COMPOSITES OF HYPERELASTIC MATERIALS WITH

PRESCRIBED LAVRENTIEV GAP FUNCTIONS

Dedicated to Walter Noll and Jerry Ericksen in Honor of Their 80th Birthdays

Victor J. Mizel Department of Mathematical Sciences Carnegie Mellon University

In 1985, J. M. Ball and V. J. Mizel raised the question of whether there exist nonlinearly elastic materials possessing a physically natural stored energy density [one which is independent of an observer’s coordinate frame (objective) and is invariant under a group of linear transformations (isotropic)] as well as physically reasonable boundary value problems for such materials, such that the infimum of the total stored energy for continuous deformations of the material meeting the boundary conditions (admissible deformations) and belonging to a Sobolev space W^{1,p_{2}} for some p_{2} > 1 is strictly greater than its infimum for those admissible deformations which belong to a Sobolev space W^{1,p_{1}} with p_{1} < p_{2}, despite the density of the former Sobolev space in the latter. The question was motivated by M. Lavrentiev’s 1926 demonstration of such a gap for 1-dimensional variational boundary value problems on a bounded interval whose smooth integrand satisfied the conditions of Tonelli’s existence theorem - as well as improved versions developed in the 1980’s. Thereafter, M. Foss demonstrated in 2000 that there are (nonphysical) model problems in which the infimum over W^{1,p} varies continuously with p. The positive (2-dimensional) resolution of the 1985 Ball/Mizel question was achieved by Foss, Hrusa and Mizel in 2003.

The present article constructs for each given positive continuous increasing function i on a compact interval [p^{0}, p^{1}] of the half axis (1, ∞) a “composite” of 2-dimensional hyperelastic materials such that for a certain continuous deformation of the composite satisfying prescribed boundary conditions the relation

(∗)  E|_{W^{1,p}} = i(p) for all p in [p^{0}, p^{1}].

for the minimal stored energy E holds. The continuous deformation u will be constructed as the sum of an infinite series of mappings associated with i(·), each of which is defined on a proper subdomain of the domain Ω_{α} of u. That is, for specified α > β > 0 (with α < 2π) we present examples in 2-dimensions of deformations u ∈ A_{1}^{α,β} = ⊕_{α′∈(0,∞]}W^{1,1}(Ω_{α′}, ℝ^{2}) where (in polar coordinates) the domains Ω_{α}, Ω_{α′} and Ω_{β} are given by Ω_{α} = {x ∈ ℝ^{2}\{(0, 0)} : r(x) < 1, θ(x) ∈ (0, α)} , Ω_{α′} = {x ∈ ℝ^{2}\{(0, 0)} : r(x) < 1, θ(x) ∈ (0, α′)} , Ω_{β} = {u ∈ ℝ^{2}\{(0, 0)} : r(u) < 1, θ(u) ∈ (0, β)} , and Γ_{3,α}, Γ_{3,α′}, Γ_{3,β} denote the curvilinear boundaries of these domains. The deformations under consideration depend on a parameter q with q ∈ [q^{0}, q^{00}] for some specified q^{00} > q^{0} > 1 and, using complex value notation, are defined by

(1)  \{ \begin{matrix}u_{P M,q}(x) = r(x)^{δq}e^{iγ_{q}θ(x)} \\ u_{AM,q}(x) = r(x)^{γ_{q}}e^{iγ_{q}θ(x)} \end{matrix}

where

γ_{q} = 1 − \frac{1}{q} − ϵ_{q}, δ_{q} = 1 − \frac{1+γ_{q}}{2q−1} = 1 − \frac{1}{q} + \frac{ϵ_{q}}{2q−1} for a small ϵ_{q} > 0. We put α′_{q} = α when γ_{q}α > β, and we put α′_{q} = β/γ_{q} otherwise, so α′_{q} < α. Thus the associated stored energy E := J_{q}[u] is expressed in terms of the first derivatives of u = u + iv [employing complex notation] with \frac{∂}{∂\overline{z}}(u + iv) = (u_{x} − v_{y}) + i(v_{x} + u_{y}) as

