Citation: Alyssa D. Lokits, Julia Koehler Leman, Kristina E. Kitko, Nathan S. Alexander, Heidi E. Hamm, Jens Meiler. A survey of conformational and energetic changes in G protein signaling[J]. AIMS Biophysics, 2015, 2(4): 630-648. doi: 10.3934/biophy.2015.4.630
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The aim of this work is to derive von-Kármán plate theory from nonlinear, three-dimensional, atomistic models in a certain energy scaling as the interatomic distance
The passage from atomistic interaction models to continuum mechanics (i.e., the limit
Our first aim is to close this gap. For thin films consisting of many atomic layers one expects the scales
By way of contrast, for ultrathin films consisting of only a few atomic layers, more precisely, if
Our third aim concerns a more fundamental modelling point of view which is based on the very low energy of the von-Kármán scaling: If the the plate is not too thick (more precisely, if
Finally, on a technical note, the proof of the our main result set forth in Section 4 elucidates the appearance and structure of the correction terms in the ultrathin film regime. Both in [18] and the present contribution, at the core of the proof lies the identification of the limiting strain, which in the discrete setting can be seen as a
This work is organized as follows: In Section 2, we first describe the atomistic interaction model and then present our results. Our main theorem, Theorem 2.1, details the
Let
More precisely, as in [18] we let
$ Z = (z^1, \dots, z^8) = \frac{1}{2}\left(−111−1−111−1−1−111−1−111−1−1−1−11111\right). $ |
Furthermore, by
$ Eatom(w)=∑x∈Λ′nW(x,→w(x)), $ | (1) |
where
As a full interaction model with long-range interaction would be significantly more complicated in terms of notation and would result in a much more complicated limit for finitely many layers, we restrict ourselves to these cell energies.
In the following we will sometimes discuss the upper and lower part of a cell separately. We write
If the full cell is occupied by atoms, i.e.,
$ W(x,\vec{w}) = {Wcell(→w)ifx3∈(εn/2,hn−εn/2),Wcell(→w)+Wsurf(→w(2))ifνn≥3andx3=hn−εn/2,Wcell(→w)+Wsurf(→w(1))ifνn≥3andx3=εn/2,Wcell(→w)+∑2i=1Wsurf(→w(i))ifνn=2,andx3=hn/2. $ |
Example 1. A basic example is given by a mass-spring model with nearest and next to nearest neighbor interaction:
$ Eatom(w)=α4∑x,x′∈Λn|x−x′|=εn(|w(x)−w(x′)|εn−1)2+β4∑x,x′∈Λn|x−x′|=√2εn(|w(x)−w(x′)|εn−√2)2. $ |
$ Wcell(→w)=α16∑1≤i,j≤8|zi−zj|=1(|wi−wj|−1)2+β8∑1≤i,j≤8|zi−zj|=√2(|wi−wj|−√2)2 $ |
and
$ Wsurf(w1,w2,w3,w4)=α8∑1≤i,j≤4|zi−zj|=1(|wi−wj|−1)2+β8∑1≤i,j≤4|zi−zj|=√2(|wi−wj|−√2)2. $ |
We will also allow for energy contributions from body forces
$ E_{\rm body}(w) = \sum\limits_{x \in \Lambda_n} w(x) \cdot f_n(x). $ |
We will assume that the
$ ∑x∈Λnfn(x)=0,∑x∈Λnfn(x)⊗(x1,x2)T=0, $ | (2) |
to not give a preference to any specific rigid motion. At last, we assume that after extension to functions
Overall, the energy is given as the sum
$ En(w)=ε3nhn(Eatom(w)+Ebody(w)). $ | (3) |
Due to the factor
Let us make some additional assumptions on the interaction energy. We assume that
$ W(A) = W(A + (c,\dots,c)) \mbox{ and } W(RA) = W(A) $ |
for any
Since our model is translationally invariant, it is then equivalent to consider the discrete gradient
$ \bar{\nabla} w(x) = \frac{1}{ \varepsilon_n} \big(w(x+ \varepsilon_n z^1) - \langle w \rangle , \dots, w(x+ \varepsilon_n z^8) - \langle w \rangle\big) $ |
with
$ \langle w \rangle = \frac{1}{8} \sum\limits_{i = 1}^8 w(x+ \varepsilon_n z^i) $ |
instead of
$ \sum\limits_{i = 1}^8 (\bar{\nabla} w(x))_{\cdot i} = 0. $ |
The bulk term is also assumed to satisfy the following single well growth condition.
(G) Assume that there is a
$ W_{\rm cell}(A) \geq c_0 {\rm{dist}}^2(A, {\rm{SO}}(3)Z) $ |
for all
In the same way as in a pure continuum approach, it is convenient to rescale the reference sets to the fixed domain
$ H_n = (10001000hn). $ |
A deformation
$ (\bar{\nabla}_n y(x))_{\cdot i} : = \frac{1}{ \varepsilon_n}(y(x' + \varepsilon_n (z^i)', x_3 + \frac{ \varepsilon_n}{h_n} z^i_3)-\langle y \rangle) = \bar{\nabla}w (H_n x) $ |
for
$ \langle y \rangle = \frac{1}{8} \sum\limits_{i = 1}^8 y(x'+ \varepsilon_n (z^i)',x_3 + \frac{ \varepsilon_n}{h_n} z^i_3 ). $ |
For a differentiable
In Section 3 we will discuss a suitable interpolation scheme with additional modifications at
$ ˜yn:=R∗nT˜˜yn−cn, $ | (4) |
which would then be close to the identity. The von-Kármán displacements in the limit will then be found as the limit objects of
$ un(x′):=1h2n∫10(˜yn)′−x′dx3,and $ | (5) |
$ vn(x′):=1hn∫10(˜yn)3dx3. $ | (6) |
To describe the limit energy, let
$ D2Wcell(Z)[A,BZ]=0,D2Wsurf(Z(1))[A′,BZ(1)]=0 $ | (7) |
for all
In particular,
$ Qcell(BZ+c⊗(1,…,1))=Qsurf(BZ(1)+c⊗(1,1,1,1))=0 $ | (8) |
for all
We introduce a relaxed quadratic form on
$ Qrelcell(A)=minb∈R3Qcell(a1−b2,…,a4−b2,a5+b2,…,a8+b2)=minb∈R3Qcell(A+(b⊗e3)Z)=minb∈R3Qcell(A+sym(b⊗e3)Z). $ |
By Assumption (G)
$ Qrelcell(A)=Qcell(A+(b(A)⊗e3)Z)=Qcell(A+sym(b(A)⊗e3)Z). $ | (9) |
Here we used (7) to arrive at the symmetric version. Furthermore, the mapping
At last, let us write
$ Q_2(A) = Q_{\rm cell}^{\rm rel} \bigg( (A000) Z \bigg), \qquad Q_{2,{\rm surf}}(A) = Q_{\rm surf} \bigg( (A000) Z^{(1)} \bigg) $ |
for any
We are now in place to state our main theorem in its first version.
