CH.1. THERMODYNAMIC FOUNDATIONS OF CONSTITUTIVE MODELLING

Computational Solid Mechanics- Xavier Oliver-UPC

1.1 Dissipation approach for constitutive modelling

Ch.1. Thermodynamical foundations of constitutive modelling

Power, , is the work done per unit of time.

In some cases, the power is an exact differential of a field, which, then is termed energy :

It will be assumed that the continuous medium obtains power from the exterior through: Mechanical Power: the work performed by the mechanical actions

(body and surface forces) acting on the medium. Thermal Power: the heat entering the medium.

Power

( )W t

( ) ( )d tW t

dt=

( )t

The external mechanical power is the work done by the body forces and surface forces per unit of time. In spatial form it is defined as:

External Mechanical Power

( )e V VP t dV dSρ

∂= ⋅ + ⋅ b v t v

d dVdt

ρ ⋅ rb= v

d dSdt

⋅ rt= v

Theorem of the expended mechanicalpower

( ) 2

Stress power Kinetic energy

1 v 2

t

e V VV V V

dP t dV dS dV dVdt

σ

ρ ρ∂

≡

= ⋅ + ⋅ = + b v t v :d

σ

PKexternal mechanical power entering the medium

( ) ( )edP t tdt σ= +P

REMARKThe stress power is the mechanical power entering the system which is not spent in changing the kinetic energy. It can be interpreted as the work done, per unit of time, by the stresses in the deformation process of the medium.A rigid solid will have zero stress power.

The external heat power is the incoming heat in the continuum medium per unit of time.

The incoming heat can be due to: Non-convective heat transfer across the

body surface (characterized by )

Internal heat sources (characterized by )

External Heat Power

non convective (conduction) incoming heatunit of time V

dS∂

= − ⋅ q n

heat generated by internal sources unit of time V

r dVρ=

heat conduction

flux vector

( , )tq x

heat source

field

( , )r tx

The external heat power is the incoming heat in the continuum medium per unit of time. It is defined as:

Where:is the heat flux per unit of spatial surface area.is an internal heat source rate per unit of mass.

External Heat Power

( ),r tx( ), tq x

The total power entering the continuous medium is:

Total Incoming Power

21 v 2input e e

V V V V Vt

dW P Q dV dV r dV dSdt

ρ ρ≡ ∂

= + = + + − ⋅ :d q nσ

( )eP t ( )eQ t

stress powerkinetic energy Internal heat source

External heat conduction

Stored mechanical power: is that part of the incoming mechanical power that can be eventually returned by the body:

Mechanical dissipation: is that part of the incoming mechanical power that is not stored (eventually can be lost)

Stored Mechanical PowerMechanical Dissipation

2

Stored mechanicalKinetic energy energy

1 v 2

t

storedV V V

d d d dP dV dVdt dt dt dt

ρ ρψ≡

= + = +

K V

K V

mech e stored

d d ddt dt dt

dD P Pdt

σ+ +

= − =

P (

V

eP

ddVdt

+ − :d

σ

) V V

VdV

d dV dVdt

ρψ

ρψ+ = −

:d

σ

0

Density of free energy Stored mechancical energy

(Helmholtz energy) unit of volume( , )tρ ψ →

→

x

Second principle of the thermodynamics Dissipation Mechanical dissipation

Thermal dissipation :

Global (integral) form of the second principle of thermodynamics

[ ( ) : ] 0mech thermV

D D D s dV V Vρ θ= + = − ψ + + ≥ ∀Δ ⊂ d

σ

eP → ++d ddt dt mechD

thermD

mechV V

D dV dVρψ= − + :d σ

thermV

D s dVρ θ= − ( , ) Density of enthropy( , ) Absolute temperature (>0)

s ttθ

→→

xx

"Localization" process

[ ( ) : ] 0ρ θ

↓

= = − ψ + + ≥ ∀Δ ⊂ dV V

D dV s dV V V σ

( )Local (differential) form of the second principle of the thermodynamics

( , ) : 0t s tρ θ= − ψ + + ≥ ∀ ∀x d x σ

Second principle of the thermodynamics Dissipation

Dissipation( , ) Density of dissipation=unit of volume

t →x

The internal energy per unit of mass (specific internal energy) is :

Taking the material time derivative,

and introducing it into the Dissipation inequality

Alternative forms of the Dissipation

Clausius-Planck Inequality in terms of the

specific internal energy

Total stored energy

Mechanical stored stored energy

Thermal stored stored energy

( , )( , ) : ( , )

( , )

u tu t s t

s tθ

θ

→= ψ + → ψ → →

xx x

x

u s sθ θ= ψ+ + s u sθ θψ + = −

( ) : 0u sρ θ= − − + ≥d σ( ) : 0sρ θ= − ψ + + ≥d σ

REMARKFor infinitessimal deformation, , the Clausius-Planck inequality becomes:

=d ε

( ) 0sρ ψ θ− + + ≥: σ ε

Dissipation in a continuum medium

The dissipation of a continuum medium is defined as:

corresponding to the Clausius-Planck Inequality.

