SHELL MODELS: OLD AND NEW Philippe G. Ciarlet City University of - PowerPoint PPT Presentation

SHELL MODELS: OLD AND NEW Philippe G. Ciarlet City University of Hong Kong In Honor of Claude Brezinski and Sebastiano Seatzu – p. 1

Outline 1. The two fundamental forms of a surface 2. Nonlinear shell theory – The classical and intrinsic approaches 3. A nonlinear Korn inequality on a surface 4. Classical linear shell theory – Korn’s inequality on a surface 5. Intrinsic linear shell theory: Compatibility conditions of Saint–Venant type In Honor of Claude Brezinski and Sebastiano Seatzu – p. 2

1. THE TWO FUNDAMENTAL FORMS OF A SURFACE α, β, . . . ∈ { 1 , 2 } i, j, . . . ∈ { 1 , 2 , 3 } Summation convention ω : open in R 2 θ : ω ⊂ R 2 → θ ( ω ) ⊂ R 3 θ is “smooth enough” surface θ ( ω ) : curvilinear coordinates y 1 , y 2 : In Honor of Claude Brezinski and Sebastiano Seatzu – p. 3

Assume θ is an immersion: ∂ α θ linearly independent in ω a 1 ∧ a 2 def def covariant basis: = ∂ α θ , = a α a 3 | a 1 ∧ a 2 | def First fundamental form: = a α · a β = ∂ α θ · ∂ β θ a αβ = ∂ α a β · a 3 = ∂ αβ θ · ∂ 1 θ ∧ ∂ 2 θ def Second fundamental form: b αβ | ∂ 1 θ ∧ ∂ 2 θ | First fundamental form: metric notions , such as lengths, areas, angles ∴ a.k.a. metric tensor ( a αβ ) : symmetric positive-definite matrix field Second fundamental form: curvature notions ( b αβ ) : symmetric matrix field In Honor of Claude Brezinski and Sebastiano Seatzu – p. 4

Z q area θ ( ω 0 ) = det( a αβ ( y ))d y ω 0 In Honor of Claude Brezinski and Sebastiano Seatzu – p. 5

s Z a αβ ( f ( t )) d f α d t ( t ) d f β length of θ ( γ ) = d t ( t )d t I Curvature of θ ( γ ) at θ ( y ) , y = f ( t ) , when θ ( γ ) lies in a plane normal to the surface θ ( ω ) at θ ( y ) : b αβ ( f ( t )) d f α d t ( t ) d f β d t ( t ) 1 R = a αβ ( f ( t )) d f α d t ( t ) d f β d t ( t ) In Honor of Claude Brezinski and Sebastiano Seatzu – p. 6

Portion of a cylinder 0 1 R cos ϕ B C B C θ : ( ϕ, z ) → R sin ϕ @ A z In Honor of Claude Brezinski and Sebastiano Seatzu – p. 7

Portion of a torus 0 1 ( R + r cos χ ) cos ϕ B C B C θ : ( ϕ, χ ) → ( R + r cos χ ) sin ϕ @ A r sin χ In Honor of Claude Brezinski and Sebastiano Seatzu – p. 8

Cartesian coordinates 0 1 x B C B C θ : ( x, y ) → y @ A p R 2 − ( x 2 + y 2 ) In Honor of Claude Brezinski and Sebastiano Seatzu – p. 9

Spherical coordinates 0 1 R cos ψ cos ϕ B C B C θ : ( ϕ, ψ ) → R cos ψ sin ϕ @ A R sin ψ In Honor of Claude Brezinski and Sebastiano Seatzu – p. 10

Stereographic coordinates 0 1 2 R 2 u B C 1 B C 2 R 2 v θ : ( u, v ) → ( u 2 + v 2 + R 2 ) @ A R ( u 2 + v 2 − R 2 ) In Honor of Claude Brezinski and Sebastiano Seatzu – p. 11