Theorem 2.1. (a) If
$ EvK(u,v,R∗):=∫S12Q2(G1(x′))+124Q2(G2(x′))+f(x′)⋅v(x′)R∗e3dx′, $ |
where
$ \liminf\limits_{n \to \infty} \frac{1}{h_n^4} E_n(y_n) \geq E_{\rm vK}(u,v, R^*). $ |
On the other hand, this lower bound is sharp, as for every
$ \lim\limits_{n \to \infty} \frac{1}{h_n^4} E_n(y_n) = E_{\rm vK}(u,v,R^*). $ |
(b) If
$ E(ν)vK(u,v,R∗)=∫S12Qrelcell((G1(x′)000)Z+12(ν−1)G3(x′))+ν(ν−2)24(ν−1)2Q2(G2(x′))+1ν−1Qsurf((G1(x′)000)Z(1)+∂12v(x′)4(ν−1)M(1))+14(ν−1)Q2,surf(G2(x))+νν−1f(x′)⋅v(x′)R∗e3dx′. $ |
Here,
$ G3(x′)=(G2(x′)000)Z−+∂12v(x′)M, $ | (10) |
$ M=(M(1),M(2))=12e3⊗(+1,−1,+1,−1,+1,−1,+1,−1), $ | (11) |
$ Z−=(−Z(1),Z(2))=(−z1,−z2,−z3,−z4,+z5,+z6,+z7,+z8). $ | (12) |
In the following we use the notation
Example 2. Theorem 2.1 applies to the interaction energy of Example 1 if
Remark 1. 1. The result in a) is precisely the functional one obtains by first applying the Cauchy-Born rule (in 3d) in order to pass from the discrete set-up to a continuum model and afterwards computing the (purely continuum)
$ W_{\rm CB}(A) = W_{\rm cell}(A Z) $ |
to the atomic interaction
$ Q_2(A) = \min\limits_{b \in \mathbb{R}^3} Q_{\rm CB} \bigg( (A000) + b \otimes e_3 \bigg). $ |
2. In contrast, for finite
3. Suppose that in addition
$ Wcell(w1,…,w8)=Wcell(Pw5,…,Pw8,Pw1,…,Pw4),Wsurf(w1,…,w4)=Wsurf(Pw1,…,Pw4), $ |
where
$ E(ν)vK(u,v)=∫S12Q2(G1(x′))+ν(ν−2)24(ν−1)2Q2(G2(x′))+18(ν−1)2Qrelcell(G3(x′))+1ν−1Q2,surf(G1(x′))+(∂12v(x′))216(ν−1)3Qsurf(M(1))+14(ν−1)Q2,surf(G2(x′))dx′=EvK(u,v)+∫S1ν−1[Q2,surf(G1(x′))+14Q2,surf(G2(x))]+18(ν−1)2[Qrelcell(G3(x′))−13Q2(G2(x′))]+116(ν−1)3(∂12v(x′))2Qsurf(M(1))dx′. $ |
4. Standard arguments in the theory of
5. For the original sequence
One physically unsatisfying aspect of Theorem 2.1 is the strong growth assumption (G) which is in line with the corresponding continuum results [13]. The problem is actually two-fold. First, typical physical interaction potentials, like Lennard-Jones potentials, do not grow at infinity but converge to a constant with derivatives going to
Contrary to the continuum case, it is actually possible to remove these restrictions in our atomistic approach. Indeed, if one assumes
In this case, growth assumptions at infinity should no longer be relevant. In fact, we can replace (G) by the following much weaker assumption with no growth at infinity and full
(NG) Assume that
$ W_{\rm cell}(A) \geq c_0 {\rm{\;dist}}^2(A, {\rm{O}}(3)Z) $ |
for all
$ W_{\rm cell}(A) \geq c_0 $ |
for all
One natural problem arising from this is that atoms that are further apart in the reference configuration can end up at the same position after deforming. In particular, due to the full
As a remedy, whenever we assume (NG), we will add a rather mild non-penetration term to the energy that can be thought of as a minimal term representing interactions between atoms that are further apart in the reference configuration. To make this precise, for small
$ E_{\rm nonpen}(w) = \sum\limits_{x,\bar{x} \in \Lambda_n} V\Big(\frac{w(x)}{ \varepsilon},\frac{w(\bar{x})}{ \varepsilon}\Big). $ |
Then,
The overall energy is then given by
$ En(w)=ε3nhn(Eatom(w)+Ebody(w)+Enonpen(w)). $ | (13) |
Theorem 2.2. Assume that
Note that in this version, we assume
In the spirit of local
$ \mathcal{S}_\delta = \{w: \Lambda_n \to \mathbb{R}^3 {\rm{\;such\;that\;}} {\rm{dist}}(\bar{\nabla} w(x),{\rm{SO}}(3)Z) < \delta {\rm{\;for\;all\;}} x \in {\Lambda'_n}^{\circ}\}, $ |
where
$ En(w)={ε3nhn(Eatom(w)+Ebody(w))ifw∈Sδ,∞else. $ | (14) |
We then have a version of the
Theorem 2.3. Assume that
$ \lim\limits_{n \to \infty} \inf \Big\{ \frac{1}{h_n^4} E_n(w) : w \in \mathcal{S}_\delta \backslash \mathcal{S}_{\delta/2} \Big\} = \infty. $ |
Remark 2. 1. For
2. To formulate it differently, if a sequence
3. As the energy only has to be prescribed in
Example 3. In the setting of Theorems 2.2 and 2.3, Example 2 can be generalized to energies of the form
$ Eatom(w)=α4∑x,x′∈Λn|x−x′|=εnV1(|w(x)−w(x′)|εn−1)+β4∑x,x′∈Λn|x−x′|=√2εnV2(|w(x)−w(x′)|εn−√2), $ |
where
We first extend a lattice deformation slightly beyond
For
$ Q_n(x) = x + (-\tfrac{ \varepsilon_n}{2}, \tfrac{ \varepsilon_n}{2})^3. $ |
and also write
On a cell that has a corner outside of
Let
$ \Omega^{\rm in}_{n} = \bigg(\bigcup\limits_{x \in {\Lambda_n'}^{\circ}} \overline{Q_{n}(x)} \bigg)^{\circ}. $ |
Recall the definition of
$ \bar{\Lambda}_n = \Lambda_n' + \{ z^1, \ldots, z^8 \}, \qquad \Omega^{\rm out}_{n} = \bigg(\bigcup\limits_{x \in \Lambda_n'} \overline{Q_{n}(x)} \bigg)^{\circ}. $ |
The (lateral) boundary cells
$ x \in \partial \Lambda_n' : = \Lambda_n' \setminus {\Lambda_n'}^{\circ}. $ |
Later we will also use the rescaled versions of these sets which are denoted
If
For every cell
As a result of this procedure,
Our modification scheme guarantees that the rigidity and displacements of boundary cells can be controlled in terms of the displacements, respectively, rigidity of inner cells, see [19,Lemmas 3.2 and 3.4]1:
1We apply these lemmas without a Dirichlet part of the boundary, i.e.,
Lemma 3.1. There exist constants
$ \sum\limits_{x \in \partial \Lambda_n'} | \bar{\nabla} w'(x) - R^* Z |^2 \le C \sum\limits_{x \in {\Lambda_n'}^{\circ}} | \bar{\nabla} w'(x) - R^* Z |^2 $ |
as well as
$ \sum\limits_{x \in \partial \Lambda_n'} {\rm{dist}}^2(\bar{\nabla} w'(x), {\rm{SO}}(3)Z) \le C \sum\limits_{x \in {\Lambda_n'}^{\circ}} {\rm{dist}}^2(\bar{\nabla} w'(x), {\rm{SO}}(3)Z). $ |
For the sake of notational simplicity, we will sometimes write
Let
Let
$ \operatorname{co} ( x, x + \varepsilon_n v^k, x + \varepsilon_n z^i, x + \varepsilon_n z^j ) $ |
with
$ ˜w(x)=−∫Q(x)˜w(ξ)dξ, $ | (15) |
$ ˜w(x+εnvk)=−∫x+εnFk˜w(ζ)dζ, $ | (16) |
for every face
For the second interpolation we first let
$ \bar{\nabla} \bar{w}(x) = \frac{1}{ \varepsilon_n} \big(\bar{w}(x+ \varepsilon_n z^1) - \langle \bar{w} \rangle , \dots, \bar{w}(x+ \varepsilon_n z^8) - \langle \bar{w} \rangle\big) $ |
with
$ \bar{\nabla} \bar{w}(\xi) = \bar{\nabla} w(x) \quad \rm{whenever} \quad \xi \in Q_n(x),\; x \in \Lambda_n'. $ |
It is not hard to see that the original function controls the interpolation and vice versa.