In terms of the Helmholtz free energy, dissipation may be written as:

Hypotheses assumed Infinitesimal deformation:

Hence, the dissipation may be written as:

( ): : 0ρ θ= − − + ≥d u s σ

( ) : 0ρ θ= − ψ + + ≥d s σ

0

( , )( )

t ttρ ρ

= =

d xx

ε( , ) x,

( )0 : 0ρ θ= − ψ + + ≥ε s σDissipation of a continuum medium in terms of the Helmholtz free energy assuming infinitesimal deformation

1.2 A thermodynamic framework for constitutive modeling

Ch.1. Thermodynamical foundations of constitutive modelling

Sets of thermo-mechanical variables

In a thermo-mechanical problem we will consider the set of all the variables of the problem:

which will be classified into:

Free variables:

which are physically observable variables, whose evolution along time is unrestricted

{ } { }v1 2 n vv ,v ,...,v v ( , ) 1,2,...,i t i n= ∈x:

{ } { }1 2, , ..., ( , ) 1, 2, ...,Fn i Ft i nλ λ λ λ= ∈x:

( , )( , ) anyii

ttt

λλ ∂= →∂

xx

Internal/Hidden variables:

are non-observable variables. Their evolution is limited along time in terms of specific evolution equations defined as:

which account for micro-structural mechanisms.

Dependent variables:

are the remaining of the variables of the problem (depending on the previous ones):

{ } { }1 2, , ..., ( , ) 1, 2, ...,In i It i nα α α α= ∈x:

{ }instantaneous

values (at time )

( , ) ( ( , ), ( , ) ) 1, 2, ...ii i I

t

t t t i nt

αα ξ∂= = ∈∂

x λ x x α

Sets of thermo-mechanical variables

{ } { }1 2, , ..., ( , ) 1, 2, ...,Dn i Dd d d d t i n= ∈x:

{ }( , ) ( , ) 1, 2, ...i i i i Dd d i nγ ϕ= → = ∈λ λ λ α α ,

= ∪ ∪ ∩ = ∩ = ∩ = ∅

Example: Problem variables:

Free variables:

Internal/Hidden variable:

Dependent variables:

( , )α γ ρ α= ε,

{ }, , , , , , ,u sρ ψ θ α= σ ε:

{ },ρ= ε:

{ }α=:

provided by theevolution equation

not depending onthe internal variable

evolution,

( , )( , , , ( , )) ( , , , )

( , )( , , , ( , ) ) ( , , ,

α

ψ ψ ρ αψ γ ρ α ρ α ρ α ψ ρ α ρ

ρ αϕ ρ α ρ α ρ α ρ α ρ

== =

= =

ε,ε, ε, ε, ε, ε

ε,ε, ε, ε, ε,

σ = σσ σ )ε

ρ=: , ,σ ε , , , , ,u sψ θ α{ }

,ρ∀ ∀ ε

1) Definition of the free variables of the problem

2) Choice of the internal variables of the problem

3) Definition of the corresponding evolution equations,

4) Postulate a specific form of the free energy:

Elements of a constitutive model

provides the constitutive equation( , , t )

through the dissispation inequality

ψ = ψ →

λ α

{ }( , ) 1, 2, ...i i Ii nα α λ α= ∈

{ }1 2, , ...,Fnλ λ λ

λ

:=

{ }1 2, , ...,Inα α α=

α

:

Example: Thermo-elastic material

Linear relation stresses-strains

1D:

3D:

Isotropic elastic material: being a second order tensor, and and

the Lamé constants.

