Code_Aster ®
Version
4.0
Titrate:
Calculation of the thermal deformation
Date:
23/12/98
Author (S):
A. Mr. DONORE
Key:
R4.08.01-A
Page:
1/6
Organization (S): EDF/IMA/MN
Handbook of Référence
R4.08 booklet: Influence thermics on mechanics
Document: R4.08.01
Calculation of the thermal deformation
Summary
This document is devoted to the presentation of the calculation of the thermal deformation. One indicates them to it
information necessary to the calculation of the thermal deformation and the various possibilities of definition of these
information by the user.
Handbook of Référence
R4.08 booklet: Influence thermics on mechanics
HI-75/98/044/A

Code_Aster ®
Version
4.0
Titrate:
Calculation of the thermal deformation
Date:
23/12/98
Author (S):
A. Mr. DONORE
Key:
R4.08.01-A
Page:
2/6
Contents
1 Introduction ............................................................................................................................................ 3
2 thermal Dilation coefficient known compared to Tref ............................................................... 4
3 Dilation coefficient known compared to a temperature T
T
def
ref. ......................................... 4
3.1 Calculation of!
(T)
I in temperatures different of Tref (except for a precision) ........................ 5
3.2 Calculation of!
(T)
I for temperatures close to Tref (except for a precision) ......................... 5
3.2.1 Calculation of (T
)
ref.
................................................................................................................ 6
Handbook of Référence
R4.08 booklet: Influence thermics on mechanics
HI-75/98/044/A

Code_Aster ®
Version
4.0
Titrate:
Calculation of the thermal deformation
Date:
23/12/98
Author (S):
A. Mr. DONORE
Key:
R4.08.01-A
Page:
3/6
1 Introduction
The values of the dilation coefficients are determined by tests of dilatometry which take place with
to leave the ambient temperature (0°C or more generally 20°C). So one lays out in general
values of the dilation coefficient defined compared to 20°C (temperature to which one supposes
null thermal deformation).
Certain studies require to take a temperature of reference different from the temperature
ambient (null thermal deformation for another temperature that the ambient temperature). It
is then necessary to carry out a change of reference mark in the calculation of the thermal deformation (equation
[éq 1-1] and appears below).
HT
HT (T)
HT (T
HT (T)
m
ref.)
m
T
Tre F
T
of F
Temperature
HT (T) = HT
HT
m (T) - m (ref.
T)
éq 1-1
HT
where
m is the measured thermal deformation (definite compared to the ambient temperature)
HT is the calculated thermal deformation (definite compared to a temperature of reference)
In Code_Aster, the thermal deformation is calculated by the expression
HT (T) =! (T) (T - ref.
T) where! (T) is the average dilation coefficient (with direction RCC_M) with
temperature T determined compared to the temperature Tref (Tref being the temperature to which one
consider that HT (ref.
T) = 0.).
Handbook of Référence
R4.08 booklet: Influence thermics on mechanics
HI-75/98/044/A

Code_Aster ®
Version
4.0
Titrate:
Calculation of the thermal deformation
Date:
23/12/98
Author (S):
A. Mr. DONORE
Key:
R4.08.01-A
Page:
4/6
2
Thermal dilation coefficient known compared to Tref
The values of the thermal dilation coefficient were determined by tests of dilatometry
carried out starting from the Tref temperature.
In this case, key word TEMP_DEF_ALPHA should not be specified in the command
DEFI_MATERIAU [U4.23.01].
The equation [éq 1-1] is simplified, since HT
m (ref.
T) = 0.
From where:
HT (T) =! (T) (T - ref.
T)
éq 2-1
and HT (ref.
T) = 0
where values of the dilation coefficient!
(T) are well informed under key word ALPHA (or
ALPHA_ *) in DEFI_MATERIAU.
3 Dilation coefficient known compared to one
temperature T T
def
ref.
The values of the thermal dilation coefficient were determined by tests of dilatometry
who took place starting from a Tdef temperature different from the temperature of Tref reference.
Indeed, in general one has the values of the dilation coefficient defined compared to
ambient temperature, 0°C or more generally 20°C, and certain studies require to take one
temperature of reference different from the ambient temperature.
It is then necessary to carry out a change of reference mark [éq 1-1].
In this case, the user informs under key word TEMP_DEF_ALPHA of the command
DEFI_MATERIAU, the value of the Tdef temperature, and under key word ALPHA (or ALPHA_ *) them
values of the dilation coefficient (T) (definite compared to the Tdef temperature). In
order AFFE_MATERIAU under key word TEMP_REF, it indicates the value of the temperature of
Tref reference.
The calculation of HT (T) is done by using the formula:
HT (T) = (T) (T - dTef) - (rTef) (ref.
T
- D
T EFF)
=!(T) (T - ref.
T)
éq 3-1
and HT (ref.
T) = 0
The calculation of HT (T) requires the preliminary calculation of the values of the function!
(T).
The function!
(T) remains defined (or well informed) for the same values of T as (T), I =,
1 NR and
keep the same attributes (even standard of interpolation (“LIN”, “LOG”) and even type of prolongation
(“CONSTANT”, “LINEAR”, “EXCLUDED”)).
Handbook of Référence
R4.08 booklet: Influence thermics on mechanics
HI-75/98/044/A

