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Code_Aster
®
Version
6.3
Titrate:
Pre and postprocessing for the “composite” hulls
Date:
13/09/02
Author (S):
P. MASSIN, F. NAGOT, F. VOLDOIRE, J.M. PROIX
Key
:
R4.01.01-B
Page
:
1/20
Manual of Reference
R4.01 booklet: Composite materials
HT-66/02/004/A
Organization (S):
EDF/AMA, SEPTEN















Manual of Reference
R4.01 booklet: Composite materials
R4.01.01 document



Pre and Postprocessing for the thin hulls
out of “composite” materials




Summary:

One extends the results of the theory of the elements of plates exposed in documentation [R3.07.03] to the case
multi-layer orthotropic materials. Documentation suggested gathers the thermal aspects and
thermo élasto-mechanics. The use of these materials is theoretically valid only in the case of one
geometrical symmetry compared to the average layer of the plate. It is thus necessary that the coupling
membrane-bending is null.
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Code_Aster
®
Version
6.3
Titrate:
Pre and postprocessing for the “composite” hulls
Date:
13/09/02
Author (S):
P. MASSIN, F. NAGOT, F. VOLDOIRE, J.M. PROIX
Key
:
R4.01.01-B
Page
:
2/20
Manual of Reference
R4.01 booklet: Composite materials
HT-66/02/004/A
Count
matters
1
Introduction ............................................................................................................................................ 3
2
Homogenized characteristics of a thin hull in thermo elasticity and thermics ............... 4
2.1
Notations - Assumptions ................................................................................................................... 4
2.2
Thermics ....................................................................................................................................... 5
2.3
Thermomechanical ......................................................................................................................... 6
3
Reference mark in the tangent plan with the hull. Matric notation ............................................................ 8
3.1
Reference mark ........................................................................................................................................... 8
3.2
Matric notation .......................................................................................................................... 9
3.2.1
Thermics .............................................................................................................................. 9
3.2.2
Thermomechanics ............................................................................................................... 10
4
Hulls made up of homogeneous layers ..................................................................................... 12
4.1
Description of the layers ............................................................................................................... 12
4.2
Thermics ..................................................................................................................................... 13
4.3
Thermomechanics ........................................................................................................................ 14
4.3.1
Relation of behavior ................................................................................................... 14
4.3.2
Transverse shearing ....................................................................................................... 15
4.3.3
Generalized efforts ............................................................................................................... 17
4.3.4
Localization of the stresses (postprocessing) ..................................................................... 18
4.3.5
Calculation of the criteria of rupture in the layers (postprocessing) ..................................... 18
5
Bibliography ........................................................................................................................................ 20
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Code_Aster
®
Version
6.3
Titrate:
Pre and postprocessing for the “composite” hulls
Date:
13/09/02
Author (S):
P. MASSIN, F. NAGOT, F. VOLDOIRE, J.M. PROIX
Key
:
R4.01.01-B
Page
:
3/20
Manual of Reference
R4.01 booklet: Composite materials
HT-66/02/004/A
1 Introduction
The modeling of the thermomechanical behavior by a theory of hulls of the structures
composed of laminated composite materials present compared to the homogeneous case isotropic one
certain number of characteristics:
·
coefficients intervening in the relations of linear behavior connecting the sizes
mechanics and thermics defined on the average surface of the hull must be calculated
starting from the space distribution in the thickness of various materials,
·
the materials constitutive of the hull are in general orthotropic:
- it is necessary to define, in each point of the average surface of the hull, a direction
material fixing the reference mark in which the relations of behavior are described,
-
the form of the anisotropy produced on the total behavior of the hull can be
unspecified,
·
finally couplings between sizes characterizing of the symmetrical phenomena and
antisymmetric compared to average surface can appear (coupling
bending-membrane, coupling temperature average average-gradient in the thickness). In
thermo_mecanic the results presented are however theoretically valid only
when the coupling membrane-bending is null,
·
the analysis of the rupture or the damage of these structures requires to return to one
level of description finer than that provided by the models of hulls: the criteria are
formulated, layer by layer in the thickness, according to the stresses
“three-dimensional”.
The preprocessing makes it possible the user “to build” the sizes intervening in the theories of
hulls starting from a simple space description of the distribution of the various materials (position,
thickness, orientation).
Postprocessing intervenes once the structural analysis completed to provide, layer by layer,
an evaluation of some criteria of rupture or damage.
The party taken here is to specify pre and postprocessings so that they are independent, in
the framework of the models of hull selected, the type of element chosen by the user to make the calculation of
structure. Indeed, numerical difficulties of the calculation of the hulls and the representation of their
geometry results in proposing according to the situations, several types of finite elements of hull or of
plate.
The note is divided into three parts. The first briefly points out the assumptions of the theory of
hull used for mechanical thermo calculations and the expressions of the coefficients homogenized with
to introduce. The second specifies the choices retained for the description of the orientation of materials by
report/ratio with the elements like some notations. The last part details the application of these choices
with the case of the hulls made up of homogeneous layers.
To allow the use of the options of calculations available in Code_Aster, it is thus
necessary to define controls the pre one and postprocessing for composite materials
laminates compatible with the existing controls.
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Code_Aster
®
Version
6.3
Titrate:
Pre and postprocessing for the “composite” hulls
Date:
13/09/02
Author (S):
P. MASSIN, F. NAGOT, F. VOLDOIRE, J.M. PROIX
Key
:
R4.01.01-B
Page
:
4/20
Manual of Reference
R4.01 booklet: Composite materials
HT-66/02/004/A
2
Homogenized characteristics of a thin hull in
thermo elasticity and in thermics
2.1
Notations - Assumptions
The hull is made up various layers of orthotropic materials parallel to laid out
surface average
(cf [Figure 2.1-a]).
2h
N
X
X
2
1
I]
, [
= -
H H
Appear 2.1-a
While noting
()
X
co-ordinates
(
)
X X
1
2
,
on
and
X
3
the normal co-ordinate on the surface
]
[
X
H H
3
-
,
,
one can define the various characteristics of materials intervening in Thermics
and in Thermo elasticity. One will suppose moreover than one of the axes of orthotropism coincides with
normal
N
at the point
()
X
with the hull
.
·
Conductivity:
() ()
K
X X
K
X X
,
,
,
3
33
3
·
Voluminal heat:
()
C X X
,
3
·
Expansion factors:
()
D
X X
,
3
·
Elastic rigidity (plane stress):
()
µ
X X
,
3
·
Rigidity of shearing:
()
3 3
3
,
X X
·
Density:
()
X X
,
3
The Greek indices traverse {1, 2}. The system
()
X
necessarily does not correspond to the axes
of orthotropism of materials in the tangent plan.
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Code_Aster
®
Version
6.3
Titrate:
Pre and postprocessing for the “composite” hulls
Date:
13/09/02
Author (S):
P. MASSIN, F. NAGOT, F. VOLDOIRE, J.M. PROIX
Key
:
R4.01.01-B
Page
:
5/20
Manual of Reference
R4.01 booklet: Composite materials
HT-66/02/004/A
2.2 Thermics
One places oneself within the framework of the thermal model of hull describes in [R3.11.01] and [bib1].
A field of temperature “hull” is represented by the three fields
(
)
T T T
m
S
I
,
,
defined on
of
the following way in the thickness
:
()
()
()
()
()
()
()
()
()
T X X
T
X
P X
T
X
P X
T
X
P X
T X
P X
J
J
J
m
S
I
,
3
1
3
3
1
3
2
3
3
3
=
=
+
+
=
éq 2.2-1
where them
P
J
are the polynomials of LAGRANGE
]
[
()
(
)
()
(
)
()
(
)
I
H H
P X
X H
P X
X
H
X H
P X
X
H
X H
= -
= -
=
+
= -
-
,
:
/
/
/
1
3
3
2
2
3
3
3
3
3
3
3
1
2
1
2
1
The interpretation of the fields
T
J
is then the following one:
() ()
T
X
T X
m
=
, 0
(temperature on the average surface of the hull),
() (
)
T
X
T X
H
S
=
+
,
(temperature on the upper surface of the hull),
() (
)
T X
T X
H
I
=
-
,
(temperature on the lower surface of the hull).
Thanks to the representation [éq 2.2-1], one calculates the bilinear form
K
T
of
(
)
T
T T
m
S
I
,
,
T
to leave
form of the 3D problem (indices
ij
take the values
m S I
,
):
()
(
)
K
With
T
B
T
D
T
ji
I
J
ji
I
J
T
,
.
.
.
.
,
,
=
+
(summation on the repeated indices),
where
is a virtual field of temperature and where
()
(
)
() ()
()
(
)
()
()
With
With
With
B
B
With
X
K
X
X
P X
P X
dx
B
X
K
X
X
P
X X
P
X
X
dx
ij
ji
ij
ij
ji
ij
I
J
I
ij
I
J
I