The components a αβ : ω → R and b αβ : ω → R of the two fundamental forms cannot be arbitrary functions : Let ( a στ ) def def def = a στ Γ αβτ = ( a αβ ) − 1 , Γ σ Γ αβτ = ∂ α a β · a τ and αβ αβ are the Christoffel symbols The functions Γ αβτ and Γ σ Then it is easy to see that: ∂ σ Γ αβτ − Γ µ ∂ ασ a β · a τ = αβ Γ στµ − b αβ b στ , ∂ σ b αβ + Γ µ ∂ ασ a β · a 3 = αβ b σµ . Besides,  ∂ ασ a β · a τ = ∂ αβ a σ · a τ ∂ ασβ θ = ∂ αβσ θ ⇐ ⇒ ∂ ασ a β = ∂ αβ a σ ⇐ ⇒ ∂ ασ a β · a 3 = ∂ αβ a σ · a 3 In Honor of Claude Brezinski and Sebastiano Seatzu – p. 12

Necessary conditions : ∂ β Γ αστ − ∂ σ Γ αβτ + Γ µ αβ Γ στµ − Γ µ ασ Γ βτµ = b ασ b βτ − b αβ b στ in ω Gauß equations ∂ β b ασ − ∂ σ b αβ + Γ µ ασ b βµ − Γ µ αβ b σµ = 0 in ω Codazzi-Mainardi equations Remarkably, these conditions are also sufficient if ω is simply-connected (see next theorem). Observe that the Christoffel symbols Γ αβτ and Γ σ αβ can be expressed solely in terms of the components of the first fundamental form: Γ αβτ = 1 Γ σ αβ = a στ Γ αβτ with ( a στ ) = ( a αβ ) − 1 2( ∂ β a ατ + ∂ α a βτ − ∂ τ a αβ ) and Consequently, the Gauß and Codazzi-Mainardi equations are (nonlinear) relations between the first and second fundamental forms . In Honor of Claude Brezinski and Sebastiano Seatzu – p. 13

def S 2 = { symmetric 2 × 2 matrices } def S 2 = { symmetric positive-definite 2 × 2 matrices } > def O 3 = { proper orthogonal 3 × 3 matrices } + FUNDAMENTAL THEOREM OF SURFACE THEORY: ω ⊂ R 2 : open, connected, simply connected. Let there be given ( a αβ ) ∈ C 2 ( ω ; S 2 > ) and ( b αβ ) ∈ C 1 ( ω ; S 2 ) satisfying the Gauß and Codazzi-Mainardi equations in ω . Then there exists θ ∈ C 3 ( ω ; R 3 ) such that: b αβ = ∂ αβ θ · ∂ 1 θ ∧ ∂ 2 θ a αβ = ∂ α θ · ∂ β θ and in ω | ∂ 1 θ ∧ ∂ 2 θ | Uniqueness holds modulo isometries of R 3 : All other solutions are: with a ∈ R 3 , Q ∈ O 3 y ∈ ω → χ ( y ) = a + Qθ ( y ) + ⇐ ⇒ ( χ , θ ) ∈ R S. Mardare (2003): ( a αβ ) ∈ W 1 ,p ( ω ; S 2 > ) and ( b αβ ) ∈ L p ( ω ; S 2 ) , p > 2 . Then θ ∈ W 2 ,p ( ω ; R 3 ) In Honor of Claude Brezinski and Sebastiano Seatzu – p. 14

2. NONLINEAR SHELL THEORY: THE CLASSICAL AND INTRINSIC APPROACHES EXAMPLES OF SHELLS: Blades of a rotor In Honor of Claude Brezinski and Sebastiano Seatzu – p. 15

Inner tube In Honor of Claude Brezinski and Sebastiano Seatzu – p. 16

Cooling tower In Honor of Claude Brezinski and Sebastiano Seatzu – p. 17

Hangar for Zeppelins (upside down) In Honor of Claude Brezinski and Sebastiano Seatzu – p. 18

