Fundamentals of computational thermo-elasticity

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Thermo-elasticity Examples

Fundamentals of computational thermo-elasticity

J. Pamin

in cooperation with

J. Jaśkowiec, P. Pluciński, R. Putanowicz, A. Stankiewicz, A. Wosatko

Cracow University of Technology, Department of Civil Engineering, Institute for Computational Civil Engineering

5 listopada 2015

J. Pamin & co. (Cracow University of Technology) Fundamentals of computational thermo-elasticity

Overview and assumptions

1 Thermo-elasticity

Thermodynamic background Theory and algorithms

2 Examples

Assumptions

small displacements and strains isotropic continuum

linear elasticity static loading

nonstationary heat transport coupling due to thermal expansion

selected material parameters temperature-dependent

Fundamentals of computational thermo-elasticity

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Thermodynamics and mechanics

Thermodynamics provides restrictions on the form of constitutive models

State variables (observable and internal), e.g.

(, κ, θ) - strain tensor, internal variable vector, temperature, resp.

A thermodynamic system is reversible if thermodynamic potentials do not depend on internal variables

Laws of thermodynamics (for elementary material volume) 1 → ˙e = σ : ˙ + r − ∇q (balance of internal energy) 2 → s _pro 0 (specific internal entropy production)

Thermodynamic background Theory and algorithms

Dissipation

Definition of energy dissipation density

D = θs _pro = D _mech + D _ther 0 (local + conductive) D = σ : ˙ − ( ˙e − θ ˙s) − q · ∇θ

θ 0 (Clausius-Duhem inequality) Helmholtz free energy ψ(, κ, θ)

e − θs = ψ Dissipation inequality for isothermal conditions

D = σ : ˙ − ˙ ψ 0 ψ = ∂ψ : ˙ + ∂κψ · ˙κ ˙ D = (σ − ∂ψ) : ˙ + K · ˙κ 0 K = −∂κψ (thermodynamic forces)

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Material models - isothermal elasticity

Hyperelasticity

σ = σ() and D = 0 Selected ψ is path-independent internal stress power W

ψ = ˙ ˙ W () = σ : ˙ (stored elastic energy) Work equals stored elastic energy

W | ^t _t

¹

0

=

Z t

1

t

₀

W ()dt = ψ( ˙ ₁ ) − ψ( ₀ )

ψ = ˙ ∂ψ

∂ : ˙

D = σ : ˙ − ∂ψ

∂ : ˙ = 0 → σ = ∂ψ

∂

˙

σ = ∂ ² ψ

∂ ⊗ ∂ : ˙ (tangent stiffness)

Non-isothermal conditions

Dissipation inequality for non-isothermal conditions D = σ : ˙ − ( ˙ ψ + s ˙ θ) − q ·∇θ

θ 0

Thermo-elasticity ψ = ˙ ∂ψ

∂ : ˙ + ∂ψ

∂θ

θ , ˙ s = − ∂ψ

∂θ , D = − q · ∇θ θ Thermo-elasto-plasticity

ψ = ψ( ^e , κ, θ), ^e = − ^p ψ = ˙ ∂ψ

∂ ^e : ˙ ^e + ∂ψ

∂κ ˙κ + ∂ψ

∂θ θ ˙ D _mech = σ : ˙ ^p + K · ˙ κ 0

D _ther = − q · ∇θ θ 0

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Temperature dependence

In principle all material

parameters (isotropy assumed) are temperature-dependent:

– density ρ

– Young modulus E – Poisson ratio ν

– expansion coefficient α – heat conductivity k – heat capacity c

For elastic-plastic materials additionally:

– yield stress σ _y

– hardening modulus h _p

Figures from (Ottosen and Ristinmaa, 2005)

Sources of nonlinearity

Arbitrary choice made for present derivation: E (θ), k(θ)

Hence, nonlinearity is due to temperature-dependence of Young modulus and heat conductivity, and possibly natural boundary conditions for the thermal subproblem (radiation).

