Generalized thermomechanics with dual internal...

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Generalized thermomechanics with dualinternal variables

Arkadi BerezovskiCentre for Nonlinear Studies, Institute of Cybernetics

Tallinn University of Technology, Tallinn, Estonia

October, 2013

Motivation

Motivation

❖ Main problem

❖ Extension ofcontinuumdescription

❖ Governingequations

Materialthermomechanics

Single internalvariable

Dual internalvariables

Generalizedcontinua

Micromorphic linearelasticity

Conclusions


Microstructured solids

Motivation

❖ Main problem






Generalizedcontinua


Conclusions


Thin-section photomicrograph of a concrete.


Motivation

❖ Main problem






Generalizedcontinua


Conclusions


The microstructure of the roller ball, which is made of cast iron. The flakes of

graphite are surrounded by ferrite, the brown is the pearlite.


Motivation

❖ Main problem






Generalizedcontinua


Conclusions


Shape-memory alloy Cu-Al-Ni.

Main problem

Motivation

❖ Main problem






Generalizedcontinua


Conclusions


● Wave propagation in solids with microstructure

✦ Diagnostic tool for material characterization

✦ Non-destructive testing of constructions

✦ Impact phenomena

✦ Moving discontinuities in solids (cracks,phase-transition fronts)

Extension of continuum description

Motivation

❖ Main problem






Generalizedcontinua


Conclusions


● Generalized continuum theoriesMindlin, Eringen, ...

● Multifield theoriesCapriz, Mariano, ...

● Internal variablesColeman-Gurtin, Rice, Maugin, ...

Governing equations

Motivation

❖ Main problem






Generalizedcontinua


Conclusions


● Phenomenology and/or homogenizationMindlin, Eringen, Capriz, ...

● Principle of virtual powerGermain, Maugin, dell’Isola, Mariano, ...

● Internal variablesColeman-Gurtin, Rice, Maugin, ...

Material thermomechanics

Motivation

Materialthermomechanics❖ Local balancelaws –Piola-Kirchhoff❖ Canonicalequations

❖ Notations❖ Inhomogeneousthermoelasticity ofconductors



Generalizedcontinua


Conclusions


Local balance laws – Piola-Kirchhoff

Motivation





Generalizedcontinua


Conclusions


● Balance of linear momentum

∂ρ0v

∂t

∣∣∣∣X

−∇R · T = f

● Balance of energy

∂H

∂t

∣∣∣∣X

−∇R · (Tv − Q) = 0

● Entropy inequality

∂S

∂t

∣∣∣∣X

+∇R · S ≥ 0, S = (Q/θ) + K

ρ0(x) is the matter density, T is the first Piola-Kirchhoff stress tensor, H = K + E,

K = 12

ρ0v2 is the kinetic energy per unit volume in the reference configuration, E is

the corresponding internal energy, Q is the material heat flux, f is body force, S is

the entropy per unit volume, θ is temperature, S is the entropy flux, and K is the

”extra entropy flux”.

Canonical equations

Motivation





Generalizedcontinua


Conclusions


● Balance of energy

∂(Sθ)

∂t

∣∣∣∣X

+∇R · Q = hint, hint := T : F −∂W

∂t

∣∣∣∣X

● Balance of linear momentum

∂P

∂t

∣∣∣∣X

−∇R · b = fint + fext + finh

● Clausius-Duhem inequality

Sθ + S∇Rθ ≤ hint +∇R · (θK)

Notations

Motivation





Generalizedcontinua


Conclusions


● Material momentum

P := −ρ0v · F

● Eshelby stress

b = − (L1R + TF) , L = K − W

● Material forces

finh :=∂L

∂X

∣∣∣∣expl

=

(1

2v2

)∇Rρ0 −

∂W

∂X

∣∣∣∣expl

fext := f · F, fint = T : (∇RF)T − ∇RW|impl

F is the deformation gradient, W is free energy per unit volume.