HOW IS A SHELL PROBLEM POSED? In Honor of Claude Brezinski and Sebastiano Seatzu – p. 19

CLASSICAL APPROACH Unknown : ϕ = ( ϕ i ) : ω → R 3 : deformation of middle surface S Boundary conditions : ϕ = θ on γ 0 (simple support), or ϕ = θ and ∂ ν ϕ = ∂ ν θ on γ 0 (clamping) ( length γ 0 > 0 ) Applied forces : ( f i ) : ω → R 3 Lamé constants of the elastic material: λ > 0 , µ > 0 4 λµ A αβστ = λ + 2 µ a αβ a στ + 2 µ ( a ασ a βτ + a ατ a βσ ) , where ( a στ ) = ( a αβ ) − 1 P There exists c 0 > 0 such that A αβστ ( y ) t στ t αβ ≥ c 0 α,β | t αβ | 2 for all y ∈ ω, ( t αβ ) ∈ S 2 Thickness of the shell: 2 ε > 0 Area element along S : √ a d y where a = det( a αβ ) P .G. Ciarlet: An Introduction to Differential Geometry with Applications to Elasticity , Springer, 2005 In Honor of Claude Brezinski and Sebastiano Seatzu – p. 20

Problem : To find ϕ : ω → R 3 such that: ϕ : ω → R 3 smooth enough ; e J ( ϕ ) = inf { J ( e ϕ ); e ϕ = θ on γ 0 } Total energy of the shell – W.T. Koiter (1966): Z a αβ − a αβ ) √ a d y ε A αβστ ( e J ( e ϕ ) = a στ − a στ )( e ◭ membrane energy 2 ω Z ε 3 b αβ − b αβ ) √ a d y ◭ flexural energy A αβστ ( e b στ − b στ )( e + 6 ω Z √ a d y, f i e ◭ forces − ϕ i ω def ◭ change of metric a αβ − a αβ = ∂ α e ϕ · ∂ β e ϕ − a αβ e tensor ϕ ∧ ∂ 2 e ϕ · ∂ 1 e ϕ def e b αβ − b αβ = ∂ αβ e ϕ | − b αβ ◭ change of curvature | ∂ 1 e ϕ ∧ ∂ 2 e tensor In Honor of Claude Brezinski and Sebastiano Seatzu – p. 21

INTRINSIC APPROACH : Another look at the energy of the shell: Z a αβ − a αβ ) √ a d y ε A αβστ ( e ◭ membrane energy J ( e ϕ ) = a στ − a στ )( e 2 ω Z ε 3 b αβ − b αβ ) √ a d y A αβστ ( e b στ − b στ )( e + ◭ flexural energy 6 ω Z √ a d y f i e ◭ forces − ϕ i ω a αβ and e Hence the fundamental forms e b αβ of the unknown surface e ϕ ( ω ) appear as natural unknowns This is the basis of the intrinsic approach : J.L. Synge & W.Z. Chien (1941); W.Z. Chien (1944) S.S. Antman (1976) W. Pietraszkiewicz (2001); S. Opoka & W. Pietraszkiewicz (2004) In Honor of Claude Brezinski and Sebastiano Seatzu – p. 22

a αβ and e But, if e b αβ are chosen as the primary unknowns: b αβ ) the integral R ϕ √ a d y taking into a αβ ) and ( e – How to express in terms of ( e ω f · e account the forces in the energy? a αβ ) and ( e b αβ ) the boundary condition , e.g., e – How to express in terms of ( e ϕ = θ on Γ 0 , that the admissible deformations must satisfy? – How to handle such expressions if minimizing sequences are considered: ϕ k → e a k e b k e e αβ − k →∞ e → a αβ and αβ − → b αβ = ⇒ e ϕ ? k →∞ a αβ and e – Constrained minimization problem : The new unknowns e b αβ must satisfy the ( highly nonlinear ) Gauß and Codazzi-Mainardi equations In Honor of Claude Brezinski and Sebastiano Seatzu – p. 23