Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 41, 3, pp. 487-508, Warsaw 2003 ASYMPTOTIC ANALYSIS OF NONLINEARLY ELASTIC SHELLS WITH VARIABLE THICKNESS Liliana Gratie Liu Bie Ju Center for Mathematical Sciences, City University of Hong Kong e-mail: mcgratie@cityu.edu.hk P.G.Ciarlet recently proposed, and justified with A. Roquefort through themethod of formal asymptotic expansions, a nonlinear shellmodel for shells with constant thickness. This model is analogous in its form to the model formerly proposed by W.T.Koiter, but is more amenable to numerical computations. In the same spirit, we propose and we justi- fy here, again by the method of formal asymptotic expansions, a more general nonlinearmodel, which is valid for shellswith variable thickness. Key words: asymptotic analysis, nonlinearly elastic shells, Koiter’s mo- del, variable thickness, energy functional, variational problems 1. Introduction and technical preliminaries In this paper, we propose and, using the method of formal asymptotic expansions, we justify a shell model ”of Koiter’s type” for nonlinearly elastic shells with variable thickness, which extends that proposed byCiarlet (2000b) for shellswith constant thickness. In doing so,we show that nonlinearly elastic shells with variable thickness have two essentially distinct limit behaviors as their thickness approaches zero, either that of a nonlinearly elastic membrane shell or that of a nonlinearly elastic flexural shell with variable thickness. Complete proofs and further details will be found in Gratie (2003). We emphasize here that ”membrane and flexural shells” represents a gene- ral terminology about shells that is commonly used in theWestern literature, as in e.g., Ciarlet (2000a). Other terminologies are often favored. In this direc- tion, the author is grateful to the referee, who pointed out that ”membrane shells” and ”flexural shells” could be equally well labeled as ”geometrically 488 L.Gratie rigid shells” and ”geometrically bendable shells”. This latter terminology is adopted in the present article. Note that, while there is a huge literature about shells with constant thick- ness (see e.g., the extensive list of references provided in Ciarlet (2000a)), comparatively much less attention has been paid to the analysis of shells with variable thickness. This problem however, was addressed in the pioneering contributions of Ladevèze (1976) and Busse (1997) for linearly elastic shells. The derivation of variational equations of our model is based on the me- thod of asymptotic expansions. We use here the well-established ”variational approach of Ciarlet” (to paraphraseGilbert andVashakmadze (2000)) and, in particular, we use the samenotations as inCiarlet (2000a). As is customary in the mathematical elasticity theory, Greek indices or exponents: α, β, µ, etc. take their values in the set {1,2}, while Latin indices or exponents: i, j, k, etc. take their values in the set {1,2,3}, andweuse the summation convention with respect to repeated indices and exponents. Let ω be a domain in R2, i.e., an open, bounded, connected subset with a Lipschitz- continuous boundary γ = ∂ω, such that the set ω is locally on one side of γ, and let y = (yα) denote a generic point in the closed set ω. The area element in ω is dy and the partial derivatives with respect to the variable y are denoted ∂α = ∂/∂yα and ∂αβ = ∂ 2/(∂yα∂yβ). The length element along the boundary γ is denoted dγ, the unit outer normal vector and the unit tangent vector along γ are respectively denoted (να) and (τα), where τ1 =−ν2, τ2 = ν1. We denote by ∂νf = να∂αf the outer normal derivative of a function f along the boundary γ, and similarly, its tangential derivative by ∂τf = τα∂αf. Sometimes, the ”horizontal” curvilinear coordinates xα will be also denoted yα. Let θ : ω ⊂ R2 → R3 be an injective and smooth enough mapping, such that the two vectors aα(y) := ∂αθ(y) are linearly independent at all points y = (x1,x2) ∈ ω. They form the covariant basis of the tangent plane to the surface S := θ(ω) at the point θ(y). On the other hand, the two vectors aα(y), defined by the relations aα(y) ·aβ(y) = δαβ , form the contravariant basis of the same tangent plane. We consider a third vector, normal to S at the point θ(y), with Euclidean norm one, defined by a3(y)=a 3(y)= a1(y)×a2(y) |a1(y)×a2(y)| The triple ( a1(y),a2(y),a3(y) ) is the contravariant basis at θ(y), and similarly, ( a1(y),a2(y),a3(y) ) is the covariant basis at the same point. Asymptotic analysis of nonlinearly elastic shells... 