Heterogeneous Thin Films of Martensitic
Materials
Y. C. Shu
Communicated by S. M ¨uller
Abstract
We study the effective behavior of heterogeneous thin films with three com-peting length scales: the film thickness and the length scales of heterogeneity and material microstructure. We start with three-dimensional nonhomogeneous nonlin-ear elasticity enhanced with an interfacial energy of the van der Waals type, and derive the effective energy density as all length scales tend to zero with given limit-ing ratios. We do not require any a priori selection of asymptotic expansion or ansatz in deriving our results. Depending on the dominating length scale, the effective en-ergy density can be identified by three procedures: averaging, homogenization and thin-film limit. We apply our theory to martensitic materials with multi-well energy density and use a model example to show that the “shape-memory behavior” can crucially depend on the ratios of these length scales. We comment on the effective conductivity of linear composites, and also on multilayers made of shape-memory and elastic materials.
1. Introduction
Martensitic thin films have recently attracted much interest because of their potential for application as microactuators [27, 28, 39, 33, 18, 17]. Martensitic ma-terials undergo a diffusionless phase transformation during which there is a sud-den change in the crystal structure at a certain temperature. The high temperature
austenite phase is cubic while the low temperature martensite phase has less
sym-metry. This gives rise to symmetry-related variants and these variants usually form microstructures or fine-scale mixtures. Crystals undergoing a thermoelastic marten-sitic transformation often exhibit the shape-memory effect. Below the transforma-tion temperature, they are extremely malleable – sustaining a huge deformatransforma-tion with strains as large as 10% under very small forces. When they are heated above the transformation temperature, the specimen springs back to its original shape as
κ
h
d
S
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Fig. 1. A heterogeneous thin film with three different length scales.
all the strain is recovered. Actuators utilizing the shape-memory effect are predicted to have the largest energy output per unit volume per cycle among a variety of com-mon actuator systems [33]. But bulk shape-memory actuators have enjoyed limited success in temperature sensitive applications because the response is slow due to thermal inertia. On the other hand, the enhanced rate of heat transfer in thin films makes these alloys ideal for microactuators, micropumps and microelectromechan-ical system (MEMS) applications.
Typically, martensitic films are polycrystalline rather than monocrystalline. A polycrystal consists of a large number of single crystal grains with different orien-tations. The behavior of a polycrystal can be very different from that of a single crystal because of the constraining effect of neighboring grains. Depending on the deposition technique, the size of grains within the film can be larger than, com-parable to or smaller than the thickness of film. Furthermore, depending on the material, the length scale of the microstructure can also be larger than, comparable to or smaller than that of grains. The behavior of the film can critically depend on the relative magnitudes of these length scales, and we seek to understand this.
Consider a heterogeneous (possibly multilayer) thin film shown in Fig. 1. It occupies a reference domain
h= {x ∈ R3: (x
1, x2) ∈ S, 0 < x3< h}, (1.1) whereS is a bounded Lipschitz domain, {x1, x2, x3} are relative to an orthonor-mal film basis{e1, e2, e3}, and h is the film thickness. Let ˜y : h → R3 be the deformation of the film. The total energy of the heterogeneous thin film is
˜e(h)[˜y] = Z h n κ2|∇2˜y|2+ ϕ∇ ˜y,x1 d , x2 d , x3 h o dx (1.2) whereϕ : M3×3× R2× (0, 1) → R is the elastic free energy density of the film andMm×nis the set of allm × n matrices. We assume that ϕ is periodic in the
in-plane variablesx1andx2with period[0, 1]2. Sod scales like the typical grain size. Further, since we wish to model martensites,ϕ(F, ·, ·, ·) may have a multi-well structure and consequently nonconvex energy densities. Note that we have included the interfacial energy of the typeκ2|∇2˜y|2. Minimizers of the energy (1.2) have oscillations on a length scale that scales withκ and hence we call κ the length scale of the microstructure. We are interested in finding the limiting behavior of the film when all length scalesκ, d and h tend to zero. Therefore, we take
κ = κ(h) > 0, d = d(h) > 0, lim
h→0κ(h) = 0, h→0limd(h) = 0, (1.3)
and assume that they have fixed limiting ratios:
α = lim h→0 κ d, β = limh→0 h d, α0= limh→0 κ h. (1.4)
In bulk materials, the homogenization of cellular elastic materials with non-convex energy densityϕ has been studied by Braides [13] and M ¨uller [36]. The same problem including the interfacial energy has been studied by Francfort &
M ¨uller [24]. However, microstructure in thin films can be significantly different
to that in bulk materials, endowing materials with dramatically distinct properties (for example, see [3]). Recently, Bhattacharya & James [9] have developed a theory of single crystal martensitic thin films which captures this effect. Related work on the modeling of thin structures with convex (quadratic) energy density includes, for example, Kohn & Vogelius [31, 32], Damlamian & Vogelius [20] and Caillerie [16]; and related problems with nonconvex energy density include, for example, Acerbi et al. [1], Le Dret & Raoult [21] and Fonseca &
Franc-fort [22]. We wish to combine homogenization with the thin-film analysis for
non-convex energies and apply it to heterogeneous martensitic films. Braides, Fonseca
& Francfort [15] have studied a similar problem withκ = 0.
Our approach is variational. We study the “variational limit” of (1.2) ash tends to zero. Since the energy defined in (1.2) scales likeh as h tends to zero, we shall be interested in the limiting energy per unit thickness, i.e.,
˜e(h)1 = 1
h˜e(h).
We expect the minimum values and the minimizers of the functional ˜e(h)1 to converge to those of a “limiting energy”˜e1(0), which we try to find. In this context, the natural tool is0-convergence as proposed by De Giorgi [25] and De Giorgi
& Franzoni [26] which under a suitable technical hypothesis is nearly identical
to that of convergence of minimizers (see also Remark 1). Using this notion, we show that the limiting energy is always given by
˜e1(0)[y] = Z S ¯ϕ ∂y ∂x1, ∂y ∂x2 dx1dx2,
where¯ϕ is the effective energy density and only depends on the in-plane gradient of deformation y and not explicitly on the position. It describes the overall behavior of
Table 1. Summary of the effective behavior of a heterogeneous thin film. A means averaging, H means homogenizing and T means thin-film limit. TH denotes that the effective energy
density¯ϕ is obtained by taking the thin-film limit first, and then homogenizing in the plane of the film. On the other hand, HT denotes homogenization first followed by the thin-film limit. Finally, a stacked symbolH
Tdenotes the simultaneous performance of these two operations.
( ? ) A T T H T H A H T Theorem 2 Theorem 3 Theorem 3 H H T Theorem 2 A A T Theorem 4 Theorem 1 Theorem 1 A T Theorem 1 A T Theorem 5 T H κ ∼ h κ ∼ h h ∼ d h d h d κ ∼ d κ d κ d
the heterogeneous thin film after taking into account the martensitic microstructure, grains and multilayers.