=σ εEW LEA L

Δ=

:=σ ε σ = εij ijkl klor

2ijkl λ μ= ⊗ + Ι1 1

ijkl μλ

Linear elastic material

{ }( ) 2

σ ε 2 ε , 1, 2, 3ij kk ij ij

tri j

λ μλ δ μ

= + = + ∈

σ ε ε

Tensor Notation (reminder)

Open product Of two first-order tensors:

Between two second order tensor:

Identity tensors First order identity tensor

Second order tensor identity tensor

,[ ]i j i ja b⊗ =a b

[ ]ijkl ij klA B⊗ =A B

[ ] { }1, 1, 2, 3

0ijij

i ji j

i jδ

== ∈ ≠

1 =

[ ] { }1 [ ] , , , 1, 2, 32 ik jl il jkijkl

I i j k lδ δ δ δ= + ∈

REMARKEinstein notation (summation of repeated indices) is considered

Adding the thermal effects (thermoelasticity)

Thermo-elastic material (reminder)

Thermal property

Thermal expansion coefficient1 2

βνα

β

=→−= →

Linear thermo-elastic material

{ }( ) 2

σ ε 2 ε ( ) , 1, 2, 3ij kk ij ij ij

tri j

λ μ β θλ δ μ β θ δ

= + − Δ = + − Δ ∈

σ ε ε 1

Coleman’s Method

Theorem

Proof: Taking

Taking

0( , , , )

( , )( , ) ( , ) ,

( , )0

0f x y

f x y g x yx y x yg

y yx y

x x=

= + ≥ =∀

0y = ( , ) 0= ≥ ∀ f x y x x

( , ) 0f x y <If taking 0x > NOT POSSIBLE( , ) 0f x y x= <

( , ) 0f x y =( , ) 0f x y >If taking 0x < ( , ) 0f x y x= <

0=x ( , ) 0= ≥ ∀ g x y y y

( , ) 0<g x yIf taking 0>y ( , ) 0g x y y= <

( , ) 0=g x y( , ) 0g x y >If taking 0y < ( , ) 0g x y y= <

NOT POSSIBLE

Elastic Material formulation

Variable sets definition Free variables:

Internal variables:

Dependent:

Potential Helmholtz free energy:

Dissipation

{ }= ε:{ } No evolution equation= ∅ →:

{ }ψ= σ ,:1

0 2( ) : :ρ ψ =ε ε ε 00 0

( )( )

( )( ) ρ ψρ ψ ρ ψ

∂ → = ∂ ∂ → = ∂

σ εσ ε σ : εε

εε : εε

0 ( sρ θ= − ψ + ) : 0+ ≥σ ε 0

( )

( )) : 0 ( ) 0

f

fρ ψ∂= =∂

∀− ≥

ε

εε (σ ε εε

Isothermal case

0

Constituve equation )

( ) :ρ ψ

=

∂= =∂

ε

ε

σ Σ(

εσε 0=

Thermo-elastic Material formulation

Variable sets definition Free variables: Internal variables:

Dependent:

Potential definition Helmholtz free energy:

Dissipation

{ }θ= ε ,:{ }= ∅:

10 02( , ) : : ( )

θρ θ β θ θ

Δψ = − −ε ε ε Ι

{ }0 0

0

( ) ( ) :

( ) ( ): :

σ θ σ θσ θθψ

ρ ψ θ ρ ψ θρ ψ θθ

∂ ∂ = ∂ ∂= → ∂ ∂ = ∂ ∂

ε ε: εεσ

ε εεε

, , + ,

, , + :

( )0 : 0ρ θ= − ψ + + ≥σ ε s

0 00

( ) ( )

( ) ( )) ( ) 0 ,

f g

s

θ θ

ρ ψ θ ρ ψ θρ θ θθ

∂ ∂= − + ≥ ∀∂ ∂

ε ε

ε ε(σ : ε − : εε

, ,

, ,

Using the Coleman’s method:

and differentiating the Helmholtz free energy:

0 0( )( ) 0g s ψ θθ ρ ρθ

∂= + =∂εε ,,

Constitutiveequations

0( )( ) 0ψ θθ ρ ∂= − =∂εε σε

f ,, 0 ( )

( )s

ρ ψ θ

ψ θθ

∂ = ∂ ∂ = − ∂

εεε

,σ

,

10 02

0 1 12 2

0

( )

:

( , ) : : ( ) :

: : ( ) : ( )

1 ( )

Tr

Tr

θρ θ β θ θ

ρ β θ β θ

βθ ρ

=

Δψ = − − →

∂ ψ = + − Δ = − Δ ∂

∂ψ = − ∂

ε

ε

ε ε ε 1 ε

ε ε 1 ε 1ε

ε

0

0

( ) :

( ) 1 ( )s Tr

ψ θρ β θ

ψ θ βθ ρ

∂ = = − Δ ∂ ∂ = − =

∂

ε 1

ε

εε

ε

,σ

,

( ) 00

( ) 0fg

θθ

= =

=

εε

,,

Thermo-elastic Material formulation

24/02/2012 MMC - ETSECCPB - UPC

END OF LECTURE 1