489 A general shell structure can be fully represented by a middle surface geometry and the thickness at each point of its middle surface. We intend to model a family of nonlinearly elastic thin shells having in common the middle surface S := θ(ω), and such that for each ”small” para- meter ε> 0, the variable thickness of each shell is defined by h(y) := 2εe(y) for all y∈ω, where e :ω→ R is a given function, which does not depend on ε.We shall assume for definite- ness that e∈W2,∞(ω).We also assume that the ”thickness function” e does not vanish in ω. Thus, there exist two positive constants e0 and e1 such that 0 0 small enough, the mapping Θe : Ω ε → R3 is injective (see Ciarlet, 2000a) and the three vectors g e,ε i (x ε) := ∂εiΘ e(xε) are linearly independent. This shows that thephysicalproblemiswell posed since the set Θe(Ω ε )doesnot interpenetrate itself. The three vectors g e,ε i (x ε) form the covariant basis (of the tangent spa- ce, here R3, to the manifold Θe(Ω ε )) at the point Θe(xε), and the three vectors gi,e,ε(xε) defined by the relations gi,e,ε(xε) · ge,εj (xε) = δij form the contravariant basis at Θe(xε). Each shell is subjected to: • a boundary condition of place along the portion Θe(γ0 × [−ε,ε]) of its lateral face Θe(γ × [−ε,ε]), where γ0 ⊂ γ with length(γ0) > 0; this means that the displacement vanishes on Θe(γ0× [−ε,ε]) • applied body forces fi,ε ∈L2(Ωε) in its interior Θe(Ωε) 490 L.Gratie • applied surface forces hi,ε ∈ L2(Γε+ ∪Γε−) on its upper and lower faces Θe(Γε+) and Θ e(Γε − ), where Γε+ :=ω×{ε} and Γε− :=ω×{−ε}. We recall now some elementary notions from differential geometry in R3. Thearea element along the surface S =θ(ω) is √ adywhere a=det{aαβ(y)}, and aαβ(y)=aα(y) ·aβ(y)= ∂αθ(y) ·∂βθ(y) are the covariant components of themetric tensor of the surface S (also named the first fundamental form of S). Similarly, the contravariant components of the metric tensor of S are defined by aαβ =aα ·aβ. Note that thematrix {aαβ(y)} is positive definite since the vectors aα(y) are assumed to be linearly independent. In particular, there exists a positive constant a0 such that 0 0, which means that η=0 on γ0. Extending the definition given inMiara (1998) and Ciarlet (2000a, Chap- ter 9), we say that a shell is a nonlinearly elastic, geometrically rigid shell with variable thickness if {η=(ηi)∈ W2,p(ω); η=0 on γ0, aeαβ(η)−aαβ =0 in ω}= {0} Note that the various regularities mentioned above or subsequently are simply chosen so that the energies (to be introduced later) are differentiable. The covariant and mixed components of the curvature tensor of S (also named the second fundamental formof the surface) are respectively definedby bαβ =a 3 ·∂βaα and bβα = aβσbσα If the two vectors aeα(η) are linearly independent in ω, we let Reαβ(η) := b e αβ(η)−bαβ where beαβ(η) := ∂αβ(θ+η e) · ae1(η)×ae2(η) |ae1(η)×ae2(η)| Asymptotic analysis of nonlinearly elastic shells... 491 Next, we define functions ηeβ//α := ∂αηβ −Γ σ αβησ− 1 e bαβη3 ηe3//α := b σ αησ +∂α (1 e η3 ) where theChristoffel symbols of the surface S are given by Γσαβ =aσ ·∂βaα. Accordingly, we can rewrite the components Geαβ(η) as Geαβ(η) := 1 2 (aeαβ(η)−aαβ)= 1 2 (ηeα//β +η e β//α+a mnηem//αη e n//β) where ai3 = a3i := δi3. The Gâteaux derivatives of each function Geαβ : W 1,4(ω) → L2(ω) are given by (Geαβ) ′(ζ)η := 1 2 [ηeα//β +η e β//α+a mn(ζem//αη e n//β + ζ e n//βη e m//α)] We assume for simplicity that the shells are made of an homogeneous iso- tropic material of Saint Venant-Kirchhoff’s type. This implies in particular that the reference configuration Θe(Ω ε ) is a natural state, i.e. stress-free. Hence, the material is characterized by its two Lamé constants λε > 0 and µε > 0, and the contravariant components aαβστ,ε of its two-dimensional ela- sticity tensor are given by aαβστ,ε := 4λεµε λε+2µε aαβaστ +2µε(aασaβτ +aατaβσ) Extending the definition given in Lods and Miara (1998) and Ciarlet (2000a, Chapter 10), we say that a nonlinearly elastic shell with the middle surface S, subjected to a boundary condition of place along the portion of its lateral face with θ(γ0) as its middle curve, where γ0 ⊂ γ and length(γ0)> 0, is a nonlinearly elastic, geometrically bendable shell with variable thickness, if the manifold W e F(ω) := {η=(ηi)∈ W2,4(ω); η= ∂νη=0 on γ0, aeαβ(η)−aαβ =0 inω} and its tangent space TζW e F(ω) := {η∈ W2,4(ω); η= ∂νη=0 on γ0, (Geαβ)′(ζ)η=0 in ω} contains nonzero functions, i.e., WeF(ω) 6= {0} and TζWeF(ω) 6= {0} at each ζ ∈ WeF(ω). 