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
Manual of Reference
R4.08 booklet: Influence thermics on mechanics
HI-75/98/044/A
Organization (S):
EDF/IMA/MN
Manual of Reference
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.
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
Manual of Reference
R4.08 booklet: Influence thermics on mechanics
HI-75/98/044/A
Contents
1 Introduction ............................................................................................................................................ 3
2 thermal Expansion factor known compared to
T
ref.
............................................................... 4
3 Expansion factor known compared to a temperature
T
T
def
ref.
......................................... 4
3.1 Calculation of
!()
T
I
in temperatures different from
T
ref.
(except for a precision) ........................ 5
3.2 Calculation of
!()
T
I
for temperatures close to
T
ref.
(except for a precision) ......................... 5
3.2.1 Calculation of
(T
)
ref.
................................................................................................................ 6
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
Manual of Reference
R4.08 booklet: Influence thermics on mechanics
HI-75/98/044/A
1 Introduction
The values of the expansion factors 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 expansion factor 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
m
HT
HT
T
T
Re F
T
of F
HT
(T)
(T)
m
(T
ref.
)
Temperature
()
()
()
HT
m
HT
m
HT
ref.
T
T
T
=
-
éq 1-1
where
m
HT
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
ref.
T
T T T
=
-
!
where
()
!
T
is the average expansion factor (with direction RCC_M) with
temperature
T
determined compared to the temperature
T
ref.
(
T
ref.
being the temperature to which one
consider that
()
HT
ref.
T
=
0.
).
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
Manual of Reference
R4.08 booklet: Influence thermics on mechanics
HI-75/98/044/A
2
Thermal expansion factor known compared to
T
ref.
The values of the thermal expansion factor were determined by tests of dilatometry
carried out starting from the temperature
T
ref.
.
In this case, the key word
TEMP_DEF_ALPHA
does not have to be specified in the control
DEFI_MATERIAU
[U4.23.01].
The equation [éq 1-1] is simplified, since
()
m
HT
ref.
T
=
0
.
From where:
()
()
(
)
HT
ref.
T
T T T
=
-
!
éq 2-1
and
()
HT
ref.
T
=
0
where values of the expansion factor
()
!
T
are well informed under the key word
ALPHA
(or
ALPHA_ *
) in
DEFI_MATERIAU
.
3 Expansion factor known compared to one
temperature
T
T
def
ref.
The values of the thermal expansion factor were determined by tests of dilatometry
who took place starting from a temperature
T
def
different from the temperature of reference
T
ref.
.
Indeed, in general one has the values of the expansion factor 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 the key word
TEMP_DEF_ALPHA
control
DEFI_MATERIAU
, the value of the temperature
T
def
, and under the key word
ALPHA
(or
ALPHA_ *
) them
values of the expansion factor
()
T
(definite compared to the temperature
T
def
). In
order
AFFE_MATERIAU
under the key word
TEMP_REF
, it indicates the value of the temperature of
reference
T
ref.
.
The calculation of
()
HT
T
is done by using the formula:
()
()
(
) () (
)
()
(
)
()
HT
def
ref.
ref.
def
ref.
HT
ref.
T
T T T
T
T
T
T T T
T
=
-
-
-
=
-
=
!
and
0
éq 3-1
The calculation of
()
HT
T
require the preliminary calculation of the values of the function
!()
T
.
The function
!()
T
remain defined (or well informed) for the same values of
T
that
()
T I
NR
,
,
=
1
and
keep the same attributes (even standard of interpolation (`
FLAX
“,”
LOG
') and even type of prolongation
(`
CONSTANT
', `
LINEAR
', `
EXCLUDED
')).
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
Manual of Reference
R4.08 booklet: Influence thermics on mechanics
HI-75/98/044/A
3.1 Calculation
of
!()
T
I
in temperatures different from
T
ref.
(with one
precision near)
One obtains the expression of
!()
T
I
by using the equation [éq 3-1].
-
=
-
-
-
-
I
T
T
Prec
T
T T
T
T
T
T
T
T
I
ref.
I
I
I
def
ref.
ref.
def
I
ref.
t.q
.
!()
() (
)
(
) (
)
éq 3.1-1
The value of the precision is:
·
that is to say specified by the user under the key word
PRECISION
key word factor
ELAS_FO
(control
DEFI_MATERIAU
[U4.23.01]),
·
that is to say equalizes to 1. : default value.
3.2 Calculation
of
!()
T
I
for temperatures close to
T
ref.
(with 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 temperature
T
ref.
.
HT
def
ref.
ref.
def
ref.
HT
def
ref.
ref.
ref.
ref.
def
ref.
(T)
(T) (T T
)
(T
) (T
T
)
(T) (T T
)
T
(T) (T T
)
(T)
(T) (T T
)
(T)
(T
)
(T
) (T
T
)
(T
)
=
-
-
-
=
-
=
-
+
=
-
+
=
-
+
!
!
!
!
from where
and thus
éq 3.2-1
The equation [éq 3.2-1] gives the expression of
()
!
T
ref.
.
It is considered that
! )
!
(T
(T
)
I
T
T
Prec
I
ref.
I
ref.
=
.
t.q
-
<
The value of the precision is:
·
that is to say specified by the user under the key word
PRECISION
key word factor
ELAS_FO
(control
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.
.
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
Manual of Reference
R4.08 booklet: Influence thermics on mechanics
HI-75/98/044/A
3.2.1 Calculation
of
(T
)
ref.
()
()
() ()
()
1st case:
-
<
-
-
-
-
I
T
T
Prec I
T
=
T
T
T
T
T
T
T
T
I
ref.
ref.
i+
ref.
i+
ref.
ref.
I
ref.
I
t.q.
and
and
1
+
I NR
1
2
1
1
1
1
éq 3.2.1-1
()
()
()
2nd case:
-
<
-
-
-
I T
T
Prec
I
T
T
T
T
T
I
ref.
ref.
ref.
I
ref.
I
t.q.
and
if
= NR
=
1
1
éq 3.2.1-2
()
()
()
3rd case:
-
<
=
-
-
I
T
T
prec
I
T
T
T
T
T
I
ref.
ref.
i+
ref.
i+
ref.
t.q.
and if
1
1
1
=
éq 3.2.1-3