mathematical problems in elasticity and plasticity - an ... elasticity and plasticity an...

Zentrum furTechnomathematik

Fachbereich 03Mathematik/Informatik

Mathematical Problemsin Elasticity and Plasticity

An Introduction

Soren Boettcher

Zentrum fur Technomathematik - Universitat Bremen

January 11, 2011

Soren Boettcher 1 / 26Elasticity Thermoelasticity Plasticity Outlook References

http://www.math.uni-bremen.de/zetem

http://www.uni-bremen.de



Contents

1 Elasticity

2 Thermoelasticity

3 Plasticity

4 Outlook

5 References






Preparations

Hooke’s law of elasticity

Tensile testing

F

A= Y

l − l0l0






PreparationsHooke’s law of elasticity: material returns to original shapeafter removing stress that made it deform

stress :=force

area

strain :=change of length

initial length(deformation)

stress = Young modulus× strain

a− a0

a0=

b − b0

b0= −ν l − l0

l0, ν ∈ (0, 0.5)






Preparations

Deformation

Definition

Ω body if Ω ⊂ Rn bounded Lipschitz domain

Definition

s : Ω× [0,T ]→ Rn deformation if

1 ∀ t ∈ [0,T ] s(·, t) : Ω→ s(Ω, t) bijective

2 s ∈ C 2([0,T ], [C 2(Ω)]n)

3 det(∇s(·, t)) > 0 ∀ t ∈ [0,T ]






PreparationsLagrangian description

x = s(ξ, t), ξ ∈ Ω, t ∈ [0,T ] given deformation

f : Ω× [0,T ]→ M , (M 6= ∅) arbitrary function

Eulerian description

f (x , t) = f (ξ, t) = f (s−1(x , t), t), (x , t) ∈⋃

t∈[0,T ]

s(Ω, t)× t

Displacement field

u(ξ, t) := s(ξ, t)− ξ






Preparations

Balance of momentum

ρ(ξ)∂2u

∂t2(ξ, t)− div((Id +∇u)S(ξ, t)) = f(ξ, t),

(ξ, t) ∈ Ω× (0,T )

ρ – density (t = 0)

S – second Piola-Kirchhoff stress tensor

f – volume density of external forces






Preparations

Generalized Hooke’s law:

S = f (E)

Green strain tensor:

E :=1

2

(∇u +∇uT +∇uT∇u

)Isotropic body:

S = 2µE + λ tr(E) Id +δE2






Preparations

physical linearization δ = 0

geometrical linearization E ≈ ε, Id +∇u ≈ Id

Linear elasticity theory:

ρ∂2u

∂t2− div(σ) = f

σ = 2µε+ λ tr(ε) Id (Cauchy stress tensor)

ε =1

2

(∇u +∇uT

)(linearized Green strain tensor)






Lame equationdynamic

ρ∂2u

∂t2− µ∆u− (λ + µ) grad(div(u)) = f in Ω× (0,T )

u(x , 0) = u0,∂u

∂t= u1 in Ω

u = 0 on Γ0 × (0,T )

σ · ν = b on Γ1 × (0,T )

static

−µ∆u− (λ + µ) grad(div(u)) = f in Ω






Lame equation

Γ0 ⊂ ∂Ω, Γ1 = ∂Ω \ Γ0

∂Ω = Γ0 displacement problem

∂Ω = Γ1 stress problem

ν – unit outward normal on Γ1

b – external force density on Γ1

µ = Y2(1+ν)

, λ = νY(1+ν)(1−2ν)

– Lame parameters

(µ – shear modulus)






Lame equation

LemmaΩ ∈ C0,1 boundedΓ0 ⊂ ∂Ω closed, meas(Γ0) > 0µ, λ ∈ L∞(Ω), ∃µ0 > 0 : µ0 ≤ µ(x), 0 ≤ λ(x) f.a.a. xV := u ∈ [W 1,2(Ω)]n : u = 0 on Γ0, H := [L2(Ω)]n

A : V → V ∗

〈Au, v〉 =

∫Ω

σ : ∇u dx

(=

∫Ω

σ : ε(u)dx

)= 2

∫Ω

µε(u) : ε(v)dx +

∫Ω

λ div(u) div(v)dx ∀u, v ∈ V

is linear, continuous, symmetric and coercive.






Lame equation

Lemma (Korn’s inequality)

‖ε(u)‖[L2(Ω)]n×n ≥ c(Ω, n)‖∇u‖[L2(Ω)]n×n ∀u ∈ V

Theorem (existence/uniqueness: static problem)

f ∈ H, b ∈ [L2(Γ1)]n

∃! u ∈ V : 〈Au, v〉 =

∫Ω

f · v dx +

∫Γ1

b · v dσ ∀ v ∈ V






Lame equation

Theorem (existence/uniqueness: dynamic problem)

f ∈ L2(0,T ;H), b = 0, u0 ∈ V , u1 ∈ H

∃! u ∈ L2(0,T ;V ) ∩ L∞(0,T ;V ), u′ ∈ L∞(0,T ;H) :

−∫ T

0ρ(u′, v′) dt +

∫ T

0〈Au, v〉 dt =

∫ T

0(f, v) dt + ρ(u1, v(0))

u(0) = u0

∀ v ∈ L2(0,T ;V ), v′ ∈ L2(0,T ;H), v(T ) = 0






Thermoelasticity

modification: adding thermal effects

S = 2µE + λ tr(E) Id +δE2 − 3Kα(θ − θ0) Id

θ – temperature

θ0 – initial temperature

K = λ + 23µ – compression modulus

α – thermal expansion coefficient






ThermoelasticityBalance of momentum

ρ(ξ)∂2u

∂t2(ξ, t)− div((Id +∇u)S(ξ, t)) = f(ξ, t)