=
=
=
=
=




,
,
,
3
3
3
3
33
3
3
3
3
3
3
éq
2.2-2
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Code_Aster
®
Version
6.3
Titrate:
Pre and postprocessing for the “composite” hulls
Date:
13/09/02
Author (S):
P. MASSIN, F. NAGOT, F. VOLDOIRE, J.M. PROIX
Key
:
R4.01.01-B
Page
:
6/20
Manual of Reference
R4.01 booklet: Composite materials
HT-66/02/004/A
The bilinear form related to voluminal heat in the problem of evolution is written:
()
()
()
() ()
MR. T
T
X
C X X P X P X dx
ij
I
J
ij
I
I
I
,
.
.
,
=
=
C
C
3
3
3
3
éq
2.2-3
2.3 Thermomechanical
One places oneself within the framework of the modeling of hull of LOVE-KIRCHHOFF (hull thin) or
REISSNER-MINDLIN (thick hull). In both cases, the sections are supposed to remain plane.
Deformations of the tangent plan with
thus express themselves, in the thickness, using the tensors of
deformations
()
E
X
, of variation of curvature
()
K
X
and of distortion
()
X
surface
[bib2]:
()
()
()
() ()
X X
E
X
X K
X
X X
X
,
,
3
3
3
3
2
=
+
=
éq
2.3-1
The material undergoing a local deformation of thermal origin given by (
T
ref.
is the temperature
of reference):
()
()
(
)
()
HT
ref.
X X
T X X
T
D
X X
,
,
,
3
3
3
=
-
The local stress field is given by the thermoelastic law in plane stresses:
(
)
µ
µ
µ
=
-
HT
maybe with the preceding model for
T
:
()
() ()
()
()
[
]
()
()
()
()
,
,
,
,
.
,
µ
µ
µ
µ
µ
µ
X X
X X
E
X
X K
X
X X
X X
T
X
P X
T
D
X X
HT
HT
J
J
J
ref.
3
3
3
3
3
1
3
3
3
=
+
-
=
-