Lemma 3.2. There exist constants
$ c|ˉ∇w(x)|2≤ε−3n∫Q|∇˜w(ξ)|2dξ≤C|ˉ∇w(x)|2. $ |
Proof. After translation and rescaling we may without loss assume that
$ \tilde{w} \mapsto |\bar{\nabla} \tilde{w}(x)| \quad \mbox{and} \quad \tilde{w} \mapsto \| \nabla \tilde{w} \|_{L^2(Q; \mathbb{R}^{3 \times 3})} $ |
are norms on the finite dimensional space of continuous mappings
Lemma 3.3. There exist constants
$ cdist2(ˉ∇w(x),SO(3)Z)≤ε−3n∫Qdist2(∇˜w(ξ),SO(3))dξ≤Cdist2(ˉ∇w(x),SO(3)Z). $ |
This is in fact [19,Lemma 3.6]. We include a simplified proof.
Proof. After translation and rescaling we may without loss assume that
$ cminR∈SO(3)‖∇˜w−R‖2L2(Q)≤∫Qdist2(∇˜w(ξ),SO(3))dξ≤CminR∈SO(3)‖∇˜w−R‖2L2(Q). $ |
By definition also
$ dist2(ˉ∇w(x),SO(3)Z)=minR∈SO(3)|ˉ∇w(x)−RZ|. $ |
The claim then follows from applying Lemma 3.2 to
For a sequence
$ \tilde{ \tilde{{y}} }_n(x) : = \tilde{w}_n(H_n x) \quad\mbox{with}\quad \tilde{\Omega}^{\rm out}_n : = H_n^{-1} \Omega^{\rm out}_n $ |
and
$ \bar{ \bar{{y}} }_n(x) : = \bar{w}_n(H_n x) \quad\mbox{with}\quad \tilde{V}^{\rm out}_n : = H_n^{-1} V^{\rm out}_n. $ |
(Later we will normalize by a rigid change of coordinates to obtain
$ \nabla_n \tilde{ \tilde{{y}} }_n(x) : = \nabla \tilde{w}_n(H_n x) \quad\mbox{and}\quad \bar{\nabla}_n \bar{ \bar{{y}} }_n(x) : = \bar{\nabla} \bar{w}_n(H_n x) $ |
for all
$ fn(x)=0forx∈ˉΛn∖Λ∘n $ | (17) |
and its the piecewise constant interpolation is
Remark 3. Suppose
Suppose
●
●
●
The same is true in case
In particular, limiting deformations do not depend on the interpolation scheme.
For the compactness we will heavily use the corresponding continuum rigidity theorem from [12,Theorem 3] and [13,Theorem 6]:
Theorem 4.1. Let
$ ‖∇ny−R‖2L2(Ω)≤CI, $ | (18) |
$ ‖R−˜R‖2L2(S)≤CI, $ | (19) |
$ ‖∇˜R‖2L2(S)≤CIh2n, $ | (20) |
$ ‖∇ny−R∗‖2L2(Ω)≤CIh2n, $ | (21) |
$ ‖R−R∗‖2Lp(S)≤CpIh2n, ∀p<∞. $ | (22) |
Crucially, none of the constants depend on
Furthermore, we will also use the continuum compactness result [12,Lemmas 4 and 5] and [13,Lemma 1,Eq. (96),and Lemma 2] based on the previous rigidity result applied to some sequence
Theorem 4.2. Let
$ ‖∇nyn−Rn‖2L2(Ω)≤Ch4n $ | (23) |
$ ‖Rn−˜Rn‖2L2(S)≤Ch4n $ | (24) |
$ ‖∇˜R‖2L2(S)≤Ch2n $ | (25) |
$ ‖∇nyn−Id‖2L2(Ω)≤Ch2n $ | (26) |
$ ∫Ω(∇nyn)12−(∇nyn)21dx=0. $ | (27) |
And, up to extracting subsequences,
$ 1h2n∫10y′n−x′dx3=:un⇀uinW1,2(S;R2), i=1,2, $ | (28) |
$ 1hn∫10(yn)3dx3=:vn→vinW1,2(S;R), $ | (29) |
$ ∇nyn−Idhn=:An→A=e3⊗∇′v−∇′v⊗e3inL2(Ω;R3×3), $ | (30) |
$ 2sym(Rn−Id)h2n→A2inLp(S;R3×3), ∀p<∞, $ | (31) |
$ RTn∇nyn−Idh2n⇀GinL2(Ω;R3×3), $ | (32) |
where the upper left
$ G″(x)=G1(x′)+(x3−12)G2(x′), $ | (33) |
with
$ symG1=12(∇′u+(∇u)T)+∇′v⊗∇′v,G2=−(∇′)2v. $ | (34) |
The following proposition allows us to apply these continuum results.
