Exercise 4.1 Displacement formulation of linear elastodynamics: strong and weak forms, Galerkin FE model General IBVP of linear elastodynamics Find u i = u i ( x , t ) =?, ε ij = ε ij ( x , t ) =?, σ ij = σ ij ( x , t ) =? satisfying in Ω: � ε ij = 1 C ijkl ε kl (anisotropy) , � � σ ij | j + f i = ̺ ¨ u i , u i | j + u j | i , σ ij = 2 2 µ ε ij + λ ε kk δ ij (isotropy) , with the initial conditions (at t = t 0 ): u i ( x , t 0 ) = u 0 u i ( x , t 0 ) = v 0 i ( x ) and ˙ i ( x ) in Ω , and subject to the boundary conditions on Γ = Γ u ∪ Γ t (Γ u ∩ Γ t = ∅ ): σ ij ( x , t ) n j = ˆ u i ( x , t ) = ˆ u i ( x , t ) on Γ u , t i ( x , t ) on Γ t . 1 Derive the displacement formulation of elasticity (Navier’s equations) 2 Derive the weak variational form of the displacement formulation 3 Derive the Galerkin FE model (i.e., M ¨ q ( t ) + K q ( t ) = Q ( t ))
Exercise 4.2 A simple thermo-elastic problem symmetry axis thermal insulation 0 . 04 T 0 + 50 K 0 . 16 aluminium 0 . 40 T 0 ̺ = 2700 kg W J m 3 , k = 238 m K , C p = 900 kg K E = 70 · 10 9 Pa , ν = 0 . 33, α = 23 · 10 − 6 1 0 . 20 K thermal insulation 0 . 80
Exercise 4.3 Using the Weak Form PDE Interface in COMSOL Multiphysics 0 . 04 aluminium cantilever plate lead 0 . 01 thickness = 0.005 Q(t) 0 . 80 E Al = 70 · 10 9 Pa , ̺ Al = 2700 kg ν Al = 0 . 33 , m 3 , E Pb = 16 · 10 9 Pa , ̺ Pb = 11340 kg m 3 , ν Pb = 0 . 44 .
Exercise 4.3 Using the Weak Form PDE Interface in COMSOL Multiphysics 0 . 04 aluminium cantilever plate lead 0 . 01 thickness = 0.005 Q(t) 0 . 80 E Al = 70 · 10 9 Pa , ̺ Al = 2700 kg ν Al = 0 . 33 , m 3 , E Pb = 16 · 10 9 Pa , ̺ Pb = 11340 kg m 3 , ν Pb = 0 . 44 . concentrated mass weak term m Pb = ̺ Pb · 0 . 04 [ m ] · 0 . 01 [ m ] 0 . 02 aluminium cantilever plate thickness = 0.005 Q(t) 0 . 80
Exercise 4.3 Using the Weak Form PDE Interface in COMSOL Multiphysics Weak form for harmonic linear elasticity � � � � ω 2 ˆ ̺ u i δ u i − σ ij δ u i | j + f i δ u i + t i δ u i = 0 Ω Ω Ω Γ t C ijkl u k | l – for anisotropic material Here: σ ij = σ ij ( u ) = � � µ u i | j + u j | i + λ u k | k δ ij – for isotropic material
Exercise 4.3 Using the Weak Form PDE Interface in COMSOL Multiphysics Weak form for harmonic linear elasticity � � � � ω 2 ˆ ̺ u i δ u i − σ ij δ u i | j + f i δ u i + t i δ u i = 0 Ω Ω Ω Γ t C ijkl u k | l – for anisotropic material Here: σ ij = σ ij ( u ) = � � µ u i | j + u j | i + λ u k | k δ ij – for isotropic material Constants: omega = 2*pi*f mu = E/2/(1+nu) lam = nu*E/(1+nu)/(1-2*nu) Domain expressions (for an isotropic material): s11 = 2*mu*u1x + lam*(u1x+u2y) s22 = 2*mu*u2y + lam*(u1x+u2y) s12 = mu*(u1y+u2x)
Exercise 4.3 Using the Weak Form PDE Interface in COMSOL Multiphysics Weak form for harmonic linear elasticity � � � � ω 2 ˆ ̺ u i δ u i − σ ij δ u i | j + f i δ u i + t i δ u i = 0 Ω Ω Ω Γ t C ijkl u k | l – for anisotropic material Here: σ ij = σ ij ( u ) = � � µ u i | j + u j | i + λ u k | k δ ij – for isotropic material Domain integrand (in Ω): omega^2*rho*( u1*test(u1) + u2*test(u2) ) - s11*test(u1x) - s12*test(u1y+u2x) - s22*test(u2y) f1*test(u1) + f2*test(u2) Neumann boundary integrand (on Γ t ): t1*test(u1) + t2*test(u2) Concentrated mass weak term: omega^2*mass*( u1*test(u1) + u2*test(u2) )
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