Polyconvexity and existence theorem for nonlinearly elastic shells

0.17 MB·

by  Sylvia Anicic

Polyconvexity and existence theorem for
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nonlinearly elastic shells
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0
2

n Sylvia Anicic
a
J

9 January 11, 2017

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P
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Abstract
.
h
We present an existence theorem for a large class of nonlinearly
t
a elastic shells. We restrict our discussion to hyperelastic materials,
m
that is to elastic materials possessing a stored energy function. We
[
define the notion of a polyconvex and orientation preserving stored

1 energy function for shells and we give an example of such a function.
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0
3
3 1 Introduction
2
0
. A shell is a three-dimensional continuum which occupies a volume contained
1
0 between two surfaces (in general parallel) close to each other. A natural way
7 todefinea shell istoconsider asurfaceS embedded inR3 andtothicken it on
1
each side. For non-extreme aspect to thickness ratios, one can use 3D FEM
:
v
codes but for ultrathin materials, such as thin polymeric films or biological
i
X
membranes, a 2D shell model is needed.
r
a In the mathematical analysis of 3D hyperelasticity, the lack of convexity
of the stored energy function stood for a long while as a major difficulty
in establishing an existence theorem until J. M. Ball was able to overcome
it in a landmark paper (Ball, 1977) by means of the weaker requirement of
polyconvexity.
The purpose of this paper, inspired by the approach of J. M. Ball, is to
present a general theorem of existence of global minimizers to a large class
of nonlinear shell models under realistic hypothesis.
To give the starting point of our result, let us briefly recall the framework
considered in the context of three-dimensional elasticity. Let Ω R3 be a
⊂
1
domain considered as the reference configuration of an elastic body. The
admissible deformations Θ : Ω R3 satisfy
→
det Θ > 0.
∇
Nowweconsider ashell withthickness 2ε > 0whosereferenceconfiguration
C
is the set
= Φ(x,z) = ϕ(x)+za (x), (x,z) Ω := ω ( ε,ε)
3
C { ∈ × − }
where ω R2 is a domain and
⊂
∂ ϕ(x) ∂ ϕ(x)
a (x) := 1 ∧ 2
3 ∂ ϕ(x) ∂ ϕ(x)
1 2
| ∧ |
is the unit normal vector to the middle surface S := ϕ(ω). We make the
realistic assumption that the deformations Θ : R3 of the shell are of the
C →
form
Θ(Φ(x,z)) = ψ(x)+za (ψ)(x), (x,z) Ω,
3
∈
where
∂ ψ(x) ∂ ψ(x)
a (ψ)(x) := 1 ∧ 2
3
∂ ψ(x) ∂ ψ(x)
1 2
| ∧ |
is the unit normal vector to the deformed middle surface Sˆ := ψ(ω) of the
shell. By letting
Ψ(x,z) := Θ Φ(x,z) = ψ(x)+za (ψ)(x),
3
◦
it follows that
det Ψ(x,z) = det Θ(Φ(x,z))det Φ(x,z)
∇ ∇ ∇
Hence, in order that the condition det Θ(Φ(x,z)) > 0 may be satisfied, we
∇
will impose
det Φ > 0 and det Ψ > 0.
∇ ∇
Thus, since
z z
det Ψ = 1 1 ∂ ψ ∂ ψ
∇ − R (ψ) − R (ψ) | 1 ∧ 2 |
(cid:18) 1 (cid:19)(cid:18) 2 (cid:19)
2
where 1/R (ψ), 1/R (ψ) arethe principal curvatures of thedeformed middle
1 2
surface, we impose
ε
∂ ψ ∂ ψ = 0 and max < 1.
1 ∧ 2 6 α∈{1,2}(cid:12)Rα(ψ)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
Notethatthecondition∂ ψ ∂ ψ = 0isalreadymentioned in(Ciarlet et al.,
1 2 (cid:12) (cid:12)
∧ 6
2013) where the authors present a notion of polyconvexity and orientation-
preserving condition for a surface. However, for a shell, this condition is not
sufficient to ensure the local injectivity of the deformation.
2 Preliminaries
In all that follows, Greek indices and exponents range in the set 1,2 while
{ }
Latin indices and exponents range in the set 1,2,3 (save when they are
{ }
used for indexing sequences). We use the Einstein summation convention
with respect to repeated indices and exponents.
