Mathematical modelling and numerical analysis of thin elastic shells

Mathematical modelling and numerical analysis of thin elastic shells Zhang Sheng Wayne State University, Michigan Zhang Sheng (szhang@wayne.edu) Thin elastic shell May, 202 / 9

Models for mechanics of structures of small thickness Timoshenko beam model, Naghdi and Koiter arch models, Reissner Mindlin plate bending model, Koiter shell model, Naghdi shell model,, These models seek minimizer u ɛ H of the functional [ ɛ 2 ] (Au, Au) U + (Bu, Bu) V F, u. 2 u general displacement kind of functions, A bending strain, B membrane/shear strain, F resultant loading functional. Zhang Sheng (szhang@wayne.edu) Thin elastic shell May, 202 2 / 9

Examples Timoshenko beam (ɛ = beam thickness): Reissner Mindlin plate bending: Naghdi shell (ɛ = shell thickness): A(θ, w) = θ, B(θ, w) = θ + w. A(θ, u, w) = 2 (θ α β + θ β α ) + 2 (bλ β u α λ + b λ αu β λ ) c αβ w, B(θ, u, w) = [ 2 (u α β + u β α ) b αβ w] [b λ β u λ + θ β + β w]. Zhang Sheng (szhang@wayne.edu) Thin elastic shell May, 202 3 / 9

Behavior when ɛ 0 Depending on whether B : H V is injective, and closed-ranged. ker B 0 for straight beams, curved arches, flat plates, parabolic or hyperbolic shells clamped on asymptotics, free elliptic shells. ker B = 0 for many cases of shells: partially clamped elliptic shells, or sufficiently clamped parabolic or hyperbolic shells. When F ker B = 0 and W (range of B in V ) is equal to V, u ɛ, the structure resists the load strongly. When F ker B 0 u ɛ ɛ 2, the structure resists the load weakly. Most shell structures are intermediate; They are moderately strong. Zhang Sheng (szhang@wayne.edu) Thin elastic shell May, 202 4 / 9

A beam subject to different loads Strong resistence u ɛ Weak resistence u ɛ ɛ 2 Zhang Sheng (szhang@wayne.edu) Thin elastic shell May, 202 5 / 9

Finding a strong structure for certain purpose For given loads, determine the shape, such that F KerB = 0. Assume that the arch is only loaded by its own weight(curved roof, e.g.). The middle curve is Y = f (X ), X [, ] on the XY -plane, with f ( ) = f () = 0. Then F KerB = 0 f (f ) 2 [ + (f ) 2 ]f = 0. The solution is f (X ) = c(cosh(x /c) cosh(/c)). The shape is a catenary. As hangs the flexible line, so but inverted will stand the rigid arch. Hooke, 675. Zhang Sheng (szhang@wayne.edu) Thin elastic shell May, 202 6 / 9

Arch strength is sensitive to its shape.4.2 0.8 0.6 0.4 0.2 0 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 Zhang Sheng (szhang@wayne.edu) Thin elastic shell May, 202 7 / 9

Some results The applicable range of Reissner Mindlin is wider than that of Kirchhoff Love (that is biharmonic when clamped). Their accuracies are the same when applicable (2003 J. Elasticity, 2006 M2NA) The bending strain in the Koiter, Naghdi, shell and arch models need to be modified (200 2008 M3AN) from 2 (θ α β + θ β α ) 2 (bλ β u λ α + b λ αu λ β ) + c αβ w to 2 (θ α β + θ β α ) + 2 (bλ β u α λ + bαu λ β λ ) c αβ w. The shear correction factor in Timoshenko Reissner Mindlin Naghdi type models needs to be reset from 5/6 to. (2008 M3AN). The model solutions of intermediate shells have asymptotic limits in any case (2006 C.R. Math), but the model solutions are not close to that of 3D elasticity (2006 M2NA). Zhang Sheng (szhang@wayne.edu) Thin elastic shell May, 202 8 / 9

Numerical method for shells p, p 2, p 3 τ n = nαa α u, u 2, w φ τ Ω Zhang Sheng (szhang@wayne.edu) Thin elastic shell May, 202 9 / 9