=
with
éq
2.3-2
Generalized efforts (bending
M
and membrane
NR
) are related to
by:
()
()
()
()
M
X
X X X dx
NR
X
X X dx
I
I
=
=



,
,
,
,
3
3
3
3
3
éq
2.3-3
so that the law of behavior of the hull is written at the point
X
:
M
P
K
Q
E
M
NR
R
E
Q
K
NR
HT
HT
µ
µ
µ
µ
µ
µ
µ
µ
=
+
+
=
+
+




éq
2.3-4
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Code_Aster
®
Version
6.3
Titrate:
Pre and postprocessing for the “composite” hulls
Date:
13/09/02
Author (S):
P. MASSIN, F. NAGOT, F. VOLDOIRE, J.M. PROIX
Key
:
R4.01.01-B
Page
:
7/20
Manual of Reference
R4.01 booklet: Composite materials
HT-66/02/004/A
where
()
()
()
P
X X dx
Q
X X dx
R
X dx
NR
dx
M
X dx
I
I
I
HT
HT
I
HT
HT
I
µ
µ
µ
µ
µ
µ
µ
µ
µ
µ

= +
= +
= +
= -
= -














3
3
2
3
3
3
3
3
3
3
3
3
éq
2.3-5
When the temperature is calculated by the model of Thermics one can express them directly
“thermal” efforts according to the three “components”
(
)
T
T
T
m
S
I
,
:
() ()
[
]
(
)
(
)
() ()
[
]
(
)
(
)
M
D
X P X X dx
T
T
DM
T
T
NR
D
X P X dx
T
T
DNN
T
T
HT
I
J
J
ref.
J
J
ref.
HT
I
J
J
ref.
J
J
ref.
µ
µ
µ
µ
= -
-
=
-
= -
-
=
-



3
3
3
3
3
3
3
éq 2.3-6
Quantities
DNN
and
DM
depend only on materials constitutive of the hull and of their
distribution.
Note:
When the provision of materials is symmetrical compared to
, certain integrals, being
summon odd terms, cancel themselves:
Q
DM
DM
DNN
µ
,
;
=
=
=
=
0
0
0
1
3
2
.
The sharp efforts and stresses shear transverse are obtained by writing of
local equilibrium equations without voluminal force:
,
J
ij
=
0
where
{}
{}
I J
,
,
1 2 3
what makes it possible to write:
()
()
()
()
V
X
M
X
X X
X
Z dz
H
X
=
= -
-
,
,
,
,
3
3
3
by using the fact that
(
)
(
)
3
3
0
X
H
X
H
,
,
+
=
- =
.
The role of the preprocessing is to calculate the various sizes
With
,
B
,
C
, P, Q, R, DM, DNN, to leave
description of the material (a number, orientation and thickness of the various layers,
local characteristics
C K
D
,
).
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Code_Aster
®
Version
6.3
Titrate:
Pre and postprocessing for the “composite” hulls
Date:
13/09/02
Author (S):
P. MASSIN, F. NAGOT, F. VOLDOIRE, J.M. PROIX
Key
:
R4.01.01-B
Page
:
8/20
Manual of Reference
R4.01 booklet: Composite materials
HT-66/02/004/A
3 Reference marks in the tangent plan with the hull. Notation
matric
3.1 Reference mark
One considers the total reference mark of the structure
(
)
X Y Z
,
: to see figure [Figure 3.1-a]. In the case of them
laminated composites the orientation of full-course is defined compared to a direction of reference
E
ref.
in the tangent plan (
T
).
This vector
E
ref.
is determined by the projection of a vector
X
1
, given by the user under the key word
ANGL_REP
of
AFFE_CARA_ELEM [U4.24.01]
, on the tangent level (
T
) in an unspecified point of
hull.
(
T
)
Tangent plan (
T
)
XYZ
total reference mark
X
1
X
1
X
Y
Z
Z
X
2
1
Y
0
E
ref.
Appear 3.1-a
The vector
X
1
is defined by the user by two directed angles:
1
: enter
0X
and
(
)
X
1 proj X Y
,
2
: enter
(
)
X
1 proj X Y
,
and
X
1
1
: fact of passing from the direction
0X
with projection in the plan
X0Y
vector
X
1
.
2
: fact of passing from this projection to
X
1
itself: to see figure [Figure 3.1-a].
Whenever in a given area of the hull, (
T
) is orthogonal with
X
1
, the user will have
to define another vector (in practice for certain meshs).
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Code_Aster
®
Version
6.3
Titrate:
Pre and postprocessing for the “composite” hulls
Date:
13/09/02
Author (S):
P. MASSIN, F. NAGOT, F. VOLDOIRE, J.M. PROIX
Key
:
R4.01.01-B
Page
:
9/20
Manual of Reference
R4.01 booklet: Composite materials
HT-66/02/004/A
For a finite element of type facets planes, contained in the tangent plan (
T
), the reference mark is defined
orthonormé
(
)
V V
1
2
,
room with the element using the classification of the nodes. For example for
triangle:
(
T
)
N3
N2
N1
V
1
V
2
E
ref.
0
Appear 3.1-b: Identifies local element
(
)
V V
1
2
,
The directed angle
(
)
0
1
,
=
O C
ref.
allows to pass from the local reference mark to the element to the reference mark of
reference.
3.2 Notation
matric
In thermics as into thermomechanical, the programming of the elements requires to express them
operators of elasticity and conduction in the local reference mark of the finite element
(
)
V V
1
2
,
. There is the practice
to simplify the representation of the tensorial sizes as follows.
3.2.1 Thermics
One represents the tensorial sizes in the reference mark
(
)
V V
1
2
,
:
()
{
} () {}
(
)
(
)
With
ij
I J
m S I
,
,
,
,
2
2
1 2
in a vectorial form with 6 vectors by taking account of symmetries [§2.2]:
()
()
With
ij
ij
ij
ij
I
J
H
H
With
With
With
P X
P X
K
K
K
dx
=