J_{q}[u] = ∫_{Ω_{α′_{q}}} ([|(cos θ(x)\frac{∂u(x)}{∂r(x)} − \frac{sin θ(x)}{r(x)} \frac{∂u(x)}{∂θ(x)} − sin θ(x)\frac{∂v(x)}{∂r(x)} − \frac{cos θ(x)}{r(x)} \frac{∂v(x)}{∂θ(x)} )

+i (cos θ(x)\frac{∂v(x)}{∂r(x)} − \frac{sin θ(x)}{r(x)} \frac{∂v(x)}{∂θ(x)} + sin θ(x)\frac{∂u(x)}{∂r(x)} + \frac{cos θ(x)}{r(x)} \frac{∂u(x)}{∂θ(x)})|^{2}]^{q}) dxdy

which leads to

(2)

J_{q}[u] = ∫_{Ωα′_{q}} [(\frac{∂u(x)}{∂r(x)} − \frac{1}{r(x)} \frac{∂v(x)}{∂θ(x)})^{2} + (\frac{1}{r(x)} \frac{∂u(x)}{∂θ(x)} + \frac{∂v(x)}{∂r(x)})^{2}]^{q} dxdy

where α′_{q} = α ∧ \frac{β}{1 − \frac{1}{q} − ϵ_{q}} for ϵ_{q} > 0 sufficiently small. Hereafter we refer to the stored energy density integrand in (2) as W_{q}(∇u), where u = (u, v).

It follows from (1) that

\begin{matrix}u_{P M,q} ∈ A_{p}^{α,β}, ∀p ∈ [1, \frac{2}{1 − δ_{q}} ) = [1, \frac{2q}{1 − ϵ_{q} \frac{q}{2q−1}} ) , \\ u_{AM,q} ∈ A_{p}^{α,β}, ∀p ∈ [1, \frac{2}{1 − γ_{q}}) = [1, \frac{2q}{1 + qϵ_{q}} ) . \end{matrix}

Thus if we put p_{q}^{∗} = \frac{2q}{1+qϵ_{q}} , p_{q}^{∗∗} = \frac{2q}{1−ϵ_{q} \frac{q}{2q−1}} it follows that p_{q}^{∗} < p_{q}^{∗∗} with

(3)  p_{q}^{∗∗} = p_{q}^{∗} + 2q (\frac{1}{1 − ϵ_{q} \frac{q}{2q−1}} − \frac{1}{1 + qϵ_{q}} ) ≈ p_{q}^{∗} + \frac{4q^{3}}{2q − 1}ϵ_{q}.

In addition (cf. [15]):

(4) ∇u_{P M,q} = r(x)^{− \frac{1+γq}{2q−1}} ( \begin{matrix}(1 − \frac{1+γ_{q}}{2q−1}) cos γ_{q}θ(x) & −γ_{q} sin γ_{q}θ(x) \\ (1 − \frac{1+γ_{q}}{2q−1}) sin γ_{q}θ(x) & γ_{q} cos γ_{q}θ(x) \end{matrix} )
det ∇u_{P M,q} = γ_{q} (1 − \frac{1+γ_{q}}{2q−1}) r(x)^{− \frac{2(1+γq)}{2q−1}} > 0

Now it is easy to verify that (cf. [15]) the form of J_{q} for each q ∈ [q^{0}, q^{00}] ⊂ (1, ∞) is given by

(5)  \{ \begin{matrix}J_{q}[u_{P M,q}] = ∫_{Ω_{α′_{q}}} [r(x)^{2(δ_{q}−1)} [(δ_{q} − γ_{q})^{2}]^{q} dxdy = \\ α′_{q} [(δ_{q} − γ_{q})^{2}]^{q} \frac{1}{2q(δ_{q} − 1) + 2} =: K_{q} ∀p ∈ [1, p_{q}^{∗∗}) \\ J_{q}[u_{AM,q}] = 0, ∀p ∈ [1, p_{q}^{∗}), \end{matrix}

whereby, as indicated in [15], the infimum of J_{q}, for q ∈ [q^{0}, q^{00}] for some q^{00} > q^{0} > 1 is given by

(6)

inf_{Ãp,α′_{q}} J_{q}[u] = J_{q}[u_{AM,q}] = 0, ∀p ∈ [1, p_{q}^{∗})

inf_{Ãp,α′_{q}} J_{q}[u] = J_{q}[u_{P M,q}] = K_{q}, ∀p ∈ [p_{q}^{∗}, p_{q}^{∗∗}) [actually, for ∀p ∈ (p_{q}^{∗}, ∞)].

