Gilles Francfort

3d-2d asymptotics for thin films

The premise of the analysis is to view thin films and their properties as limit behavior of their 3-d analogue as the ``thickness'' becomes vanishingly small. The analysis is at present restricted to the purely elastic framework although it allows for large deformations -which seems necessary in thin film technology- and arbitrary ``elastic'' constitutive laws. The upper and lower surfaces of the film or of the film/substrate complex remain traction free (free floating setting). The method consists in setting up an appropriate 3d variational problem and passing to the $\Gamma$-limit as the thickness tends to $0$.

This is the object of various collaborations with K. Bhattacharya (Caltech), A. Braides (SISSA) and I. Fonseca (CMU).

The schematics is as follows: a domain $\Omega^\varepsilon = \{(x_\alpha, x_3^\varepsilon); x_\alpha \in \omega, x_3^\v... ...[-\varepsilon f^\varepsilon((x_\alpha), \varepsilon f^\varepsilon((x_\alpha)]\}$, where $\omega$ is a two-dimensional section and $0 < \beta \leq f^\varepsilon((x_\alpha) \leq 1$ is considered ( $f^\varepsilon(x_\alpha)$ translates the possible non-flatness of the upper and lower profiles). Its associated bulk energy is

$$\int_{\Omega^\varepsilon} \hat{W}^\varepsilon(x_\alpha,x_3^\varepsilon; D_\alpha v,D_3 v) \; dx_\alpha dx_3^\varepsilon.$$

A rescaling of that domain is implemented upon setting $x_3 = \frac{x_3^\varepsilon} {\varepsilon}$. Then, the bulk energy becomes

$$\varepsilon E_\varepsilon(v,\omega) = \varepsilon \int_{\omeg... ...alpha,x_3; D_\alpha v ,1/ \varepsilon D_3 v) \; dx_\alpha dx_3,$$

where $\omega^\varepsilon$ is the rescaled version of $\Omega^\varepsilon$ and $W^\varepsilon(x_\alpha, x_3; \xi) = \hat{W}^\varepsilon(x_\alpha, \varepsilon x_3; \xi)$. The goal is to compute the following:

$$J(v,\omega) = \inf_{\{\varepsilon\}} \; \liminf_\varepsilon ... ... v^\varepsilon \to v \quad\mbox{in the appropriate topology}\}.$$

The generic result is that

$$J(v,\omega) = \left\{\begin{array}{ll}\int_\omega W(x_\alpha;... ...on $x_3$,}\\ +\infty,\quad \mbox{otherwise.} \end{array}\right.$$

Well, not quite correct as stated but it will be the case in all quoted applications.

The energy $W$ is then computed in several settings:

• (1) $f^\varepsilon(x_\alpha) = 1$ and $W^\varepsilon(x_\alpha,x_3; \xi) = W(x_3;\xi)$; this is the generalization of prior work by LE DRET-RAOULT.
• (2) $f^\varepsilon(x_\alpha) = f(\frac{x_\alpha}{\varepsilon})$ and $W^\varepsilon(x_\alpha,x_3;\xi) = W(\frac{x_\alpha}{\varepsilon},x_3;\xi)$
• (3) $f^\varepsilon(x_\alpha) = 1$ and optimal design/damage: two phase cylindrical mixtures with local volume fraction $\theta(x_\alpha)$ are considered; the following is computed:
  
  $$\hskip-1cm J(v,\omega) = \inf_{\{\varepsilon\}} \; \liminf_\v... ...uad\mbox{in the proper topology}\} \end{array}\right.\right\},$$
  
  
  where
  
  $$E_\varepsilon(v^\varepsilon,\chi^\varepsilon,\omega) = \int_{... ...ilon) W_2)(D_\alpha v ,1/ \varepsilon D_3 v) \; dx_\alpha dx_3.$$
  
  

• (4) $f^\varepsilon(x_\alpha) = 1$ and debonding film/substrate: a surface energy of the form $\varepsilon^{\alpha-1} \int_\omega \psi(u^+-u^-) \, dx_\alpha$ is added to the bulk energy. Only specific $\alpha$'s and $\psi$'s will be considered.