The three-dimensional Euclidian space is identified with R3 by choosing
an origin and a Euclidian basis. Vector and tensor fields are denoted by
boldface letters. The Euclidian norm, the inner product, the vector product
and the tensor product of two vectors u and v in R3 are respectively denoted
u , u v, u v and u v. The sets of all n n real matrices and of all
| | · ∧ ⊗ ×
m n real matrices are respectively denoted Mn and Mm×n. The notation
×
A designate the Frobenius norm of a real matrix A Mm×n defined by
| | ∈
A := tr(ATA)1/2.
| |
A domain ω R2 is a bounded, connected, open set with a Lipschitz-
⊂
continuous boundary γ := ∂ω, the set ω being locally on the same side of γ.
A generic point in the set ω is denoted by x = (x ) and partial derivatives,
α
in the classical or distributional sense, are denoted ∂ := ∂/∂x .
α α
The notation Lp(ω;R3) with 1 6 p < denotes the space of vector fields
∞
ξ = (ξ ) : ω R3 with components ξ in the usual Lebesgue space Lp(ω). It
i i
→
is equipped with the norm
1/p
ξ := ξ(x) pdx for any ξ Lp(ω;R3).
Lp
k k | | ∈
(cid:18)Zω (cid:19)
Likewise, the notation W1,p(ω;R3) denotes the space of vector fields ξ =
(ξ ) : ω R3 with components ξ in the usual Sobolev space W1,p(ω). It is
i i
→
3
equipped with the norm
1/p
2
kξkW1,p := kξkpLp + k∂αξkpLp for any ξ ∈ W1,p(ω;R3).
!
α=1
X
The space W1,∞(ω;R3) denotes the space of vector fields ξ = (ξ ) : ω R3
i
→
with components ξ in the usual Sobolev space W1,∞(ω). The space W1,∞(ω)
i
consists of those Lipschitz continuous functions on ω.
Strong and weak convergences are respectively denoted and ⇀.
→
3 Definition of a G1 shell
The purpose of this section is to define the regularity of the shell that we
consider. To this end, we define a G1 shell which has been first introduced in
(Anicic, 2001) and (Anicic, 2003). The term G1 stands for First-Order Geo-
metric Continuity. This regularity allows us to take into account curvature
discontinuities of S aswell asthe tangent plane continuity even if thetangent
vectors are not continuous. Hence, if we consider a surface which is defined
via smooth patches, we are only lead to match the unit normal vectors on
their interfaces and not the tangent vectors. This makes for great versatility
in practice. Besides, this regularity does not involve any Christoffel symbols.
The middle surface of the reference configuration of a shell is denoted by
S := ϕ(ω) where
ϕ W1,∞(ω;R3). (1)
∈
The two vectors a := ∂ ϕ L∞(ω;R3) span the tangent plane to the
α α
∈
surface S. We suppose that ϕ satisfy the additional assumption
a a
ess inf a a > 0 and a := 1 ∧ 2 W1,∞(ω;R3), (2)
ω | 1 ∧ 2| 3 a1 a2 ∈
| ∧ |
where a is the unit normal vector to the surface S.
3
Thecovariant componentsa L∞(ω)ofthefirst fundamental formand
αβ
∈
b L∞(ω) of the second fundamental form of S are respectively defined
αβ
∈
by
a := a a and b := a ∂ a = a ∂ a .
αβ α β αβ α β 3 β α 3
· − · − ·
The area element along S is √adx, where
a := det(a ) = a a 2 L∞(ω).
αβ 1 2
| ∧ | ∈
4
Since
ess inf a(x) > 0
ω
the inverse of the matrix (a ), which we denote by (aαβ), belongs to L∞(ω).
αβ
The contravariant basis aα L∞(ω;R3) is then defined by letting
∈
aα = aαβa
β
and then satisfy
aα a = δα,
· β β
where δα is the Kronecker symbol.
β
The mixed components bβ L∞(ω) of the second fundamental form are
α ∈
defined by
bβ := b aρβ.
α αρ
The mean curvature H L∞(ω) and the gaussian curvature K L∞(ω) are
∈ ∈
respectively defined by
1 1 1 1 1
H := (b1 +b2) = + and K := b1b2 b2b1 = ,
2 1 2 2 R R 1 2 − 1 2 R R
(cid:18) 1 2(cid:19) 1 2
where the invariants 1/R are the principal curvatures of the middle surface
α
S of the shell.
The reference configuration of a shell with thickness 2ε > 0 is the set
Φ(x,z) := ϕ(x)+za (x); (x,z) Ω := ω ( ε,ε) .
