Continuum Mechanics VI. Typical Problems of Elasto-Statics 1. Elasto

Continuum mechanics VI. Typical problems of elasto-statics

AleˇsJanka

oﬃce Math 0.107 [email protected] http://perso.unifr.ch/ales.janka/mechanics

Apr 13, 2011, Universit´ede Fribourg

AleˇsJanka VI. Problems of elasto-statics 1. Elasto-statics in small deformations, linear material

Let us formulate the simplest problem of elasticity in 3D: given a computational domain Ω IR3, ﬁnd a displacement ﬁeld ⊂ u :Ω IR3 such that: → Force equilibria: symmetric Euler stress-tensor τ, τ ij = F i ie. div τ = F in Ω −∇j − ij ijk` Constitutive law: Hooke’s law, τ = E ek`, e.g. 2µ τ = 2µe + λ tr(e) Id where λ = K − 3 Kinematic equation: Cauchy strain tensor e: 1 1 e = [ u + u ] ie. e = u + ( u)T in Ω k` 2 ∇k ` ∇` k 2 ∇ ∇ Boundary conditions h i u = u¯ on Γ ∂Ω (Dirichlet-type), D ⊂ τ n = g on Γ = ∂Ω Γ (Neumann-type). · N \ D AleˇsJanka VI. Problems of elasto-statics 1.1. Weak formulation of elasto-statics: howto

a) Non-homogenous Dirichlet condition homogeneous: → Instead of ﬁnding u [H1(Ω)]3 with u = u¯ on Γ , let us oﬀset u ∈ D by the known u¯ (extended somehow onto the whole Ω).

u = u¯ + δu in Ω, with δu = 0 on ΓD Define the appropriate functional subspace of [H1(Ω)]3 by V = w [H1(Ω)]3 : w = 0 on Γ . 0 { ∈ D } Instead of finding u, find δu V , u = u¯ + δu and ∈ 0 b) Multiply force equilibria by any test function v V to get ∈ 0 force equilibria in weak form

div τ v dx = F v dx v V . − · · ∀ ∈ 0 ZΩ ZΩ ie. (component-wise)

τ ij v dx = F i v dx v V . −∇j i i ∀ ∈ 0 ZΩ ZΩ AleˇsJanka VI. Problems of elasto-statics 1.1. Weak formulation of elasto-statics: howto

c) Apply Green’s Theorem (integration by parts)

τ ij v dx τ ij n v dΓ = F i v dx v V · ∇j i − j i i ∀ ∈ 0 ZΩ Z∂Ω ZΩ d) Split to and and apply boundary conditions: ∂Ω ΓD ΓN R R Rv = 0 on ΓD , ij i τ nj = g on ΓN .

ij i i to get τ j vi dx = F vi dx + g vi dΓ. Ω · ∇ Ω ΓN Z ij Z1 ij ji Z e) Use symmetry of τ = 2 τ + τ to get ij 1 ij 1 ji τ (u) j vi dx = τ (u) j vi dx + τ (u) j vi dx Ω · ∇ 2 Ω · ∇ 2 Ω · ∇ Z Z Z τ ij (u) v ·∇i j 1 = τ ij (u) ( v + v ) dx| {z } · 2 ∇j i ∇i j ZΩ eij (v) AleˇsJanka |VI. Problems{z of elasto-statics} 1.1. Weak formulation of elasto-statics: howto

f) Use constitutive law to express τ: ﬁnd δu V , so that u = u¯ + δu and ∈ 0 E ijk` e (u) e (v) dx = F i v dx + g i v dΓ, v V k` ij i i ∀ ∈ 0 ZΩ ZΩ ZΓN

g) Use linearity of e(u) to separate u¯ from the unknown δu: ﬁnd δu V so that ∈ 0 E ijk`e (δu) e (v) dx = F i v dx+ g i v dΓ E ijk` e (u¯) e (v) dx v V k` ij i i − k` ij ∀ ∈ 0 ΩZ ΩZ ΓZN ΩZ

AleˇsJanka VI. Problems of elasto-statics 2. Nonlinear elasto-statics: classiﬁcation

Origins of nonlinearity of elasticity problems: Geometrical nonlinearity: also called kinematic nonlinearity for large displacements and/or large deformations Material nonlineraity: also called physical nonlinearity due to constitutive laws (nonlinear materials). Classiﬁcation: Material nonlinearity + geometrical linearity: use “small deformations” theory, iterate on material nonlinearity Large displacements + small deformations: big shifts and rotations as a rigid body, but small deformations. Use nonlinear kinematics (Green or Almansi strain). Linear constitutive law (e.g. Hooke’s law) can be used. Large displacements + large deformations: need nonlinear kinematics (Green or Almansi strain) and nonlinear constitutive law (non-linear material).

