The linearized Kirchhoff theory for plates with incompatible prestrain

In this paper, we derive a linearized Kirchhoff model from three dimensional nonlinear elastic energy of plates with incompatible prestrain as its thickness $h$ tends to zero and its elastic energy scales like $h^{\beta}$ with $2<\beta<4.$ The incompatible prestrain is given as a Riemannian metric $G(x')$ in the three dimensional thin plate which only depends on mid-plate of the thin plates. The problem is studied rigorously by using a variational approach and establishing the $\Gamma-$ limit of the non-Euclidean version of the nonlinear elasticity functional when the gauss curvature of the mid-plate $(\Omega, g=G_{2\times2})$ is always positive, negative or zero.