3
{ ∈ × − }
The tangent vectors are respectively defined by
g := ∂ Φ = a +z∂ a .
α α α α 3
The two vectors g form a basis of the tangent plane of S; its contravariant
α
basis gα is defined by
gα g = δα,
· β β
where δα is the Kronecker symbol. Hence
β
det Φ(x,z) = g (x,z) g (x,z) a (x)
∇ 1 ∧ 2 · 3
= (1 2H(x)z +K(x)z2) a(x)
−
z z
p
= 1 1 a(x).
− R (x) − R (x)
(cid:18) 1 (cid:19)(cid:18) 2 (cid:19)
p
5
In addition to the hypothesis (1)-(2) and in order that gα L∞(Ω;R3), we
∈
also impose that ϕ and ε satisfy the following assumption:
ess inf det Φ > 0. (3)
Ω ∇
To sum up, equivalently to the hypothesis (1)-(2)-(3), we will suppose
that ϕ G1 where
∈
G1 := ϕ W1,∞(ω;R3); ess inf a a > 0,
1 2
( ∈ ω | ∧ |
a a ε
1 ∧ 2 W1,∞(ω;R3), max < 1 .
|a1 ∧a2| ∈ α∈{1,2}(cid:12)Rα(cid:12)∞,ω )
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
4 An existence theorem
In this section, we define a notion of polyconvex and orientation-preserving
stored energy function for a shell and we establish an existence theorem for
the minimization problem of a nonlinearly elastic shell.
Let ω be a domain in R2. For a given deformation ψ W1,p(ω;R3),
∈
p > 2, we denote
a(ψ) := det(a (ψ)) = ∂ ψ ∂ ψ 2
αβ 1 2
| ∧ |
where a (ψ) := ∂ ψ ∂ ψ and if ∂ ψ ∂ ψ = 0 a.e. in ω, we denote
αβ α β 1 2
· ∧ 6
∂ ψ ∂ ψ
a (ψ) := 1 ∧ 2
3 ∂ ψ ∂ ψ
1 2
| ∧ |
the unit normal vector to the deformed surface ψ(ω).
If a (ψ) W1,p(ω;R3), we denote
3
∈
1 1 1 1
H(ψ) := + and K(ψ) :=
2 R (ψ) R (ψ) R (ψ)R (ψ)
(cid:18) 1 2 (cid:19) 1 2
themeanandgaussiancurvatureofthedeformedsurfaceψ(ω)where1/R (ψ)
1
and 1/R (ψ) are the principal curvatures, namely the two eigenvalues of the
2
matrix (bβ(ψ)) defined as
α
bβ(ψ) := b (ψ)aρβ(ψ), b (ψ) := ∂ a (ψ) ∂ ψ,
α αρ αρ − α 3 · ρ
with the matrix (aαβ(ψ)) the inverse of the matrix (a (ψ)).
αβ
6
Theorem 1. Let ω be a domain in R2 and let γ be a non-empty relatively
0
open subset of γ := ∂ω. For p > 2 and q > 1, we define the functional
I : Vε R + by letting
→ ∪{ ∞}
Vε := ψ W1,p(ω;R3); a(ψ) Lq(ω), a(ψ) = 0 a.e. in ω,
{ ∈ ∈ 6
a (ψ) W1,p(ω;R3), max ε/R (ψ) , ε/R (ψ) < 1 a.e. in ω,
3 p 1 2
∈ {| | | |}
ψ = ϕ and a (ψ) = a dγ-a.e. in γ
3 3 0
}
and for each ψ Vε,
∈
I(ψ) := W(x,ψ)dx L(ψ,a (ψ)),
3
−
Zω
where L is a continuous linear form over the space W1,p(ω;R3) W1,p(ω;R3)
×
and W : ω Vε R is a function with the following properties:
× →
(a) Polyconvexity: For almost all x ω, there exist a convex function
∈
W(x, ) : M R where
· →
M := (A,B,a,b,c) (M )2 R3; a b > 0 and a 2 b +c > 0
3×2
{ ∈ × −| | − | | }
such that for almost all x ω
∈
W(x,ψ) = W x, ψ(x), a (ψ)(x), 1,εH(ψ(x)),ε2K(ψ(x)) a(ψ(x)) .
3
∇ ∇
(b) Measur(cid:16)ability: The function W(cid:0)( ,A,B,a,b,c) : ω R (cid:1)ispmeasurab(cid:17)le
· →
for all (A,B,a,b,c) M.