For generalization refer e.g. to (Ottosen and Ristinmaa, 2005)

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Thermoelasticity

Voigt’s notation:

σ = [σ _x , σ _y , σ _z , τ _xy , τ _xz , τ _yz ] ^T

= [ _x , _y , _z , γ _xy , γ _xz , γ _yz ] ^T Assumed b, ˆ t independent of temperature

Equilibrium

L ^T σ + b = 0 in V σn = ˆ t on S _t u = ˆ u on S _u

Linear elasticity + thermal expansion (θ = T − T ₀ )

= ^e + ^θ , σ = E( − ^θ ), = Lu

^θ = αθΠ, Π = [1 1 1 0 0 0] ^T , E = E(θ)

Momentum balance

Weak form at t + ∆t (u = ˆ u, v _u = 0 on S _u ) Z

V

(Lv _u ) ^T σ dV = Z

V

v _u ^T b dV + Z

S

_t

v ^T _u ˆ t dS ∀v _u

σ ^t+∆t = σ ^t + ∆σ

∆σ = lim

i →∞ ∆σ _i , ∆σ _{i +1} = ∆σ _i + d σ, σ ^t+∆t _i = σ ^t + ∆σ _i Z

V

(Lv _u ) ^T d σ dV = W ^t+∆t _ext − W ^t+∆t _int,i

W ^t+∆t _int,i = Z

V

(Lv _u ) ^T σ ^t+∆t _i dV

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Momentum balance

Linearized weak form

d σ = E d − d ^θ + dE − ^θ

d E = dE

dθ d θ, d ^θ = αΠ d θ, s _θ = dE

dθ ( − αθΠ) − αEΠ d σ = E d + s _θ d θ

d = L d u Z

V

(Lv _u ) ^T E L d u dV + Z

V

(Lv _u ) ^T s _θ d θ dV = W ^t+∆t _ext − W ^t+∆t _int,i

Heat transport

Assumed ρ, r , ˆ q independent of temperature Heat transport

c ˙ θ + ∇ ^T q = r in V q ^T n = ˆ q on S _q

θ = ˆ θ on S _θ Fourier’s law

q = −Λ∇θ, Λ = Λ(θ)

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Heat transport

Weak form at t + ∆t (θ = ˆ θ, v _θ = 0 on S _θ ) Z

V

v _θ c ˙ θ dV − Z

V

(∇v _θ ) ^T q dV = Z

V

v _θ r dV − Z

S

q

v _θ ˆ q dS ∀v _θ

θ ^t+∆t = θ ^t + ∆θ Generalized midpoint rule

(1 − γ) ˙ θ ^t + γ ˙ θ ^t+∆t = θ ^t+∆t − θ ^t

∆t For γ = 1 backward Euler for time integration

θ ˙ ^t+∆t = ∆θ

∆t Z

V

v _θ c ∆θ

∆t dV − Z

V

(∇v _θ ) ^T q ^t+∆t dV = Q ^t+∆t _ext

Heat transport

Weak form at t + ∆t (θ = ˆ θ, v _θ = 0 on S _θ )

∆θ = lim

i →∞ ∆θ _i , ∆θ _{i +1} = ∆θ _i + d θ, θ _i ^t+∆t = θ ^t + ∆θ _i q ^t+∆t = q ^t + ∆q

∆q = lim

i →∞ ∆q _i , ∆q _{i +1} = ∆q _i + d q, q ^t+∆t _i = q ^t + ∆q _i Linearization:

d q = − dΛ

dθ ∇θ d θ − Λ∇( d θ)

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Heat transport

Linearized weak form

1 ∆t Z

V

v _θ c d θ dV + Z

V

(∇v _θ ) ^T dΛ

dθ ∇θ d θ dV + +

Z

V

(∇v _θ ) ^T Λ∇( d θ) dV = Q ^t+∆t _ext + Q ^t+∆t _int,i

Q ^t+∆t _int,i = Z

V

(∇v _θ ) ^T q ^t+∆t _i dV − 1

∆t Z

V

v _θ c∆θ _i dV

Discretization and coupling

u, v _u using N _u , u = N _u ˇ u, B _u = LN _u θ, v _θ using N _θ , θ = N _θ θ, B ˇ _θ = ∇N _θ Substitution leads to:

"