Inhomogeneous thermoelasticity ofconductors

Motivation





Generalizedcontinua


Conclusions


● Free energyW = W(F, θ, X)

● Equations of state

T =∂W

∂F, S = −

∂W

∂θ

● Material force and heat source

fint = fth = S∇Rθ,

hint = hth = Sθ

Inhomogeneous thermoelasticity ofconductors

Motivation





Generalizedcontinua


Conclusions


● Canonical equations (no body force)

∂(Sθ)

∂t

∣∣∣∣X

+∇R · Q = hth,

∂P

∂t

∣∣∣∣X

−∇R · b = fth + finh

Single internal variable

Motivation


Single internalvariable❖ Single internalvariable (Maugin,2006)

❖ Non-zero extraentropy flux

❖ Necessarymodifications❖ Evolution equationfor internal variable


Generalizedcontinua


Conclusions


Single internal variable (Maugin, 2006)

Motivation






Generalizedcontinua


Conclusions


● Free energyW = W(F, θ, α,∇Rα)


T =∂W

∂F, S = −

∂W

∂θ, A := −

∂W

∂α, A := −

∂W

∂∇Rα

● Material force and heat source

fint = fth + fintr, hint = hth + hintr

Non-zero extra entropy flux

Motivation






Generalizedcontinua


Conclusions


● Extra entropy flux (Maugin, 1990)

K = −θ−1A : α

● Canonical balance laws

∂P

∂t

∣∣∣∣X

−∇R · b = fth + fintr

∂(Sθ)

∂t

∣∣∣∣X

+∇R · Q = hth + hintr

● Thermal source terms

fth = S∇Rθ, hth = Sθ

Necessary modifications

Motivation






Generalizedcontinua


Conclusions


● ”Internal force”

A ≡ −δW

δα:= −

(∂W

∂α−∇R ·

∂W

∂(∇Rα)

)= A −∇R · A

● Heat flux

S = θ−1Q, Q = Q − A : α

● Eshelby tensor

b = −(L1R + TF − A : (∇Rα)T)

● Intrinsic source terms

fintr = A : (∇Rα)T, hintr = A : α

Evolution equation for internal variable

Motivation






Generalizedcontinua


Conclusions


● Dissipation inequality

Φ = hintr − S∇Rθ ≥ 0, hintr := A : α

● Isothermal case (θ = const)

A = kα

k ≥ 0 ⇒ Φ = kα : α ≥ 0

Example of evolution equation

Motivation






Generalizedcontinua


Conclusions


● Free energy

W = W(..., α,∇Rα) = f (..., α) +1

2D(∇α)2

● ”Internal force”

A := −

(∂W

∂α−∇R ·

∂W

∂(∇Rα)

)= D∇2α − f ′(α)

● Ginzburg-Landau or Allen-Cahn equation

A = kα ⇒ kα = D∇2α − f ′(α)

Dual internal variables

Motivation



Dual internalvariables❖ Dual internalvariables❖ Non-zero extraentropy flux

❖ Necessarymodifications❖ Evolutionequations

❖ A nondissipativecase❖ Hyperbolicevolution equation

Generalizedcontinua


Conclusions


Dual internal variables

Motivation






Generalizedcontinua


Conclusions


● Free energy

W = W(F, θ, α,∇Rα, β,∇Rβ)


T =∂W

∂F, S = −

∂W

∂θ, A := −

∂W

∂α, A := −

∂W

∂∇Rα,

B := −∂W

∂β, B := −

∂W

∂∇Rβ

Non-zero extra entropy flux

Motivation






Generalizedcontinua


Conclusions


● Extra entropy flux

K = −θ−1A : α − θ−1B : β

● Canonical balance laws

∂P

∂t

∣∣∣∣X

−∇R · b = fth + fintr

∂(Sθ)