In the following, we give a non-technical description of our main results which are summarized in Table 1. The most important finding is that the effective energy density ¯ϕ crucially depends on the limiting ratios of these three length scales. 1. Strong interfacial energy (κ >> d). Assume ϕ = ϕ(F,x1
d,xd2). Our Theorem 1
shows that the effective energy density ¯ϕ is obtained by averaging the micro-scopic energyϕ over the period, then passing to the thin-film limit. It costs materials more energy to form microstructures within each grain as a result of strong interfacial energy. Material is internally stressed. The result is also true ifϕ = ϕ(F,x1
d,xd2,xh3) and if κ >> d and κ >> h.
2. Flat grains (d >> h). Assume ϕ = ϕ(F,x1
d,xd2). If the length scale of the
microstructure is much smaller than that of grains (i.e., ifκ << d), then The-orem 2(i) shows that the elastic energy dominates the interfacial energy and materials can form microstructures freely. As a result, the macroscopic energy density ¯ϕ is impervious to the presence of interfacial energy. Further, ¯ϕ is ob-tained by taking the thin-film limit first, and then homogenizing in the plane of the film. The thin-film limit says that only the in-plane compatibility is im-portant and this allows a wider class of microstructures to be formed in thin films than in bulk materials. On the other hand, if the length scales of grains and microstructure are of the same order of magnitude (i.e., ifκ ≈ d), Theorem 2(ii) shows that the interfacial energy explicitly contributes to the effective energy
density ¯ϕ. Materials can form only a limited amount of microstructure because of competing energies between elastic energy and interfacial energy.
3. Comparable grains (d ≈ h). Assume ϕ = ϕ(F,x1
d,xd2,xh3). Our Theorem 3
gives the expression of effective energy density ¯ϕ when all length scales are comparable. This case apparently has no simple explanation since the averaging, homogenizing and thin-film limit are taken into account together.
4. Long grains (d << h). Assume ϕ = ϕ(F,x1
d,xd2). Theorem 1 includes the case d << κ. On the other hand, if κ = 0 or κ << d, Theorem 5 says that the
effec-tive energy density is obtained by homogenizing the bulk material, then passing to the thin-film limit. Finally, ifκ and d are of the same order of magnitude and both are much smaller thanh (κ ≈ d << h), we conjecture that the effective energy density is obtained by taking averaging and bulk homogenization first, and then passing to the thin-film limit.
5. Multilayers (κ versus h). Assume ϕ = ϕ(F,x3
h). In such a situation, only two
physical parametersκ and h are relevant. Our Theorem 4 gives the expression of ¯ϕ containing through-the-thickness variations.
We apply our results in Section 7. We use examples to show that the macroscopic behavior of films can significantly depend on the limiting ratios of these length scales. In our first example, we are interested in the shape-memory behavior of a polycrystalline martensitic film. Shape-memory materials are modeled with a multi-well energy densityϕ, each well representing a phase or variant. The relaxation ofϕ has the degeneracy, i.e., Qϕ = 0 on a set S. This set S contains all strains recoverable on heating in a single crystal. Similarly, the strains recoverable on heating in a polycrystal are contained in the set P on which ¯ϕ vanishes [11]. While the setS can be obtained in most martensitic materials, the set P is rather difficult to calculate. The estimation of this set P in bulk martensitic materials have been studied in [10, 11, 37]. We extend this framework to thin films. Our result shows that, for strong interfacial energy (i.e.,κ >> d), the shape-memory behavior is expected to be negligible in general polycrystals since materials cannot form microstructures within each grain to accommodate deformation. On the other hand, for small interfacial energy (i.e.,κ << d), materials can form microstructures freely and our model example shows that this setP significantly depends on the limiting ratio ofhd. We further consider cubic-monoclinic shape-memory thin films. We show that recoverable strains in thin films with flat columnar grains (d >> h) differs from (are larger than) those with long columnar grains (d << h). We also establish that films made by sputtering can recover only relatively small strains in Ti-Ni and other common shape-memory alloys.
Next, we consider effective conductivity of linear composites. We show that in general the effective conductivity of composites made of anisotropic materials can depend on the ratiohd. We also provide bounds to estimate it in our model example. We compare this result with the optimal bounds of Damlamian & Vogelius [20]. Finally, we consider a multilayered thin film made of a finite number of al-ternating layers of a martensitic material and a purely elastic material. We find quite different behavior when κh tends to zero and infinity. We conclude that such
a multilayered thin film provides an opportunity to design materials with unusual transformation properties.
2. Preliminaries
It is convenient to work on a fixed domain instead of a varying domainh, so we introduce the following change of variables:
zp = (z1, z2) = xp= (x1, x2), z3= 1
hx3, x ∈ h, (2.1) and set
1= S × (0, 1). (2.2)
With each deformation˜y : h→ R3we associate a deformation y: 1→ R3via
y(z(x)) = ˜y(x), x ∈ h.
We use the notation∇pfor the gradient in the plane of the film, i.e.,
∇py= y,1⊗ e1+ y,2⊗ e2, and y,1 = ∂z∂y 1 = ( ∂y1 ∂z1, ∂y2 ∂z1, ∂y3 ∂z1)
T; etc. We now change variables in 1
h˜e(h)using
(1.2) and (2.1) and get
˜e(h)1 [y] := 1 h˜e(h)[˜y] = Z 1 κ2 |∇2 py|2+ 2 h2|∇py,3| 2+ 1 h4|y,33| 2 (2.3) + ϕ y,1|y,2|1 hy,3, zp d , z3 dz.
We have used the notation
F= (f1|f2|f3) = f1⊗ e1+ f2⊗ e2+ f3⊗ e3 for F∈ M3×3.
We assume the energy densityϕ satisfies the following conditions: 1. ϕ(F, z) is Carath´eodory and nonnegative.
2. Periodicity in the plane of the film:ϕ(F, zp, z3) is periodic in the in-plane vari-able zp = (z1, z2) with period [0, 1]2for all F∈ M3×3andz3∈ (0, 1). 3. Growth and coercivity conditions:
c1(|F|p− 1) 5 ϕ(F, z) 5 c2(|F|p+ 1) (2.4) for all F∈ M3×3and for a.e. z= (zp, z3) ∈ R2× (0, 1).
4. Lipschitz condition:
|ϕ(F, z) − ϕ(G, z)| 5 c2(1 + |F|p−1+ |G|p−1)|F − G| (2.5) for all(F, G) ∈ M3×3× M3×3and for a.e. z= (zp, z3) ∈ R2× (0, 1). Above, 0< c15 c2andp satisfies 1 < p < ∞.