Proposition 1. In the setting of Theorem 2.1, consider a sequence
$ En(wn)≤Ch4n $ | (35) |
Then,
$ 0≤I(˜˜yn)=∫Ωdist2(∇n˜˜yn,SO(3))dx≤Ch4n. $ | (36) |
Here,
In the setting of Theorem 2.3 the statement remains is true as well, while in the setting of Theorem 2.2 (36) is still true but now
Proof. Rescaling the
Take
$ \frac{ \varepsilon_n^3}{h_n}\sum\limits_{ x \in \tilde{\Lambda}_n'} \lvert \bar{\nabla}_n \bar{ \bar{{y}} }_n (x) - R^*_n Z \rvert^2 \leq C\int_\Omega \lvert \nabla_n \tilde{ \tilde{{y}} }(x) - R^*_n \rvert^2 \,dx \leq C \frac{\mathcal{I}_n}{h_n^2}. $ |
A standard discrete Poincaré-inequality then shows
$ \frac{ \varepsilon_n^3}{h_n}\sum\limits_{ x \in {\tilde{\Lambda}_n}^{\circ}} \Big\lvert \bar{ \bar{{y}} }_n(x) - R^*_n (x′hx3) - \bar{c}_n \Big\rvert^2 \leq\frac{ \varepsilon_n^3}{h_n}\sum\limits_{ x \in \tilde{\Lambda}_n'} \lvert \bar{\nabla}_n \bar{ \bar{{y}} }_n (x) - R^*_n Z \rvert^2 \leq C \frac{\mathcal{I}_n}{h_n^2} $ |
for a suitable
$ ε3nhnEbody(wn)=ε3nhn∑x∈˜Λn∘fn(x′)⋅yn(x)=ε3nhn∑x∈˜Λn∘fn(x′)⋅(ˉˉyn(x)−R∗n(x′hx3)−ˉcn). $ |
Using
$ \Big\lvert \frac{ \varepsilon_n^3}{h_n} E_{\rm body}(w_n) \Big\rvert \leq C \sqrt{\mathcal{I}_n} h_n^2. $ |
On the other hand, due to
$ ε3nhnEatom(wn)≥c0ε3nhn∑x∈(˜Λn′)∘dist2(ˉ∇nyn(x),SO(3)Z)≥cε3nhn∑x∈˜Λ′ndist2(ˉ∇nˉˉyn(x),SO(3)Z)≥cIn. $ |
Hence,
$ 0≤In≤Cε3nhnEatom(wn)≤Ch4n+Cε3nhn|Ebody(wn)|≤Ch4n+C√Inh2n. $ |
We thus have
$ 0≤In≤Ch4n. $ |
All these statements remain true in the setting of Theorem 2.3 as the Assumptions
Now, consider the setting of Theorem 2.2 with Assumption
$ 0 \leq W_{\rm cell}(\bar{\nabla} w(x)) \leq C \frac{h_n^5}{ \varepsilon_n^3} $ |
for every
$ 0 \leq V\Big(\frac{w_n(\bar{x})}{ \varepsilon_n},\frac{w_n(\bar{ \bar{{x}} })}{ \varepsilon_n}\Big) \leq C \frac{h_n^5}{ \varepsilon_n^3} $ |
for all
$ \bar{\nabla} w_n(x) \in U {\rm{\;for\;all\;}} x \in {\Lambda'_n}^{\circ} $ |
and
$ |wn(ˉx)−wn(ˉˉx)|>εnδ $ | (37) |
for all
$ {\rm{dist}}^2(\bar{\nabla} w_n(x), {\rm{O}}(3)Z) \leq C \frac{h_n^5}{ \varepsilon_n^3}. $ |
Again, for
$ {\rm{dist}}^2(\bar{\nabla} w_n(x), {\rm{O}}(3)Z) = \lvert \bar{\nabla} w_n(x) - QZ \rvert^2 \leq C \frac{h_n^5}{ \varepsilon_n^3}, $ |
and
$ {\rm{dist}}^2(\bar{\nabla} w_n(x'), {\rm{O}}(3)Z) = \lvert \bar{\nabla} w_n(x') + Q'Z \rvert^2 \leq C \frac{h_n^5}{ \varepsilon_n^3}, $ |
with
$ \bar{\nabla} w_n(x') (0, b)^T = \bar{\nabla} w_n(x) (b, 0)^T $ |
for all
$ |w(x′+εnz1)−w(x+εnz5)|=|w(x+εnz5)−w(x+εnz1)+w(x′+εnz5)−w(x′+εnz1)|≤εn(|Qz5−Qz1−Q′z1+Q′z5|+Ch5nε3n)=εn(|(Q−Q′)e3|+Ch5nε3n)≤εnCh5nε3n≤δεn $ |
for
That means, we have
$ ε3nhn∑x∈Λ′n∘dist2(σnˉ∇wn(x),SO(3)Z)≤Ch4n $ |
for an
$ \int_\Omega {\rm{dist}}^2( \nabla_n \tilde{ \tilde{{y}} }(x), {\rm{SO}}(3)Z)\,dx \leq C h_n^4. $ |
Now we can directly apply Theorems 4.1 and 4.2 for the continuum objects
$ un⇀uinW1,2(S;R2),vn→vinW1,2(S;R). $ | (38) |
For later we also introduce
We will also use the following finer statement.
Proposition 2. In the setting of Theorem 4.2, applied to
$ 1h2n((˜yn)′−x′)=:ˆun⇀ˆuinW1,2(Ω;R2), $ | (39) |
$ 1hn(˜yn)3=:ˆvn⇀ˆvinW1,2(Ω), $ | (40) |
where
$ ˆu(x)=u(x′)−(x3−12)∇′v(x′), $ | (41) |
$ ˆv(x)=v(x′)+(x3−12). $ | (42) |
Proof. According to Korn's inequality
$ ‖ˆun‖W1,2(Ω;R2)≤C(‖sym∇′ˆun‖L2(Ω;R2×2)+‖∂ˆun∂x3‖L2(Ω;R2)+|∫Ωskew∇′ˆundx|+|∫Ωˆundx|). $ |
According to Theorem 4.2,
$ ∂(ˆun)i∂x3=1hn(∇n˜yn−Id)i3, $ |
$ ∫10ˆundx3⇀uinW1,2(S;R2), $ |
by (28) and
$ ∂(ˆun)i∂x3=1hn(∇n˜yn−Id)i3→−∂v∂xiinL2(Ω), $ |
for
(26) and (29) in Theorem 4.2 also show that
$ \int_0^1 \hat{v}_n \,dx_3 \to v. $ |
As a first consequence, we will now describe the limiting behavior of the force term
Note that the forces considered are a bit more general than in [13].
Proposition 3. Let
$ \frac{ \varepsilon_n^3}{h_n^5} E_{\rm body}(y_n) \to {∫Sf(x′)⋅v(x′)R∗e3dx′,ifνn→∞,νν−1∫Sf(x′)⋅v(x′)R∗e3dx′,ifνn=νconstant, $ |
as
Proof. In terms of the extended and interpolated force density we have
$ ε3nh5nEbody(yn)=1h4n∫˜Voutnˉfn(x)⋅ˉˉyn(x)dx=1h4n∫˜Voutnˉfn(x)⋅(ˉˉyn(x)−R∗n(x′0)−R∗ncn)dx=∫˜Voutnh−3nR∗nTˉfn(x)⋅h−1n(ˉyn−(x′0))dx. $ |
By Proposition 2,
$ ε3nh5nEbody(yn)→∫ΩR∗Tf(x)⋅ˆv(x)e3dx=∫Ωf(x′)⋅v(x′)R∗e3dx′ $ |
if
$ ε3nh5nEbody(yn)→1ν−1ν−1∑j=0∫SR∗Tf(x′)⋅ˆv(x′,jν−1)e3dx′=νν−1∫Sf(x′)⋅v(x′)R∗e3dx′ $ |
with an analogous argument for the last step.