492 L.Gratie Note that the admissible displacement field must satisfy in this case the ”two-dimensional boundary conditions of strong clamping” along the curve θ(γ0), i.e. not only the points and the tangents spaces (as for the weaker bo- undary conditions of clamping ηi = ∂νη3 = 0 on γ0), but also the vectors tangent to the coordinate lines of the deformed and undeformedmiddle surfa- ces coincide along the curve θ(γ0). This remark emphasizes the essential role played by the set θ(γ0) for determining the type of a shell. 2. Two-dimensional variational scaled problems for geometrically rigid and bendable, nonlinearly elastic shells with variable thickness In this section, we convert into ”the displacement approach” the two- dimensional equations of nonlinearly elastic, geometrically rigid and geome- trically bendable shells with variable thickness, as they were identified by Roquefort (2001, Chapter 4), through ”the deformation approach”. Our aim is to study the behavior of the displacement field uεig i,e,ε :Ω ε → R 3 that the shell undergoes the influence of the applied forces as ε → 0, by means of the method of formal asymptotic expansions. The unknown in the three-dimensional formulation is the vector field uε = (uεi) : Ω ε → R3, where the functions uεi : Ω ε → R represent the covariant components of the displacement field of the shell. This method relies in particular on two essential guiding rules: no restric- tion should be put on the applied forces and the linearization of any nonlinear equation found in this process should provide an equation from the linear theory (”linearization requirement”). Thefirst task in the asymptotic analysis consists in transforming the three- dimensional problems Pe(Ωε) (for a geometrically rigid or geometrically ben- dable shell) into ”asymptotically equivalent” problems posed over a domain independent of ε. More specifically, we let Ω :=ω×]−1,1[ Γ0 := γ0× [−1,1] Γ+ :=ω×{1} Γ− :=ω×{−1} where x = (x1,x2,x3) denotes a generic point in the closure Ω of the fixed domain Ω, and ∂i := ∂/∂xi. We make then the change of the variable from Asymptotic analysis of nonlinearly elastic shells... 493 the fixed domain Ω to the domain Ωε through the bijection πε :x=(x1,x2,x3)∈Ω→πε(x1,x2,x3)= (xε1,xε2,xε3)= = (x1,x2,εx3)=x ε ∈Ωε where the coordinate x3 ∈ [−1,1] is the scaled transverse variable. The rela- tions between the first derivatives with respect to the variable xε ∈ Ωε and the derivatives of the same order with respect to the scaled variable belonging to the fixed domain x∈Ω are ∂εα = ∂α and ∂ ε 3 = 1 ε ∂3 The scaled unknown u(ε) = (ui(ε)) : Ω → R3 satisfies the scaled three- dimensional nonlinear variational problem Pe(ε;Ω) of a shell with variable thickness (in Section 4 of Roquefort (2001), it is derived by means of the deformations of the middle surface). To begin the asymptotic analysis, we firstwrite the scaled unknown as a formal expansion in terms of powers of the thickness (considered as usually as a ”small” parameter) u(ε)= ε−ku−k+ ...+ε−2u−2+ε−1u−1+u0+εu1+ε2u2+ ... for some integer k­ 0. Given a function v :ω×]−1,1[→ R3, let v :ω→ R3 represent its average defined by the integral v(y) := 1 2 1 ∫ −1 v(y,x3) dx3 Thenwe have (note that it can be proved as inMiara (1998) that there are no negative powers, i.e. the first nonzero term of the formal series is indeed u0): Theorem 2.1. Consider a family of nonlinearly elastic, geometrically rigid shellswithnonvanishingvariable thickness h(y)= 2εe(y), e∈W2,∞(ω), with the same middle surface S = θ(ω) and with each subjected to a boundary condition of place along a portion of their lateral face having the same curve θ(γ0) as their middle line. Assume that the scaled unk- nown u(ε) = (ui(ε)) satisfying the scaled three-dimensional variational problem Pe(ε;Ω) admits a formal asymptotic expansion of the form u(ε)=u0+εu1+ε2u2+ ... 