=




-
11
22
12
3
3
11
22
12
3
.
.
where
K
=




K
K
K
11
22
12
indicate the thermal vector conductivity built using the tensor
K
K
0
0
0 0
33


(cf [§2.1]),
and of
()
P X
I
3
, polynomials of LAGRANGE in the thickness. One makes in the same way for
B C
ij
ij
,
.
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Code_Aster
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Version
6.3
Titrate:
Pre and postprocessing for the “composite” hulls
Date:
13/09/02
Author (S):
P. MASSIN, F. NAGOT, F. VOLDOIRE, J.M. PROIX
Key
:
R4.01.01-B
Page
:
10/20
Manual of Reference
R4.01 booklet: Composite materials
HT-66/02/004/A
While placing itself in the reference mark of the element
(
)
V V
1
2
,
, the matrix of passage is used
()
P
K
m
tensor
of conductivity
K
K
K
K
=




11
22
12
of
(
)
V V
1
2
,
towards the reference mark associated with
E
ref.
[bib3]:
()
P
K
m
C
S
CS
S
C
CS
CS CS C
S
=
-
-
-




2
2
2
2
2
2
2
2
where
()
()
C
S
=
=
cos
sin
0
0
It results from it that the matrix from passage
P
K
m
()
-
1
tensor of conductivity of the reference mark associated with
E
ref.
towards
(
)
V V
1
2
,
is given by:
()
P
K
m
C
S
CS
S
C
CS
CS
CS C
S
-
=
-
-
-




1
2
2
2
2
2
2
2
2
where
()
()
C
S
=
=
cos
sin
0
0
3.2.2 Thermomechanics
One also represents in a vectorial form in the reference mark
(
)
V V
1
2
,
:
·
on the one hand, normal stresses
11
22
,
, shearing
12
in the plan and it
transverse shearing
13
and
23
:


=




=




11
22
12
13
23
,
·
in addition, corresponding deformations:

=




=


=
11
22
12
13
23
, 12
2
12
12
who break up with the generalized deformations of membrane
E
and of bending
K
:
()
()
()
()
()
()
()
()
()
X
U
X
X
U
X
X
X
X
T
T
HT
HT
X
ref.
3
3
3
3
3
3
3
3
=
-
=
+
=
-




with
E
K
D
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Code_Aster
®
Version
6.3
Titrate:
Pre and postprocessing for the “composite” hulls
Date:
13/09/02
Author (S):
P. MASSIN, F. NAGOT, F. VOLDOIRE, J.M. PROIX
Key
:
R4.01.01-B
Page
:
11/20
Manual of Reference
R4.01 booklet: Composite materials
HT-66/02/004/A
for an ordinate
X
H H
3
-
]
, [
, and:
E
K
D
=




=




=




E
E
E
K
K
K
D
D
D
11
22
11
11
22
11
11
22
11
,
,
where D is the vector associated with the expansion factors thermal.
The vector forced
is obtained using the matrix of rigidity (3 X 3):
()
(
)
=
.
R
U
HT
-
with
R
, opposite of the matrix of flexibility (see in [§4.3]).
While placing itself in the reference mark of the element
(
)
V V
1
2
,
, the matrix of passage is used
()
P
m
tensor
deformations
=




11
22
12
of
(
)
V V
1
2
,
towards the reference mark associated with
E
ref.
[bib3]:
()
P
m
C
S
CS
S
C
CS
CS
CS C
S
=
-
-
-




2
2
2
2
2
2
2
2
where
()
()
C
S
=
=
cos
sin
0
0
While placing itself in the reference mark of the element
(
)
V V
1
2
,
, the matrix of passage is used
()
P
2
m
tensor
deformations
1
2

