The energy density of non simple materials grade two thin films via a Young measure approach. (English) Zbl 1129.74028
In this article dimension reduction is used to derive the form of the free energy of a thin film made of a non simple material of grade two. Let \(\Omega_\varepsilon=\omega\times (-\frac\varepsilon2,\frac\varepsilon2)\), where \(\omega\subset\mathbb{R}^2\) is open, bounded and connected, \(\varepsilon\) is a small positive number. The energy of the body is
\[
I_\varepsilon:u\in W^{2,p}(\Omega_\varepsilon)\to \int_{\Omega_\varepsilon} W(D^2 u)dx
\]
which should be minimized over
\[
{\mathcal B}_\varepsilon = \left\{u\in W^{2,p}(\Omega_\varepsilon\mathbb{R}^3): u(x)=x \;\text{on}\;\partial \omega\times(-\frac\varepsilon2,\frac\varepsilon2) \right\},
\]
where \(W: Sym(\mathbb{R}^3)\to\mathbb{R}\) is a bulk energy density. The goal is to study the \(\Gamma-\)limit of this minimization problem for \(\varepsilon\to 0\). Using Young measures the authors obtain in the limit a Cosserat model for the thin film.
Reviewer: Hans-Dieter Alber (Darmstadt)
MSC:
74K35 | Thin films |
49J45 | Methods involving semicontinuity and convergence; relaxation |
74G65 | Energy minimization in equilibrium problems in solid mechanics |