∈
(c) Coerciveness: There exist constants C > 0 and C such that
1 2
W(x,ψ) > C ψ p + a (ψ) p +a(ψ)q/2 +C
1 3 2
{|∇ | |∇ | }
for all ψ Vε and almost all x ω.
∈ ∈
(d) Orientation-preserving condition:
W(x,ψ) as 1 2εH(ψ(x))+ε2K(ψ(x)) a(ψ(x)) 0+
→ ∞ { − } →
and W(x,ψ) as 1+2εH(ψ(x))+ε2K(ψ(x)) pa(ψ(x)) 0+
→ ∞ { } →
for all ψ Vε and almost all x ω. p
∈ ∈
Assume that infψ∈VεI(ψ) < , then there exists at least one function
∞
η Vε such that
∈
I(η) = inf I(ψ).
ψ∈Vε
7
Proof. (i)The integrals W(x,ψ)dx are well definedfor all ψ Vε. First,
ω ∈
wenotethattheset Misaconvex opensubset of(M )2 R3. Furthermore,
3×2
R ×
each ψ Vε satisfy a(ψ(x)) > 0 and ε/R (ψ(x)) < 1 for almost all x ω,
α
∈ | | ∈
hence
1 ε ε
εH(ψ(x)) = + < 1,
| | 2 R (ψ(x)) R (ψ(x))
(cid:12) 1 2 (cid:12)
(cid:12) (cid:12) ε ε
1 2εH(ψ(x))+(cid:12) ε2K(ψ(x)) = 1 (cid:12) 1 > 0,
(cid:12) (cid:12)
− − R (ψ(x)) − R (ψ(x))
(cid:18) 1 (cid:19)(cid:18) 2 (cid:19)
ε ε
1+2εH(ψ(x))+ε2K(ψ(x)) = 1+ 1+ > 0.
R (ψ(x)) R (ψ(x))
(cid:18) 1 (cid:19)(cid:18) 2 (cid:19)
Thus,
εH(ψ(x)) a(ψ(x)) < a(ψ(x))
| |
and 2 εH(ψ(x))p a(ψ(x)) <p1+ε2K(ψ(x)) a(ψ(x))
| |
p (cid:16) (cid:17)p
almost everywhere in ω. It follows that almost everywhere in ω,
ψ(x), a (ψ)(x), 1,εH(ψ(x)),ε2K(ψ(x)) a(ψ(x)) M.
3
∇ ∇ ∈
Furthe(cid:0)rmore, for almost a(cid:0)ll x ω, the function(cid:1)Wp(x, ) : (cid:1)M R in
∈ · →
continuous (as a convex and real-valued function on a convex open sub-
set of a finite-dimensional space (Ciarlet, 2013, Theorem 2.17-1)). For all
(A,B,a,b,c) M, the function W( ,A,B,a,b,c) : ω R is measur-
∈ · →
able, and M is a Borel set. Therefore the function W : ω M R is
× →
a Carath´eodory function, and consequently the function
x ω W(x, ψ(x), a (ψ)(x),α(x),β(x),γ(x)) R
3
∈ → ∇ ∇ ∈
with α(x) := a(ψ(x)), β(x) := εH(ψ(x))α(x) and γ(x) := ε2K(ψ(x))α(x)
is measurable for each ψ Vε. The function W being in addition bounded
p ∈
from below (by the coerciveness inequality), the integral
W(x,ψ)dx = W(x, ψ(x), a (ψ)(x),α(x),β(x),γ(x))dx
3
∇ ∇
Zω Zω
is thus a well-defined extended real number in the interval [C area ω, ] for
2
∞
each ψ Vε.
∈
8
(ii) We find a lower bound for I(ψ) when ψ Vε.
∈
From the assumed coerciveness (c) of the function W and the assumed
continuity of the linear form L, we infer that there exists a constant C > 0
3
such that
I(ψ) > C ψ p + a (ψ) p +a(ψ)q/2 dx+C area ω
1 3 2
{|∇ | |∇ | }
Zω
C ( ψ + a (ψ) ) for all ψ Vε.
3 1,p,ω 3 1,p,ω
− k k k k ∈
Combining the boundary conditions ψ = ϕ and a (ψ) = a on γ with the
3 3 0
generalized Poincar´e inequality, we thus conclude that there exist constants
C > 0 and C such that
4 5
I(ψ) > C ψ p + a (ψ) p + a(ψ) q +C for all ψ Vε.