In view of the constancy of “J_{q}[u_{P M,q}]”(·) we will henceforth express this function in terms of the characteristic function I_{[c,∞)} of an associated half axis. Now suppose that [p^{0}, p^{1}] ⊂ (1, ∞) is a prescribed closed interval and that i : [p^{0}, p^{1}] → ℝ is a prescribed positive increasing continuous function. Our goal is to produce a “composite hyperelastic material of type I” such that its minimal stored energy function E satisfies

(∗∗)  E|_{W^{1,p}} = i(p) ∀p ∈ [p^{0}, p^{1}].

The strategy to be utilized involves constructing an infinite series of functions with terms of the form

\frac{1}{q_{k}} ∫_{Ω_{α′}} (|∇u_{P M,q_{k}}|^{2})^{q_{k}} dx dy q_{k} ∈ [q^{0}, q^{00}] ⊂ (1, ∞),

where it is known (cf. [15]) that for each q > 1

“ ∫_{Ω_{α′}} |∇u_{P M,q}|^{2q} dx dy”(·)

is constant along the half axis p ∈ [q, ∞). Thus, our method is along the lines of an exercise in chapter 7 of Principles of Mathematical Analysis (3rd edition) by W. Rudin. Utilizing the fact that u_{P M,q} is a solution of the Euler-Lagrange system for our variational problem we obtain (7)

inf_{Ã_{p,α′_{q}}} J_{q}[u] = “J_{q}[u_{P M,q}]” = K_{q}, ∀p ∈ [p_{q}^{∗}, ∞);
inf_{Ãp,α′_{q}} J_{q}(u) = J_{q}[u_{AM,q}] = 0,  ∀p ∈ [1, p_{q}^{∗}) where p_{q}^{∗} = \frac{2q}{1 + qϵ_{q}} .

We proceed as follows under the assumption (to avoid a trivial conclusion): (#) r = i(p_{q^{0}}^{∗}) < i(p_{q^{00}}^{∗}) := r + S with r ≥ 0, S > 0.

Step 1 Select an integer n_{0} > 1 and let q_{0,1} be the smallest value in [q^{0}, q^{00}] satisfying

(i)_{0} r + \frac{1}{2^{n_{0}}} SI (·)_{[p_{q_{0,1}}^{∗},∞)} ≡ i(p_{q_{0,1}}^{∗}),

(ii)_{0} Next let q_{0,2} denote the smallest value in [q_{0,1}, q^{00}] satisfying

r + \frac{2}{2^{n_{0}}} SI (·)_{[p_{q_{0,2}}^{∗},∞)} ≡ i(p_{q_{0,2}}^{∗}),

(iii)_{0} Next let q_{0,3} denote the smallest value in [q_{0,2}, q^{00}] satisfying

r + \frac{3}{2^{n_{0}}} SI (·)_{[p_{q_{0,3}}^{∗},∞)} ≡ i(p_{q_{0,3}}^{∗})

and continue this procedure until we arrive at (2^{n_{0}} − 1)_{0}. Let q_{0,2^{n_{0}}−1} denote the smallest value in [q_{0,2^{n_{0}}−2}, q^{00}] satisfying

r + \frac{2^{n_{0}}−1}{2^{n_{0}}}SI (·)_{[p_{q_{0,2^{n_{0}} −1}}^{∗},∞)} ≡ i(p_{q_{0},2^{n_{0−1}}}^{∗}).

