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Arkadi Berezovski Generalized thermomechanics with dual internal variables – 1 / 44
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
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 2 / 44
Microstructured solids
Motivation
❖ Main problem
❖ Extension ofcontinuumdescription
❖ Governingequations
Materialthermomechanics
Single internalvariable
Dual internalvariables
Generalizedcontinua
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 3 / 44
Thin-section photomicrograph of a concrete.
Microstructured solids
Motivation
❖ Main problem
❖ Extension ofcontinuumdescription
❖ Governingequations
Materialthermomechanics
Single internalvariable
Dual internalvariables
Generalizedcontinua
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 4 / 44
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.
Microstructured solids
Motivation
❖ Main problem
❖ Extension ofcontinuumdescription
❖ Governingequations
Materialthermomechanics
Single internalvariable
Dual internalvariables
Generalizedcontinua
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 5 / 44
Shape-memory alloy Cu-Al-Ni.
Main problem
Motivation
❖ Main problem
❖ Extension ofcontinuumdescription
❖ Governingequations
Materialthermomechanics
Single internalvariable
Dual internalvariables
Generalizedcontinua
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 6 / 44
● 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
❖ Extension ofcontinuumdescription
❖ Governingequations
Materialthermomechanics
Single internalvariable
Dual internalvariables
Generalizedcontinua
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 7 / 44
● Generalized continuum theoriesMindlin, Eringen, ...
● Multifield theoriesCapriz, Mariano, ...
● Internal variablesColeman-Gurtin, Rice, Maugin, ...
Governing equations
Motivation
❖ Main problem
❖ Extension ofcontinuumdescription
❖ Governingequations
Materialthermomechanics
Single internalvariable
Dual internalvariables
Generalizedcontinua
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 8 / 44
● 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
Single internalvariable
Dual internalvariables
Generalizedcontinua
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 9 / 44
Local balance laws – Piola-Kirchhoff
Motivation
Materialthermomechanics❖ Local balancelaws –Piola-Kirchhoff❖ Canonicalequations
❖ Notations❖ Inhomogeneousthermoelasticity ofconductors
Single internalvariable
Dual internalvariables
Generalizedcontinua
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 10 / 44
● 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
Materialthermomechanics❖ Local balancelaws –Piola-Kirchhoff❖ Canonicalequations
❖ Notations❖ Inhomogeneousthermoelasticity ofconductors
Single internalvariable
Dual internalvariables
Generalizedcontinua
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 11 / 44
● 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
Materialthermomechanics❖ Local balancelaws –Piola-Kirchhoff❖ Canonicalequations
❖ Notations❖ Inhomogeneousthermoelasticity ofconductors
Single internalvariable
Dual internalvariables
Generalizedcontinua
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 12 / 44
● 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
Materialthermomechanics❖ Local balancelaws –Piola-Kirchhoff❖ Canonicalequations
❖ Notations❖ Inhomogeneousthermoelasticity ofconductors
Single internalvariable
Dual internalvariables
Generalizedcontinua
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 13 / 44
● 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
Materialthermomechanics❖ Local balancelaws –Piola-Kirchhoff❖ Canonicalequations
❖ Notations❖ Inhomogeneousthermoelasticity ofconductors
Single internalvariable
Dual internalvariables
Generalizedcontinua
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 14 / 44
● Canonical equations (no body force)
∂(Sθ)
∂t
∣∣∣∣X
+∇R · Q = hth,
∂P
∂t
∣∣∣∣X
−∇R · b = fth + finh
Single internal variable
Motivation
Materialthermomechanics
Single internalvariable❖ Single internalvariable (Maugin,2006)
❖ Non-zero extraentropy flux
❖ Necessarymodifications❖ Evolution equationfor internal variable
Dual internalvariables
Generalizedcontinua
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 15 / 44
Single internal variable (Maugin, 2006)
Motivation
Materialthermomechanics
Single internalvariable❖ Single internalvariable (Maugin,2006)
❖ Non-zero extraentropy flux
❖ Necessarymodifications❖ Evolution equationfor internal variable
Dual internalvariables
Generalizedcontinua
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 16 / 44
● Free energyW = W(F, θ, α,∇Rα)
● Equations of state
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
Materialthermomechanics
Single internalvariable❖ Single internalvariable (Maugin,2006)
❖ Non-zero extraentropy flux
❖ Necessarymodifications❖ Evolution equationfor internal variable
Dual internalvariables