4{k k1,p,ω k 3 k1,p,ω k k0,q,ω} 5 ∈
(iii) We show that if (ηk) is a sequenpce with ηk Vε for all k for which
∈
there exist η W1,p(ω;R3), κ W1,p(ω;R3), (ξ ,ξ ,ξ ) (Lq(ω;R3))3 and
∈ ∈ 1 2 3 ∈
(α ,α ,α ) (Lq(ω))3 such that
1 2 3
∈
ηk ⇀ η in W1,p(ω;R3), a (ηk) ⇀ κ in W1,p(ω;R3)
3
∂ ηk ∂ ηk ⇀ ξ in Lq(ω;R3), a(ηk) ⇀ α in Lq(ω)
1 ∧ 2 1 1
H(ηk)∂ ηk ∂ ηk ⇀ ξ in Lq(ω;Rp3), H(ηk) a(ηk) ⇀ α in Lq(ω)
1 ∧ 2 2 2
K(ηk)∂ ηk ∂ ηk ⇀ ξ in Lq(ω;R3), K(ηk)pa(ηk) ⇀ α in Lq(ω)
1 ∧ 2 3 3
p
then almost everywhere in ω
κ = a (η), max ε/R (η) , ε/R (η) 6 1,
3 1 2
{| | | |}
ξ = ∂ η ∂ η, ξ = H(η)∂ η ∂ η, ξ = K(η)∂ η ∂ η
1 1 ∧ 2 2 1 ∧ 2 3 1 ∧ 2
α = a(η), α = H(η) a(η) and α = K(η) a(η).
1 2 3
p p p
To prove this assertion, we begin by showing that κ = a (η). By the Rellich-
3
Kondraˇsov compact imbedding theorem W1,p(ω;R3) ⋐ Lr(ω;R3) for all r
with 1 6 r < (Adams and Fournier, 2003, p. 168),
∞
1 1
a (ηk) κ in Lp′(ω;R3), + = 1
3 → p p′
and a (ηk) κ in L2(ω;R3).
3
→
9
Hence
∂ ηk a (ηk) ⇀ ∂ η κ in L1(ω)
α 3 α
· ·
and a (ηk) 2 κ 2 in L1(ω).
3
| | → | |
Since for all k, ∂ ηk a (ηk) = 0 and a (ηk) = 1, it follows that
α 3 3
· | |
∂ η κ = 0 and κ = 1.
α
· | |
In order to prove that κ = a (η), it remains to show that
3
∂ η ∂ η κ > 0 a.e. on ω.
1 2
∧ ·
In order to simplify the notations, we define for all ϕ W1,p(ω;R3) and all
1 ∈
ϕ W1,p(ω;R3)
2 ∈
1
[ϕ ,ϕ ] := (∂ ϕ ∂ ϕ +∂ ϕ ∂ ϕ ) Lp/2(ω;R3).
1 2 2 1 1 ∧ 2 2 1 2 ∧ 2 1 ∈
We now notice that [ϕ ,ϕ ] can be rewritten in the following form
1 2
1
[ϕ ,ϕ ] = ∂ (ϕ ∂ ϕ +ϕ ∂ ϕ )+∂ (∂ ϕ ϕ +∂ ϕ ϕ ) .
1 2 4 { 1 1 ∧ 2 2 2 ∧ 2 1 2 1 1 ∧ 2 1 2 ∧ 1 }
Hence, if(ϕk,ϕk)isasequence ofW1,p(ω;R3), p > 2,whichconvergesweakly
1 2
to (l ,l ) W1,p(ω;R3), then by the Rellich-Kondraˇsov compact imbedding
1 2
∈
theorem W1,p(ω;R3) ⋐ Lr(ω;R3) for all r with 1 6 r < , it follows that
∞
for all α 1,2
∈ { }
1 1
ϕk l in Lp′(ω;R3), + = 1.
α → α p p′
Therefore,
ϕk,ϕk ⇀ [l ,l ] in ′(ω;R3).
1 2 1 2 D
Now by applying this r(cid:2)esult to(cid:3) the sequence (ηk) which converges weakly to
η in W1,p(ω;R3), it follows that
[ηk,ηk] := ∂ ηk ∂ ηk ⇀ [η,η] := ∂ η ∂ η in ′(ω;R3)
1 2 1 2
∧ ∧ D
Hence
ξ = ∂ η ∂ η and ∂ ηk ∂ ηk ⇀ ∂ η ∂ η in Lq(ω;R3).
1 1 ∧ 2 1 ∧ 2 1 ∧ 2
10