It is readily seen that the function

f^{0} = r + ∑_{k=1}^{2^{n_{0}}−1} \frac{k}{2^{n_{0}}} SI (·)_{[p_{q_{0,k},∞)}^{∗}}

is a step function on [p^{0}, p^{1}] which agrees with i(·) at the points {p_{q_{0,k}}^{∗}} and that

(C_{0})  0 ≤ i(·) − f^{0}(·) ≤ \frac{1}{2^{n_{0}}} S;

Step 2 Select an integer n_{1} > n_{0} and let q_{1,1} be the smallest value in [q^{0}, q^{00}] satisfying

(i)_{1} r + \frac{1}{2^{n_{1}}} SI (·)_{[p_{q_{1,1}}^{∗},∞)} ≡ i(p_{q_{1,1}}^{∗})

(ii)_{1} Next let q_{1,2} denote the smallest value in [q_{1,1}, q^{00}] satisfying

r + \frac{2}{2^{n_{1}}} SI (·)_{[p_{q_{1,2}}^{∗},∞)} ≡ i(p_{q_{1,2}}^{∗})

(iii)_{1} Next let q_{1,3} denote the smallest value in [q_{1,2}, q^{00}] satisfying r + \frac{3}{2^{n_{1}}} SI_{[p_{q_{1,3}}^{∗},∞)}(·) ≡ i(P_{q_{1,3}}^{∗}), and continue this process until arriving at step (2^{n_{1}} − 1)_{1}: Let q_{1,2^{n_{1}}−1} denote the smallest value in [q_{0,2^{n_{1}}−2}, q^{00}] satisfying

r + \frac{2^{n_{1}}−1}{2^{n_{1}}}SI (·)_{[p_{q_{1},2^{n_{1}} −1}^{∗},∞)} ≡ i(p_{q_{1,2^{n_{1}} −1}}^{∗}).

It is clear that the function f^{1} = r + ∑_{k=1}^{2^{n_{1}}−1} \frac{k}{2^{n_{1}}} SI (·)_{p_{q_{1,k}}^{∗} ,∞} is a step function on [p^{0}, p^{1}] such that f^{1} ≥ f^{0} and f^{1} agrees with i(·) at the points p_{q_{1,k}}^{∗} and satisfies

(C_{1})  0 ≤ i(·) − f^{1}(·) ≤ \frac{1}{2^{n_{1}}} S.

By continuing in this fashion we produce a sequence of step functions f^{j}(·) which converge uniformly to i(·) on [p^{0}, p^{1}] and for which the associated points p_{q_{s,j}}^{∗} are densely distributed outside intervals of constancy of the monotone continuous function i(·). Now the results described in [14] and [15] imply that for each q > 1 the associated “pseudominimizer” is given by

u_{P M,q} = r(x)^{\frac{2q−2−γq}{2q−1}} ( \begin{matrix}cos γ_{q}θ(x) \\ sin γ_{q}θ(x) \end{matrix} ) = r(x)^{1− \frac{1+γq}{2q−1}} ( \begin{matrix}cos γ_{q}θ(x) \\ sin γ_{q}θ(x) \end{matrix} )

which [avoiding complex-valued notation] we abbreviate as

(8)

u_{P M,q} = r(x)^{δ_{q}} ( \begin{matrix}cos γ_{q}θ(x) \\ sin γ_{q}θ(x) \end{matrix} ) , with

γ_{q} = 1 − \frac{1}{q} − ϵ_{q}, δ_{q} = 1 − \frac{1 + γ_{q}}{2q − 1} = 1 − \frac{1}{q} + \frac{ϵ_{q}}{2q − 1}, for a small ϵ_{q} > 0.

Thus

∇u_{P M,q} = (r(x))^{\frac{−(1+γq)}{2q−1}} ( \begin{matrix}δ_{q} cos γ_{q}θ(x) & −γ_{q} sin γ_{q}θ( x) \\ δ_{q} sin γ_{q}θ(x) & γ_{q} cos γ_{q}θ(x) \end{matrix} )

whence by equation (2) for u = u + iv we have

(9)

\begin{aligned}W_{q}(∇u_{P M,q}) = & [( \frac{∂u(x)}{∂r(x)} − \frac{1}{r(x)} \frac{∂v(x)}{∂θ(x)} )^{2} + ( \frac{1}{r(x)} \frac{∂u(x)}{∂θ(x)} + \frac{∂v(x)}{∂r(x)} )^{2}]^{q} . \\ & = [||∇u||^{2} − 2det ∇(u)]^{q} = r(x)^{\frac{−2q(1+γq)}{2q−1}} (δ_{q} − γ_{q})^{2q} . \end{aligned}