Code_Aster ®
Version
4.0
Titrate:
Calculation of the thermal deformation
Date:
23/12/98
Author (S):
A. Mr. DONORE
Key:
R4.08.01-A
Page:
5/6
3.1 Calculation
of!
(T)
I in temperatures different of Tref (to one
precision near)
One obtains the expression of!
(T)
I by using the equation [éq 3-1].
I t.q. Ti - Tref Prec
(T) (T
I
I - Tdef) - (T
) (T
ref.
ref. - T
)
éq 3.1-1
!
(T
def
I) =
Ti - Tref
The value of the precision is:
· that is to say specified by the user under key word PRECISION of the key word factor ELAS_FO
(command DEFI_MATERIAU [U4.23.01]),
· that is to say equalizes to 1. : default value.
3.2 Calculation
of!
(T)
I for temperatures close to Tref (to one
precision near)
One cannot use the equation [éq 3-1] directly. One derives the equation [éq 3-1] compared to
temperature and one take the value of derived at the Tref temperature.
HT (T) = (T) (T - of
T F) - (R
T EFF) (ref.
T
- D
T EFF) = (!T) (T - ref.
T
)
HT
from where
= (T) (

T - of
T F) + (T) =! (T) (

T - R
T EFF) + (!T)

éq 3.2-1
T
and thus! ref.
(T
) = (R
T EFF) ref.
(T
- D
T EFF) + ref.
(T
)
The equation [éq 3.2-1] gives the expression of!
(Tref).
It is considered that!
(T) =!(T) I t.q. T - T
< Prec
I
ref.
I
ref.
The value of the precision is:
· that is to say specified by the user under key word PRECISION of the key word factor ELAS_FO
(command DEFI_MATERIAU [U4.23.01]),
· that is to say equalizes to 1. : default value.
Also, to calculate!
(T)
I it is necessary as a preliminary to calculate (T
)
ref.
.
Handbook of Référence
R4.08 booklet: Influence thermics on mechanics
HI-75/98/044/A

Code_Aster ®
Version
4.0
Titrate:
Calculation of the thermal deformation
Date:
23/12/98
Author (S):
A. Mr. DONORE
Key:
R4.08.01-A
Page:
6/6
3.2.1 Calculation
of
(T)
ref.
1st case: t.q I -
I
T
Tref < Prec
and I and
1 I NR

1 Ti+1 - T
T
ref.
ref. -

T
éq 3.2.1-1
(
i-1
T

ref.)
()

() () ()
= 2
+
Ti+1 - T
T
ref.
ref. - T


i-1

2nd case: I T
t.q I - Tref < Prec and
if
I = NR
Tref - T
éq 3.2.1-2
(
I 1
Tref)
() (-)
=
Tref - Ti-1
3rd case: I t.q. Ti - Tref < prec and if I = 1
Ti+1 - T
éq 3.2.1-3
(
ref.
Tref)
() ()
=
Ti+1 - Tref
Handbook of Référence
R4.08 booklet: Influence thermics on mechanics
HI-75/98/044/A

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