AleˇsJanka VI. Problems of elasto-statics 2. Nonlinear elasto-statics (Lagrange formulation)

Force equilibria:

T j` uk + δk = F k in Ω −∇` ∇j j 0 0 h i Kinematic equation: Green strain 1 ε (u) = u + u + uk u ij 2 ∇j i ∇i j ∇i · ∇j k Nonlinear constitutive law: nonlinear material ∂W T ij = = T ij (ε) ∂εij Boundary conditions

u = u¯ on Γ ∂Ω (Dirichlet-type), D ⊂ 0 Id + u T n = σ n = g on Γ = ∂Ω Γ (Neumann-type). ∇ · 0 · 0 N 0 \ D AleˇsJanka VI. Problems of elasto-statics 2.1 Weak formulation of elasto-statics (Lagrange formul.)

Same steps as in Section 1.1.:

a)-b) ﬁnd δu V s.t. u = u¯ + δu and ∈ 0 j` k k k ` T j u + δj vk dx = F0 vk dx , v V0. Ω −∇ ∇ Ω ∀ ∈ Z 0 h i Z 0 c) Apply Green’s Theorem (integration by parts):

T j` uk +δk v dx T j` uk +δk n0 v dΓ = F k v dx ∇j j ∇` k − ∇j j ` k 0 k ΩZ0 ∂ZΩ0 ΩZ0

d) Split to and , apply boundary conditions v V0 ∂Ω0 ΓD ΓN ∈ on Γ and σ n = g on Γ to get D R · 0 R NR T j` uk +δk v dx = F k v dx+ g k v dΓ v V ∇j j ∇` k 0 k k ∀ ∈ 0 ΩZ0 ΩZ0 ΓZN

AleˇsJanka VI. Problems of elasto-statics 2.1 Weak formulation of elasto-statics (Lagrange formul.)

j` 1 j` `j e) Use symmetry of T = 2 T + T T j` uk +δk v dx = T j` uk v dx + T k` v dx ∇j j ∇` k ∇j ∇` k ∇` k ΩZ0 ΩZ0 ΩZ0 uk v + v uk v + v = T j` ∇j ∇` k ∇j k ∇` dx + T k` ∇` k ∇k ` dx 2 2 ΩZ0 ΩZ0 v + v + uk v + uk v = T ij ∇i j ∇j i ∇i ∇j k ∇j ∇i k dx 2 ΩZ0 Dεij (u,v)

Here, we have introduced | {z } 1 Dε (u, v) = v + v + uk v + uk v ij 2 ∇i j ∇j i ∇i ∇j k ∇j ∇i k

How does it relate to εij (u)?

AleˇsJanka VI. Problems of elasto-statics 2.1 Weak formulation of elasto-statics (Lagrange formul.)

f) Use constitutive law to express T ij : T ij = ∂Ψ(u) ∂εij ﬁnd δu V so that u = u¯ + δu and ∈ 0 ∂Ψ(u) Dε (u, v) dx = F k v dx + g k v dΓ ∂ε ij 0 k k ZΩ0 ij ZΩ0 ZΓN

f’) Dεij (u, v) to εij (u): directional (Gˆateaux)derivative: What is the variation of εij (u+α v), seen as a function of α IR? Get the slope at u, ie. at α = 0: ∈ d [εij (u+α v)]α 0 = dα →

d 1 = (u +αv ) + (u +αv ) + (u +αv ) (uk +αv k ) dα2∇i j j ∇j i i ∇i k k · ∇j  u uk +α( uk v + uk v )+α2 v k v   ∇i k ·∇j ∇j ·∇i k ∇i ·∇j k ∇j ·∇i k α 0 1    → = v + v + uk v + uk | v = D{zε (u, v) } 2 ∇i j ∇j i ∇i · ∇j k ∇j · ∇i k ij AleˇsJanka VI. Problems of elasto-statics 2.1 Weak formulation of elasto-statics (Lagrange formul.)

f”) Directional derivative DΨ(u, v) of Ψ(u) at u along the direction v: use chain rule to diﬀerentiate Ψ(ε(u)):

d ∂Ψ(u) [Ψ(u+αv)]α 0 = Dεij (u, v) dα → ∂εij

T ij

f?) Introduce the total potential energy:| {z }

Π(u) = Ψ(u) dx F k u dx + g k u dΓ − 0 k k ZΩ0 ZΩ0 ZΓN strain energy work by external forces | {z } | {z }

AleˇsJanka VI. Problems of elasto-statics 2.1 Weak formulation of elasto-statics (Lagrange formul.)