=


13
23
of
(
)
V V
1
2
,
towards the reference mark associated with
E
ref.
:
()
P
2
m
C
S
S C
= -




where
()
()
C
S
=
=
cos
sin
0
0
In the same way, while placing itself in the reference mark of the element
(
)
V V
1
2
,
, the matrix of passage
()
P
m
tensor
stresses

=




11
22
12
of
(
)
V V
1
2
,
towards the reference mark associated with
E
ref.
is worth:
()
P
m
C
S
CS
S
C
CS
CS CS C
S
=
-
-
-




2
2
2
2
2
2
2
2
where
()
()
C
S
=
=
cos
sin
0
0
It results from it that the form of the matrix of passage of the reference mark associated with
E
ref.
towards the reference mark of
the element
(
)
V V
1
2
,
for the stresses above is such as:
()
()
P
P
P
m
m
T
(m)
-
=
-
=
1
0
(
)
. This
property will be particularly useful in the continuation of the talk.
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Code_Aster
®
Version
6.3
Titrate:
Pre and postprocessing for the “composite” hulls
Date:
13/09/02
Author (S):
P. MASSIN, F. NAGOT, F. VOLDOIRE, J.M. PROIX
Key
:
R4.01.01-B
Page
:
12/20
Manual of Reference
R4.01 booklet: Composite materials
HT-66/02/004/A
4
Hulls made up of homogeneous layers
4.1
Description of the layers
One considers the hull made up of a stacking of
NR
couch
layers (parallel with the tangent plan) in
the thickness
]
, [
-
H H
constituted each one of one of
M
to subdue
orthotropic homogeneous materials (hull
laminated [Figure 4.1-a]).
-
H
+
H
2h
X
3
E
N
Appear 4.1-a
A layer N is defined by:
·
its thickness
E
N
with the ordinates of the interfaces lower and higher:
X
H
E
X
X
E
N
J
N
N
N
J
N
3
1
3
3
1
1
1
-
-
=
-
= - +
=
+
;
;
·
the constitutive material
m
, and its physical characteristics,
·
the angle
N
first direction of orthotropism (noted L) in the tangent plan (
T
) by
report/ratio with the direction of reference
E
ref.
(see figure [Figure 4.1-b]).
Note:
In the case of a layer made up of fibers in a matrix of resin, first direction
of orthotropism corresponds to the direction of fibers.
E
ref.
L
N
0
(T)
V
1
V
2
L
T
Be reproduced 4.1-b: On orthotropic layer
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Code_Aster
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Pre and postprocessing for the “composite” hulls
Date:
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Author (S):
P. MASSIN, F. NAGOT, F. VOLDOIRE, J.M. PROIX
Key
:
R4.01.01-B
Page
:
13/20
Manual of Reference
R4.01 booklet: Composite materials
HT-66/02/004/A
4.2 Thermics
The expression of the vectors
()
{
}
(
)
With
ij
I J
m S I
I
J
,
,
,
2
defined in [§3.2.1] is obtained from
conductivities
K
m
material
m
constituting the layers
N
.
In the cases of orthotropism
(
)
L T
,
material
m
, the coefficients of conductivity are:
()
K
L T
L
T
K
K
,
=




0
In the case of a transverse isotropic material the coefficient
K
33
is equal to
K
T
.
To have the expression of
With
ij
in the reference mark of the element
(
)
V V
1
2
,
one must apply rotation
following, of the reference mark of orthotropism towards the reference mark of the element, as clarified with [§3]:
()
(
)
K
m
L
T
L T
K
K
K
C
S
S
C
CS
CS
K
K
=




=
-






11
22
12
2
2
2
2
,
with
(
)
(
)
C
S
I
I
cos
sin
=
+
=
+
0
0
Vectors
With
ij
can then express itself by integration in the thickness of the contributions of
sleep:
() ()
()
=
-
=
couch
N
N
NR
N
X
X
m
J
I
ij
dx
X
P
X
P
1
3
3
3
3
1
3
.
.
.
.
K
With
éq
4.2-1
Terms
() {}
(
)
J
I
J
I
ij
,
3
,
2
,
2
B
are:
()
()
()
=
-
=
couch
N
N
NR
N
X
X
m
J
I
ij
dx
K
X
X
P
X
X
P
1
3
33
3
3
3
3
3
1
3
.
.
.
B
In the same way for
C
ij
:
() ()
()
=
-
=
couch
N
N
NR
N
X
X
m
J
I
ij
dx
C
X
P
X
P
1
3
3
3
3
1
3
.
.
.
C
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Code_Aster
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Titrate:
Pre and postprocessing for the “composite” hulls
Date:
13/09/02
Author (S):
P. MASSIN, F. NAGOT, F. VOLDOIRE, J.M. PROIX
Key
:
R4.01.01-B
Page
:
14/20
Manual of Reference
R4.01 booklet: Composite materials
HT-66/02/004/A
4.3 Thermomechanics
4.3.1 Relation of behavior
In the case of the laminated hulls, it is shown that the relation between the deformations
and the stress
in the layer “
N
“depends on the constants of orthotropic material”
m
“:
That is to say:
() () () () () ()
() ()


m
T
T
ml
L
mZ
T
mZ
L
m
T
L
m
T
L
m
T
T
ml
L
D
D
G
G
G
E
E
,
,
,
,
,
,
elastic coefficients
expansion factors
In the axes of orthotropism (
L, T
) of material
m
, the matrix of flexibility
S
express yourself by:
() ()
()
m
T
L
T
T
T
T
TL
T
T
LT
L
L
T
L
m
G
E
E
E
E














