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Generalized problem of thermoelasticity GNIII

 Student:

Е. S. Kosyanenko

Saint-Petersburg 2014

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Introduction , h – heat flux, - thermal conductivity, - temperature.* * 1h

LS theory: was obtained hyperbolic equation of extending the heat, based on the new law of heat conduction:

* (2)relh h GL theory is characterized in that heat flux vector in a thermoelastic body h is depends on the rate of change of the absolute temperature θ and temperature gradient:

* (3)h b GN theory combines the properties of both classical thermoelastisity and non-dissipative thermoelasticity. It was formulated in 3 different versions of the thermodynamic: GNI, GNII, GNIII.

GNIII is characterized by that the heat flux vector is linearly depends on the temperature gradient and gradient of the thermacy:

* , (4)h

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Complete system of relations of GNIII-theory

* (5)h

0(6)div u

( ) ( ) (7)s tr h

2 ( ) (8)Tu u

( )(9)envn h

Λ – thermal conductivity rate , ϑ – thermacy, ρ – density of medium, u – displacement vector, ψ – density of Helmholtz free energy(per unit volume), s – entropy density, ξ – internal entropy production (ξ≥0), σ – heat transfer coefficient, θenv – environment temperature.

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Linear coupled equations of motion and heat conduction

* 0

( ) 0(10)

0

u u u

k u

* 0

( 2 ) 0(11)

0

u u

k u

* * *

* * * *

0(12)

0

u u

u

2

2, (13)

( 2 ) ( 2 )

bc c

a d

2* * *

0 *

, , , 1(14)k

d c a bk

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Solution

0

0

( , )(13)

( , )

x iwt

x iwt

u x t u e e

T x t T e e

(12)i , λ – wave vector

4 4 2 2(1 ) (1 )(14)iw w w i w

2 2

2

2

(15)p

w

qw

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Results of calculations

2 2

2 (1 2 )( ) (16)

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q pw p

p q p

2 2 2 2 2 2 21

( ) ( 2 ( ) (1 2 ) ( ) ((1 ) 4( 1 ) 4 ) 2 ( 3 ( 1 2 ))))(17)2 2

q p p p p p

2 24( ) (18)

2

p p qp w

2 2

2( ) (19)

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qwp

p p q

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Plots

Pic. 1. Plot q(p) Pic. 2. Plot γ(w)

Pic. 3. Plot w(w/δ)

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Conclusion 1. Studying GNIII-theory. 2. Formulation of the problem. 3. Dimensionless of the equations. 4. Parameterization of the system of equations. 5. Results of calculation is close system of equations. 6. Obtaining plots.

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Thank you for attention