Weak formulation and its equivalent minimization problem: The weak formulation: find u u¯ + V such that ∈ 0 ∂Ψ(u) Dε (u, v) dx = F k v dx + g k v dΓ v V ∂ε ij 0 k k ∀ ∈ 0 ZΩ0 ij ZΩ0 ZΓN simplifies to finding u u¯ + V such that ∈ 0 DΠ(u, v) = 0 v V ∀ ∈ 0

This is in fact the optimality condition for an equivalent minimization problem: ﬁnd u u¯ + V such that ∈ 0 Π(u) min . →

AleˇsJanka VI. Problems of elasto-statics 2.1 Weak formulation of elasto-statics (Lagrange formul.)

g) Linearize around the known state u,¯ express the problem in the unknown δu V : ﬁnd u u¯ + V such that ∈ 0 ∈ 0 ∂Ψ(u) Dε (u, v) dx = F k v dx + g k v dΓ v V ∂ε ij 0 k k ∀ ∈ 0 ZΩ0 ij ZΩ0 ZΓN with limited Taylor expansion around u¯: ∂Ψ(u) ∂Ψ(u¯) ∂2Ψ(u¯) = + Dεk`(u¯, δu) + o(δu). ∂εij ∂εij ∂εij ∂εk` ·

T ij (u¯) E ijk`(u¯) and | {z } | {z } 2 Dεij (u, v) = Dεij (u¯, v) + D εij (u¯, v, δu) + o(δu) where 2 d D εij (u¯, v, δu) = [Dεij (u¯ + αδu, v)]α 0 dα → 1 = δuk v + δuk v 2 ∇i · ∇j k ∇j · ∇i k AleˇsJanka VI. Problems of elasto-statics 2.1 Weak formulation of elasto-statics (Lagrange formul.) g) Linearize around the known state u,¯ express the problem in the unknown δu V : ﬁnd δu V such that ∈ 0 ∈ 0 T ij (u¯)+E ijk`(u¯) Dε (u¯, δu) Dε (u¯, v)+D2ε (u¯, v, δu) dx k` · ij ij ΩZ0 = F k v dx + g k v dΓ + o(δu). v V 0 k k ∀ ∈ 0 ΩZ0 ΓZN Collect known terms onto the right-hand side, neglect o(δu) terms:

Linearized problem for δu (one iteration of Newton’s method): Given initial guess u¯, ﬁnd a correction δu V for which ∈ 0 ijk` ij 2 E (u¯) Dεk`(u¯, δu) Dεij (u¯, v)+T (u¯) D εij (u¯, v, δu) dx Z Ω0 = F k v dx + g k v dΓ T ij (u¯)Dε (u¯, v) v V . 0 k k − ij ∀ ∈ 0 ΩZ0 ΓZN ΩZ0

R(u¯,v)...nonlinear residual AleˇsJanka VI. Problems of elasto-statics | {z } 3. Incompressible linear elasto-statics

So far, we have treated compressible materials, ie. bulk modulus K < , Poisson’s ratio ν [ 1, 0.5). ∞ ∈ − Consider now an incompressible linear material, ie. K = , ∞ ν = 0.5. The “compressible” formulation of Section 1. does not have a sense, because λ = in Hooke’s law. We need to reformulate the ∞ elasticity problem!

AleˇsJanka VI. Problems of elasto-statics 3.1. Incompressible linear elasto-statics

Hooke’s law written in deviators: hydrostatic pressure p:

im τ˜ = 2 µ e˜im E p = s = K e = (3 λ + 2 µ) e = e − `` 1 2 ν − Cauchy stress:

τ ij =τ ˜ij + s δij =τ ˜ij p δij − Mixed formulation: 2 equations for 2 unknowns (u, p): force equilibria and volume/pressure relation:

∂ τ˜ij p δ + ρf i = 0 ∂y j − ij p e + = 0 `` K This form is meaningful even if K (incompressible limit)! → ∞ AleˇsJanka VI. Problems of elasto-statics 3.1. Incompressible linear elasto-statics

Mixed formulation: ﬁnd (u, p) such that ∂ 1 2 µ e e δ p δ = ρ f i −∂y j ij − 3 `` ij − ij p e + = 0 in Ω `` K + boundary conditions on ∂Ω.

In global notation: Stokes problem: ﬁnd (u, p) such that 2µ div 2 µ e tr(e)Id + p = F − − 3 ∇ τ˜ p div(u) + = 0 in Ω, | {z } K u = u¯ on Γ ∂Ω, D ⊂ τ˜ p Id n = g on Γ = ∂Ω Γ . − · N \ D AleˇsJanka VI. Problems of elasto-statics