∂t

∣∣∣∣X

+∇R · Q = hth + hintr

Necessary modifications

Motivation






Generalizedcontinua


Conclusions


A ≡ −δW

δα:= −

(∂W

∂α−∇R ·

∂W

∂(∇Rα)

)= A −∇R · A

B ≡ −δW

δβ:= −

(∂W

∂β−∇R ·

∂W

∂(∇Rβ)

)= B −∇R · B

● Heat flux

S = θ−1Q, Q = Q − A : α − B : β

● Eshelby tensor

b = −(L1R + TF − A : (∇Rα)T − B : (∇Rβ)T)

● Intrinsic force

fintr := A : ∇Rα + B : ∇Rβ

Evolution equations

Motivation






Generalizedcontinua


Conclusions


● Dissipation inequality

Φ = hintr − S∇Rθ ≥ 0

● Isothermal case

Φ = hintr = A : α + B : β ≥ 0

● Evolution equations for internal variables

(α

β

)=

(L11 L12

L21 L22

)(A

B

)

A nondissipative case

Motivation






Generalizedcontinua


Conclusions


● Antisymmetric case (L12 = −L21, L11 = L22 = 0) ⇒

hintr = A : (L11A) + B : (L22B) = 0

● Evolution equations

(α

β

)=

(0 L12

−L12 0

)(A

B

)

orα = L12B

β = −L12 A

Hyperbolic evolution equation

Motivation






Generalizedcontinua


Conclusions


● Simple case

B = 0, B := −∂W

∂β= −β ⇒ B = −β

● Evolution equations

α = −L12β, β = −L12 A

● Hyperbolic evolution equation for the primary internalvariable

α = L12(L12A) = (L12L12)A

α = −(L12L12)

(∂W

∂α−∇R ·

∂W

∂(∇Rα)

)

= (L12L12) (A −∇R · A)

Generalized continua

Motivation




Generalizedcontinua❖ Generalizedcontinua❖ Generalizedcontinua❖ Higher order andhigher gradetheories


Conclusions



Motivation






Conclusions


Continuousmedium

Localaction

Nonlocalaction


Motivation






Conclusions


Continuousmedium

Localaction

Simplematerials

Nonsimplematerials

Nonlocalaction

Nonlocal theory:Integral formulation

(Eringen 1976)


Motivation






Conclusions


Continuousmedium

Localaction

Simplematerials

Cauchycontinuum

(1823)

Nonsimplematerials

Higherordermedia

Highergrademedia

Nonlocalaction


(Eringen 1976)


Motivation






Conclusions


Continuousmedium

Localaction

Simplematerials

Cauchycontinuum

(1823)

Nonsimplematerials

Higherordermedia

Cosserat (1909)

Micromorphic(Eringen,

Mindlin 1964)

Highergrademedia

Second gradient(Mindlin & Eshel 1968)

Gradient of internalvariable (Maugin 1990)

Nonlocalaction


(Eringen 1976)

Source: S. Forest. Generalized continua. In: Encyclopedia of Materials: Science

and Technology. Updates. (Buschow, K., Cahn, R., Flemings, M., Ilschner, B.,

Kramer, E., Mahajan, S., eds.), pp. 1-7 (Elsevier, Oxford, 2005).

Higher order and higher grade theories

Motivation






Conclusions


Higher order theories Higher grade theories

Multipolar theoryui , εij, ηijk = ε jk,i , ηijkl = εkl,ij , ...