Thus

J_{q}[u_{P M,q}] = ∫_{Ωα′_{q}} [(\frac{∂u(x)}{∂r(x)} − \frac{1}{r(x)} \frac{∂v(x)}{∂θ(x)})^{2} + (\frac{1}{v(x)} \frac{∂u}{∂θ(x)} + \frac{∂r(x)}{∂r(x)})^{2}]^{q}dxdy

so that by (8)

(10)  \begin{matrix}E(u_{P M,q}) = “ ∫_{Ω_{α′_{q}}} W_{q}(∇u_{P M,q})dxdy” = \\ (δ_{q} − γ_{q})^{2_{q}}α′_{q} (2 − \frac{1}{q} + \frac{2}{2q − 1}ϵ_{q})^{−1} =: M_{q} with α′_{q} > \frac{β}{γ_{q}} , \end{matrix}

whence α′_{q} < α for ϵ_{q} > 0 small enough.

It follows from (9) that for all p ≥ \frac{2}{1+γ_{q}} one has

E(u_{P M,q})|_{W^{1,p}} ≡ M_{q}.

We now consider a final issue in demonstrating the continuity of the mapping from the composite to the region Ω_{β}. For all exponents p_{q_{j,k}}^{∗} ∈ [p^{0}, p^{1}] which arose in connection with the construction associated with i(·), we wish to ascertain those values q ∈ [q^{0}, q^{00}] which are associated with these exponents. Recall that for a given q ≥ q^{0} > 1 we have γ_{q} = 1 − \frac{1}{q} − ϵ_{q}, δ_{q} =

1 − \frac{1}{q} (1 − \frac{ϵ_{q}}{2q−1}) for some ϵ_{q} > 0 so that u_{P M,q} = r(x)^{δ_{q}} ( \begin{matrix}cos γ_{q}θ(x) \\ sin γ_{q}θ(x) \end{matrix} )

must be such that the p values in [p^{0}, p^{1}] for which E|_{W^{1,p}} ≠ 0 must include p_{q}^{∗} = \frac{2}{1−γ_{q}} = \frac{2q}{1+qϵ_{q}}. Thus for a given p_{l}^{∗} ∈ [p^{0}, p^{1}] one obtains

(11)  q_{p_{l}^{∗}} = \frac{p_{l}^{∗}}{2} (1 − \frac{p_{l}^{∗}}{2} ϵ_{q})^{−1} so that q_{p_{l}^{∗}} = \frac{p_{l}^{∗}}{2 (1 − \frac{p_{l}^{∗}}{2} ϵ_{q_{p^{∗}_{l}}})}

This ensures that on Γ_{3,α} αq_{p_{l}^{∗}} > β, so that on choosing α′_{q}γ_{q_{p^{∗}_{l}}} = β one finds α′_{q} < α. It follows that W_{q}(∇u_{P M,q}) ∈ L^{1} if and only if γ_{q} = 1 − \frac{1}{q} − ϵ_{q}, δ_{q} = 1 − \frac{1}{q} + \frac{ϵ_{q}}{2q−1}, whence δ_{q} − γ_{q} = \frac{2q}{2q−1}ϵ_{q}. Thus E|_{W^{1,p}} ≠ 0 for all those values γ_{q_{l}} with p_{q_{l}}^{∗} = \frac{1}{1− \frac{1}{q−1} ϵ} corresponding to q_{p_{l}^{∗}} ∼ \frac{p_{l}^{∗}}{2} (1 + ϵ_{q_{p^{∗}_{l}}}). This ensures that p_{l} ↦ q_{p_{l}}^{∗} is a continuous mapping from Γ_{3,α′} onto Γ_{3,β}. Thus [for the case where i(·) has no interval of constancy] the mapping described here when applied to the dense sequence of exponents in [p^{0}, p^{1}] yields a dense sequence on a subinterval of [q^{0}, q^{00}], from which the continuity result for the mapping from the composite to Ω_{β} follows. [An adaptation to the case where i(·) has intervals of constancy can be easily achieved.]

