Summary: The authors J. R. Willis [Adv. Appl. Mech. 21, 1–78 (1981; Zbl 0476.73053)] and M. J. Miksis [SIAM J. Appl. Math. 43, 1140–1155 (1983; Zbl 0521.73095)] considered a composite formed by a periodic mixed in prescribed proportion of a two homogeneous dielectric materials whose respectively energy density are \(W_1(Z)=\frac{\alpha_{1}}{2}|z|^2\), \(W_2(Z)=\frac{a_{2}}{2}|Z|^2+\frac{\gamma}{4}|Z|^4\), being \(0<\alpha_1<\alpha_2\), \(\gamma>0\). Willis gave a lower bound on the effective energy density, but his method failed to give an upper bound. The difference between Milkis work and ours is that Milkis gives a selfconsistent asymptotic expansion for the effective dielectric constant when the microstructure geometry is fixed and the non-linear phase has very low volume fraction. By contrast, we will give bounds on the effective energy density for the same material with arbitrary geometry and volume fractions \(\theta_1\), \(\theta_2\), valid for any spatially periodic microstructure.
This work gives not only lower and upper bound of that composite but also of a more general class considering \(W_1(Z)=\frac{\alpha_{1}}{2}|Z|^2\), \(W_2(Z)=\frac{\alpha_{2}}{2}|Z|^2+\frac{\gamma}{p}|Z|^p\) being \(0<\alpha_1<\alpha_2\), \(\gamma>0\) and \(p>2\).
Moreover, we will prove that our bounds converge, as \(\gamma\rightarrow 0^+\), to the optimal bounds of the effective energy density \(\widetilde{W}_L\) of the considered Linear Composites, that is when \(\gamma=0\). In the article [K. A. Lurie and A. V. Cherkaev, Proc. R. Soc. Edinb., Sect. A, Math. 99, 71–87 (1984; Zbl 0564.73079)] it has been proved that the optimal bounds of the linear-isotropic case (this is when \(\gamma=0\)) are expressed in the form \[ (\widetilde{W}_L-W_1)^\ast (\eta) \leq A(\eta),\qquad (W_2-\widetilde{W}_L)^\ast (\eta) \leq B(\eta), \] while in our composite, the bounds of the isotropic case will be expressed in the form \[ \begin{aligned} (\widetilde{W}-W_1)^\ast (\eta) \leq A(\eta)-\gamma \mathcal L(\eta) + o(\gamma^2)|\eta|^{2p-4},\\ (W_0+W_2-\widetilde{W})^\ast(\eta) \leq B(\eta)-\gamma \mathcal U (\eta) + o(\gamma^2)|\eta|^{2p-4}\end{aligned} \] where \(W_0(\xi)=\frac{\gamma 2^{p-1}}{p}|\xi|^p\).
Moreover, we will give bounds to the anisotropic case. This article is a generalization of the particular case \(p=4\), which is called the Willis Composite (this particular case was first studied by [Willis, loc. cit.] and [Miksis, loc. cit.] and later was completed by [G. Tepedino, Bounds on the effective energy density of nonlinear composites. New York: Courant Institute of Mathematical Sciences (PhD Thesis) (1988)].