K _i K _{θ ,i} 0 C + H _i

# "

d ˇ u d ˇ θ

#

=

"

f ^t+∆t _ext − f ^t+∆t _int,i h ^t+∆t _ext − h ^t+∆t _int,i

#

where K

i

=

Z

V

B ^T _u E

i

B u dV , K

_{θ ,i}

= Z

V

B ^T _u s

_θ,i

N

_θ

dV

C = 1

∆t Z

V

N ^T

_θ

cN

θ

dV , H

i

= Z

V

B ^T

_θ

_dΛ dθ

i

∇θ

i

N

θ

+ Λ

i

B

θ

dV

f

^t+∆t

_int,i = Z

V

B ^T _u σ

^t+∆t_i

dV , h

^t+∆t

_int,i = Z

V

N ^T

_θ

c ∆θ

i

∆t dV − Z

V

B ^T

_θ

q

^t+∆t_i

dV

Staggered algorithm possible

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Square configuration under thermal load

q

n

= 0

q

n

= 0

q

n

= 0 T ( t )

t T

1 Mechanics:

– density ρ = 2

– Young modulus E = 20000 – Poisson ratio ν = 0.2 – yield stress σ _y = 300

– hardening modulus E _T = 2000

Heat flow:

– initial temperature T ₀ = 0 – heat conductivity k = 10 – heat capacity c = 1

– expansion coefficient α = 0.01

Square configuration under thermal load (FEAP)

Equivalent stress (elasticity)

Evolution of equivalent stress (FEAP)

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Square configuration under thermal load (FEAP)

Equivalent stress (elasticity)

Square configuration under thermal load (FEAP)

Equivalent stress (elasticity)

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Square configuration under thermal load (FEAP)

Equivalent stress (elasticity)

Square configuration under thermal load (FEAP)

Equivalent stress (elasticity)

(12)

Square configuration under thermal load (FEAP)

Equivalent stress (elasticity)

Square configuration under thermal load (FEAP)

Equivalent stress (elasticity)

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Square configuration under thermal load (FEAP)

Equivalent stress (elasticity)

Square configuration under thermal load (ANSYS)

Equivalent stress (elasticity)

coarse mesh dense mesh

Caution: sensitivity of results to space and time discretization!

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Square configuration under thermal load (ANSYS)

Equivalent stress (plasticity)

coarse mesh dense mesh

Caution: locking in plasticity must be prevented!

Square configuration under thermal load (ANSYS)

Plastic strain measure

coarse mesh dense mesh

Question: what if we need full coupling due to plastic dissipation and thermal softening?

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FEMDK (Tochnog kernel): heat induced deformation

Problem setup

0.9 1.0

1.0

00 00 00 00 00 00 00 00

11 11 11 11 11 11 11 11

00 00 00 00 00 00 00 00

11 11 11 11 11 11 11 11

0000 0000 0000 0000 1111 1111 1111 1111

000 111

00 00 00 00 00 00 00 00

11 11 11 11 11 11 11 11

00 00 00 00 00 00 00 00

11 11 11 11 11 11

11 11 000 000 000 111 111 111

0000 0000 0000 1111 1111 1111

000 000 111 111

00 00 00 11 11 11

convection to environment T

t T(t) free to

move

fixed fixed

T(0) = 0 T(2) = 1600 insulation

insulation

Material properties (SI Units): Steel AISI-304, density ρ = 8030.0, Young modulus E = 1.93e11, Poisson ratio ν = 0.29, yield stress σ

y

= 24.3e6, linear expansion coefficient α = 1.78e − 5, heat conductivity k = 16.3, convection coefficient h c = 10.0, environment temperature T ambient = 0.0.

Fundamentals of computational thermo-elasticity

Fundamentals of computational thermo-elasticity

J. Pamin

in cooperation with

J. Jaśkowiec, P. Pluciński, R. Putanowicz, A. Stankiewicz, A. Wosatko

Cracow University of Technology, Department of Civil Engineering, Institute for Computational Civil Engineering

5 listopada 2015

Overview and assumptions

1 Thermo-elasticity

Thermodynamic background Theory and algorithms

2 Examples

Assumptions

small displacements and strains isotropic continuum

linear elasticity static loading

nonstationary heat transport coupling due to thermal expansion

selected material parameters temperature-dependent

Thermodynamics and mechanics

Thermodynamics provides restrictions on the form of constitutive models

State variables (observable and internal), e.g.

