Download - Generalized problem of thermoelasticity GNIII 1 Student: Е. S. Kosyanenko Saint-Petersburg 2014
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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