494 L.Gratie Then, to free the applied forces from any restriction and to satisfy the linearization requirement, the Lamé constants and contravariant compo- nents of the applied loading must be of the form λε =λ µε =µ fi,ε(xε)= fi,0(x) for xε =πε(x)∈Ωε hi,ε(xε)= εhi,1(x) for xε =πε(x)∈Γε+∪Γε− where the constants λ > 0, µ > 0 and the scaled functions fi,0(x)∈ L2(Ω), hi,1(x)∈L2(Γ+∪Γ−) are independent of ε. Under these hypotheses, the leading term u0 is independentof the trans- verse variable x3 and its average ζ0 := (ζ0i )= 1 2 1 ∫ −1 u0 dx3 =u 0 satisfies the scaled two-dimensional variational problem PeM(ω) of a non- linearly elastic, geometrically rigid shell with variable thickness: Find ζ0 ∈ WM(ω) := {η∈ W1,4(ω); η=0 on γ0} such that ∫ ω aαβστGeστ(ζ 0)[(Geαβ) ′(ζ0)η]e √ a dy= ∫ ω pi,0ηie √ a dy for all η=(ηi)∈ WM(ω), where, for any ζ,η∈ W1,4(ω) Geαβ(η) := 1 2 [aeαβ(η)−aαβ] = 1 2 (ηeα//β +η e β//α+a mnηem//αη e n//β) ηeβ//α := ∂αηβ −Γ σ αβησ− 1 e bαβη3 ηe3//α := b σ αησ+∂α (1 e η3 ) (Geαβ) ′(ζ)η := 1 2 [ηeα//β +η e β//α+a mn(ζem//αη e n//β + ζ e n//αη e m//α] aαβστ := 4λµ λ+2µ aαβaστ +2µ(aασaβτ +aατaβσ) pi,0 := 1 ∫ −1 fi,0 dx3+h i,1 + +h i,1 − h i,1 + =h i,1(·,+1) hi,1 − =hi,1(·,−1) � Asymptotic analysis of nonlinearly elastic shells... 495 Let the scaled two-dimensional energy jeM : WM(ω)→ R of a nonlinearly elastic, geometrically rigid shell with variable thickness be defined by jeM(η)= 1 8 ∫ ω aαβστ [aeστ(η)−aστ ][aeαβ(η)−aαβ]e √ a dy− ∫ ω pi,0ηie √ a dy= = 1 2 ∫ ω aαβστGeστ(η)G e αβ(η)e √ a dy− ∫ ω pi,0ηie √ a dy The functional jeM is differentiable over the Sobolev space W 1,4(ω), hence also over its subspace WM(ω), and ζ 0 ∈ WM(ω) is a solution to thevariational problem PeM(ω) of Theorem 2.1 if and only if it is a stationary point of the functional jeM over the space WM(ω),whichmeans that (j e M) ′(ζ0)= 0.Hence, particular solutions to the problem PeM(ω) can be obtained by solving the minimization problem: Find ζ ∈ WM(ω) such that jeM(ζ)= inf η∈WM(ω) jeM(η) where the scaled unknown is the two-dimensional displacement vector field ζ = (ζi) and ζi are the covariant components of the displacement ζia i : ω → R3 of the points of the middle surface S = θ(ω). More preci- sely, ζi(y)a i(y) is the displacement of the point θ(y)∈S. Thus we emphasize that, as expected for shells with nonconstant thick- ness, the specific computation leads to the fact that the ”thickness function” e :ω→ R appears in the energy functional. Consider next the case of geometrically bendable shells. Theorem 2.2. Assume that the manifold WeF(ω) defined in Section 1 con- tains nonzero elements and possesses nonzero tangent vectors at each of its points. Consider a family of nonlinearly elastic, geometrically ben- dable shells, with the nonvanishing variable thickness h(y) = 2εe(y), where e∈W2,∞(ω). Assume that they all have the samemiddle surface S = θ(ω) and that they are subjected to a boundary condition of place along a portion of their lateral face having the same curve θ(γ0) as their middle line. Finally, assume that the scaled unknown u(ε) = (ui(ε)) appearing in the scaled three-dimensional variational problem Pe(ε;Ω) admits a formal asymptotic expansion of the form u(ε)=u0+εu1+ε2u2+ ... 496 L.Gratie Let the Lamé constants be independent of ε, i.e. λε = λ and µε = µ. Then, assuming that no restriction can be put on the applied forces involved into the equations verified by the leading term u0, and that the linearization requirement must be satisfied, their components must be scaled as follows fi,ε(xε)= ε2fi,2(x) for all xε =πε(x)∈Ωε hi,ε(xε)= ε3hi,3(x) for all xε =πε(x)∈Γε+∪Γε− where the functions fi,2(x) ∈ L2(Ω) and hi,3(x) ∈ L2(Γ+ ∪Γ−) are independent of ε. This being the case, the leading term u0 : Ω → R3 is independent of the transverse variable x3 and its average ζ 0 := (ζ0i )= 1 2 1 ∫ −1 u 0 dx3 satisfies the following scaled two-dimensional variational problem PeF(ω) of a nonlinearly elastic, geometrically bendable shell with variable thickness: Find ζ0 ∈ WeF(ω)= {η∈ W2,4(ω); η= ∂νη=0 on γ0, Geαβ(η)= 0 in ω} such that 1 3 ∫ ω aαβστReστ(ζ 0)[(Reαβ) ′(ζ0)η]e3 √ a dy= ∫ ω pi,2ηie √ a dy for all η=(ηi)∈T ζ0W e F (ω), with Tζ0W e F(ω) := {η∈ W2,4(ω); η= ∂νη=0 on γ0, (Geαβ) ′(ζ0)η=0 in ω} where Reαβ(η) := b e αβ(η)− bαβ beαβ(η) := ∂αβ(θ+η e) · ae1(η)×ae2(η) |ae1(η)×ae2(η)| Asymptotic analysis of nonlinearly elastic shells... 