In view of the brevity of the description provided for the construction which was referred to as a “final issue” it seems worthwhile to discuss matters in a bit more detail. First, given a small number e_{1} > 0 it follows from our construction of the step functions f^{0}, f^{1}, … approximating the monotone continuous function i(·) on [p^{0}, p^{1}] (cf. (C_{0}), (C_{1})) that for some sufficiently large k the step function f^{k} will satisfy the requirement

(12) f^{k}(p) = ∑_{l=1}^{2^{k}−1}(r + \frac{l}{2^{n_{k}}} SI (p)_{[p_{qk,l}−∞)} = i(p) for p_{qk,1} ∈ [p^{0}, p^{1}], 1 ≤ l ≤ 2^{k} −1

as well as

0 ≤ i(·) − f^{k}(·) ≤ \frac{1}{2^{n_{k}}} S in [p^{0}, p^{1}], q_{k,l+1} − q_{k,l} < e_{1}

Since by (8)

u_{P M,q} = r(x)^{δ_{q}} ( \begin{matrix}cos γ_{q}θ(x) \\ sin γ_{q}θ(x) \end{matrix} ) , ∇u_{P M,q} = r(x)^{δq−1} \binom{δ_{q} cos γ_{q}θ(x) − γ_{q} sin γ_{q}θ(x)}{δ_{q} sin γ_{q}θ(x) γ_{q} cos γ_{q}θ(x)}

with γ_{q} = 1 − \frac{1}{2} − ϵ_{q}, δ_{q} = 1 − \frac{1 + γ_{q}}{2q − 1} = 1 − \frac{1}{q} + \frac{ϵ_{q}}{2q − 1}, for some ϵ_{q} > 0, (9) yields

W_{q}(∇u_{P M,q}) = [||∇u_{P M,q}||^{2}−2det ∇u_{P M,q}]^{q} = r(x)^{2q(δ_{q}−1)}(δ_{q}−γ_{q})^{2q} ∈ L^{1}(Ω_{α′}; Ω_{β})

so that (10) yields

(13)

J_{q}[u_{pM,q}] = α′_{q} \frac{2q − 1}{2qϵ_{q}} (\frac{2q}{2q − 1}ϵ_{q})^{2q} = α′_{q} (\frac{2q}{2q − 1}ϵ_{q})^{2q−1} =:K_{q}. Thus by (∗∗) and (7)

E|_{W^{1,p}} = “J_{q}[u_{P Mq}](p)” = K_{1} for all p ∈ [p_{q}^{∗}, p_{q}^{∗∗}) [actually for all p ∈ [p_{q}^{∗}, ∞)]

We now consider one further item. For those exponents p_{q_{k,l}} ∈ [p^{0}, p^{1}] associated with the entries q_{k,l} ∈ [q^{0}, q^{00}] involved in constructing the function f^{k} we find p ≥ p_{q}^{∗} = \frac{2q}{1+qϵ_{q}} = \frac{2}{1−γ_{q}} so that \frac{1+ϵ_{q_{k,l}}}{q_{k,l}} < \frac{2}{p_{q_{k,l}}}

This leads to q_{kl} ≈ \frac{p_{q_{k,l}}}{2} so that the q values corresponding to the given p values are close to one another – thus ensuring the claimed continuity of the q mappings.

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ACKNOWLEDGEMENTS

First, I want to express my appreciation to Moshe Marcus for his en-

couragement in my efforts to pursue this line of research. I also want to

express my appreciation to Mikil Foss for some challenging questions and to

Bill Hrusa for his effort to educate me in the study of nonlinear elasticity.

Finally, I want to express my appreciation to Walter Rudin, who as a C.L.E.

Moore Instructor at MIT introduced me long ago to the beauty of rigorous

abstract analysis, and who through his textbooks has succeeded in educating

large numbers of scientists in the utility of abstract analysis.