Generalizedcontinua
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 17 / 44
● 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
Materialthermomechanics
Single internalvariable❖ Single internalvariable (Maugin,2006)
❖ Non-zero extraentropy flux
❖ Necessarymodifications❖ Evolution equationfor internal variable
Dual internalvariables
Generalizedcontinua
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 18 / 44
● ”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
Materialthermomechanics
Single internalvariable❖ Single internalvariable (Maugin,2006)
❖ Non-zero extraentropy flux
❖ Necessarymodifications❖ Evolution equationfor internal variable
Dual internalvariables
Generalizedcontinua
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 19 / 44
● 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
Materialthermomechanics
Single internalvariable❖ Single internalvariable (Maugin,2006)
❖ Non-zero extraentropy flux
❖ Necessarymodifications❖ Evolution equationfor internal variable
Dual internalvariables
Generalizedcontinua
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 20 / 44
● 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
Materialthermomechanics
Single internalvariable
Dual internalvariables❖ Dual internalvariables❖ Non-zero extraentropy flux
❖ Necessarymodifications❖ Evolutionequations
❖ A nondissipativecase❖ Hyperbolicevolution equation
Generalizedcontinua
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 21 / 44
Dual internal variables
Motivation
Materialthermomechanics
Single internalvariable
Dual internalvariables❖ Dual internalvariables❖ Non-zero extraentropy flux
❖ Necessarymodifications❖ Evolutionequations
❖ A nondissipativecase❖ Hyperbolicevolution equation
Generalizedcontinua
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 22 / 44
● Free energy
W = W(F, θ, α,∇Rα, β,∇Rβ)
● Equations of state
T =∂W
∂F, S = −
∂W
∂θ, A := −
∂W
∂α, A := −
∂W
∂∇Rα,
B := −∂W
∂β, B := −
∂W
∂∇Rβ
Non-zero extra entropy flux
Motivation
Materialthermomechanics
Single internalvariable
Dual internalvariables❖ Dual internalvariables❖ Non-zero extraentropy flux
❖ Necessarymodifications❖ Evolutionequations
❖ A nondissipativecase❖ Hyperbolicevolution equation
Generalizedcontinua
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 23 / 44
● 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
Materialthermomechanics
Single internalvariable
Dual internalvariables❖ Dual internalvariables❖ Non-zero extraentropy flux
❖ Necessarymodifications❖ Evolutionequations
❖ A nondissipativecase❖ Hyperbolicevolution equation
Generalizedcontinua
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 24 / 44
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
Materialthermomechanics
Single internalvariable
Dual internalvariables❖ Dual internalvariables❖ Non-zero extraentropy flux
❖ Necessarymodifications❖ Evolutionequations
❖ A nondissipativecase❖ Hyperbolicevolution equation
Generalizedcontinua
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 25 / 44
● 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
Materialthermomechanics
Single internalvariable
Dual internalvariables❖ Dual internalvariables❖ Non-zero extraentropy flux
❖ Necessarymodifications❖ Evolutionequations
❖ A nondissipativecase❖ Hyperbolicevolution equation
Generalizedcontinua
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 26 / 44
● 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
Materialthermomechanics
Single internalvariable
Dual internalvariables❖ Dual internalvariables❖ Non-zero extraentropy flux
❖ Necessarymodifications❖ Evolutionequations
❖ A nondissipativecase❖ Hyperbolicevolution equation
Generalizedcontinua
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 27 / 44
● 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
Materialthermomechanics
Single internalvariable
Dual internalvariables
Generalizedcontinua❖ Generalizedcontinua❖ Generalizedcontinua❖ Higher order andhigher gradetheories
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 28 / 44
Generalized continua
Motivation
Materialthermomechanics
Single internalvariable
Dual internalvariables
Generalizedcontinua❖ Generalizedcontinua❖ Generalizedcontinua❖ Higher order andhigher gradetheories
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 29 / 44
Continuousmedium
Localaction
Nonlocalaction
Generalized continua
Motivation
Materialthermomechanics
Single internalvariable
Dual internalvariables
Generalizedcontinua❖ Generalizedcontinua❖ Generalizedcontinua❖ Higher order andhigher gradetheories
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 30 / 44
Continuousmedium
Localaction
Simplematerials
Nonsimplematerials
Nonlocalaction
Nonlocal theory:Integral formulation
(Eringen 1976)
Generalized continua
Motivation
Materialthermomechanics
Single internalvariable
Dual internalvariables
Generalizedcontinua❖ Generalizedcontinua❖ Generalizedcontinua❖ Higher order andhigher gradetheories
