Advances in Applied Mechanics

5.1 Physically Nonlinear Elastic Microstructures

5.1.1 Hierarchies of Mesoscale Bounds

Consider physically nonlinear elastic materials in the range of infinitesimal strains, described by the constitutive law

$\begin{array}{l}\hfill \mathit{\sigma }=\sigma (\varepsilon )=\frac{\partial w(\varepsilon )}{\partial \varepsilon }\varepsilon =\varepsilon (\sigma )=\frac{\partial w*(\mathit{\sigma })}{\partial \mathit{\sigma }},\end{array}$

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where the energy densities are related by w* = σ : ɛ − w; w is a statistically homogeneous and ergodic field. The Hill–Mandel condition, and its implication for the type of admissible boundary conditions, is

$\begin{array}{l}\hfill \overline{\mathit{\sigma }:d\varepsilon }=\overline{\mathit{\sigma }}:d\overline{\varepsilon }\iff {\int }_{\partial B_{\delta }}(\mathbf{t}-\overline{\mathit{\sigma }}\cdot \mathbf{n})\cdot (d\mathbf{u}-d\overline{\varepsilon }\cdot \mathbf{x})dS=0,\end{array}$

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where, again by mean strain and stress theorems, ...