Koiter shell model The model seeks u, u 2, w such that a αβλγ ρ λγ (u 3, w)ρ αβ (y, z) + ɛ 2 a αβλγ γ λγ (u, w)γ αβ (y, z) = (p α y α + p 3 z) (y, z) PDE system, fourth order in w, third order in u, u 2 Zhang Sheng (szhang@wayne.edu) Thin elastic shell May, 202 0 / 9

Koiter shell model The model seeks u, u 2, w such that a αβλγ ρ λγ (u 3, w)ρ αβ (y, z) + ɛ 2 a αβλγ γ λγ (u, w)γ αβ (y, z) = (p α y α + p 3 z) (y, z) PDE system, fourth order in w, third order in u, u 2 γ αβ (u, w) = 2 (u α β + u β α ) b αβ w change of metric tensor membrane strain ρ αβ (u, w) = 2 αβ w Γγ αβ γw + b γ α β u γ + b γ αu γ β + b γ β u γ α c αβ w change of curvature tensor bending strain Zhang Sheng (szhang@wayne.edu) Thin elastic shell May, 202 0 / 9

On the shell mid-surface Basis vectors a α = φ/ x α, a α a β = δβ α, a 3 = a 3 = (a a 2 )/ a a 2. Metric tensor, curvature tensor, and the third fundamental form a αβ = a α a β, a αβ = a α a β, b αβ = a 3 β a α, c αβ = bαb γ γβ. Christoffel symbols Covariant derivatives Γ γ αβ = aγ β a α. σ αβ γ = γ σ αβ + Γ α γλ σλβ + Γ β γτ σ ατ, τ γ α β = βτ γ α + Γ γ λβ τ λ α Γ τ αβ τ γ τ, u α β = β u α Γ γ αβ u γ, u α β = β u α + Γ α γβ uγ. Zhang Sheng (szhang@wayne.edu) Thin elastic shell May, 202 / 9

Constitutive tensor and Green s theorem a αβδγ = µ(a αδ a βγ + a βδ a αγ ) + 2µλ 2µ + λ aαβ a δγ. Here, λ and µ are the Lamé constants of the elastic material. The compliance tensor a αβδγ is defined such that Green s theorem on surface: σ αβ = a αβδγ γ δγ γ αβ = a αβδγ σ δγ. τ u α α = τ u α n α. The left integral is taken with respect to surface area, and the right with respect to arc length. Zhang Sheng (szhang@wayne.edu) Thin elastic shell May, 202 2 / 9

Koiter model in mixed form with a splitted membrane 3 a αβλγ ρ λγ (u, w)ρ αβ (y, z) + + M αβ γ αβ (y, z) = a αβλγ γ λγ (u, w)γ αβ (y, z) (p α y α + p 3 z) (y, z) H M αβ = ɛ 2 a αβλγ γ λγ (u, w) membrane stress. Here, ɛ is defined by ɛ 2 = ɛ 2, henceforth denoted by ɛ. Zhang Sheng (szhang@wayne.edu) Thin elastic shell May, 202 3 / 9

Koiter model in mixed form with a splitted membrane 3 a αβλγ ρ λγ (u, w)ρ αβ (y, z) + + M αβ γ αβ (y, z) = a αβλγ γ λγ (u, w)γ αβ (y, z) (p α y α + p 3 z) (y, z) H M αβ = ɛ 2 a αβλγ γ λγ (u, w) membrane stress. Here, ɛ is defined by ɛ 2 = ɛ 2, henceforth denoted by ɛ. This is a stabilization technique used by, for example, Arnold Brezzi element for Naghdi shell. Zhang Sheng (szhang@wayne.edu) Thin elastic shell May, 202 3 / 9

Variational principle defined on pw discontinuous functions a(u, w; v, z) + b(m; v, z) = f; v, z v, z b(u, w; N ) ɛ 2 c(m, N ) = 0 N. This is defined on a space of discontinuous piecewise smooth functions, subordinated to a triangulation T h of Ω. The bilinear forms are b(n ; u, w) = N αβ γ αβ (u, w) T h [{N αβ }] [[u α ]] nβ Ẽ h c(m; N ) = a αβλγ M αβ N λγ T h and Zhang Sheng (szhang@wayne.edu) Thin elastic shell May, 202 4 / 9