-
-
=
1
0
0
0
1
0
1
,
S
with
T
T
TL
L
LT
E
E
=
Rigidity
()
()
1
-
=
m
m
S
being:
() ()
()
m
T
L
T
L
L
T
T
T
T
L
L
T
T
T
LT
T
L
L
T
L
L
TL
T
L
L
T
L
L
T
L
m
G
E
E
E
E
















-
-
-
-
=
0
0
0
.
1
.
1
.
0
.
1
.
.
1
,
Rigidity in transverse shearing is expressed for its part in the following way:
() ()
()
m
Z
T
Z
L
T
L
m
G
G


=
0
0
,
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Code_Aster
®
Version
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Titrate:
Pre and postprocessing for the “composite” hulls
Date:
13/09/02
Author (S):
P. MASSIN, F. NAGOT, F. VOLDOIRE, J.M. PROIX
Key
:
R4.01.01-B
Page
:
15/20
Manual of Reference
R4.01 booklet: Composite materials
HT-66/02/004/A
While placing itself in the reference mark of the element
(
)
V V
1
2
,
, the matrix of passage is used
()
P
m
tensor
deformations defined in [§3] of
(
)
V V
1
2
,
towards the reference mark of orthotropism:
()
P
m
C
S
CS
S
C
CS
CS CS C
S
=
-
-
-




2
2
2
2
2
2
2
2
where
(
)
(
)
C
S
I
I
=
+
=
+
cos
sin
0
0
In the same way the vector dilation is expressed in the reference mark
(
)
V V
1
2
,
:
()
(
)
(
)
D
P
m
m
L
TT
L T
L L
T T
L T
D
D
D
D
D
C
S
S
C
CS
CS
D
D
=




=




=
-








-
11
22
12
1
2
2
2
2
0
2
2
,
,
One thus has in the layer
N
(material:
m
), in
X
3
:
()
()
()
()
()
(
)
()
()
()
()
(
)
()
()
(
)
N
L T
m
HT
T
m
L T
m
HT
m
HT
m
U
U
U
=
-
=
-
=
-
-
P
P
P
P
1
.
.
.
.
.
.
,
,
with:
()
()
(
)
U
E
E
E
X
K
K
K
D
D
D
T X
T
HT
ref.
=




+




=




-
11
22
12
3
11
22
12
11
22
12
3
and
.
Note:
In the code, one chose to carry out the passage of the reference mark of orthotropism to the reference mark of the element in
two stages. A first stage relates to the passage of the reference mark of orthotropism to the definite reference mark
by
ANGL_REP
. Data of
DEFI_MATERIAU
are thus transformed at the time of this first
passage. One treats then equivalent material as one would do it with elements of
conventional plates.
The processing of thermal dilation is made in the form of a contribution to the second
member of the matric equation to solve resulting from the principle of virtual work. This contribution
is written:
()
()
()
HT N
T
m
L T
L
TT
D
T
D
T
= -






P.
.
,

0
.
4.3.2 Shearing
transverse
Rigidity in transverse shearing of each layer is written in the reference mark
(
)
V V
1
2
,
the same one
way that dilation:
()
(
)
()
()
()
m
T
m
m
m
V V
P
P
1
2
2
2
,
.
.
=
with
()
P
2
m
C
S
S C
= -




stamp passage vectorial of
(
)
V V
1
2
,
towards the reference mark of orthotropism.
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Code_Aster
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Titrate:
Pre and postprocessing for the “composite” hulls
Date:
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Author (S):
P. MASSIN, F. NAGOT, F. VOLDOIRE, J.M. PROIX
Key
:
R4.01.01-B
Page
:
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Manual of Reference
R4.01 booklet: Composite materials
HT-66/02/004/A
Rigidity in transverse shearing total of the hull
[]
R
C
calculated so as to be equal to
that given by the law of three-dimensional elasticity [bib2], the matrix
[]
R
C
is defined so that
surface density of transverse energy of shearing
U
2
obtained for a distribution
three-dimensional of the stresses
13
and
23
that is to say identical to that associated the model of plate of
Noted REISSNER-MINDLIN
U
2
.
()
[]
{}
[]
{}
[]
{}
U
D
U
D
D
m
H
H
C
H
H
C
H
H
1
1
3
13
23
2
1
3
1
3
1
2
1
2
1
2
=
=
=
=








-
-
-
-
-
-
V R
V
H
with the equilibrium equations:
(
)
(
)
13
11 1
12 2
3
23
12 1
22 2
3
3
3
= -
+
= -
+