497 aαβστ := 4λµ λ+2µ aαβaστ +2µ(aασaβτ +aατaβσ) pi,2 = 1 ∫ −1 fi,2 dx3+h i,3 + +h i,3 − h i,3 + =h i,3(·,+1) hi,3 − =hi,3(·,−1) � For the sequel, we need to recast the two-dimensional variational problem PeF(ω) as aminimization problem. To this end, let the scaled two-dimensional energy of a nonlinearly elastic, geometrically bendable shell with variable thickness jeF : W e F(ω)→ R be defined by jeF(η)= 1 6 ∫ ω aαβστ [beστ(η)− bστ ][beαβ(η)−bαβ]e3 √ a dy− ∫ ω pi,2ηie √ a dy= = 1 6 ∫ ω aαβστReστ(η)R e αβ(η)e 3 √ a dy− ∫ ω pi,2ηie √ a dy The functional jeF is differentiable over the space W e F , and ζ 0 is a solution to thevariational problem PeF(ω) ofTheorem2.2 if andonly if it is a stationary point of functional jeF over the space W e F , which means that (j e F) ′(ζ0) = 0. Hence, particular solutions to problem PeF(ω) can be obtained by solving the minimization problem: Find ζ ∈ WeF(ω) such that jeF(ζ)= inf η∈W e F (ω) jeF(η) 3. A two-dimensional nonlinear shell model of Koiter’s type with variable thickness Koiter’s approach to nonlinear, constant thickness, shell theory is based upon two a priori assumptions (see Koiter, 1966): • The first one is of a geometrical nature and it asserts that the normals to the middle surface stay normal to the deformed middle surface and the distance of any point on these normals to themiddle surface remains constant (the Kirchhoff-Love assumption). 498 L.Gratie • The second one is of a mechanical nature and it consists in assuming that the state of stress inside the shell is planar and the stresses parallel to themiddle surface vary linearly across the thickness. This assumption was justified in the fundamental work of John (1965). Using these assumptions,Koiter showed that the displacement field across the thickness of the shell can be completely expressed in terms of the displace- mentfieldof themiddle surface, andhedetermineda two-dimensional problem for finding this field. By analogy, the strain energy for our shell model of Koiter’s type with variable thickness could thus be simply the sum of the strain energy of a non- linearly elastic, geometrically rigid shell and that of a nonlinearly elastic, geo- metrically bendable shell, both with variable thickness h(y) = 2εe(y), where e∈W2,∞(ω). The unknown vector field ζε = (ζεi ) : ω → R3, where the functions ζεi : ω → R are the covariant components of the displacement field ζεiai of themiddle surface should thus solve the following two-dimensional variational problem P ε,e K (ω) for an ad hoc p> 2: Find ζ ε ∈ WK(ω)= {η∈ W2,p(ω); η= ∂νη=0 on γ0} such that ε ∫ ω aαβστGeστ(ζ ε)[(Geαβ) ′(ζε)η]e √ a dy+ + ε3 3 ∫ ω aαβστReστ(ζ ε)[(Reαβ) ′(ζε)η]e3 √ a dy= ∫ ω pi,εηie √ a dy where pi,ε := ε ∫ −ε fi,ε dxε3+h i,ε + +h i,ε − h i,ε + =h i,ε(·,+1) hi,ε − =hi,ε(·,−1) or, equivalently, the covariant components of the displacement field of the surface S, should be a stationary point of the energy functional defined by j ε,e K (η)= ε 2 ∫ ω aαβστGeστ(η)G e αβ(η)e √ a dy+ + ε3 6 ∫ ω aαβστReστ(η)R e αβ(η)e 3 √ a dy− ∫ ω piηie √ a dy Asymptotic analysis of nonlinearly elastic shells... 499 Unfortunately, the functions beαβ(η) are not defined at those points of ω where the two vectors aeα(η)= ∂α(θ+ηαa α+ 1 e η3a 3) are collinear. Hence, it appears the difficulty of choosing the right manifold for minimizing the energy. To avoid this ambiguity, we replace in the strain energy the functions Reαβ(η) := b e αβ(η)−bαβ by the new functions (by analogy with Thm. 10.3-2, Ciarlet (2000a); see also Ciarlet (2000b)) R #,e αβ (η) := 1√ a ∂αβ(θ+η e) · {ae1(η)×ae2(η)}− bαβ whichhave the advantage of beingwell defined for all smooth enough fields ηe, irrespective of whether or not the two vectors aeα(η) are collinear in a subset of ω. Obviously, R #,e αβ ≡Reαβ, when η = (ηi) is such that aeαβ(η)−aαβ = 0 in ω. Consequently, the energy functional now takes the form j e,ε K (η)= ε 2 ∫ ω aαβστGeστ(η)G e αβ(η)e √ a dy+ + ε3 6 ∫ ω aαβστR#,eστ (η)R #,e αβ (η)e 3 √ a dy− ∫ ω piηie √ a dy Theminimization problemwill be: Find ζ ε ∈ WK(ω)= {η∈ W2,p(ω), p> 2; η= ∂νη=0 on γ0} such that ε ∫ ω aαβστGeστ(ζ ε)[(Geαβ) ′(ζε)η]e √ a dy+ + ε3 3 ∫ ω aαβστR#,eστ (ζ ε)[(R #,e αβ ) ′(ζε)η]e3 √ a dy= ∫ ω piηie √ a dy where (R #,e αβ ) ′(ζ)η := 1√ a ∂αβ(θ+ζ e) · {ae1(ζ)×∂2(ηe)+∂1(ηe)×ae2(ζ)}+ + 1√ a ∂αβ(η e) · {ae1(ζ)×ae2(ζ)} 500 L.Gratie with a e α(ζ) := ∂α ( θ+ ζαa α+ 1 e ζ3a 3 ) Once this variational problem for a shell with variable thickness is written in extenso (see supra), its specific form suggests that we once again use the ansatz of the formal asymptotic method in order to justify it. 4. Asymptotic analysis of Koiter’s model of shells with variable thickness This section shows that the leading term ζ0 of the formal asymptotic expansion of the two-dimensional scaled unknown ζ(ε) satisfies ad hoc limit two-dimensional nonlinear equations that are exactly either the geometrically rigid or the geometrically bendable equations found in Section 2, according to which family of shells we would consider. To this end,wewill identify inTheorem4.4 and inTheorem4.7 two classes of variational problems that the leading term ζ0 should verify, according to specific assumptions on the geometry of the middle surface S = θ(ω) of the shell, specific boundary conditions, and to specific powers of ε that affect the components of the applied forces. We now carry out the formal asymptotic analysis for themodel of Koiter’s type for shells with variable thickness, introduced in the previous section. More specifically, our objective is to study the behavior as ε → 0 of a two- dimensional displacement field ζε that satisfies the problems P ε,e K (ω). Our first task consists in ”scaling” the problems P ε,e K (ω); accordingly, we let Ω=ω×]−1,1[ Γ+ =ω×{+1} Γ− =ω×{−1} and with each point x ∈ Ω, we associate the point xε ∈ Ωε through the bijection πε :x=(x1,x2,x3)∈Ω→ xε =(xεi)= (x1,x2,εx3)∈Ω ε Theorem 4.1. On the assumptions that there exist functions fi(ε)∈L2(Ω), hi(ε)∈L2(Γ+∪Γ−) and pi(ε)∈L2(ω), independent of ε, such that fi(ε)(x) := fi,ε(xε) for all xε =πεx∈Ωε hi(ε)(x) :=hi,ε(xε) for all xε =πεx∈Γε+∪Γε− εpi(ε)(y) := pi,ε(y) for all y∈ω Asymptotic analysis of nonlinearly elastic shells... 501 the scaled unknown ζ(ε) := ζε satisfies the following two-dimensional scaled variational problem PeK(ε;ω), for the shell model with variable thickness: Find ζ(ε)= ( ζi(ε) ) ∈ WK(ω)= {η∈ W2,p(ω), p> 2; η= ∂νη=0 on γ0} such that ∫ ω aαβστGeστ(ζ(ε))[(G e αβ) ′(ζ(ε))η]e √ a dy+ + ε2 3 ∫ ω aαβστR#,eστ (ζ(ε))[(R #,e αβ ) ′(ζ(ε))η]e3 √ a dy= ∫ ω pi(ε) ·ηie √ a dy for all η=(ηi)∈ WK(ω), where pi(ε) := 1 ∫ −1 fi(ε) dx3+ 1 ε hi+(ε)+ 1 ε hi − (ε) hi+(ε) =h i(ε)(·,+1) hi − (ε)=hi(ε)(·,−1) � According to theproceduresetupbyMiara (1998), ourasymptotic analysis will be guided by two requirements: • we do not wish to retain limit equations where restrictions must be imposed on the applied force densities in order that these equations possess solutions • by linearizationwith respect to theunknown,we shouldfind theproblem solved by the leading term of the linear theory (”linearization require- ment”); in other words, taking formal limits as ε → 0 and linearizing should commute. Remark. As Roquefort (2001, Chapter 4) noticed, the order of the forces is the same for shells with constant thickness as for shells with variable thickness. To begin with, we have the following analog of Theorem 5.1 from Ciarlet and Roquefort (2001). 502 L.Gratie Theorem 4.2. Assume that for some integer N, the scaled solution ζ(ε) of the above problem PeK(ε;ω) admits a formal asymptotic expansion in the form of polynomial ansatz ζ(ε) = ε−Nζ−N + ...+ε−1ζ−1+ε0ζ0+ε1ζ1+ ... such that ζ−N =(ζ−Ni )∈ WK(ω) and ζ −N 6=0. Then N ¬ 0, which means that the first nonzero term has to be ε0ζ0 = ζ0. � We focus now on the variational problems solved by the leading term ζ0. The formal asymptotic expansion of the scaled unknown ζ(ε)= ζ0+εζ1+ ... induces the following expansions (the leading terms G0,eστ and H 0,e αβ(η) are given in the statement of the next theorem) Geστ(ζ(ε)) =G 0,e στ + ... (G e αβ) ′(ζ(ε))η=H 0,e αβ(η)+ ... The smallest power of ε found in the left-hand side of the variational equ- ations in the problem PeK(ε;ω) is then ε 0; accordingly, we have to choose pi(ε)= ε0pi,0 = pi,0, where the new scaled functions pi,0 ∈L2(ω) are indepen- dent of ε. Cancelling the factor of ε0 in the