To show the lower bounds in our
$ ˉGn:=1h2n(RTnˉ∇nˉyn−Z). $ |
By Proposition 1
$ \frac{1}{h_n^2} (R_n^T \nabla_n \tilde{y}_n - {\rm{Id}}) \rightharpoonup G {\rm{\;in\;}} L^2(\Omega; \mathbb{R}^{3 \times 3}), $ |
where
For the discussion of discrete strains, recall that we defined
$ Z−=(−z1,−z2,−z3,−z4,+z5,+z6,+z7,+z8),M=12e3⊗(+1,−1,+1,−1,+1,−1,+1,−1). $ |
We define a projection
$ Pf(x) = \mathit{{\rlap{-} \smallint }}_{(k-1)/(\nu-1)}^{k/(\nu-1)} f(x',t) \, dt \qquad {\rm{\;if\;}} \qquad \tfrac{k-1}{\nu-1} \le x_3 < \tfrac{k}{\nu-1} $ |
in case
Proposition 4. Let
$ \bar{G}_n \rightharpoonup \bar{G} : = {GZ,ifνn→∞,PGZ+12(ν−1)G3,ifνn≡ν∈N, $ |
in
Proof. The compactness follows from Theorem 4.2. On a subsequence (not relabeled) we thus find
$ RnˉGn=1h2n(ˉ∇nˉyn−RnZ)⇀ˉG. $ |
We have
$ \lim\limits_{n \to \infty} \frac{1}{h_n^2} (R_n^T \nabla_n \tilde{y}_n - {\rm{Id}}) = \lim\limits_{n \to \infty} \frac{1}{h_n^2} (\nabla_n \tilde{y}_n - R_n) = G, $ |
weakly in
In order to discuss the discrete strains in more detail, we separate affine and non-affine contributions. We say that a
We begin by identifying the easier to handle affine part of the limiting strain. By construction we have
$ \bar{\nabla}_n \bar{y}_n (x) b^1 = \frac{1}{2 \varepsilon_n} \big( (y_2 + y_3 + y_6 + y_7) - ( y_1 + y_4 + y_5 + y_8) \big), $ |
where
$ ˉ∇nˉyn(x)b1=2εn−∫x+{−εn2}×(−εn2,εn2)×(−εn2hn,εn2hn)˜yn(ξ+εne1)−˜yn(ξ)dξ=2−∫˜Qn(x)∂1˜yn(ξ)dξ. $ |
Analogous arguments yield
$ ˉ∇nˉyn(x)b2=2−∫˜Qn(x)∂2˜yn(ξ)dξandˉ∇nˉyn(x)b3=2hn−∫˜Qn(x)∂3˜yn(ξ)dξ. $ |
By
$ Pn[RnˉGn]bi=2h2nPn[∂i˜yn−Rnei]⇀2PGei=PGZbi,i=1,2 $ |
and
$ Pn[RnˉGn]b3=2h2nPn[h−1n∂3˜yn−Rne3]⇀2PGe3=PGZb3. $ |
In summary we get that for every affine
$ ˉGb=PGZb. $ | (43) |
For the discussion of the non-affine part of the strain we fix a non-affine
$ \bar{\nabla}^{\rm 2dim} f (x) : = \frac{1}{ \varepsilon_n} \Big( f(x' + \varepsilon_n (z^i)', x_3) - \frac{1}{4} \sum\limits_{j = 1}^4 f(x' + \varepsilon_n (z^j)', x_3) \Big)_{i = 1,2,3,4}. $ |
The idea is now to separate differences into in-plane and out-of-plane differences, as all in-plane differences are infinitesimal, while out-of-plane differences stay non-trivial if
Using
$ ˉ∇nˉyn(x)=(ˉ∇2dimnˉyn(x−εn2hne3),ˉ∇2dimnˉyn(x+εn2hne3))+12hn−∫˜Qn(x)∂3˜yn(ξ)dξ⊗(−1,−1,−1,−1,+1,+1,+1,+1) $ |
we find
$ RnˉGn(x)b=1h2nˉ∇nˉyn(x)b=1h2n(ˉ∇2dimnˉyn(x+εn2hne3)−ˉ∇2dimnˉyn(x−εn2hne3))b(2) $ | (44) |
$ +1h2nˉ∇2dimnˉyn(x−εn2hne3)(b(1)+b(2)), $ | (45) |
where we have used that
First consider the term (45). Since
$ 1h2neTi∫Ωˉ∇2dimn(ˉyn−¯id)(x−εn2hne3)(b(1)+b(2))φ(x)dx=1h2neTi∫Ω(ˉyn−¯id)(x−εn2hne3)(ˉ∇2dimn)∗φ(x)(b(1)+b(2))dx→−∫Ωˆui(˜x)∇′φ(x)Z2dim(b(1)+b(2))dx=0, $ | (46) |
where, either
For the third component, we instead have
$ 1h2neT3∫Ωˉ∇2dimnˉyn(x−εn2hne3)(b(1)+b(2))φ(x)dx=1hnεn(νn−1)eT3∫Ω(ˉ∇2dimnˉyn(x−εn2hne3)−∇′nˉyn(x−εn2hne3)Z2dim)(b(1)+b(2))φ(x)dx=1(νn−1)εn∫Ω(ˉyn)3(x−εn2hne3)hn((ˉ∇2dimn)∗φ(x)+∇′nφ(x)Z2dim)(b(1)+b(2))dx. $ |
Now,
$ 1εn((ˉ∇2dimn)∗φ(x)+∇′nφ(x)Z2dim)→(12∇′2φ(x)[(zi)′,(zi)′]−184∑j=1∇′2φ(x)[(zj)′,(zj)′])i=1,...,4 $ |
uniformly. Therefore, (40) gives
$ 1h2neT3∫Ωˉ∇2dimnˉyn(x−εn2hne3)(b(1)+b(2))φ(x)dx→0, $ | (47) |
if
$ 1h2neT3∫Ωˉ∇2dimnˉyn(x−εn2hne3)(b(1)+b(2))φ(x)dx→1(ν−1)∫Ωˆv(x′,⌊(ν−1)x3⌋ν−1)(12∇′2φ(x)[(zi)′,(zi)′])i=1,...,4(b(1)+b(2))dx=1(ν−1)∫Ω(12∇′2v(x′)[(zi)′,(zi)′])i=1,...,4(b(1)+b(2))φ(x)dx, $ | (48) |
where we have used that
We still need to find the limit of (44). For any test function
$ ∫Ω1h2n(ˉ∇2dimnˉyn(x+εn2hne3)−ˉ∇2dimnˉyn(x−εn2hne3))b(2)⋅φ(x)dx=εnhn∫Ω1εnhn(ˉyn(x+εn2hne3)−ˉyn(x−εn2hne3))⋅(ˉ∇2dimn)∗Pnφ(x)b(2)dx=1h2n∫Ω−∫˜Qn(x)(ˉyn(ξ+εn2hne3)−ˉyn(ξ−εn2hne3))dξ⋅(ˉ∇2dimn)∗Pnφ(x)b(2)dx=εnh3n∫Ω∂3˜yn(x)⋅(ˉ∇2dimn)∗Pnφ(x)b(2)dx=εnhn∫ΩPnAn(x)e3⋅(ˉ∇2dimn)∗φ(x)b(2)dx. $ |