Micromorphic theoryui, ψij, ηijk = ψjk,i

Second strain gradient theoryui , εij , ηijk = ε jk,i , ηijkl = εkl,ij

Microstretch theoryui, φi , χ

Strain gradient theoryui, εij, ηijk = ε jk,i

Micropolar theoryui , φi , κij = φj,i

Couple stress theoryui , εij , 2κij = ǫkljul,ki

Classical continuum theoryui, εij


Motivation






Conclusions











ψij = ui,j


Motivation






Conclusions











ψij = ui,j

2φk = ǫijkuj,i

Micromorphic linear elasticity

Motivation




Generalizedcontinua


❖ Mindlin theory(Mindlin, 1964)

❖ Centrosymmetric,isotropic materials

❖ Equations ofmotion

❖ Rearrangement

❖ Microdeformationas an internalvariable

Conclusions


Mindlin theory (Mindlin, 1964)

Motivation




Generalizedcontinua





❖ Rearrangement


Conclusions


● Classical strain tensor

εij ≡1

2

(∂iuj + ∂jui

)

● Relative deformation tensor

γij ≡ ∂iuj − ψij

● Microdeformation gradient

κijk ≡ ∂iψjk

● Cauchy stressσij ≡

∂W

∂εij= σji

● Relative stressτij ≡

∂W

∂γij

● Double stressµijk ≡

∂W

∂κijk

Centrosymmetric, isotropic materials

Motivation




Generalizedcontinua





❖ Rearrangement


Conclusions


● Cauchy stress

σij ≡∂W

∂εij= σji = λδijεkk + 2µεij + g1δijγkk + g2(γij + γji)

● Relative stress

τij ≡∂W

∂γij= g1δijεkk + 2g2εij + b1δijγkk + b2γij + b3γji

● Double stress

µijk ≡∂W

∂κijk

Equations of motion

Motivation




Generalizedcontinua





❖ Rearrangement


Conclusions


● Equations of motion (no body forces)

ρuj = ∂i

(σij + τij

)

1

3ρ′d2

ijψik = ∂iµijk + τjk

ρ′d2ij is a microinertia tensor.

Two balance laws: balances of linear momentum at macro-and microscales.

Rearrangement

Motivation




Generalizedcontinua





❖ Rearrangement


Conclusions


● Distortion ∂jui, microdeformation tensor ψji,microdeformation gradient κijk ≡ ∂iψjk

● Modified stresses

σ′ij ≡

∂W

∂(∂iuj), τ′

ij ≡∂W

∂ψij

The double stress remains unchanged.● Modified Cauchy stress

σ′ij ≡

∂W

∂(∂iuj)= λδij∂kuk + µ(∂iuj + ∂jui)+

+ g1δij(∂kuk − ψkk) + g2

(∂iuj − ψij + ∂jui − ψji

)+

+ b1δij(∂kuk − ψkk) + b2(∂iuj − ψij) + b3(∂jui − ψji)

Rearrangement

Motivation




Generalizedcontinua





❖ Rearrangement


Conclusions


● Modified relative stress

τ′ij ≡

∂W

∂ψij= −g1δij∂kuk − g2(∂iuj + ∂jui)−

− b1δij(∂kuk − ψkk)− b2(∂iuj − ψij)− b3(∂jui − ψji)

● Equations of motion

ρuj = ∂iσ′ij,

1

3ρ′d2

ijψik = ∂iµijk − τ′jk

Microdeformation as an internal variable

Motivation




Generalizedcontinua





❖ Rearrangement


Conclusions


● In the non-dissipative case, the evolution equation for theinternal variable α

α = (L12L12)A = (L12L12)

(−

∂W

∂α+∇ ·

∂W

∂(∇α)

)

● In terms of components of the microdeformation tensor ψij

(L12L12)−1ij ψik =

(−

∂W

∂ψjk+∇ ·

∂W

∂(∇ψjk)

)= ∂iµijk − τ′

jk

● Balance of microimomentum in Mindlin theory

1

3ρ′d2

ijψik = ∂iµijk − τ′jk

Conclusions

Motivation




Generalizedcontinua


Conclusions


Conclusions

Motivation




Generalizedcontinua


Conclusions


● Wave propagation in solids with microstructure

● Internal variable approach to generalizedthermomechanics

● Dual internal variables for microstructure influence

● Coupling between microdeformation andmicrotemperature

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