For any y∈ W1,p(1, R3), we extend the functional ˜e(h)1 [y] to
e(h)1 [y] =
˜e(h)1 [y] if y∈ W2,2(1, R3),
+∞ otherwise. (2.6)
Now our goal is to compute the0-limit of e1(h)ash, d and κ tend to zero with fixed limiting ratios (1.4). To this purpose, we recall that
Definition 1. A familye(h)1 of functionals onW1,p(1, R3) (1 < p < ∞) is said to0-converge (in the weak W1,p(1, R3) topology) to e1(0)if and only if
(I) every sequence y(h)with
y(h)* y in W1,p(1, R3) as h → 0,
satisfies the “lower bound” lim inf
h→0 e (h)
1 [y(h)] = e(0)1 [y];
(II) for every y∈ W1,p(1, R3), there exists a sequence y(h)called the “recovery sequence” such that
y(h)* y in W1,p(1, R3) as h → 0 and lim h→0e (h) 1 [y(h)] = e(0)1 [y].
Remark 1. The limiting functionale(0)1 is, by construction, lower semicontinuous with respect to weak convergence in W1,p(1, R3) [14] and, therefore, attains its minimum value due to the coercivity condition (2.4). Further, using the fact that the L2 norm is sequentially lower semicontinuous and Rellich’s compact-ness theorem, one can show thate(h)1 admits a minimum for any fixedh > 0 (cf.
Francfort & M ¨uller [24]). Therefore, minimizers ofe(h)1 converge to those of
e(0)1 by the fundamental theorem of0-convergence (see, for example, Braides &
In the following, we will show that, for any y∈ W1,p(1, R3), e(h)1 0-converges to a functionale(0)1 of the form
e1(0)[y] = Z S¯ϕ(∇py) dzp if y∈ VS, +∞ otherwise, (2.7)
where ¯ϕ will be determined explicitly and VS is defined by
VS = {y : y ∈ W1,p(1, R3) and y,3= 0 for a.e. z in 1} (2.8) which is canonically isomorphic toW1,p(S, R3). The following lemma is the first step towards proving (2.7); with it we only need to compute the0-limit of e(h)1 for
y∈ VS.
Lemma 1. Lete1(h)be defined by (2.6) and assume y /∈ VS. Then, lim infh→0e(h)1
[y(h)] = +∞ for any sequence y(h)such that y(h)* y in W1,p(1, R3) as h → 0.
Proof. We prove it by contradiction. Suppose there exists a sequence y(h) converg-ing weakly to y inW1,p(1, R3) with lim infh→0e(h)1 [y(h)] = M finite. Therefore, there exists a subsequence y(h)(not relabeled) such that
e(h)1 [y(h)] → M < +∞ ash → 0.
By coercivity, 1hy,3(h)is bounded inLp(1, R3) and this implies
y(h),3 → 0 strongly in Lp(1, R3) (2.9) ash → 0. Since y(h)converges weakly to y inW1,p(1, R3) as h tends to zero, this gives
y(h),3 * yh ,3 inLp(1, R3). (2.10) Combining (2.9) and (2.10), we have y,3 = 0 a.e. by the uniqueness of the weak limit. Thus y∈ VS, which contradicts the assumption, and this completes the proof.
ut
3. Strong interfacial energy
Theorem 1. Lete(h)1 ande(0)1 be defined by (2.6) and (2.7). Then,e(h)1 0-converges to the functionale1(0)if (i) ϕ = ϕ(F,zdp), κd → ∞ as h → 0, and ¯ϕ( ¯F) = Q ˜ϕ0( ¯F), ˜ϕ0( ¯F) = inf b∈R3 ˜ϕ( ¯F|b), (3.1) ˜ϕ(F) = Z Zϕ(F, zp) dzp,
whereQ ˜ϕ0 is the lower quasi-convex envelope of ˜ϕ0, ¯F ∈ M3×2 andZ =
(ii) ϕ = ϕ(F,zdp, z3), κd → ∞,κh → ∞ as h → 0, and ¯ϕ is given by (3.1)1,
(3.1)2with(3.1)3replaced by
˜ϕ(F) = Z
Z×(0,1)ϕ(F, z) dz; (3.2)
(iii)ϕ = ϕ(F, z3), κh → ∞ as h → 0, and ¯ϕ is given by (3.1)1,(3.1)2with(3.1)3
replaced by
˜ϕ(F) = Z 1
0
ϕ(F, z3) dz3. (3.3)
Remark 2. It is clear that ˜ϕ(F) enjoys the same growth and coercivity conditions
(2.4) and is continuous by virtue of the Lipschitz condition (2.5) onϕ. It follows that
˜ϕ0given by(3.1)2is well defined and the infimum is achieved. Further, Proposition 1 of Le Dret & Raoult [21] shows that ˜ϕ0( ¯F) satisfies the growth and coercivity estimates (2.4) and is continuous.
Proof of Theorem 1. We begin with case (i):ϕ = ϕ(F,zdp) andκd → ∞ as h → 0.
We first construct a recovery sequence for any y∈ VS. Recalling Remark 2 and invoking the relaxation theorem due to Dacorogna [19] we find a sequence y(δ) which converges weakly to y inW1,p(S, R3) such that
Z
S ˜ϕ0(∇py
(δ)) dzp→Z
SQ ˜ϕ0(∇py) dzp asδ → 0. (3.4)
Since the infimum of ˜ϕ0is achieved (see Remark 2), an argument like that used by
Le Dret & Raoult [21] shows that for each element of the sequence y(δ), there exists a measurable b(δ)∈ Lp(S, R3) such that
˜ϕ0(∇py(δ)) = ˜ϕ(∇py(δ)|b(δ)). (3.5) Further, we may also assume at the moment that both y(δ)(zp) and b(δ)(zp) are smooth functions because of the Lipschitz character of∂S (see Remark 3). Define
y(δ,h)= y(δ)(zp) + hb(δ)(zp)z3 (3.6) and substitute it intoe(h)1 . We have
e(h)1 [y(δ,h)] = Z 1 n κ2|∇2 py(δ)+ h∇p2b(δ)z3|2+ 2|∇pb(δ)|2 + ϕ∇py(δ)+ h∇p(b(δ)z3) | b(δ), zp d o dz. (3.7) The first term of the integrand,κ2
|∇2
py(δ)+ h∇p2b(δ)z3|2+ 2|∇pb(δ)|2
, vanishes for any fixedδ since κ(h) → 0 as h → 0. Therefore, using the Lipschitz condition
(2.5) onϕ, we get e(h)1 [y(δ,h)]−→h Z S Z Zϕ(∇py (δ)(zp)|b(δ)(zp), ˆzp) d ˆzpdzp = Z S ˜ϕ(∇py (δ)(zp)|b(δ)(zp)) dzp = Z S ˜ϕ0(∇py (δ)(z p)) dzp. (3.8)
Above in (3.7), we have approximated(∇py(δ)| b(δ)) by a piecewise constant
ele-ment inLp(S, R9), passed to the limit as in (3.8) using the Lemma A.1 by Ball
& Murat [7]1, and then use the estimate (2.5) onϕ again to complete the whole argument. Recalling (3.4) gives us lim sup δ→0 lim suph→0 e (h) 1 [y(δ,h)] = e(0)1 [y]. (3.9) Now appealing to the standard diagonalization argument of Attouch [4, Corollary 1.16] yields a sequence y(δ(h))that converges weakly to y inW1,p(1, R3) as h → 0 and satisfies
lim
h→0e (h)
1 [y(δ(h))] = e(0)1 [y]. (3.10) To complete the proof, we need to establish the lower bound. Let y(h) h* y ∈ VS inW1,p(1, R3). We may assume that lim inf
h→0e1(h)[y(h)] is finite; else the result follows. We may also restrict ourselves to a subsequence y(h)(not relabeled) which achieves the lim inf.