Here the penultimate step is true by our specific choice of interpolation to define
$ limn→∞∫Ω1h2n(ˉ∇2dimnˉyn(x+εn2hne3)−ˉ∇2dimnˉyn(x−εn2hne3))b(2)⋅φ(x)dx=−1ν−1∫ΩPA(x)e3⋅∇′φ(x)Z2dimb(2)dx=1ν−1∫Ω(∂1v(x′),∂2v(x′),0)∇′φ(x)Z2dimb(2)dx=−1ν−1∫Ω(∇′2v(x′)Z2dimb(2)0)⋅φ(x)dx. $ | (49) |
Summarizing (46), (47), (48), and (49), we see that for non-affine
$ ˉGb=(−1ν−1∇′2v(x′)Z2dimb(2)1ν−1∑4i=112∇′2v(x′)[(zi)′,(zi)′](b(1)+b(2))i)=(−12(ν−1)∇′2v(x′)Z2dim(b(2)−b(1))12(ν−1)∑8i=1∇′2v(x′)[(zi)′,(zi)′]bi)−18(ν−1)Δv(x′)8∑j=1bje3 $ |
as
Elementary computations show that for the affine basis vectors
$ Z2dim((bk)2−(bk)1)=0 $ |
and also
$ 8∑i=1∇′2v(x′)[(zi)′,(zi)′]bki−14Δv(x′)8∑j=1bkj=0. $ |
Thus combining with (43), for every
$ \bar{G}b = G Z b $ |
if
$ \bar{G}b = P G Z b + (−12(ν−1)∇′2v(x′)Z2dim(b(2)−b(1))12(ν−1)∑8i=1∇′2v(x′)[(zi)′,(zi)′]bi) - \frac{1}{8(\nu-1)} \Delta v(x') \sum\limits_{j = 1}^8 b_j e_3. $ |
if
$ ˉG=PGZ−12(ν−1)(∇′2v(x′)000)Z−+12(ν−1)e3⊗(∇′2v(x′)[(zi)′,(zi)′])i=1,…,8−18(ν−1)Δv(x′))e3⊗(1,…,1). $ |
with
$ \nabla'^2 v(x') [(z^i)',(z^i)'] = {14(∂11v(x′)+2∂12v(x′)+∂22v(x′))ifi∈{1,3,5,7},14(∂11v(x′)−2∂12v(x′)+∂22v(x′))ifi∈{2,4,6,8}, $ |
with
$ ˉG=PGZ−12(ν−1)(∇′2v(x′)000)Z−+12(ν−1)∂12v(x′)M. $ |
Last, we note that subsequences were indeed not necessary, as the limit is characterized uniquely.
Having established convergence of the strain, the
Proof of the
$ W_{\rm cell}'(A) = {Wcell(A),ifdist(A,SO(3)Z)<δ,dist2(A,SO(3)Z),ifdist(A,SO(3)Z)≥δ. $ |
Furthermore, in view of Proposition 3 it suffices to establish the lower bound for
Assume that
$ \sup\limits_n E_n (y_n) < \infty $ |
so that by Proposition 1 its modification and interpolation
$ ˉGn:=1h2n(RTnˉ∇nˉyn−Z). $ |
By frame indifference and nonnegativity of the cell energy we have
$ ε3nh5nEn(yn)≥ε3nh5n∑x∈(˜Λ′n)∘W((x′,hnx3),ˉ∇nˉyn(x))=1h4n∫ΩinnW(εn(⌊x1εn⌋+12,⌊x2εn⌋+12,⌊hnx3εn⌋+12),Z+h2nˉGn(x))dx. $ |
First assume that
$ ε3nh5nEn(yn)≥1h4n∫Ωχn(x)Wcell(Z+h2nˉGn(x))dx=∫Ω12Qcell(χn(x)ˉGn(x))−h−4nχn(x)ω(|h2nˉGn(x)|)dx, $ |
where
$ \omega(t) : = \sup \big\{ | \tfrac{1}{2} Q_{\rm cell}(F) - W_{\rm cell}(Z + F) | : F \in \mathbb{R}^{3 \times 8} {\rm{\;with\;}} |F| \le t \big\} $ |
so that
$ \chi_n (h_n^2 \bar{G}_n)^{-2} \omega(h_n^2 \bar{G}_n) \to 0 $ |
uniformly,
$ h_n^{-4} \chi_n \omega \big( h_n^2 \bar{G}_n \big) = \bar{G}_n^2 \chi_n (h_n^2 \bar{G}_n)^{-2} \omega(h_n^2 \bar{G}_n) \to 0 {\rm{\;in\;}} L^1(\Omega; \mathbb{R}^{3 \times 8}). $ |
Moreover,
$ lim infn→∞ε3nh5nEn(yn)≥12∫ΩQcell(ˉG(x))dx≥12∫ΩQrelcell(ˉG(x))dx=12∫ΩQrelcell((G1(x′)+(x3−12)G2(x′)000)Z)dx. $ |
Integrating the last expression over
$ lim infn→∞ε3nh5nEn(yn)≥EvK(u,v). $ |
Now suppose that
$ ω(t):=sup{|12Qcell(F)−Wcell(Z+F)|:F∈R3×8with|F|≤t}+2sup{|12Qsurf(F)−Wsurf(Z(1)+F)|:F∈R3×4with|F|≤t} $ |
so that still
$ lim infn→∞ε3nh5nEn(yn)≥12∫ΩQcell(ˉG(x))dx+12(ν−1)∫SQsurf(ˉG(1)(x′,12(ν−1)))+Qsurf(ˉG(2)(x′,2ν−32ν−2))dx, $ |
where we have used that
$ ˉG(1)(x′,12ν−2)=−∫1ν−10G(x′,x3)dx3Z(1)+12(ν−1)G(1)3(x′),ˉG(2)(x′,2ν−32ν−2)=−∫1ν−2ν−1G(x′,x3)dx3Z(2)+12(ν−1)G(2)3(x′). $ |
The bulk part is estimated as
$ 12∫12−12Qcell(ˉG(x))dx3≥12(ν−1)ν−1∑k=1Qrelcell((sym(PG″)(x′,2k−12ν−2)000)Z+12(ν−1)G3(x′))=12(ν−1)ν−1∑k=1Qrelcell((symG1(x′)+2k−ν2ν−2G2(x′)000)Z+12(ν−1)G3(x′))=12(ν−1)ν−1∑k=1[Qrelcell((symG1(x′)000)Z+12(ν−1)G3(x′))+(2k−ν)2(2ν−2)2Qrelcell((G2(x′)000)Z)]=12Qrelcell((symG1(x′)000)Z+12(ν−1)G3(x′))+ν(ν−2)24(ν−1)2Qrelcell((G2(x′)000)Z), $ |
where we have used that
For the surface part first note that by (8), for any
$ Qsurf(AZ(1)+B)=Qsurf(AZ(1)+B+(a3⋅⊗e3−e3⊗a3⋅)Z(1)+(a⋅3+a3⋅)⊗(1,1,1,1))=Qsurf((A″000)Z(1)+B)=Qsurf((symA″000)Z(1)+B), $ |
where
$ Qsurf(AZ(2)+B)=Qsurf(AZ(1)+a⋅3⊗(1,1,1,1)+B)=Qsurf((symA″000)Z(1)+B). $ |
It follows that