(, κ, θ) - strain tensor, internal variable vector, temperature, resp.

A thermodynamic system is reversible if thermodynamic potentials do not depend on internal variables

Laws of thermodynamics (for elementary material volume) 1 → ˙e = σ : ˙ + r − ∇q (balance of internal energy) 2 → s pro ­ 0 (specific internal entropy production)

Dissipation

Definition of energy dissipation density

D = θs pro = D mech + D ther ­ 0 (local + conductive) D = σ : ˙ − ( ˙e − θ ˙s) − q · ∇θ

θ ­ 0 (Clausius-Duhem inequality) Helmholtz free energy ψ(, κ, θ)

e − θs = ψ Dissipation inequality for isothermal conditions

D = σ : ˙ − ˙ ψ ­ 0 ψ = ∂ψ : ˙ + ∂κψ · ˙κ ˙ D = (σ − ∂ψ) : ˙ + K · ˙κ ­ 0 K = −∂κψ (thermodynamic forces)

Material models - isothermal elasticity

Hyperelasticity

σ = σ() and D = 0 Selected ψ is path-independent internal stress power W

ψ = ˙ ˙ W () = σ : ˙ (stored elastic energy) Work equals stored elastic energy

W | t t

=

Z t

t

W ()dt = ψ( ˙ 1 ) − ψ( 0 )

ψ = ˙ ∂ψ

∂ : ˙

D = σ : ˙ − ∂ψ

∂ : ˙ = 0 → σ = ∂ψ

∂

˙

σ = ∂ 2 ψ

∂ ⊗ ∂ : ˙ (tangent stiffness)

Non-isothermal conditions

Dissipation inequality for non-isothermal conditions D = σ : ˙ − ( ˙ ψ + s ˙ θ) − q ·∇θ

θ ­ 0

Thermo-elasticity ψ = ˙ ∂ψ

∂ : ˙ + ∂ψ

∂θ

θ , ˙ s = − ∂ψ

∂θ , D = − q · ∇θ θ Thermo-elasto-plasticity

ψ = ψ( e , κ, θ),  e =  −  p ψ = ˙ ∂ψ

∂ e : ˙ e + ∂ψ

∂κ ˙κ + ∂ψ

∂θ θ ˙ D mech = σ : ˙ p + K · ˙ κ ­ 0

D ther = − q · ∇θ θ ­ 0

Temperature dependence

In principle all material

parameters (isotropy assumed) are temperature-dependent:

– density ρ

– Young modulus E – Poisson ratio ν

– expansion coefficient α – heat conductivity k – heat capacity c

For elastic-plastic materials additionally:

– yield stress σ y

– hardening modulus h p

Figures from (Ottosen and Ristinmaa, 2005)

Sources of nonlinearity

Arbitrary choice made for present derivation: E (θ), k(θ)

Hence, nonlinearity is due to temperature-dependence of Young modulus and heat conductivity, and possibly natural boundary conditions for the thermal subproblem (radiation).

For generalization refer e.g. to (Ottosen and Ristinmaa, 2005)

Thermoelasticity

Voigt’s notation:

σ = [σ x , σ y , σ z , τ xy , τ xz , τ yz ] T

 = [ x ,  y ,  z , γ xy , γ xz , γ yz ] T Assumed b, ˆ t independent of temperature

Equilibrium

L T σ + b = 0 in V σn = ˆ t on S t u = ˆ u on S u

Linear elasticity + thermal expansion (θ = T − T 0 )

 =  e +  θ , σ = E( −  θ ),  = Lu

(, κ, θ) - strain tensor, internal variable vector, temperature, resp.