For anyδ > 0 let S0⊂⊂ S with |S\S0| < δ. Define
Pd = {zp ∈ dZ2: z p+ dZ ⊂ S0}, Sd = [ zp∈Pd (zp+ dZ), 1,d = Sd× (0, 1) and 0= S0× (0, 1). ClearlySd⊂ S0. For eachˆz in 1,ddefine
Y(h)(ˆzp, ˆz3) = 1 d2 Z zp+dZ ∇py(h)|1 hy(h),3 d ˜zp, ˆzp∈ zp+ dZ, zp∈ Pd. (3.11) One can check easily that
kY(h)kLp(1,d)5 ∇py(h)|1 hy(h),3 Lp(1,d). (3.12) 1 Suppose 15 p 5 ∞. Let g(x) ∈ Lp
loc(Rm) be [0, 1]m−periodic. Then g(xε) converges
Using the Poincar´e inequality for each small square inSdat fixedz3, summing all such squares, and integrating overz3from 0 to 1, one can deduce that
Y(h)−∇py(h)|1 hy(h),3 2 L2(1,d) 5 C d κ 2Z 1,dκ 2 |∇2 py(h)|2+ 1 h2|∇py (h) ,3 |2 dz (3.13) whereC is some constant that does not depend on h. Using the fact thatdκ → 0 as
h → 0 and the finiteness of lim infh→0e(h)1 [y(h)], we have
Y(h)−∇py(h)|1 hy,3(h) L2(1,d)→ 0 as h → 0. (3.14)
Thus, we can apply Egoroff’s theorem to assert the existence of a measurable subset
A of 0such that, for sufficiently smallh, A ⊂ 1,d, |0\A| < δ and
Y(h)− ∇py(h)|1 hy(h),3 → 0 uniformly on A (3.15)
as h → 0 for some subsequence
Y(h)− (∇py(h)|1hy(h),3 )
(not relabeled). Us-ing the Lipschitz condition (2.5), (3.12), (3.15) and the uniform boundedness of
k(∇py(h)|h1y(h),3 )kLp(1), we have Z Aϕ Y(h),zp d dz − Z Aϕ ∇py(h)|1 hy,3(h), zp d dz → 0 as h → 0. (3.16) LetAz3 be the projection of the slice ofA at the constant z3, i.e.,
Az3 = {(z1, z2) : (z1, z2, z3) ∈ A}. (3.17) Also, pick anyˆzp ∈ Pdand let
QS= ˆzp+ dZ, and Q = {(zp, z3) : zp ∈ QS, z3∈ (0, 1)}, (3.18) and notice that Y(h) is constant over QS for any fixedz3 ∈ (0, 1). Thus, using Fubini’s theorem, we have
Z A∩Qϕ Y(h),zp d dz = Z 1 0 Z QS∩Az3 ϕY(h),zp d dzpdz3 and Z QS∩Az3 ϕY(h),zp d dzp = Z QS ϕY(h),zp d dzp− Z QS\Az3 ϕY(h),zp d dzp = Z QS ˜ϕ(Y(h)) dzp− c2(1 + |Y(h)|p) |QS\Az3| (3.19)
where we have used the second inequality in (2.4). Using the other inequality in (2.4), we have Z QS∩Az3 ϕY(h),zp d dzp= c1(|Y(h)|p− 1) |QS∩ Az3|. (3.20)
Combining (3.19) and (3.20), we obtain
Z QS∩Az3 ϕY(h),zp d dzp= Z QS µ(d)(z) ˜ϕ(Y(h)) dzp− 2 c 2 Z QS\Az3 µ(d)(z) dzp, (3.21) where µ(d)(z) = c1|QS∩ Az3| c1|QS∩ Az3| + c2|QS\Az3| = c1− R ˆzp+dZχAz3(˜zp)d ˜zp (c1− c2) − R ˆzp+dZχAz3(˜zp)d ˜zp+ c2 for z∈ Q (3.22) andχAz3 is the characteristic function of the setAz3. Integrating (3.21) overz3from 0 to 1 and summing the same equation over allˆzpinPd, gives
Z Aϕ Y(h),zp d dz = Z 1,dµ (d)(z) ˜ϕ(Y(h)) dz − 2 c 2 Z 1,d\Aµ (d)(z) dz. (3.23) Invoking the Lebesgue point theorem on (3.22) asd → 0 as h → 0 for each fixed
z3, we have µ(d)(zp, z3) → c1χAz3(zp) c1χAz3(zp) + c2(1 − χAz3(zp)) = χAz3(zp) a.e. on 0, (3.24) and (3.23) becomes lim inf h→0 Z Aϕ Y(h),zp d dz = lim inf h→0 Z Aµ (d)(z) ˜ϕ(Y(h)) dz. (3.25)
Recalling (3.16), (3.25), (3.15), and the fact thatϕ is nonnegative, we obtain lim inf h→0 Z ϕ ∇py(h)|1 hy(h),3 , zp d dz = lim inf h→0 Z Aϕ ∇py(h)|1 hy(h),3 , zp d dz = lim inf h→0 Z Aϕ Y(h),zp d dz = lim inf h→0 Z Aµ (d)(z) ˜ϕ(Y(h)) dz = lim inf h→0 Z Aµ (d)(z) ˜ϕ∇ py(h)|1 hy,3(h) dz = lim inf h→0 Z Aµ (d)(z) ˜ϕ 0(∇py(h)) dz. (3.26)
Egoroff’s theorem tells us that there exists a measurable subset A0 ⊂ A with
|A\A0| < δ0such that for some subsequence (not relabeled)
µ(d)→ χAz3 ≡ 1 uniformly on A0
asd → 0 as h → 0. Therefore, for any η > 0, we have
lim inf h→0 Z Aµ (d)(z) ˜ϕ 0(∇py(h)) dz = lim inf h→0 Z A0µ (d)(z) ˜ϕ 0(∇py(h)) dz = lim inf h→0 Z A0(1 − η)Q ˜ϕ0(∇py (h)) dz. (3.27) If we defineG : W1,p(1, R3) → R by G(ˆy) = Z 1Q ˜ϕ0(∇pˆy)dz
and set8 : M3×3→ R to be 8(f1|f2|f3) = Q ˜ϕ0(f1|f2). Since Q ˜ϕ0is quasiconvex, it can be shown [21] that8 is also quasiconvex, bounded below by −c1, and satisfies growth and coercivity conditions similar to (2.4). Then G is sequentially lower semicontinuous onW1,p(1, R3) (see Acerbi & Fusco [2]). Applying this result toR1(1 − η)χA0Q ˜ϕ0(∇py(h))dz, we have lim inf h→0 Z A0(1 − η)Q ˜ϕ0(∇py (h))dz =Z A0(1 − η)Q ˜ϕ0(∇py)dz.
