Elastic Wrinkling Instability

Thin film on soft substrate: wavelength selection and post-buckling

Compression ε: 0.00 Wavelength λ*: — px Amplitude: 0 px Mode: flat
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Wrinkling instability (Cerda & Mahadevan 2003): A stiff thin film (modulus E_f, thickness h) on a soft elastic substrate (modulus E_s) develops periodic wrinkles when compressed beyond a critical strain ε_c. The selected wavelength minimizes total energy = bending energy (∝ E_f h³/λ²) + substrate deformation energy (∝ E_s λ). The optimal wavelength is λ* = 2πh(E_f/3E_s)^(1/3), independent of strain. The critical strain is ε_c = (3E_s/E_f)^(2/3)/4. Above ε_c, amplitude grows as A ∝ h√(ε/ε_c − 1). Post-buckling: at large ε, the pattern localizes into a fold (period-doubling bifurcation). Applications: skin wrinkles, flexible electronics, morphogenesis.