CH.1. THERMODYNAMIC FOUNDATIONS OF CONSTITUTIVE MODELLING. Computational Solid Mechanics- Xavier Oliver-UPC

Transcription

1 CH.1. THERMODYNAMIC FOUNDATIONS OF CONSTITUTIVE MODELLING Computational Solid Mechanics- Xavier Oliver-UPC

2 1.1 Dissipation approach for constitutive modelling Ch.1. Thermodynamical foundations of constitutive modelling

3 Power () Power, W t, 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 t : () () = d t W() t dt 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.

4 External Mechanical Power 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: e () P t = ρ b v dv + t v ds V V dr ρb dv dt = v dr t ds dt = v

5 Theorem of the expended mechanical power d 1 2 Pe () t = ρbv dv + ds v dv dv V tv = ρ V dt + 2 σ :d Vt V V external mechanical power entering the medium P t e d = +P σ dt () () t Kinetic energy K Stress power P REMARK The 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. σ

6 External Heat 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 qx (, t) ) heat conduction flux vector non convective (conduction) incoming heat unit of time = qn V ds Internal heat sources (characterized by r( x, t) ) heat generated by internal sources unit of time = ρ rdv V heat source field

7 External Heat Power The external heat power is the incoming heat in the continuum medium per unit of time. It is defined as: Where: q x,t r ( ) ( x, t) is the heat flux per unit of spatial surface area. is an internal heat source rate per unit of mass.

8 Total Incoming Power The total power entering the continuous medium is: () Q Pe t e () t d 1 = + = ρ v + σ :d + ρ q n 2 Winput Pe Qe dv dv r dv ds dt 2 Vt V V V V kinetic energy stress power Internal heat source External heat conduction

9 Stored Mechanical Power Mechanical Dissipation Stored mechanical power: is that part of the incoming mechanical power that can be eventually returned by the body: d 1 v 2 d dk dv Pstored = dv dv dt ρ + 2 dt ρψ = + dt dt Vt V V ρψ( x, t) 0 Kinetic energy K Stored mechanical energy V Mechanical dissipation: is that part of the incoming mechanical power that is not stored (eventually can be lost) d d d D = P P = + dv ( mech e stored dt σ :d + ) = dv ρψ dt V dt σ :d V V d+ P d d σ + dt dt dt Density of free energy (Helmholtz energy) P e Stored mechancical energy V unit of volume ρψ dv dv

10 Second principle of the thermodynamics Dissipation Mechanical dissipation P + + e d dt d dt D mech D = ρψ dv + σ dv mech :d V V Thermal dissipation : D therm Dtherm = ρθ s s( x, t) Density of enthropy dv θ ( x, t) Absolute temperature (>0) V D = Dmech + Dtherm = [ ρ( ψ + s θ) + σ : d] dv 0 ΔV V V Global (integral) form of the second principle of thermodynamics

11 Second principle of the thermodynamics Dissipation Dissipation ( x, t) Density of dissipation= unit of volume D= dv = [ ρ( ψ + s θ) + σ : d] dv 0 ΔV V V V "Localization" process (,) x t = ρ( ψ+ s θ) + σ : d 0 x t Local (differential) form of the second principle of the thermodynamics

12 Alternative forms of the Dissipation The internal energy per unit of mass (specific internal energy) is : u(,) x t u(,): x t =ψ+ sθ ψ(,) x t sθ (,) x t Total stored energy Taking the material time derivative, u= ψ+ s θ + sθ ψ + s θ= u θs and introducing it into the Dissipation inequality ( s θ) σ : d 0 = ρ ψ + + Mechanical stored stored energy Thermal stored stored energy ( u θs) σ : d 0 = ρ + REMARK For infinitessimal deformation, d = ε, the Clausius-Planck inequality becomes: ρψ ( + s θ) + σ: ε 0 Clausius-Planck Inequality in terms of the specific internal energy

13 Dissipation in a continuum medium The dissipation of a continuum medium is defined as: corresponding to the Clausius-Planck Inequality. ( θ) : = ρ u s + σ : d 0 In terms of the Helmholtz free energy, dissipation may be written as: ( s θ) σ : d 0 = ρ ψ + + Hypotheses assumed Infinitesimal deformation: d( x, t ) = ε ( x, t) ρ( x, t) = ρ0 Hence, the dissipation may be written as: Dissipation of a continuum medium in terms = ρ ( ) 0 ψ + s θ + σ : ε 0 of the Helmholtz free energy assuming infinitesimal deformation

14 1.2 A thermodynamic framework for constitutive modeling Ch.1. Thermodynamical foundations of constitutive modelling

15 Sets of thermo-mechanical variables In a thermo-mechanical problem we will consider the set of all the variables of the problem: { v } 1,v 2,...,vn v i( x, t) i { 1,2,..., nv} : = v which will be classified into: Free variables: { λ, 1 λ,..., 2 λ } (, ) { 1, 2,..., n λi x t i nf} : = which are physically observable variables, whose evolution along time is unrestricted λi ( x, t) λi ( x, t) = any t F

16 Sets of thermo-mechanical variables Internal/Hidden variables: { α, 1 α,..., 2 α } (, ) { 1, 2,..., n α i x t i ni} : = I are non-observable variables. Their evolution is limited along time in terms of specific evolution equations defined as: α i ( x, t) α = = ξ ( (, t), (, t) ) i 1,2,... n which account for micro-structural mechanisms. Dependent variables: are the remaining of the variables of the problem (depending on the previous ones): instantaneous values (at time t ) { } i i I t λ x α x { d, 1 d,..., 2 d } (, ) { 1, 2,..., n di x t i nd} : = D { } d = γ ( λ, α) d = ϕ ( λ, α, λ ) i 1, 2,... n i i i i D = = = =