By lettingδ0andη tend to zero, we have
lim inf h→0 Z Aµ (d)(z) ˜ϕ 0(∇py(h))dz = Z AQ ˜ϕ0(∇py)dz.
Combining this with (3.26) yields
lim inf h→0 Z Aϕ ∇py(h)|1 hy(h),3 , zp d dz = Z AQ ˜ϕ0(∇py)dz.
Using the fact thatQ ˜ϕ0(∇py) belongs to L1(1) and |1\A| < 2δ, we obtain the desired lower bound by lettingδ → 0.
We now consider case (ii):ϕ = ϕ(F,zdp, z3) andκd → ∞,κh → ∞ as h → 0. We can construct the recovery sequence in a way similar to the previous case without any difficulty. The proof of the lower bound is also similar, except we have to replace (3.11) by Y(h)(ˆzp) = 1 d2 Z 1 0 Z zp+dZ ∇py(h)|1 hy(h),3 d ˜zpd ˜z3, ˆzp∈ zp+dZ, zp∈ Pd.
Note that (3.12) remains valid, but (3.13) becomes Y(h)−∇py(h)|1 hy(h),3 2 L2(1,d) 5 C d κ 2Z 1,dκ 2 |∇2 py(h)|2+ 1 h2|∇py (h) ,3 |2 dz +C h κ 2Z 1,dκ 2 1 h2|∇py (h) ,3 |2+ 1 h4|y (h) ,33|2 dz
whereC is some constant that does not depend on h. Since bothκd → ∞,κh → ∞ ash → 0, we obtain (3.14). The rest of the proof is similar and we omit it here.
Finally, case (iii) (ϕ = ϕ(F, z3) andκh → ∞ as h → 0) follows from case (ii).
ut
Remark 3. In (3.6), we have assumed that y(δ)and b(δ)are smooth functions to al-low the second derivative. Indeed, if y(δ)∈ W1,p(S, R3) and b(δ)∈ Lp(S, R3), the bounded Lipschitz domain permits the existence of sequences y(δ,ε)∈ C∞( ¯S, R3) and b(δ,ε)∈ C0∞(S, R3) such that
y(δ,ε)→ y(δ) strongly in W1,p(S, R3)
b(δ,ε)→ b(δ) strongly in Lp(S, R3) asε → 0. Then, (3.6) is replaced by
y(δ,ε,h)(z) = y(δ,ε)(zp) + hb(δ,ε)(zp)z3 and (3.9) now becomes
lim sup
δ→0 lim supε→0 lim suph→0 e (h)
1 [y(δ,ε,h)] = e(0)1 [y].
Appealing to the already quoted diagonalization argument, we find that there exists a recovery sequence labeled only in terms ofh and thus (3.10).
4. Film thickness much smaller than heterogeneity
Theorem 2. Supposeϕ = ϕ(F,zdp),κd → α andhd → 0 as h → 0. Let ¯F ∈ M3×2, Z = (0, 1)2ande(h)
1 be defined by (2.6). Then,e(h)1 0-converges to the functional
e(0)1 defined by (2.7) if (i) α = 0 and ¯ϕ( ¯F) = inf k∈N inf ω∈W1,p 0 (kZ) − Z kZϕ0( ¯F + ∇pω, zp)dzp, (4.1) ϕ0( ¯F0, zp0) = inf b∈R3ϕ( ¯F 0|b, z0 p); (4.2)
(ii) α > 0 and ¯ϕ( ¯F) = inf k∈Nω∈W1,pinf 0 (kZ)∩W2,20 (kZ) b∈Lp(kZ)∩W01,2(kZ) − Z kZ n α2|∇2 pω|2+ 2|∇pb|2 + ϕ( ¯F + ∇pω|b, zp) o dzp; (4.3) (iii)α = ∞ and ¯ϕ( ¯F) = Q ˜ϕ0( ¯F), ˜ϕ0( ¯F) = inf b∈R3 ˜ϕ( ¯F|b), ˜ϕ(F) = Z Zϕ(F, zp)dzp
whereQ ˜ϕ0is the lower quasi-convex envelope of ˜ϕ0. Note that we have used the notation−R· · · = ||1 R· · · .
Remark 4. It can be shown that the effective energy densityϕ satisfies the growth
and Lipschitz condition for 05 α 5 ∞, i.e.,
c1(| ¯F|p− 1) 5 ¯ϕ( ¯F) 5 c2(| ¯F|p+ 1) (4.4)
| ¯ϕ( ¯F) − ¯ϕ( ¯G)| 5 c20(1 + | ¯F|p−1+ | ¯G|p−1)| ¯F − ¯G| (4.5)
for all( ¯F, ¯G) ∈ M3×2× M3×2. Indeed, consider 0< α < ∞. The upper bound is obtained by setting ω = b = 0 in (4.3) and using (2.4) on ϕ. To show the lower bound, note that for everyε > 0, there exists k ∈ N, ω ∈ W01,p(kZ, R3) ∩
W2,2 0 (kZ, R 3) and b ∈ Lp(kZ, R3) ∩ W1,2 0 (kZ, R 3) such that ¯ϕ( ¯F) 5 − Z kZ n α2|∇2 pω|2+ 2|∇pb|2 + ϕ( ¯F + ∇pω|b, zp) o dzp 5 ¯ϕ( ¯F) + ε. (4.6) Using (2.4) onϕ yields ¯ϕ( ¯F) + ε = c1 − Z kZ| ¯F + ∇pω|b| pdzp− 1 = c1 − Z kZ| ¯F + ∇pω| pdzp− 1 = c1 −ZkZ( ¯F + ∇pω) dzp p − 1 = c1(| ¯F|p− 1)
since| · |pis convex and| ¯F|b|3×3= | ¯F|3×2for all b∈ R3. To prove the Lipschitz condition, we choose the sameω and b as the test functions for ¯ϕ( ¯G). We have
¯ϕ( ¯G) − ¯ϕ( ¯F) 5 − Z kZ (ϕ( ¯G+ ∇pω|b, zp) − ϕ( ¯F + ∇pω|b, zp)) dzp+ ε 5 C1+ 1 k2 ¯G+ ∇pω|b p Lp + 1 k2 ¯F+ ∇pω|b p Lp p−1 p ¯ G− ¯F + ε. (4.7)
Now invoking the growth conditions onϕ, ¯ϕ and (4.6), we get
C 1 k2 ¯F+ ∇pω|b p Lp− 1 5 − Z ϕ( ¯F + ∇pω|b, zp) dzp 5 ¯ϕ( ¯F) + ε 5 c2(| ¯F|p+ 1) + ε (4.8) and 1 k2 ¯G+ ∇pω|b p Lp 5 C | ¯G − ¯F|p+ 1 k2 ¯F+ ∇pω|b p Lp . (4.9) Combining (4.7)–(4.9) gives us ¯ϕ( ¯G) − ¯ϕ( ¯F) 5 c0 2(1 + | ¯F|p−1+ | ¯G|p−1)| ¯F − ¯G| + ε. (4.10) We have the desired inequality asε → 0. The opposite inequality can be obtained by interchanging ¯F and ¯G. The caseα = 0 and α = ∞ can be treated similarly. Proof of Theorem 2. The caseα = ∞ is a corollary of Theorem 1. The proof for
finiteα = 0 consists of two parts. First, we prove the 0-limit in the case where the limit function is affine (Part A). We then prove the general case by approximating an arbitrary function by piecewise affine functions (Part B).