$ Qsurf(ˉG1(x′,12ν−2))=Qsurf((symG1(x′)−ν−22ν−2G2(x′)000)Z(1)+12(ν−1)G(1)3(x′))=Qsurf((symG1(x′)−12G2(x′)000)Z(1)+∂12v(x′)4(ν−1)M(1)),Qsurf(ˉG2(x′,2ν−32ν−2))=Qsurf((symG1(x′)+ν−22ν−2G2(x′)000)Z(1)+12(ν−1)G(2)3(x′))=Qsurf((symG1(x′)+12G2(x′)000)Z(1)+∂12v(x′)4(ν−1)M(1))), $ |
and so
$ Qsurf(ˉG1(x′,12ν−2))+Qsurf(ˉG2(x′,2ν−32ν−2))=2Qsurf((symG1(x′)000)Z(1)+∂12v(x′)4(ν−1)M(1))+12Qsurf((G2(x′)000)Z(1)), $ |
Adding bulk and surface contributions and integrating over
$ lim infn→∞ε3nh5nEn(yn)≥∫S12Qrelcell((symG1(x′)000)Z+12(ν−1)G3(x′))+ν(ν−2)24(ν−1)2Qrelcell((G2(x′)000)Z)+1ν−1Qsurf((symG1(x′)000)Z(1)+∂12v(x′)4(ν−1)M(1))+14(ν−1)Qsurf((G2(x′)000)Z(1))dx′=E(ν)vK(u,v). $ |
Note that in the Theorem 2.1 the skew symmetric part of
Without loss of generality we assume that
If
$ yn(x)=(x′hnx3)+(h2nu(x′)hnv(x′))−h2n(x3−12)((∇′v(x′))T0)+h3nd(x′,x3) $ | (50) |
for all
We let
In order to estimate the energy of
$ ˉDiyn(x)=1εn[yn(ˆx+εn((ai)′,h−1nai3))−yn(ˆx)], $ |
where for
$ \hat{x} = \big( \varepsilon_n \lfloor \tfrac{x_1}{ \varepsilon_n} \rfloor, \varepsilon_n \lfloor \tfrac{x_2}{ \varepsilon_n} \rfloor, \tfrac{\lfloor (\nu_n-1) x_3 \rfloor}{\nu_n-1} \big), $ |
so that
$ ˉDiyn(x)=ˉ∂iˉyn(x)−ˉ∂1ˉyn(x)andˉ∂iˉyn(x)=ˉDiyn(x)−188∑j=1ˉDjyn(x). $ | (51) |
In particular, if
$ ˉ∂iˉyn(x)=Fai−188∑j=1Faj=F(ai−12(1,1,1)T)=Fzi $ | (52) |
and so
For
$ ˉDiyn(x)=∇′yn(ˆx)(ai)′+h−1n∂3yn(ˆx)ai3+εn2(∇′)2yn(ˆx)[(ai)′,(ai)′]+εnh−1n2∑j=1∂j3yn(ˆx)aijai3+εnh−2n2∂33yn(ˆx)(ai3)2+ε2n6∇3((yn)1(ζ1εn),(yn)2(ζ2εn),(yn)2(ζ2εn))T[((ai)′,h−1nai3),((ai)′,h−1nai3),((ai)′,h−1nai3)] $ |
for some
$ ˉDiyn(x)=((Id2×20)+(h2n∇′u(ˆx′)hn∇′v(ˆx′))−h2n(ˆx3−12)(∇′(∇′v(ˆx′))T0)+h3n∇′d(ˆx))(ai)′+h−1n((0hn)+0−h2n((∇′v(ˆx′))T0)+h3n∂3d(ˆx))ai3+εnhn2(0(∇′)2v(ˆx′)[(ai)′,(ai)′])+O(εnh2n)−εnhn(∇′(∇′v(ˆx′))T0)(ai)′ai3+O(εnh2n)+εnhn2∂33d(ˆx)(ai3)2+ε2n6∂333(d1(ζ1εn),d2(ζ2εn),d3(ζ3εn))T(ai3)3+O(ε2nhn). $ |
It follows that
$ ˉDiyn(x)=(Id3×3+hn(hn∇′u(ˆx′)−(∇′v(ˆx′))T∇′v(ˆx′)0)−h2n(ˆx3−12)((∇′)2v(ˆx′)000)+h2n(03×2∂3d(ˆx)))ai+εnhn((−(∇′)2v(ˆx′)(ai)′ai312(∇′)2v(ˆx′)[(ai)′,(ai)′])+12∂33d(ˆx)(ai3)2)+ε2n6∂333(d1(ζ1εn),d2(ζ2εn),d3(ζ3εn))T(ai3)3+O(εnh2n+ε2nhn). $ |
We define the skew symmetric matrix
$ B(ˆx)=(h2n2(∇′u(ˆx′)−(∇′u(ˆx′))T)−hn(∇′v(ˆx′))Thn∇′v(ˆx)0)+h2n2(02×2∂3d′(ˆx)−(∂3d′(ˆx))T0), $ |
where we have written
$ e−B(ˆx)=Id3×3−B(ˆx)+12B2(ˆx)+O(|B(ˆx)|3)=Id3×3−hn(02×2−(∇′v(ˆx′))T∇′v(ˆx′)0)−h2n2(∇′u(ˆx′)−(∇′u(ˆx′))T+∇′v(ˆx′)⊗∇′v(ˆx′)∂3d′(ˆx)−(∂3d′(ˆx))T|∇′v(ˆx′)|2)+O(|hn|3). $ |
Now compute
$ \begin{array}{l} e^{-B(\hat{x})} \bar{D}_i y_n(x) = \bar{D}_i y_n(x) - h_n \left(\begin{array}{cc} 0_{2 \times 2} & -\left(\nabla^{\prime} v\left(\hat{x}^{\prime}\right)\right)^{T} \\ \nabla^{\prime} v\left(\hat{x}^{\prime}\right) & 0 \end{array}\right)\\ \qquad \qquad \left(\mathrm{Id}_{3 \times 3}+h_{n}\left(02×2−(∇′v(ˆx′))T∇′v(ˆx′)0\right)\right) a^{i} \\ \qquad \qquad-\frac{h_{n}^{2}}{2}\left(∇′u(ˆx′)−(∇′u(ˆx′))T+∇′v(ˆx′)⊗∇′v(ˆx′)∂3d′(ˆx)−(∂3d′(ˆx))T|∇′v(ˆx′)|2\right) a^{i} \\ \qquad + O(h_n^3 + \varepsilon_n h_n^2 + \varepsilon_n^2 h_n) \\ = \bigg( {\rm{Id}}_{3\times3} + h_n^2 (sym∇′u(ˆx′)+12∇′v(ˆx′)⊗∇′v(ˆx′)0012|∇′v(ˆx′)|2) \\ \qquad - h_n^2 (\hat{x}_3 - \tfrac{1}{2}) ((∇′)2v(ˆx′)000) + h_n^2 (02×212∂3d′(ˆx)12(∂3d′(ˆx))T∂3d3(ˆx)) \bigg) a^i \\ \qquad + \varepsilon_n h_n \left( (−(∇′)2v(ˆx′)(ai)′ai312(∇′)2v(ˆx′)[(ai)′,(ai)′]) + \tfrac{1}{2} \partial_{33} d(\hat{x}) (a^i_3)^2 \right) \\ \qquad + \frac{ \varepsilon_n^2}{6} \partial_{333} \big( d_1(\zeta^1_{ \varepsilon_n}), d_2(\zeta^2_{ \varepsilon_n}), d_3(\zeta^3_{ \varepsilon_n}) \big)^T (a^i_3)^3 + O(h_n^3 + \varepsilon_n h_n^2 + \varepsilon_n^2 h_n). \end{array} $ |
(53) |
Here, the error term is uniform in
We can now conclude the proof of Theorems 2.1, 2.2 and 2.3.