Laws of thermodynamics (for elementary material volume) 1 → ˙e = σ : ˙ + r − ∇q (balance of internal energy) 2 → s _pro 0 (specific internal entropy production)

D = θs _pro = D _mech + D _ther 0 (local + conductive) D = σ : ˙ − ( ˙e − θ ˙s) − q · ∇θ

θ 0 (Clausius-Duhem inequality) Helmholtz free energy ψ(, κ, θ)

D = σ : ˙ − ˙ ψ 0 ψ = ∂ψ : ˙ + ∂κψ · ˙κ ˙ D = (σ − ∂ψ) : ˙ + K · ˙κ 0 K = −∂κψ (thermodynamic forces)

σ = σ() and D = 0 Selected ψ is path-independent internal stress power W

ψ = ˙ ˙ W () = σ : ˙ (stored elastic energy) Work equals stored elastic energy

W | ^t _t

W ()dt = ψ( ˙ ₁ ) − ψ( ₀ )

∂ : ˙

D = σ : ˙ − ∂ψ

∂ : ˙ = 0 → σ = ∂ψ

∂

σ = ∂ ² ψ

∂ ⊗ ∂ : ˙ (tangent stiffness)

Dissipation inequality for non-isothermal conditions D = σ : ˙ − ( ˙ ψ + s ˙ θ) − q ·∇θ

θ 0

∂ : ˙ + ∂ψ

ψ = ψ( ^e , κ, θ), ^e = − ^p ψ = ˙ ∂ψ

∂ ^e : ˙ ^e + ∂ψ

∂θ θ ˙ D _mech = σ : ˙ ^p + K · ˙ κ 0

D _ther = − q · ∇θ θ 0

– yield stress σ _y

– hardening modulus h _p

σ = [σ _x , σ _y , σ _z , τ _xy , τ _xz , τ _yz ] ^T

= [ _x , _y , _z , γ _xy , γ _xz , γ _yz ] ^T Assumed b, ˆ t independent of temperature

L ^T σ + b = 0 in V σn = ˆ t on S _t u = ˆ u on S _u

Linear elasticity + thermal expansion (θ = T − T ₀ )

= ^e + ^θ , σ = E( − ^θ ), = Lu

^θ = αθΠ, Π = [1 1 1 0 0 0] ^T , E = E(θ)

Weak form at t + ∆t (u = ˆ u, v _u = 0 on S _u ) Z

(Lv _u ) ^T σ dV = Z

v _u ^T b dV + Z

v ^T _u ˆ t dS ∀v _u

σ ^t+∆t = σ ^t + ∆σ

i →∞ ∆σ _i , ∆σ _{i +1} = ∆σ _i + d σ, σ ^t+∆t _i = σ ^t + ∆σ _i Z

(Lv _u ) ^T d σ dV = W ^t+∆t _ext − W ^t+∆t _int,i

W ^t+∆t _int,i = Z

(Lv _u ) ^T σ ^t+∆t _i dV

d σ = E d − d ^θ + dE − ^θ

dθ d θ, d ^θ = αΠ d θ, s _θ = dE

dθ ( − αθΠ) − αEΠ d σ = E d + s _θ d θ

d = L d u Z

(Lv _u ) ^T E L d u dV + Z

(Lv _u ) ^T s _θ d θ dV = W ^t+∆t _ext − W ^t+∆t _int,i

c ˙ θ + ∇ ^T q = r in V q ^T n = ˆ q on S _q

θ = ˆ θ on S _θ Fourier’s law

Weak form at t + ∆t (θ = ˆ θ, v _θ = 0 on S _θ ) Z

v _θ c ˙ θ dV − Z

(∇v _θ ) ^T q dV = Z

v _θ r dV − Z

v _θ ˆ q dS ∀v _θ

θ ^t+∆t = θ ^t + ∆θ Generalized midpoint rule

(1 − γ) ˙ θ ^t + γ ˙ θ ^t+∆t = θ ^t+∆t − θ ^t

θ ˙ ^t+∆t = ∆θ

v _θ c ∆θ

(∇v _θ ) ^T q ^t+∆t dV = Q ^t+∆t _ext

Weak form at t + ∆t (θ = ˆ θ, v _θ = 0 on S _θ )

i →∞ ∆θ _i , ∆θ _{i +1} = ∆θ _i + d θ, θ _i ^t+∆t = θ ^t + ∆θ _i q ^t+∆t = q ^t + ∆q

i →∞ ∆q _i , ∆q _{i +1} = ∆q _i + d q, q ^t+∆t _i = q ^t + ∆q _i Linearization:

v _θ c d θ dV + Z

(∇v _θ ) ^T dΛ