Part A. Suppose y= ¯Fzpwith ¯F∈ M3×2. We begin by constructing a recovery sequence for the case whereα > 0. It follows from the definition of ¯ϕ that there exist sequencesk(ε) ∈ N, ω(ε) ∈ W01,p(k(ε)Z, R3) ∩ W02,2(k(ε)Z, R3), and b(ε)∈
Lp(k(ε)Z, R3) ∩ W1,2 0 (k(ε)Z, R3) such that 1 k(ε)2 Z k(ε)Z n α2|∇2 pω(ε)|2+ 2|∇pb(ε)|2 + ϕ( ¯F + ∇pω(ε)| b(ε), zp) o dzp → ¯ϕ( ¯F) (4.11) as ε → 0. We use ω(ε) and b(ε) to construct our recovery sequence. Unfortu-nately, b(ε)may not be smooth enough to allow second differentiation. However, an approximation argument similar to Remark 3 shows that we may assume that
b(ε)∈ C0∞(k(ε)Z, R3). Define y(h,ε)= ¯Fzp+ d ω(ε) zp d + hb(ε)zdpz3, (4.12)
whereω(ε)and b(ε)are extended periodically in the plane of the film sinceω(ε)=
∂ω(ε)/∂n = b(ε)= ∂b(ε)/∂n = 0 on ∂k(ε)Z in the sense of trace. Clearly, y(h,ε) h→ ¯Fzpfor eachε > 0. Substituting (4.12) into e(h)1 defined by (2.6), we have
e(h)1 [y(h,ε)] = Z 1 κd∇2 pω(ε)+hd ·κd∇p2b(ε)z3 2+ 2 κd∇pb(ε) 2 ! +ϕ ¯F + ∇pω(ε)+hd∇pbz3| b(ε), zp d dz. (4.13) If we assume thathd → 0 andκd → α as h → 0, and impose the Lipschitz condition (2.5) onϕ, we get e(h)1 [y(h,ε)] → |1| k(ε)2 Z k(ε)Z n α2|∇2 pω(ε)|2+ 2|∇pb(ε)|2 +ϕ ¯F + ∇pω(ε)| b(ε), zp odzp (4.14)
ash → 0. Above, we have applied the property of mean value2to (4.13) in deriving (4.14). Then, using (4.11) gives
lim sup ε→0 lim suph→0 e (h) 1 [y(h,ε)] = | 1| ¯ϕ( ¯F) = |S| ¯ϕ( ¯F) = e(0) 1 [y]. (4.15) Recalling (4.12) and (4.15) and appealing to the standard diagonalization argument, we find that there exists a sequence ˆy(h) = y(h,ε(h)) that converges weakly to
y= ¯FzpinW1,p(1, R3) and satisfies lim
h→0e (h)
1 [ˆy(h)] = e1(0)[y].
The case ofα = 0 is similar. Indeed, an argument similar to that in Remark 2 shows that (4.2) is well defined and the infimum is also achieved. From the definition of ¯ϕ, there exists sequences k(ε)∈ N, ω(ε)∈ W01,p(k(ε)Z, R3) such that
1 k(ε)2 Z k(ε)Zϕ0 ¯F + ∇pω(ε), zp dzp→ ¯ϕ( ¯F)
asε → 0. Following an argument like the one used in (3.5), we find measurable functions b(ε)∈ Lp(k(ε)Z, R3) such that
ϕ0( ¯F + ∇pω(ε), zp) = ϕ( ¯F + ∇pω(ε)| b(ε), zp) (4.16) for almost all zp ∈ k(ε)Z. The rest of the proof follows similarly.
Setω(h)= y(h)− ¯Fzp. A slight refinement (see Lemma 2.1 in M ¨uller [36])
shows that the recovery sequence y(h)can be constructed such that
ω(h)= ∂ω(h)
∂n = 0 (4.17)
on6 = ∂S × (0, 1) in the sense of trace.
We now turn to the lower bound when y is an affine function andα > 0. We assume thatS is a square domain with side length s and
ω(h)= y(h)− ¯Fzp* 0 in Wh 1,p(1, R3),
ω(h)= ∂ω(h) ∂n = 0
(4.18)
on6 = ∂S × (0, 1) in the sense of trace. We may also assume that lim infh→0e(h)1
[y(h)] is finite; otherwise the proof is trivial. Choose k ∈ N to be smallest integer
such thatk d = s + d. We can find a square ˜S(d)with the side lengthkd such that
S ⊂ ˜S(d) and the corners of ˜S(d)are indZ2, i.e., ˜S(d) = d(z0
p+ kZ) for some zp0∈ Z2. Now extendingω(h)to ˜S(d)× (0, 1) by ˜ω(h)= ω(h), for z ∈ S × (0, 1), 0, for z∈ ( ˜S(d)\S) × (0, 1), we have e(h)1 [y(h)] = Z ˜S(d)×(0,1) κ2 ∇2 p˜ω(h) 2 +h22 ∇p˜ω(h),3 2+h14 ˜ω(h),33 2 + ϕ ¯F + ∇p˜ω(h)|h1˜ω,3,zdp dz − Z ( ˜S(d)\S)×(0,1)ϕ ¯F|0,zp d dz = I1− I2.
The second integral −I2 = −c2(1 + | ¯F|p)| ˜S(d)\S| converges to zero since
| ˜S(d)\S| 5 (s + 2d)2− s2 → 0 as d tends to zero as h tends to zero. Chang-ing variables zp 7→ d(zp0+ ˆzp) and z3 7→ dhˆz3, using the periodicity ofϕ and Fubini’s theorem, we have
I1= d2·d h · Z h d 0 Z kZ α2 ∇2 pˆω(h) 2 + 2 ∇pˆω(h),3 2 + ϕ¯F + ∇pˆω(h)| ˆω(h),3 , ˆzp d ˆzpd ˆz3 where ˆω(h)(ˆzp, ˆz3) = 1 d ˜ω(h) d ˆzp,dhˆz3 . (4.19)
Notice that for almost every ˆz3 ∈ (0,dh), the function ˆω(h)ˆz
3 (ˆzp) = ˆω
(h)(ˆz p, ˆz3) belongs toW01,p(kZ, R3) ∩ W02,2(kZ, R3). Similarly, for almost every ˆz3∈ (0,hd),
ˆω(h),3 belongs toLp(kZ, R3) ∩ W01,2(kZ, R3). It follows that
I1= d2· d h· Z h d 0 k2 ¯ϕ( ¯F) dz 3= (dk)2¯ϕ( ¯F) = s2¯ϕ( ¯F).
