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

Organization (S): EDF/AMA, SEPTEN

Handbook of Référence
R4.01 booklet: Composite materials
R4.01.01 document

Pre and Post-processing 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-inflection is null.
Handbook of Référence
R4.01 booklet: Composite materials
HT-66/02/004/A

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

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 marks 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 constraints (postprocessing) ..................................................................... 18
4.3.5 Calculation of the criteria of rupture in the layers (postprocessing) ..................................... 18
5 Bibliography ........................................................................................................................................ 20
Handbook of Référence
R4.01 booklet: Composite materials
HT-66/02/004/A

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

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
inflection-membrane, coupling temperature average average-gradient in the thickness). In
thermo_mecanic the results presented are however theoretically valid only
when the coupling membrane-inflection 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 constraints
“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 commands the pre one and postprocessing for composite materials
laminates compatible with the existing commands.
Handbook of Référence
R4.01 booklet: Composite materials
HT-66/02/004/A

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

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]).

N
2h
x2
X

1
I =] - H, [
H

Appear 2.1-a

By noting (X) the co-ordinates (X, X out of and X
1
2)
3 the normal co-ordinate on the surface
X] - H, H, one can define the various characteristics of materials intervening in Thermique
3
[
and in Thermo- elasticity. One will suppose moreover than one of the axes of orthotropism coincides with
normal N at the item (X) with the hull.

· Conductivity: K
(X, X

3), k33 (X, X



3)
· Voluminal heat: C (X, X
3)
· Dilation coefficients: D
(X X

,

3)
· Elastic rigidity (plane constraint): µ (X, X
3)
· Rigidity of shearing:
X, X
3 3 (
3)
· 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.
Handbook of Référence
R4.01 booklet: Composite materials
HT-66/02/004/A

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Version
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Titrate:
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Author (S):
P. MASSIN, F. NAGOT, F. VOLDOIRE, J.M. PROIX Key
:
R4.01.01-B Page
: 5/20

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 (Tm T S Ti
,
,) definite on
the following way in the thickness:

3
T (X, X =
=
+
+
éq 2.2-1
3)
T J (X) P
m
S
I


J (x3)
T (X) P1 (x3) T (X) P2 (x3) T (X) P3 (x3)
J = 1


2
where the P are the polynomials of LAGRANGE
I =] - H, [
H:
P
1
1 (x3) =
- (X/H
3
)
J
X
P
3
1

2 (x3) =
(+ X/H
3
)
2 H
X
P
3
1
3 (x3) = -
(- X/H
3
)
2 H
The interpretation of the fields T J is then the following one:

T m (X) = T (X,)
0
(temperature on the average surface of the hull),


T S (X) = T (X, +h

)
(temperature on the upper surface of the hull),

Ti (X) = T (X, - H

)
(temperature on the lower surface of the hull).

Thanks to the representation [éq 2.2-1], one calculates bilinear form KT
m
S
I

of (T, T, T)
T to be left
form of the 3D problem (the indices ij take the values m, S
, I):

K T (T,) = (A ji .Ti
J
.

+ B ji. Ti J

.

,

(summation on the repeated indices),

,
) D

where is a virtual field of temperature and where


Aij = A ji = Aij, Bij = B ji




Aij (X) = K

,

éq
2.2-2
I (X
X

3) Pi (x3) Pj (x3) dx3


P
P
Bij

(X) = K

,
I 33 (X
X

3)
I (x3) J (x3) dx

X
3
3
x3
Handbook of Référence
R4.01 booklet: Composite materials
HT-66/02/004/A

Code_Aster ®
Version
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Titrate:
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Date:
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Author (S):
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:
R4.01.01-B Page
: 6/20

The bilinear form related to voluminal heat in the problem of evolution is written:

M (T,)
ij
= C. Ti J
.




éq
2.2-3
ij
C (X) = C
,
I
(X X

3) Pi (x3) pi (x3) dx3

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.