17 Example: Problem variables: : = { ρ, σε,, u, ψ, s, θα, } Free variables: : = { ρ, ε} ρ, ε Internal/Hidden variable: Dependent variables: := { α } { } := ρ, σε,, u, ψ, s, θ, α provided by the evolution equation α = γ ( ρ, ε, α ) ψ = ψ ( ρ, ε, α ) ψ = γ ( ρ, ε, α, ρ, ε, α( ρ, ε, α)) = ψ( ρ, ε, α, ρ, ε) not depending on the internal variable evolution, α σ=σ( ρ, ε, α) σ = ϕ( ρ, ε, α, ρ, ε, α( ρ, ε, α ) ) = σ ( ρ, ε, α, ρ, ε)

18 Elements of a constitutive model 1) Definition of the free variables of the problem := { λ, 1 λ,..., 2 λ n } F 2) Choice of the internal variables of the problem 3) Definition of the corresponding evolution equations, λ { α, 1 α,..., 2 α ni } : = α { n } α = α ( λ, α) i 1, 2,... i i I 4) Postulate a specific form of the free energy: ψ=ψ( λ, α,t) provides the constitutive equation through the dissispation inequality

19 Example: Thermo-elastic material Linear elastic material Linear relation stresses-strains 1D: W A = E ΔL L σ = Eε 3D: σ = : ε or σ = ε ij ijkl kl Isotropic elastic material: being ijkl the Lamé constants. ijkl = λ1 1+ 2μΙ λ a second order tensor, and and σ = λtr( ε) + 2με σij = λεkkδij + 2με ij i, j 1,2,3 μ { }

20 Tensor Notation (reminder) Open product Of two first-order tensors: Between two second order tensor: Identity tensors First order identity tensor 1 i = j 1 = δ ij ij = 0 i j i, j 1,2,3 Second order tensor identity tensor [ a b] i, j =ab i j [ A B] ijkl =A ij B kl [] { } 1 [ ik jl il jk ],,, 1,2,3 2 [ I] = δ δ + δ δ i j k l { } ijkl REMARK Einstein notation (summation of repeated indices) is considered

21 Thermo-elastic material (reminder) Linear thermo-elastic material Adding the thermal effects (thermoelasticity) σ = λtr( ε) + 2με βδθ1 σij = λεkkδij + 2με ij ( β Δθ) δij i, j 1,2,3 β = 1 2ν α = β Thermal property Thermal expansion coefficient { }

22 Coleman s Method Theorem f(, x y) = 0 (, xy,, x y) = f(, x y) x+ g(, x y) y 0 xy, g(, xy) = 0 Proof: Taking y = 0 = f (, xyx ) 0 x Taking If f(, x y ) < 0 taking x > 0 = f(, x y) x < 0 If f(, x y ) > 0 taking x < 0 = f(, x y) x < 0 x = 0 g = (, xyy ) 0 y If gxy (, ) < 0 taking y > 0 = gxyy (, ) < 0 If gxy> (, ) 0 taking y < 0 = gxyy (, ) < 0 NOT POSSIBLE f(, x y ) = 0 NOT POSSIBLE gxy (, ) = 0

23 Elastic Material formulation Variable sets definition Free variables: Internal variables: Dependent: Potential Helmholtz free energy: Dissipation : = { ε} : = { } No evolution equation : = { σ,ψ } ρ ψ ( ε) = ε : : ε Isothermal case = ρ ( ψ+ sθ 0 ) + σ : ε 0 ρψ 0 () ε σ = = : ε ε = 0 Constituve equation σ= Σ( ε) σε () σε () σ = :ε ε ρψ() ρψ = ε ε 0 0 () ε ρψ 0 :ε ρ ψ() ε (σ ): ε ε 0 ε f ( ε ) 0 f () ε 0 = =

24 Thermo-elastic Material formulation Variable sets definition Free variables: Internal variables: Dependent: Potential definition : = { ε,θ } : = { } σ( ε, θ) σ( ε, θ) σ = :ε + : θ ε θ : = { σ, ψ} ρψ 0 ( θ) ρψ 0 ( θ) ρψ 0 = ε, : ε, ε + : θ ε θ 1 Helmholtz free energy: ρ 0ψ ( ε, θ ) = 2 ε : : ε β ( θ θ0) Ι Δθ Dissipation = ρ0 ψ + sθ + σ : ε 0 ( ) ρψ 0 ( ε, θ) ρψ 0 ( ε, θ) = (σ ): ε ( 0s + ): 0 ε, ε θ f( ε, θ) g( ε, θ) ρ θ θ

25 Thermo-elastic Material formulation Using the Coleman s method: ( ) ( θ) ρ ψ ε, f ε, = σ θ 0 = 0 ρψ 0 ( ε, θ) σ = ε ε ψ ( ε, θ) ψ ( ε, θ) g( ε, θ) = ρ0s+ ρ0 = 0 s = θ θ f ( ε, θ) = 0 = 0 g( ε, θ) = 0 and differentiating the Helmholtz free energy: 1 ρ Constitutive 0ψ ( ε, θ ) = 2 ε : : ε β ( θ θ0) 1 : ε ( ) equations Δθ Tr ε ρ 0ψ ψ( ε, θ) 1 1 = 2 ε : + 2 : ε ( βδ θ ) 1 = : ε ( βδθ ) 1 σ = ρ0 = : ε β Δ θ1 ε ε = : ε ψ( θ) 1 ψ 1 ε, s= = β Tr() ε = β Tr( ε) θ ρ0 θ ρ0

26 END OF LECTURE 1 24/02/2012 MMC - ETSECCPB - UPC