Wrinkling of a thin sheet under uni-axial tensile strain

A thin isotropic elastic sheet of thickness , width and length ( ) made of a material with Young's modulus and Poisson's ratio remains clamped at two sides along while it is subjected to longitudinal stretching strain along by applying a tensile stress . So long as, the strain is less than a critical strain, , the sheet stretches uniformly, however, at , the stretching results in appearance of wrinkles along . The wrinkles appear as the clamped boundaries prevent the sheet from contracting locally at their vicinity by creating a bi-axial state of stress. The bi-axial stress is tensile near the clamped boundary and compressive slightly away from it. When , the sheet buckles to accommodate the in-plane strain incompatibility generated via the Poisson effect.

Bending energy due to short wavelength wrinkles in the direction,

(18.1)

And the strain due to extension along direction,

(18.2)

Hence, the stretching energy is written as,

(18.3)