Balance of energy

ρ(ξ)ce∂θ

∂t(ξ, t)− div(κ∇θ(ξ, t)) = θ

∂S

∂θ:∂E

∂t+ r(ξ, t),

(ξ, t) ∈ Ω× (0,T )

ce – specific heat (at constant deformation)κ – heat conductivityr – volume density of heat supply






Thermoelasticitylinearization classical linear thermoelasticity

ρ∂2u

∂t2− div(2µε+ λ tr(ε) Id) + grad(3Kαθ) = f

ρce∂θ

∂t− div(κ∇θ) + 3Kαθ0 div

(∂u

∂t

)= r in Ω× (0,T )

u(x , 0) = u0,∂u

∂t= u1, θ(x , 0) = θ0 in Ω

u = 0 on Γ0 × (0,T )

σ · νΓ1 = 0 on Γ1 × (0,T )

κ∇θ · ν∂Ω = δθ(θ − θΓ) on ∂Ω× (0,T )






Thermoelasticity

δθ – heat-exchange coefficient

θΓ – temperature of the surrounding medium

Vu := V , Hu := H , Vθ := W 1,2(Ω), Hθ := L2(Ω)

Theorem (existence/uniqueness: dynamic problem)

ρ, µ, λ, ce , κ, θ0, δθ ∈ R+, θΓ ∈ L∞(∂Ω), u0 ∈ Vu, u1 ∈ Hu,f, f ′ ∈ L2(0,T ;Hu), r , r ′ ∈ L2(0,T ;V ∗θ )

∃! weak solution (u, θ) ∈ L2(0,T ;Vu)× L2(0,T ;Vθ) with

u′ ∈ L2(0,T ;Vu), u′′ ∈ L2(0,T ;Hu), θ′ ∈ L2(0,T ;Vθ)






Introduction to PlasticityPreliminary observation: consider a metal rod (e.g. steel)that is subjected to a traction force σ and that in consequenceundergoes relative elongation ε






Elastic Perfectly Plastic Case(Prandtl-Reuss law)

Generalization: plasticity describes the deformation of amaterial undergoing non-reversible changes of shape inresponse to applied forces

(deviatoric) stress exceeds a critical value material willundergo plastic deformation; critical stress can be tensileor compressive

plastic deformation is isochoric or volume preserving

hydrostatic pressure (i.e. isotropic stress) no plasticdeformation

material behaviour in plasticity is scleronomic






Elastic Perfectly Plastic Case

Formulation of the material law:

small deformation: ε = ε(u) = εe + εp

volume preserving: tr(εp) = 0

Hooke’s law: σ = 2µεe + λ tr(εe) Id

yield criterion: F (σ,σ0) =

√2

3σ∗ : σ∗︸︷︷︸

=:σvM

−σ0 ≤ 0,

F – yield functionσ∗ := σ − 1

3 tr(σ) Id (deviatoric stress)σvM – von Mises stressσ0 – yield stress







Formulation of the material law:

increase of plastic strain proportional to deviatoric stress:

ε′p = Λσ∗ (σ∗ = 2µ(ε∗ + εp))

in which (Λ – plastic multiplier)

Λ = 0, if F (σ,σ0) < 0

Λ ≥ 0, if F (σ,σ0) = 0 (yield condition)

(stresses with F (σ,σ0) > 0 are not allowed)







Lemma (equivalent variational inequality)

(ε′p, τ − σ

)≤ 0 ∀ τ ∈ KF , KF :=

τ ∈ R3×3

sym : F (τ ,σ0) ≤ 0

Lemma (equivalent differential inclusion)

Find σ∗ ∈ W 1,2(0,T ;K ) s.t.

σ∗t (t) + ∂χK (σ∗(t)) 3 2µε∗(ut(t)) f.a.a. t, σ∗(0) = 0

K := τ ∈ L2(Ω)3×3 : τ (x) ∈ KF a.e.







Mathematical formulation of elastoplasticity

ρutt − div(2µε+ λ tr(ε) Id) = − div(2µεp) + f

σ∗t + ∂χK (σ∗) 3 2µε∗(ut)

σ∗(0) = 0

plus initial and boundary conditions

Theorem (existence/uniqueness: elastoplasticity)

suitable assumptions ∃! weak solution (u,σ∗) ∈ W 1,2(0,T ;V )×W 1,2(0,T ;K )






Outlook

existence/uniqueness results

additional regularity results

numerical simulations

material parameters

thermoelastoplasticity

(isotropic/kinematic) hardening

phase transitions

transformation-induced plasticity

application: SFB 570 – Distortion Engineering






ReferencesAltenbach/Altenbach, Continuum Mechanics

Betten, Continuum Mechanics

Braess, Finite Elements

Chandrasekharaiah/Debnath, Continuum Mechanics

Ciarlet, Mathematical Elasticity

Dautray/Lions, Mathematical Analysis and Numerical Methods for Science andTechnology, Vol. 1/5

Duvant/Lions, Inequalities in Mechanics and Physics

Gurtin, Introduction to Continuum Mechanics

Han/Reddy, Plasticity

Haupt, Continuum Mechanics and Theory of Materials

Marsden/Hughes, Mathematical Foundations of Elasticity

Necas/Hlavacek, Mathematical Theory of elastic and elastoplastic bodies

Temam, Mathematical Problems in Plasticity

Valent, Boundary Value Problems of Finite Elasticity

Zeidler, Nonlinear functional analysis and its applications, Vol. 4




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