# Plates with Incompatible Prestrain Seminar - Mathias Schaeffner (Wuerzburg University, Germany)

## Event Date:

Friday, June 5, 2015 - 3:00pm to 4:00pm

## Event Location:

• 4607B South Hall

Free

## Event Contact:

Carlos Garcia-Cervera

Email: cgarcia@math.ucsb.edu

Phone: 8055638873

• Applied Math/PDE Seminar

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\$.