strength (MPa) — for margin

Bonded-Joint Mismatch Stress

ε=(α₁−α₂)ΔT; σ = Δα·ΔT / (1/E₁ + 1/E₂)

Material 1

Material 2

α₁ (ppm/K)

α₂ (ppm/K)

E₁ (GPa)

E₂ (GPa)

ΔT (K)

Stoney Thin-Film Stress

σ_f = −E_f(α_f−α_s)ΔT/(1−ν_f); wafer bow from κ.

Film: α_f (ppm/K)

E_f (GPa)

ν_f

h_f (µm)

α_s (ppm/K)

E_s (GPa)

h_s (µm)

ΔT (K)

Wafer Ø (mm)

Formula reference & material data

ΔL=αL₀ΔT; constrained σ=−EαΔT; bonded σ=Δα·ΔT/(1/E₁+1/E₂).

Timoshenko κ = 6(α₁−α₂)ΔT(1+m)² / {h[3(1+m)²+(1+mn)(m²+1/mn)]}, m=h₁/h₂, n=E₁/E₂, h=h₁+h₂.

Stoney σ_f=−E_f(α_f−α_s)ΔT/(1−ν_f), κ=6E_f h_f(α_f−α_s)ΔT/(E_s h_s²(1−ν_f)).

Solder γ=(α_sub−α_die)ΔT·D/(2h), Coffin-Manson N_f=C·(Δγ)^−k (k≈1.9). C is an empirical fit — calibrate to test data.

CTE/modulus library drawn from the vault's material tables; values are representative.