Thus, we have shown that lim infh→0e(h)1 [y(h)] = s2¯ϕ( ¯F) = e(0)1 [y] which is the desired lower bound.
For the general domainS, assume the sequence y(h)satisfies (4.18). Consider a squareQ which contains S. Using the fact that the recovery sequence can be obtained such that (4.17) is satisfied for the domain(Q\S) × (0, 1), we can also obtain the lower bound.
Now letS be any open bounded Lipschitz domain and let y(h) h* y = ¯Fzpin
W1,p(1, R3). No further assumption such as (4.18) is imposed on y(h). We use the argument of De Giorgi [25] (see also Francfort & M ¨uller [24]) to obtain
the lower bound. FixS0open and compactly contained inS. Let
R =1
2dist(S0, ∂S). For any strictly positive integerν, define
Si = zp∈ S : dist(zp, S0) < i νR , 1 5 i 5 ν,
and scalar functionsηi(zp) ∈ C0∞(S) such that
05 ηi 5 1, ηi = 1 in Si−1 andηi = 0 in S \ Si, |∇pηi| 5 ν+1R and|∇p2ηi| 5 (ν+1R )2. (4.20)
Moreover, let1i = Si× (0, 1) for i = 0, · · · , ν and set
y(h)i = ¯Fzp+ ηi(y(h)− ¯Fzp).
Then, for eachi, y(h)i converges weakly to y= ¯FzpinW1,p(1, R3) as h tends to zero and(y(h)i − ¯Fzp) satisfies (4.18). Therefore, it follows from the previous result that lim inf h→0 e (h) 1 [y(h)i ; 1] = e(0) 1 [y; 1]. (4.21) Now e(h)1 [y(h)i ; 1] = e(h) 1 [y(h); 1 i−1] + e(h)1 [y(h)i ; 1 i\1i−1] + e(h)1 [y; 1\1 i] 5 e(h)1 [y(h); 1] + e(h) 1 [y(h)i ; 1 i\1i−1] +c2(1 + | ¯F|p) |1\10|. (4.22)
Using the growth condition (2.4) onϕ and the definition of ηi(zp), e1(h)[y(h)i ; 1 i\1i−1] 5 C Z 1 i\1i−1 κ2 |∇2 py(h)|2+ 2 h2|∇py (h) ,3 |2+ 1 h4|y (h) ,33|2 + ν + 1 R 4 |y(h)− ¯Fzp|2+ 4 ν + 1 R 2 ∇py(h)− ¯F 2 +2 h2 ν + 1 R 2 |y,3(h)|2 ! + 1+ | ¯F|p+ |∇py(h)− ¯F|p+ ν + 1 R p |y(h) − ¯Fzp|p+ 1hy(h),3 p dz. (4.23)
Notice that we have used the inequalityPNi=1|ai|
2
5 NPNi=1|ai|2in deriving (4.23). Since y(h) h* ¯Fzp inW1,p(1, R3), this implies y(h) h→ ¯Fzp inLp(1;
R3) by Rellich’s compactness theorem and
Z 1 i\1i−1 ν + 1 R p |y(h)− ¯Fzp|pdz → 0 as h → 0. (4.24)
By the assumptions of finiteness of lim infh→0e(h)1 [y(h)] and non-negativity of ϕ, it is concluded that κh∇y(h),3
L2(1)is uniformly bounded inh. Further since
Z1 κ hy(h),3 dz 5 Cκ 1 hy(h),3 Lp(1) ,
andh1y(h),3 is uniformly bounded inLp(1; R3) due to coercivity of ϕ, the Poincar´e inequality implies
κ
hy(h),3 * 0 in W
1,2(1; R3) as h → 0.
Similarly, κ∇2y(h) L2(1)is uniformly bounded inh. Using the Poincar´e inequal-ity twice implies that
κy(h)* 0 in W2,2(1; R3) as h → 0, from which it is deduced that
Z 1 i\1i−1 κ2 |y(h)− ¯Fzp|2+ |∇ py(h)− ¯F|2+ |1 hy,3(h)|2 dz → 0, (4.25)
ash → 0. Collecting (4.21) to (4.25) gives
e(0)1 [y] 5 lim infh→0 e(h)1 [y(h)] + c2(1 + | ¯F|p)|1\10|
+ C Z 1 i\1i−1 κ2 |∇2 py(h)|2+ 2 h2|∇py (h) ,3 |2+ 1 h4|y (h) ,33|2 + 1+ | ¯F|p+ |∇py(h)− ¯F|p+ |1 hy(h),3 |p dz. (4.26) Summing (4.26) overi = 1, · · · , ν and dividing by ν gives
e1(0)[y] 5 lim inf h→0 e (h) 1 [y(h)] + c2(1 + | ¯F|p)| 1\1 0| +C ν Z 1 κ2 |∇2 py(h)|2+ 2 h2|∇py (h) ,3 |2+ 1 h4|y (h) ,33|2 + 1+ | ¯F|p+ |∇py(h)− ¯F|p+ |1 hy(h),3 |p dz. (4.27)
Recall the assumption of finiteness of lim infh→0e1(h)[y(h)] and note that k∇py(h)−
¯FkLp(1) andk1hy,3(h)kLp(1) are uniformly bounded inh since y(h) * ¯Fzp in
W1,p(1, R3) and ϕ enjoys the coercivity (2.4). This concludes the proof by letting
ν → +∞ and 1 0→ 1.
The proof for the case whereα = 0 is almost exactly the same except we use
ϕ ¯F + ∇p˜ω(h)|1 h˜ω,3, zp d = ϕ0 ¯F + ∇p˜ω(h),zp d after (4.19).
Part B. In the case where y is the piecewise affine function, the proof for the
lower bound is obvious. The recovery sequence can also be constructed by virtue of (4.17).
For general y ∈ VS, the existence of a recovery sequence can be deduced as follows. The Lipschitz boundary∂S of the film guarantees the existence of a sequence of piecewise affine functions y(δ)such that
y(δ)→ y in W1,p(S, R3) asδ → 0. (4.28) For each piecewise affine function y(δ), there exists a recovery sequence y(h,δ)such that y(h,δ)* y(δ) inW1,p(1, R3) ande(h)1 [y(h,δ)] → e(0)1 [y(δ)] (4.29) ash → 0. Define f (h, δ) = e(h)1 [y(h,δ)] − e (0) 1 [y] + y(h,δ)− y Lp(1) 5 e(h)1 [y(h,δ)] − e(0)1 [y(δ)] + e(0)1 [y(δ)] − e1(0)[y] + y(h,δ)− y(δ) Lp(1)+ y(δ)− y Lp(1). (4.30)
The Lipschitz condition of (4.5) for the homogenized energy density ¯ϕ implies that e(0)1 [y(δ)] − e (0) 1 [y] → 0 as δ → 0.