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 31 / 44
Continuousmedium
Localaction
Simplematerials
Cauchycontinuum
(1823)
Nonsimplematerials
Higherordermedia
Highergrademedia
Nonlocalaction
Nonlocal theory:Integral formulation
(Eringen 1976)
Generalized continua
Motivation
Materialthermomechanics
Single internalvariable
Dual internalvariables
Generalizedcontinua❖ Generalizedcontinua❖ Generalizedcontinua❖ Higher order andhigher gradetheories
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 32 / 44
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
Nonlocal theory:Integral formulation
(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
Materialthermomechanics
Single internalvariable
Dual internalvariables
Generalizedcontinua❖ Generalizedcontinua❖ Generalizedcontinua❖ Higher order andhigher gradetheories
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 33 / 44
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
Higher order and higher grade theories
Motivation
Materialthermomechanics
Single internalvariable
Dual internalvariables
Generalizedcontinua❖ Generalizedcontinua❖ Generalizedcontinua❖ Higher order andhigher gradetheories
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 34 / 44
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
ψij = ui,j
Higher order and higher grade theories
Motivation
Materialthermomechanics
Single internalvariable
Dual internalvariables
Generalizedcontinua❖ Generalizedcontinua❖ Generalizedcontinua❖ Higher order andhigher gradetheories
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 35 / 44
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
ψij = ui,j
2φk = ǫijkuj,i
Micromorphic linear elasticity
Motivation
Materialthermomechanics
Single internalvariable
Dual internalvariables
Generalizedcontinua
Micromorphic linearelasticity
❖ Mindlin theory(Mindlin, 1964)
❖ Centrosymmetric,isotropic materials
❖ Equations ofmotion
❖ Rearrangement
❖ Microdeformationas an internalvariable
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 36 / 44
Mindlin theory (Mindlin, 1964)
Motivation
Materialthermomechanics
Single internalvariable
Dual internalvariables
Generalizedcontinua
Micromorphic linearelasticity
❖ Mindlin theory(Mindlin, 1964)
❖ Centrosymmetric,isotropic materials
❖ Equations ofmotion
❖ Rearrangement
❖ Microdeformationas an internalvariable
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 37 / 44
● 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
Materialthermomechanics
Single internalvariable
Dual internalvariables
Generalizedcontinua
Micromorphic linearelasticity
❖ Mindlin theory(Mindlin, 1964)
❖ Centrosymmetric,isotropic materials
❖ Equations ofmotion
❖ Rearrangement
❖ Microdeformationas an internalvariable
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 38 / 44
● 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
Materialthermomechanics
Single internalvariable
Dual internalvariables
Generalizedcontinua
Micromorphic linearelasticity
❖ Mindlin theory(Mindlin, 1964)
❖ Centrosymmetric,isotropic materials
❖ Equations ofmotion
❖ Rearrangement
❖ Microdeformationas an internalvariable
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 39 / 44
● 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
Materialthermomechanics
Single internalvariable
Dual internalvariables
Generalizedcontinua
Micromorphic linearelasticity
❖ Mindlin theory(Mindlin, 1964)
❖ Centrosymmetric,isotropic materials
❖ Equations ofmotion
❖ Rearrangement
❖ Microdeformationas an internalvariable
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 40 / 44
● 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
Materialthermomechanics
Single internalvariable
Dual internalvariables
Generalizedcontinua
Micromorphic linearelasticity
❖ Mindlin theory(Mindlin, 1964)
❖ Centrosymmetric,isotropic materials
❖ Equations ofmotion
❖ Rearrangement
❖ Microdeformationas an internalvariable
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 41 / 44
● 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
Materialthermomechanics
Single internalvariable
Dual internalvariables
Generalizedcontinua
Micromorphic linearelasticity
❖ Mindlin theory(Mindlin, 1964)
❖ Centrosymmetric,isotropic materials
❖ Equations ofmotion
❖ Rearrangement
❖ Microdeformationas an internalvariable
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 42 / 44
● 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
Materialthermomechanics
Single internalvariable
Dual internalvariables
Generalizedcontinua
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 43 / 44
Conclusions
Motivation
Materialthermomechanics
Single internalvariable
Dual internalvariables
Generalizedcontinua
Micromorphic linearelasticity
Conclusions
Arkadi Berezovski Generalized thermomechanics with dual internal variables – 44 / 44
● 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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