There are a number of phenomena when thin plates become prestrained in an incompatible manner so that there is no stress-free configuration. In this talk, I present a joint work with K. Bhattacharya and M. Lewicka, where we study the effective elastic behavior of such prestrained plates, with a prestrain which is independent of thickness and uniform through the thickness. We model such plates as 3D elastic bodies with a prescribed stress-free state characterized by a Riemannian metric $G$ with the above properties, and seek the limiting behavior as the thickness goes to zero. More precisely: Let $Omegasubset mathbb R^2$ be a bounded domain and consider the elastic energy

$$E^h(u)=frac{1}{h^3}int_{Omega_h}W(nabla u(x) A^{-1}(x))dx,$$

of a deformation $uin W^{1,2}(Omega^h,mathbb R^3)$, where $Omega^h=Omegatimes(-frac{h}{2},frac{h}{2})$, $A=sqrt{G}$ and $W$ an elastic energy density. Using $Gamma$-convergence, we derive a Kirchhoff-type bending theory as the limit $hto0$. We show that there are metrics $G$ which are not immersible, but have zero bending energy; that is $0<inflimits_u E^h(u)$ $forall h>0$, but $limlimits_{hto0}inflimits_u E^h(u)=0$.