It follows that lim supδ→0 lim suph→0 f (h, δ) = 0. A standard diagonalization argument establishes the existence of a recovery sequence.
It remains to prove the lower bound for general y ∈ VS. Let y(h) * y in W1,p(1, R3) as h → 0. Without loss of generality, we may assume that lim infh→0e(h)1 [y(h)] is finite. First, the regularity of ∂S permits the existence of a sequenceω(h)∈ C∞( ¯S) such that
ω(h)→ y in W1,p(S, R3),
κ∇2
pω(h)→ 0 in L2(S, R12) (4.31)
ash → 0 (cf. Francfort & M ¨uller [24]). For any δ > 0, there exists a partition
{Si} of S into open sets such that X i Z Si |∇py− ¯Fi|pdzp < δ with ¯Fi = − Z Si ∇pydzp. (4.32)
Let˜y(h)= ¯Fizp+ y(h)− ω(h)for z∈ 1i = Si× (0, 1). Clearly ˜y(h)* ¯Fizpin
W1,p(1
i, R3) as h → 0. Using the previous result for piecewise affine functions,
we have after summation, lim inf h→0 X i e1(h)[˜y(h); 1 i] = X i e1(0)[ ¯Fizp; Si]. (4.33)
Notice that from (4.31), we have
κ∇2 p˜y(h) L2(1 i) − κ∇2 py(h) L2(1 i) 5 κ(∇2 p˜y(h)− ∇p2y(h)) L2(1 i) = κ∇2 pω(h) L2(1 i) → 0 (4.34)
ash → 0. Using (4.32), (4.34) and the Lipschitz conditions on ϕ and ¯ϕ, we obtain
e(h) 1 [y(h); 1] −X i e(h)1 [˜y(h); 1 i] 5 C · δ1 p (4.35) e(0) 1 [y] − X i e(0)1 [ ¯Fizp; Si] 5 C · δ1 p (4.36)
for sufficiently smallh. Collecting (4.33), (4.35) and (4.36) concludes the proof.
5. Film thickness comparable to heterogeneity
Theorem 3. Supposeϕ = ϕ(F,zdp, z3), κd → α and hd → β > 0 as h → 0. Let
¯F ∈ M3×2and e(h)
1 be defined by (2.6). Thene(h)1 0-converges to the functional
e(0)1 defined by (2.7) if (i) α = 0 and ¯ϕ( ¯F) = inf k∈N inf ω∈Aβk− Z βkϕ ¯F + ∇pω | ω,3, zp,zβ3 dz, (5.1) where βk = kZ × (0, β), Z = (0, 1)2, 6β = ∂kZ × (0, β), (5.2) ˜ Aβk = {ω : ω ∈ W1,p(β k, R3), ω |6β = 0}; (5.3) (ii) α > 0 and ¯ϕ( ¯F) = inf k∈N inf ω∈Aβk− Z βk α2|∇2ω|2+ ϕ ¯F + ∇pω | ω,3, zp,zβ3 dz (5.4) where Aβk = {ω : ω ∈ W1,p(β k, R3) ∩ W2,2(βk, R3), ω |6β = ∂ω ∂n|6β = 0}; (5.5) (iii)α = ∞ and ¯ϕ( ¯F) = Q ˜ϕ0( ¯F), ˜ϕ0( ¯F) = inf b∈R3 ˜ϕ( ¯F|b), ˜ϕ(F) = Z Z×(0,1)ϕ(F, z) dz
whereQ ˜ϕ0is the lower quasi-convex envelope of ˜ϕ0.
The proof of Theorem 3 for finiteα = 0 is very similar to that of Theorem 2. If hd = β, we construct a recovery sequence for an affine function y = ¯Fzpusing the scaling y(h)= ¯Fzp+ dω zp d , hz3 d , (5.6)
whereω(z) ∈ Aβk. The proof of lower bound also follows exactly that of Theorem 2 by using the same scaling (5.6). The caseα = ∞ is a corollary of Theorem 1(ii) sinceκh =κd dh → ∞ as h → 0 in this case.
We have a similar theorem when the in-plane heterogeneity vanishes.
Theorem 4. Supposeϕ = ϕ(F, z3) and κh → α0ash → 0. Let ¯F ∈ M3×2ande(h)1
(i) α0= 0 and ¯ϕ( ¯F) = inf k∈Nωinf∈ ˜A1 k − Z 1 k ϕ( ¯F + ∇pω | ω,3, z3) dz (5.7)
where1kand ˜A1kare defined by (5.2) and (5.3);
(ii) α0> 0 and ¯ϕ( ¯F) = inf k∈Nωinf∈A1 k − Z 1 k n α02 |∇2ω|2+ ϕ( ¯F + ∇ pω | ω,3, z3) o dz (5.8)
where1kandA1kare defined by (5.2) and (5.5);
(iii)α0= ∞ and ¯ϕ( ¯F) = Q ˜ϕ0( ¯F), ˜ϕ0( ¯F) = inf b∈R3 ˜ϕ( ¯F|b), ˜ϕ(F) = Z 1 0 ϕ(F, z3) dz3,
whereQ ˜ϕ0is the lower quasi-convex envelope of ˜ϕ0.
Remark 5. If our film is homogeneous and the interfacial energy is negligible
(α = α0 = 0), all our results coincide with that of Le Dret & Raoult [21], i.e.,
¯ϕ( ¯F) = Qϕ0( ¯F). This is obvious in Theorem 1 and Theorem 2, but not in Theorem 3 and Theorem 4. So we explain this in some detail. Consider a homogeneous film with energy densityϕ = ϕ(F) and let
ϕ0( ¯F) = inf
b∈R3ϕ( ¯F|b).
Assumeϕ satisfies (2.4) and (2.5). A similar argument used in Remark 2 shows that ϕ0 is well defined and the infimum is achieved. Further,ϕ0also enjoys the growth and coercivity estimates (2.4). Hence,W1,pquasi-convexification is equal toW1,∞quasi-convexification andQϕ0can be expressed as
Qϕ0( ¯F) = inf ˆ ω∈W1,p 0 (Z) − Z Zϕ0( ¯F + ∇pˆω) dzp, (5.9)
whereZ = (0, 1)2. On the other hand, for homogeneous films, ¯ϕ defined in (5.1) becomes ¯ϕ( ¯F) = inf k∈N inf ω∈ ˜Aβk− Z βkϕ( ¯F + ∇pω | ω,3) dz (5.10)
for any finiteβ > 0. We wish to show ¯ϕ = Qϕ0. First, it is clear that
¯ϕ( ¯F) = Qϕ0( ¯F).
To prove the reverse inequality, notice that there exist sequences of ˆωδ ∈ W01,p
(Z, R3) and bδ∈ Lp(Z, R3) such that
− Z Zϕ0( ¯F + ∇pˆω δ) dz p= − Z Zϕ( ¯F + ∇pˆω δ|bδ)dz p→ Qϕ0( ¯F) (5.11)