-
-
,
,
,
,
D
D
H
X
H
X
and conditions:
0
13
23
=
=
for
X
H
3
= ±
.
Plane stresses
11
22
12
,
,
express themselves according to the resulting efforts while making
the assumption of pure bending and absence of coupling membrane/bending. It results from it that:
()
X
X
X
m
3
3
3
1
=
-
.
()
.
()
P
M
and
()
With
P
X
X
m
3
3
1
=
-
()
()
where
P
is the matrix of rigidity of bending of the whole of multi-layer defined by [éq 2.3-5].
These calculations, as well as the following are to be carried out in a single reference mark. One chooses in
Code_Aster the intrinsic reference mark with the element. It is thus necessary to transform matrix A in this reference mark.
One has then:
()
{}
()
()
{}
X
X
X
3
1
3
2
3
=
+
D
V D
with
V
=
+
+
=
-
-
M
M
M
M
M
M
M
M
M
M
11 1
12 2
12 1
22 2
11 1
12 2
12 1
22 2
22 1
11 2
,
,
,
,
,
,
,
,
,
,
;
;
;
;
and
D
D
1
11
33
13
32
31
23
22
33
2
11
33
13
32
12
31
31
23
33
22
32
21
2
2
2
2
2
2
3
3
=
-
+
+
+
+




=
-
-
-
-
-




-
-
Z
With
With
With
With
With
With
With
With
dz
Z
With
With
With
With
With
With
With
With
With
With
With
With
dz
X
H
X
H
U
1
is thus written:
U
T
1
11
12
12
22
1
2
=




V
C
C
C
C
V
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Code_Aster
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Titrate:
Pre and postprocessing for the “composite” hulls
Date:
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Author (S):
P. MASSIN, F. NAGOT, F. VOLDOIRE, J.M. PROIX
Key
:
R4.01.01-B
Page
:
17/20
Manual of Reference
R4.01 booklet: Composite materials
HT-66/02/004/A

with
()
()
()
3
2
1
2
4
4 22
3
2
1
1
4
2 12
3
1
1
1
2
2 11
D
D
D
H
H
m
T
H
H
m
T
H
H
m
T
D
D
C
D
D
C
D
D
C
-
-
×
-
-
×
-
-
×
=
=
=
from where
{}
,
0
22
12
12
1
11
2
1
V
V
C
C
C
H
C
V
=








-
=
-
T C
U
U
the solution is thus proposed
H
C
C
=
-
11
1
.
The coefficients of transverse correction of shearing correspond to the report/ratio of the terms of
C
H
with the integral on the thickness of the laminate of the terms of
()
m
.
4.3.3 Efforts
generalized
The efforts generalized defined in [§1.3] and put in a vectorial form are obtained by integration
in the thickness of the hull by summoning the contributions of the layers (thickness
1
3
3
-
-
=
N
N
N
X
X
E
):
()
()
=
=
-
-
=
=




=
=
=




=
L
NR
N
X
X
N
L
NR
N
X
X
N
couch
N
N
couch
N
N
dx
dx
NR
NR
NR
dx
X
dx
X
M
M
M
1
3
3
12
22
11
1
3
3
3
3
12
22
11
3
1
3
3
1
3
.
.
.
.
.
.
NR
M
If one expresses like previously (with
m
material of the layer
N
):
()
()
()
()
(
)
(
)
ref.
T
X
T
X
m
m
N
-
-
+
=
3
3
.
.
D
K
E
one can note the efforts generalized in the form: (cf [§1.3])
E
R
K
Q
NR
NR
E
Q
K
P
M
M
.
.
.
.
+
=
-
+
=
-
HT
HT
with
P Q R
,
,
matrices 3 X 3 being expressed by:
()
()
()
()
()
()
()
()
()
(
)
()
=
=
-
=
-
=
=
-
=
=
-
=


-
=
=


-
=
=
-
-
couch
couch
couch
couch
N
N
couch
couch
N
N
NR
N
N
m
NR
N
N
N
m
NR
N
N
N
m
NR
N
X
X
m
NR
N
N
N
m
NR
N
X
X
m
E
X
X
X
X
dx
X
X
X
dx
X
1
1
1
3
3
1
2
1
3
2
3
1
3
3
1
3
1
3
3
3
1
3
2
3
.
.
.
2
1
.
.
.
3
1
.
.
3
1
3
3
1
3
R
Q
P
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Code_Aster
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Titrate:
Pre and postprocessing for the “composite” hulls
Date:
13/09/02
Author (S):
P. MASSIN, F. NAGOT, F. VOLDOIRE, J.M. PROIX
Key
:
R4.01.01-B
Page
:
18/20
Manual of Reference
R4.01 booklet: Composite materials
HT-66/02/004/A
sharp effort
V
is obtained by derivation of the moment [§4.3.2].
The generalized efforts of thermal origin are calculated directly:
()
()
(
)
()
()
()
(
)
()
M
D
NR
D
HT
m
ref.
m
X
X
N
NR
HT
m
ref.
m
X
X
N
NR
X
T X
T
dx
T X
T
dx
N
N
couch
N
N
couch
=
-
=
-
-
-
=
=
.
.
.
.
.
.
.
3
3
3
1
3
3
1
3
1
3
3
1
3
4.3.4 Localization of the stresses (postprocessing)
Conversely, following a calculation by finite element and of obtaining the deformations
E
and variations
of curvature
K
, one can then calculate the stress field
()
(
)
N
couch
N
NR
=
1,
in each
lay down element.
It is necessary to calculate in each layer
()
N
, the matrix
()
m
and terms
()
(
)
()
T X
T
ref.
m
3
-
.
D
(cf [§3.2]) (
m
chechmate
N
=
represent the characteristics material of the layer
N).
Stresses
with an ordinate
]
[
X
X
X
N
N
3
3
1
3
-
,
in the layer (
N
) are then:
()
()
()
()
()
(
)
[
]
N
m
m
ref.
X
X
T X
T
3
3
3
=
+
-
-
.
.
E
K
D
and transverse shearing:
()
()
()
()
N
X
D X
D X
3
1
3
2
3
=
+
.
.
V
éq
4.3.4-1
Note:
In the code postprocessings of the elements of plates are generally defined in
identify associated with
ANGL_REP
. The stresses in the intrinsic reference mark of the element are thus
brought back in the reference mark of the variety. One a:


11
22
12
2
2
2
2
2
2
11
22
12
2
2




=
+
-
-
+
-








eref
N
C
S
CS
S
C
CS
CS
CS C
S
where
()
()
[]
(
)
C
S
=
=
cos
sin
0
0
cf § 4.1
where
0
is the angle enters
V
1
and
E
ref.

4.3.5 Calculation of the criteria of rupture in the layers (postprocessing)
The limiting values of breaking stresses depend on material of the layer, the direction and of
feel stress (for a group of elements corresponding to the same field material):
era)
1
with
E
orthogonal
direction
(2nd
era)
1
with
E
orthogonal
direction
(2nd
fibers)
feel
:
E
orthotropi
direction
(1ère
fibers)
feel
:
E
orthotropi
direction
(1ère
LT
feel
in
NT
cisailleme
in
limit
:
T
feel
in
N
compressio
in
limit
:
T
feel
in
traction
in
limit
:
L
feel
in
N
compressio
in
limit
:
L
feel
in
traction
in
limit
:
chechmate
S
Y
Y
X
X
N
background image
Code_Aster
®
Version
6.3
Titrate:
Pre and postprocessing for the “composite” hulls
Date:
13/09/02
Author (S):
P. MASSIN, F. NAGOT, F. VOLDOIRE, J.M. PROIX
Key
:
R4.01.01-B
Page
:
19/20
Manual of Reference
R4.01 booklet: Composite materials
HT-66/02/004/A
It is necessary to calculate the stresses in the reference mark of the layer (defined by the axes
of orthotropism) starting from the stresses in the reference mark of the element:
the angle enters
V
1
and
E
ref.
is
0
, and that enters
E
ref.
and identifies it orthotropism is
N
:


L
T
L T
N
N
C
S
CS
S
C
CS
CS
CS C
S




=
+
-
-
+
-








2
2
2
2
2
2
11
22
12
2
2
where
(
)
(
)
[]
(
)
C
S
N
N
=
+
=
+
cos
sin
0
0
cf § 4.1
Maximum criterion of stress:
The 5 following criteria are calculated by layer: (
N
NR couch
=
-
1,
)
()
(
)
()
(
)
()
(
)
()
(
)
()
(
)
()
(
)
()
(
)
()
(
)
()
(
)
N
N
N
N
N
N
N
N
N
N
N
N
N
N
chechmate
T
L
T
chechmate
T
T
chechmate
T
L
chechmate
L
L
chechmate
L
S
Y
Y
X
X
0
0
0
0
<
>
<
>
if
if
if
if
Criterion of TSAI-HILL:
This criterion is written in each layer in the following way:
()
(
)
()
()
(
)
()
(
)
()
(
)
2
2
2
2
2
.
2
2
N
N
N
N
chechmate
N
LT
chechmate
N
T
chechmate
N
T
N
L
chechmate
N
L
TH
S
Y
X
X
C
+
+
-
=
The material is broken when
C
TH
1
.
Values
X
and
Y
are replaced by
X
and
Y
when stresses
()
()
(
)
L N
T N
,
corresponding are negative.
background image
Code_Aster
®
Version
6.3
Titrate:
Pre and postprocessing for the “composite” hulls
Date:
13/09/02
Author (S):
P. MASSIN, F. NAGOT, F. VOLDOIRE, J.M. PROIX
Key
:
R4.01.01-B
Page
:
20/20
Manual of Reference
R4.01 booklet: Composite materials
HT-66/02/004/A
5 Bibliography
[1]
S. ANDRIEUX, F. VOLDOIRE: “Formulation of a model of thermics for the hulls
thin " - Note HI-71/7131 - 1990, to also see: [R3.11.01].
[2]
J.L. BATOZ, G. DHATT: “Modeling of the structures by finite elements” - Flight 2 Beams and
plates - HERMES 1990.
[3]
J.R. TO BORE: “Elasticity”. Kluwer academic publishers.