The simplest consistent a-form a(u, w; v, z) [ ] = T h 3 aαβλγ ρ λγ (u, w)ρ αβ (v, z) + a αβλγ γ λγ (u, w)γ αβ (v, z) Ẽ h 3 aαβλγ [{ρ λγ (u, w)}] [[z,α + 2bαv σ σ ]] nβ Ẽ h 3 aαβλγ [{ρ λγ (v, z)}] [[w,α + 2bαu σ σ ]] nβ + Ẽ h 3 [{aαβλγ [ρ λγ (u, w)] β }] [[z]] nα + Ẽ h 3 [{aαβλγ [ρ λγ (v, z)] β }] [[w]] nα a αβγδ [{γ γδ (u, w)}] [[v α ]] nβ a αβγδ [{γ γδ (v, z)}] [[u α ]] nβ. Ẽ h Ẽ h Zhang Sheng (szhang@wayne.edu) Thin elastic shell May, 202 5 / 9

A consistent a-form with penalty a(u, w; v, z) = a(u, w; v, z) + C ẽ Ẽh + C ẽ Ẽh he a αβ [[u α ]][[v β ]] ẽ he a αβ [[w,α ]][[z,β ]] + C ẽ ẽ Ẽh he 3 [[w]][[z]]. ẽ Ẽ h Zhang Sheng (szhang@wayne.edu) Thin elastic shell May, 202 6 / 9

A consistent a-form with penalty a(u, w; v, z) = a(u, w; v, z) + C ẽ Ẽh + C ẽ Ẽh he a αβ [[u α ]][[v β ]] ẽ he a αβ [[w,α ]][[z,β ]] + C ẽ ẽ Ẽh When C is sufficiently big, in a finite element space he 3 [[w]][[z]]. ẽ Ẽ h a(u h, w h ; uh, w h ) (u h, w h ) 2 := u h 2,T h + w h 2 2,T h + he [[u hα ]] 2 + he [[w h,β ]] 2 + ẽ ẽ ẽ Ẽh ẽ Ẽh ẽ Ẽh This triple-bar norm is shell geometry independent. he 3 [[w]] 2. ẽ Zhang Sheng (szhang@wayne.edu) Thin elastic shell May, 202 6 / 9

Finite element model A finite element model is obtained by restricting the mixed formulation of the Koiter model on a space of piecewise polynomials. It seeks piecewise polynomials u h, w h, M h such that a(u h, w h ; v h, z h ) + b(m h ; v h, z h ) = f; v h, z h v h, z h b(u h, w h ; N h ) ɛ 2 c(m h, N h ) = 0 N h. This finite element model has a unique solution. It is consistent with the Koiter shell model. For the primary variables, such a finite element model is always stable, but it could be inaccurate due to a violation of an inf-sup condition. Such methods often fails to approximate the scaled membrane stress. Zhang Sheng (szhang@wayne.edu) Thin elastic shell May, 202 7 / 9

A finite element method that is uniformly accurate We take membrane stress tensor components M h,αβ as continuous piecewise linear, tangential displacement vector components u h,α as discontinuous piecewise quadratics, normal deflection scalar w h as discontinuous piecewise cubics. When the triangulation is shape regular, but not necessarily quasi-uniform, we have the error estimates for the primary variables: (u ɛ u h, w ɛ w h ) + ɛ τ Th τ Th h 6 τ ( b αβ 2 3,,τ + Γ λ αβ 2 2,,τ ) /2 ( ) /2 hτ 4 u ɛ 2 3,τ + hτ 4 w ɛ 2 4,τ + hτ 4 M ɛ 2 2,τ. Zhang Sheng (szhang@wayne.edu) Thin elastic shell May, 202 8 / 9

Where the shell is flat, there is no restriction on the mesh size. Where the shell is more curved, the elements need to be smaller. If the Christoffel symbols Γ γ αβ are piecewise linears and the curvature tensor b αβ are piecewise quadratics (of the coordinates), we have the optimal order of convergence without any restriction on the mesh: (u ɛ u h, w ɛ w h ) τ Th ( ) /2 hτ 4 u ɛ 2 3,τ + hτ 4 w ɛ 2 4,τ + hτ 4 M ɛ 2 2,τ. Zhang Sheng (szhang@wayne.edu) Thin elastic shell May, 202 9 / 9