new formulation of the problem PeK(ε;ω) in terms of ”leading terms”, immediately gives the equations that should hold for all η=(ηi)∈ WM(ω) ∫ ω aαβστG0,eστH 0,e αβ(η)e √ a dy= ∫ ω pi,0ηie √ a dy hence we obtain the following result. Theorem 4.3. Assume that the scaled displacement can be written as ζ(ε)= ζ0+εζ1+ε2ζ2+ ... and that the leading term of this formal asymptotic expansion satisfies ζ0 ∈ WK(ω). Then, in order that the leading term ζ0 may be computed without any restriction on the applied forces and in order that the linearization re- quirement is satisfied, we must have fi,ε(xε)= fi,0(x) for all xε =πεx∈Ωε hi,ε(xε)= εhi,1(x) for all xε =πεx∈Γε+∪Γε− Asymptotic analysis of nonlinearly elastic shells... 503 with functions fi,0 ∈L2(Ω),hi,1 ∈L2(Γ+∪Γ−), independent of ε; more specifically, the functions involved in the right-hand side of the problem must be of the final form pi(ε)= pi,0 with pi,0 ∈L2(ω) Moreover, the leading term ζ0 solves the variational equation: Find ζ0 ∈ WM(ω) := {η∈ W1,4(ω); η=0 on γ0} such that ∫ ω aαβστG0,eστH 0,e αβ(η)e √ a dy= ∫ ω pi,0ηie √ a dy for all η=(ηi)∈ WM(ω), where G 0,e αβ := 1 2 (ζ 0,e α//β + ζ 0,e β//α +amnζ 0,e m//α ζ 0,e n//β ) H 0,e αβ(η) := 1 2 [ηeα//β +η e β//α+a mn(ζ0m//αη e n//β + ζ 0 n//βη e m//α)] ηeβ//α := ∂αηβ −Γ σ αβησ − 1 e bαβη3 ηe3//α := b σ αησ+∂α (1 e η3 ) pi,0 := 1 ∫ −1 fi,0 dx3+h i,1 + +h i,1 − h i,1 + =h i,1(·,+1) hi,1 − =hi,1(·,−1) � We now reformulate Theorem 4.3 in a form more close to the variational problem found in Theorem 2.1. Theorem 4.4. Consider a family of nonlinearly elastic, geometrically rigid shells with variable thickness h(y) = 2εe(y), and with the same mid- dle surface S = θ(ω). Assume that they satisfy a boundary condition of place along a portion of their lateral face with the same middle cu- rve θ(γ0), and that they are subjected to the same applied forces as in Theorem 4.3. 504 L.Gratie Then the leading term ζ0 :ω→ R3 of the formal asymptotic expansion of the scaled displacement ζ(ε) = ζ0 + εζ1 + ε2ζ2 + ... satisfies the following scaled two-dimensional variational equations of a nonlinearly elastic, geometrically rigid shell: Find ζ 0 ∈ WM(ω) := {η∈ W1,4(ω); η=0 on γ0} such that ∫ ω aαβστGeστ(ζ 0)[(Geαβ) ′(ζ0)η]e √ a dy= ∫ ω pi,0ηie √ a dy for all η=(ηi)∈ W(ω). � Let us now consider the other case. Combining the linearization require- ment and the presentation fromCiarlet (2000a; Sections 3.4 and 10.1), we get the following result. Theorem 4.5. Assume that the scaled solution ζ(ε) of the problem PeK(ε;ω) admits a formal asymptotic expansion of the form: ζ(ε) = ζ0+ε1ζ1+..., with the leading term satisfying ζ0 ∈ WK(ω). In addition, assume that the manifold Me0(ω) has the properties Me0(ω)= {η∈ W2,p(ω), p> 2; η=0 on γ0, Geαβ(η)= 0 in ω} 6= {0} T ζM e 0(ω)= {η∈ W2,p(ω), p> 2; η=0 on γ0, (Geαβ) ′(ζ)η=0 in ω} 6= {0} at each ζ ∈ Me0(ω). Then pi,0 = 0 (the functions pi,0 ∈ L2(ω) are defined in Theorem 4.3), and Geαβ(ζ 0) vanish in ω, hence ζ0 ∈Me0(ω). � The next result is the final step in the asymptotic analysis of our mo- del of Koiter’s type for nonlinearly elastic, geometrically bendable shells with variable thickness. Theorem 4.6. Assume that Me0(ω) 6= {0}, T ζMe0(ω) 6= {0} for all ζ ∈Me0ω), and that the scaled unknown ζ(ε) of the problem PeK(ε;ω) admits the formal asymptotic expansion Asymptotic analysis of nonlinearly elastic shells... 505 ζ(ε)= ζ0+εζ1+ε2ζ2+ ... with ζ0,ζ1 ∈ WK(ω) and ζ2 ∈ W2,p(ω). Then, in order that the leading term ζ0 may be computed without any restriction on the applied forces and in order that the linearization re- quirement be satisfied, we must have fi,ε(xε)= ε2fi,2(x) for all xε =πεx∈Ωε hi,ε(xε)= ε3hi,3(x) for all xε =πεx∈Γε+∪Γε− with functions fi,2 ∈ L2(Ω), hi,3 ∈ L2(Γ+ ∪ Γ−) independent of ε; more specifically, the functions involved in the right-hand side of the variational equations must be of the form pi(ε)= ε2pi,2 with pi,2 ∈L2(ω) Then, the leading term ζ0 satisfies the following variational problem: Find ζ0 ∈ WeF(ω)= {η∈ W2,4(ω); η= ∂νη=0 on γ0, Geαβ(η)= 0 in ω} such that 1 3 ∫ ω aαβστR0,eστS 0,e αβ(η)e 3 √ a dy= ∫ ω pi,2ηie √ a dy for all η=(ηi)∈T ζ0W e F (ω), where Tζ0W e F(ω) := {η∈ W2,4(ω); η= ∂νη=0 on γ0, (Geαβ) ′(ζ0)η=0 in ω} denotes the tangent space to the manifold WeF(ω) at ζ 0 pi,2 := 1 ∫ −1 fi,2 dx3+h i,3 + +h i,3 − h i,3 + =h i,3(·,+1) hi,3 − =hi,3(·,−1) R0,eστ =R #,e στ (ζ 0) S0,eαβ(η)= (R #,e αβ ) ′(ζ0)η � 506 L.Gratie Let us now recast the above result in a form more reminiscent of that of Theorem 2.2. Theorem 4.7. Consider a family of nonlinearly elastic, geometrically ben- dable shells with variable thickness h(y) = 2εe(y). Assume that they have the samemiddle surface S =θ(ω), they satisfy a boundary condi- tion of place along a portion of their lateral face with the same middle curve θ(γ0), and they are subjected to the same applied forces as in Theorem 4.6. Then, the leading term ζ0 : ω → R3 of the asymptotic series associa- ted with the scaled displacement field ζ(ε) solves the following scaled two-dimensional variational problem PeF(ω) of a nonlinearly elastic, geo- metrically bendable shell: Find ζ 0 ∈ WeF(ω)= {η∈ W2,4(ω); η= ∂νη=0 on γ0, Geαβ(η)= 0 in ω} such that 1 3 ∫ ω aαβστR#,eστ (ζ 0)[(R #,e αβ ) ′(ζ0)η]e3 √ a dy= ∫ ω pi,2ηie √ a dy for all η=(ηi)∈T ζ0W e F (ω). � 5. Concluding remarks • The variational problems found in Theorems 4.4 and 4.7 can be equ- ivalently expressed in terms of energy functionals and, moreover, to be physically meaningful, these variational problems can be ”de-scaled”. • If e(y)≡ 1 for all y ∈ ω, then we recover the equations for shells with constant thickness 2ε, proposed by Ciarlet (2000b). • The main conclusion is that this new model of Koiter’s type has two advantages: firstly, the strain energy has no longer a possibly vanishing denominator, and secondly one does not have to know in advance if the shell is a geometrically rigid shell or a geometrically bendable shell, since it will automatically adjust itself to the appropriate model, for small enough values of the parameter ε. Asymptotic analysis of nonlinearly elastic shells... 507 Acknowledgement This work was partially supported by Research Project No. 9380037 from City University of Hong Kong, whose contribution is gratefully acknowledged. References 1. Busse S., 1997, Sur quelques questions en théorie de coques en coordonnées curvilignes, Doctoral Dissertation, Université Pierre et Marie Curie, Paris 2. Ciarlet P.G., 2000a, Mathematical Elasticity, Vol. III: Theory of Shells, North-Holland, Amsterdam 3. Ciarlet P.G., 2000b, Un modéle bi-dimensionnel non linéaire de coque ana- logue á celui deW.T. Koiter,C. R. Acad. Sci. Paris Sér. I, 331, 405-410 4. Ciarlet P.G., Roquefort A., 2001, Justification of a two-dimensional nonlinear shell model of Koiter’s type, Chinese Annals of Mathematics, 22B, 129-144 5. Gilbert R.P., Vashakmadze T.S., 2000, A two-dimensional nonlinear theory of anisotropic plates, Mathematical and Computer Modelling, 32, 855-875 6. Gratie L., 2003, Two-dimensional nonlinear shell model of Koiter’s type with variable thickness,Math. Mech. Solids (to appear) 7. John F., 1965, Estimates for the derivatives of the stresses in a thin shell and interior shell equations,Comm. Pure Appl. Math., 18, 235-267 8. KoiterW.T., 1966,On thenonlinear theoryof thin elastic shells,Proc.Konik. Ned. Akad. Wetensch., B 69, 1-54 9. Ladevèze P., 1976, Justification de la théorie linéaire des coques élastiques, J. Mécanique, 15, 813-856 10. Lods V., Miara B., 1998, Nonlinearly elastic shell models II, The flexural model,Arch. Rational Mech. Anal., 142, 355-374 11. Miara B., 1998, Nonlinearly elastic shell models I, The membrane model, Arch. Rational Mech. Anal., 142, 331-353 12. Roquefort A., 2001, Sur quelques questions liées aux modéles non linéaires de coques minces, Ph.D. Thesis, Université Pierre et Marie Curie, Paris 508 L.Gratie Asymptotyczna analiza nieliniowo sprężystych powłok o zmiennej grubości Streszczenie W pracy odniesiono się do nieliniowego modelu powłoki o stałej grubości, który uprzednio został zweryfikowany za pomocą metody formalnych rozwinięć asympto- tycznych. Jego konstrukcjamawłaściwości analogicznedo innychmodeli spotykanych w literaturze, ale jest bardziej dogodna przy zastosowaniu symulacji numerycznej. W tym samym duchu zaprezentowanow pracymetodę formalnych rozwinięć asymp- totycznych do zbudowania bardziej ogólnego modelu o podobnym charakterze, ale dotyczącego powłoki o zmiennej grubości. Manuscript received December 3, 2002; accepted for print April 14, 2003