A short note on the derivation of the elastic von Kármán shell theory

Abstract

We derive the Γ-limit of scaled elastic energies h⁻⁴E^h(u^h) associated with deformations u^h of a family of thin shells ${S^h} = \left\{ {z = x + t\vec n\left( x \right);x \in S, - g_1^h\left( x \right) < t < g_2^h\left( x \right)} \right\}$ . The obtained von Kármán theory is valid for a general sequence of boundaries g ^h₁ , g ^h₂ converging to 0 in an appropriate manner as h vanishes. Our analysis relies on the techniques and extends the results in [10] and [11].