Homogenization of periodic multi-dimensional structures: The linearly elastic/perfectly plastic case (Q2747825)

In the paper the authors consider the following homogenization problem. Given a nonnegative 1-periodic Radon measure \(\mu\) on \(\mathbb{R}^n\), normalized with \(\mu([0, 1)^n)= 1\), and taken \(\mu_\varepsilon(B)= \varepsilon^n \mu(B/\varepsilon)\), the following sequence of functionals is considered: NEWLINE\[NEWLINEF_\varepsilon(u,\Omega)= \int_\Omega f\Biggl({x\over\varepsilon}, {dEu\over d\mu_\varepsilon}\Biggr) d\mu_\varepsilon,NEWLINE\]NEWLINE where \(f(x,z)\) is 1-periodic in \(x\) and satisfies a growth condition of order \(p\geq 1\). Notice that the functional \(F_\varepsilon\) is defined for every function \(u\) whose strain tensor \(Eu={1\over 2}(Du+ {^tDu})\) is absolutely continuous with respect to \(\mu_\varepsilon\), and \(dEu/d\mu_\varepsilon\) denotes the density of the measure \(Eu\) with respect to \(\mu_\varepsilon\).NEWLINENEWLINENEWLINEThe main result is a homogenization theorem which gives the \(\Gamma\)-limit of \(F_\varepsilon\) (as \(\varepsilon\to 0\)) in the form NEWLINE\[NEWLINEF_{\text{hom}}(u,\Omega)= \int_\Omega f_{\text{hom}}(Eu) dx\qquad (\text{whenever }Eu\in L^1(\Omega)),NEWLINE\]NEWLINE where NEWLINE\[NEWLINEf_{\text{hom}}(z)= \lim_{T\to+\infty} \Biggl[\inf\Biggl\{T^{-n} \int_{[0,T)^n} f\Biggl(x,{dEu\over d\mu}\Biggr) d\mu:u-zx\text{ is }T\text{-periodic}\Biggr\}\Biggr].NEWLINE\]NEWLINE A similar representation for \(F_{\text{hom}}\) holds when \(Eu={\mathcal E}u dx+ E^su\) is a measure.NEWLINENEWLINENEWLINEThe paper ends with some examples which show that a different scaling of the functionals \(F_\varepsilon\) could lead to a limit functional of a nonlocal type.