Proof of the
We first specialize now to the case
$ G(x)=G1(x′)+(x3−12)G2(x′)=sym∇′u(x′)+12∇′v(x′)⊗∇′v(x′)−(x3−12)(∇′)2v(x′). $ | (54) |
choosing
$ d0(x′)=argminb∈R3Qcell[(G1(x′)0012|∇′v(x′)|2)Z+(b⊗e3)Z],d1(x′)=argminb∈R3Qcell[(G2(x′)000)Z+(b⊗e3)Z] $ | (55) |
according to (9), from (52) and (53) we obtain
$ e−B(ˆx)ˉ∇ˉyn(x)=(Id3×3+h2n(G(ˆx)0012|∇′v(ˆx′)|2)+h2nsym((d0(ˆx)+(ˆx3−12)d1(ˆx))⊗e3))Z+O(h3n+εnhn) $ |
and, Taylor expanding
$ 12Qrelcell((G000)Z)=12Q2(G). $ |
This shows that
$ limn→∞h−4nEn(yn)=12∫SQ2(G(x))dx=∫S12Q2(G1(x′))+124Q2(G2(x′))dx′=EvK(u,v) $ |
and thus finishes the proof in case
Now suppose that
$ (2G2(ai)′ai3−(ai)′TG2(ai)′)i=1,…,8=(00000−2f11−2f11−2f12−2f1200000−2f21−2f21−2f22−2f220f11∑μ,νfμνf220f11∑μ,νfμνf22), $ |
and hence, with
$ (2G2(ai)′ai3−(ai)′TG2(ai)′)i=1,…,8−(e3⊗b−b⊗e3)A=(0000f11+f12−f11+f12−f11−f12+f11−f120000f21+f22−f21+f22−f21−f22f21−f220−f120−f210−f120−f21),=(G2000)(Z+Z−)+12f12(2M−e3⊗(1,…,1))=(G2000)A−12b⊗(1,…,1)+(G2000)Z−+f122(2M−e3⊗(1,…,1)). $ |
This shows that
$ (−(∇′)2v(ˆx′)(ai)′ai312(∇′)2v(ˆx′)[(ai)′,(ai)′])i=1,…,8=12(e3⊗b−b⊗e3+(G2000))A−14(b+e3)⊗(1,…,1)+12(G2000)Z−+12f12M. $ |
We define the affine part of the strain
$ e−B(ˆx)ˉ∇ˉyn(x)=[Id3×3+h2n(G(ˆx′,ˆx3+12(ν−1))0012|∇′v(ˆx′)|2)+h2nsym(∂3d(ˆx))⊗e3)+h2n2(ν−1)(e3⊗b(ˆx′)−b(ˆx′)⊗e3)]Z+h2n2(ν−1)G3(ˆx′)+O(h3n)+[εnhn2∂33d(ˆx)+ε2n6∂333(d1(ζ1εn),d2(ζ2εn),d3(ζ3εn))T]⊗(z13,…,z83), $ |
where we have used (52) and (51).
We set
$ d0(x′)=argmind∈R3Qcell[(G1(x′)0012|∇′v(x′)|2)Z+sym(d⊗e3)Z+12(ν−1)G3(x′)],d1(x′)=argmind∈R3Qcell[(G2(x′)000)Z+sym(d⊗e3)Z] $ |
according to (9) and define
$ d(x′,j−1ν−1+t)=d(x′,j−1ν−1)+td0(x′)+t2j−ν2(ν−1)d1(x′)ift∈[j−1ν−1,jν−1], $ | (56) |
for
$ \partial_3 d(x) = d_0(x') + \tfrac{2j-\nu}{2(\nu-1)} d_1(x') = d_0(x') + (\hat{x}_3 - \tfrac{1}{2} + \tfrac{1}{2(\nu-1)}) d_1(x') $ |
since
$ 12Qrelcell((G1(x′)+2j−ν2(ν−1)G2(x′)000)Z+12(ν−1)G3(x′)) $ |
for each
$ 1h4n∫˜ΩoutnWcell(ˉ∇ˉyn(x))dx→∫S12Qrelcell((G1(x′)000)Z+12(ν−1)G3(x′))+ν(ν−2)24(ν−1)2Qrelcell((G2(x′)000)Z)dx′. $ | (57) |
For the surface part we write
$ x \mapsto h_n^{-4} W_{\rm surf} ([\bar{\nabla}\bar{y}_n(x)]^{(1)}) = h_n^{-4} W_{\rm surf} ([e^{-B(\hat{x})} \bar{\nabla}\bar{y}_n(x)]^{(1)}), $ |
converge uniformly to
$ 12Qsurf((G1(x′)−ν−22(ν−1)G2(x′)000)Z+12(ν−1)G3(x′))=12Qsurf((symG1(x′)−12G2(x)000)Z(1)+∂12v(x′)4(ν−1)M(1)). $ |
Similarly, the mappings
$ x \mapsto h_n^{-4} W_{\rm surf} ([\bar{\nabla}\bar{y}_n(x)]^{(2)}) = h_n^{-4} W_{\rm surf} ([e^{-B(\hat{x})} \bar{\nabla}\bar{y}_n(x)]^{(2)}), $ |
converge uniformly to
$ 12Qsurf((symG1(x′)+12G2(x)000)Z(1)+∂12v(x′)4(ν−1)M(1)). $ |
So with
$ 1h4n(ν−1)∫SoutnWsurf([ˉ∇ˉyn(x′,12(ν−1))](1))+Wsurf([ˉ∇ˉyn(x′,2ν−32(ν−1))](2))dx′→∫S1ν−1Qsurf((symG1(x′)000)Z(1)+∂12v(x′)4(ν−1)M(1))+14(ν−1)Qsurf((G2(x)000)Z(1))dx′. $ | (58) |
Summarizing (58) and (57), we have shown that
$ limn→∞h−4nEn(yn)=limn→∞ε3nh−5n∑x∈˜Λ′nW(x,ˉ∇yn(x))=E(ν)vK(u,v) $ |
as
Proof of the energy barrier in Theorem 2.3. If a sequence of
$ {\rm{dist}}^2(\bar{\nabla} w_n(x),{\rm{SO}}(3)Z) \le C E_{\rm atom} (w_n) \le C h_n^5 \varepsilon_n^{-3} = C (\nu_n-1)^5 \varepsilon_n^2, $ |
which tends to
This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under project number 285722765, as well as the Engineering and Physical Sciences Research Council (EPSRC) under the grant EP/R043612/1.
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