On the derivation of homogenized bending plate model

Igor Velčić

doi:10.1007/s00526-014-0758-1

Calculus of Variations and Partial Differential Equations

July 2015, Volume 53, Issue 3–4, pp 561–586 | Cite as

On the derivation of homogenized bending plate model

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Igor VelčićEmail author

Article

First Online: 29 June 2014

Received: 30 November 2013

Accepted: 22 May 2014

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Abstract

We derive, via simultaneous homogenization and dimension reduction, the $\Gamma $-limit for thin elastic plates of thickness $h$ whose energy density oscillates on a scale $\varepsilon (h)$ such that $ \varepsilon (h)^2 \ll h\ll \varepsilon (h)$. We consider the energy scaling that corresponds to Kirchhoff’s nonlinear bending theory of plates.

Mathematics Subject Classification

35B27 49J45 74E30 74Q05

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Notes

Acknowledgments

The work on this paper was mainly done while the author was affiliated with Peter Hornung’s group in Max Planck Institute for Mathematics in the Sciences in Leipzig, Germany. The author was supported by Deutsche Forschungsgemeinschaft grant no. HO-4697/1-1.

Appendix

Let us recall some well-known properties of two scale convergence. We refer to [2, 13, 23] for proofs in the standard two scale setting and to [14] for the easy adaption to the notion of two scale convergence considered here.

Lemma 5.1

(i)

Any sequence that is bounded in $L^2(\tilde{\Omega })$ admits a two scale convergent subsequence.
(ii)

Let $\widetilde{f}\in L^2(\tilde{\Omega }\times Y)$ and let $(f^h)_{h>0}\subset L^2(\tilde{\Omega })$ be such that $f^h\mathop {\rightharpoonup }\limits ^{2,0}\widetilde{f}$. Then $f^h\rightharpoonup \int _Y\widetilde{f}(\cdot ,y)\,dy$ weakly in $L^2(\tilde{\Omega })$.
(iii)

Let $f^0 \in L^2(\tilde{\Omega })$ and $(f^h)_{h>0} \subset L^2(\tilde{\Omega })$ be such that $f^h\rightharpoonup f^0$ weakly in $L^2(\tilde{\Omega })$. Then (after passing to subsequences) we have $f^h\mathop {\rightharpoonup }\limits ^{2,0}f^0(x)+\widetilde{f}$ for some $\widetilde{f}\in L^2(\tilde{\Omega }\times Y)$ with $\int _Y\widetilde{f}(\cdot ,y)\,dy=0$ almost everywhere in $\tilde{\Omega }$. $\widetilde{f}$ is uniquely characterized by the fact that $\int _{\tilde{\Omega }} f^h \psi (x,\tfrac{x}{\varepsilon (h)}) \,\mathrm {d}x \rightarrow \int _{\tilde{\Omega }} \int _{\mathcal {Y}} \widetilde{f}(x,y) \psi (x,y) \,\mathrm {d}y\,\mathrm {d}x$, for every $\psi \in C_0^\infty (\tilde{\Omega },\mathring{C}^\infty (\mathcal {Y}))$.
(iv)

Let $f^0 \in H^1(\tilde{\Omega })$ and $(f^h)_{h>0} \subset H^1(\tilde{\Omega })$ be such that $f^h\rightarrow f^0$ strongly in $L^2(\tilde{\Omega })$. Then $f^h\xrightarrow {2,0}f^0$, where we extend $f^0$ trivially to $\tilde{\Omega }\times Y$.
(v)

Let $f^0$ and $f^h\in H^1(\tilde{S})$ be such that $f^h\rightharpoonup f^0$ weakly in $H^1(\tilde{S})$. Then (after passing to subsequences)
$$\begin{aligned} \nabla ' f^h\mathop {\rightharpoonup }\limits ^{2,0}\nabla 'f^0+\nabla _y\phi , \end{aligned}$$
for some $\phi \in L^2(\tilde{S}, H^1(\mathcal Y))$.

The following lemma gives us the characterization of two scale limits of scaled gradients when $h \ll \varepsilon (h)$. For the proof of it see [14, Theorem 6.3.3].

Lemma 5.2

Let $u^0 \in H^1(\tilde{\Omega },\mathbb R^3)$ and $(u^h)_{h>0}\subset H^1(\tilde{\Omega },\mathbb R^3)$ be such that $u^h\rightharpoonup u^0$ weakly in $H^1(\tilde{\Omega },\mathbb R^3)$ and $\liminf _{h\rightarrow 0}\Vert \nabla _h u^h\Vert _{L^2}<\infty $. Under the assumption $\lim _{h\rightarrow 0} \tfrac{h}{\varepsilon (h)}=0$ there exist $\phi \in L^2(\tilde{S},\mathring{H}^1(\mathcal Y,\mathbb R^3))$, $d \in L^2(\tilde{S},L^2(I\times \mathcal {Y},\mathbb R^3))$ such that (after passing to subsequences)

$$\begin{aligned} \nabla _h u^h \mathop {\rightharpoonup }\limits ^{2,0}(\nabla 'u^0\,,\,0)+(\nabla _y\phi \,,\,d). \end{aligned}$$

(58)

Here $u^0$ is the weak limit of $u^h$ i.e. $\int _I u^h$ in $H^1(\tilde{\Omega },\mathbb R^3)$ i.e. $H^1(\tilde{S},\mathbb R^3)$.

The following two lemmas are proved in [22]. The first one tells us that two scale limit, like weak limit, behaves nicely when we multiply it with a bounded sequence which converges in measure. The second one tells us that if we have a sequence of vector fields that is closer to a sequence of gradients than the order of oscillations, then the form of the oscillatory part of its two scale limit is the second gradient, like the two scale limit of gradients. This is also valid under subtle assumption that the sequence of gradients itself converges only strongly in $L^2$ (but not necessarily weakly in $H^1$).

Lemma 5.3

Let $(u^h)_{h>0}$ be a bounded sequence in $L^2(\tilde{\Omega })$ which two scale converges to $u_0(x,y) \in L^2(\tilde{\Omega } \times \mathcal {Y})$. Let $(v^h)_{h>0}$ be a sequence bounded in $L^\infty (\tilde{\Omega })$ which converges in measure to $v_0 \in L^{\infty }(\tilde{\Omega })$. Then $v^h u^h \mathop {\rightharpoonup }\limits ^{2,0}v_0(x) u_0(x,y)$.

Lemma 5.4

Let $(u^h)_{h>0}$ be a sequence which converges strongly to $u$ in $H^1(\tilde{\Omega })$ and $(v^h)_{h>0}$ be a sequence which is bounded in $H^1(\tilde{\Omega },\mathbb R^n)$ such that for each $h>0$

$$\begin{aligned} \Vert \nabla u^h-v^h\Vert _{L^2} \le C\eta (h), \end{aligned}$$

(59)

for some $C>0$ and $\eta (h)$ which satisfies $\lim _{h \rightarrow 0} \tfrac{\eta (h)}{\varepsilon (h)}=0$. Then for any subsequence of $(\nabla v^h)_{h>0}$ which converges two scale there exists a unique $v \in L^2(\Omega ,\mathring{H}^2(\mathcal Y))$ such that $\nabla v^h \mathop {\rightharpoonup }\limits ^{2,0}\nabla ^2 u(x)+\nabla ^2_y v(x,y)$.

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Igor Velčić
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1.Faculty of Electrical Engineering and ComputingUniversity of ZagrebZagrebCroatia

On the derivation of homogenized bending plate model

Abstract

Mathematics Subject Classification

Notes

Acknowledgments

References

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