The deformations of the tangent plan with are thus expressed, in the thickness, using the tensors of
deformations E
(X


), of variation of curvature K
(X

) and of distortion (
X) of surface
[bib2]:



X
(
X, X =
+
=
éq
2.3-1
3)

E (X) x3
K (X) 3 (X, x3) ()


2

The material undergoing a local deformation of thermal origin given by (T ref. is the temperature
of reference):

HT
ref.
(
X, X =
-

3)
(T (X, x3) T) D (X, x3)

The local stress field is given by the thermoelastic law in plane constraints:

=
HT
µ (µ - µ)

maybe with the preceding model for T:


HT
(X,

X =
+
-
3)
µ (X,

x3) [ (X) X K
3
µ (
X) µ (X, x3)]
3


éq
2.3-2
with HT
J
ref.


µ (X,

X =
-
3)
T (X). Pj (x3) T
(X, x3)
j=1


The generalized efforts (inflection M and membrane NR) are related to by:

M

(X

=
)
,
,
I
(X X

3) X dx

3
3
éq
2.3-3
NR


(X

=
)
,
,
I
(X X

3) dx3

so that the law of behavior of the hull is written as in point X:


M = Pµ K + Qµ E + M
µ
µ
HT

éq
2.3-4
NR
= Rµ E + Qµ K + NR


µ
µ
HT
Handbook of Référence
R4.01 booklet: Composite materials
HT-66/02/004/A

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Author (S):
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:
R4.01.01-B Page
: 7/20

where

µ
= +
(X 2
3) X
dx
I

3
3

µ

= +
(x3) X dx
I

3
3


µ
= +
(x3) dx éq
2.3-5
I
3
NR
µ
HT

= -
dx
HT
I
µ
3
M
µ
HT

= -
X dx
HT

I
µ
3
3

When the temperature is calculated by the model of Thermique one can express them directly
efforts “Thermiques” according to the three “components” (Tm T S Ti
,):

M
J
ref.

J
ref.
= -
3
3
3
3
-
=
-
HT

[
D
µ
µ (X) P (X) X dx
I
J
] (T T) DMj (T T)

éq 2.3-6
NR
J
ref.

J
ref.
= -
3
3
3
-
=
-
HT

[
D
µ
µ (X) P (X) dx
I
J
] (T T) DNj (T T)

The 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:
µ



Q
=,
0 DM
= DM = 0; DNN = 0.
1
3
2

The sharp efforts and stresses shear transverse are obtained by writing of
local equilibrium equations without voluminal force:

ij = 0 where {I, J} {, 12,}
3
, J

what makes it possible to write:

V (X

) = M, (X)

3 (
X
X, X
3



3
= -

)
,
,
-
(X Z

) dz
H


by using the fact that 3 (
3

X, + H) = (X, - H) = 0.
The role of the preprocessing is to calculate various sizes A, 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
).
Handbook of Référence
R4.01 booklet: Composite materials
HT-66/02/004/A

Code_Aster ®
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:
R4.01.01-B Page
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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 in the tangent plan (
ref.
T).

This vector E
is determined by the projection of a vector X, given by the user under the key word
ref.
1
ANGL_REP of AFFE_CARA_ELEM [U4.24.01], on the tangent level (T) in an unspecified point of
hull.

Z
X1
X1
Z
2 Y
0
1
(T)
X
eréf
Y
Tangent plan (T)
XYZ locates total
X

Appear 3.1-a

Vector X is defined by the user by two directed angles:
1

: between 0X and X

1
1 proj (X, Y)
: between X
and X
2
1 proj (X, Y)
1

: fact of passing from the direction 0X to projection in plan X0Y of vector X.
1
1
: fact of passing from this projection to X itself: to see figure [Figure 3.1-a].
2
1

Whenever in a given zone of the hull, (T) is orthogonal with X, the user will have
1
to define another vector (in practice for certain meshs).
Handbook of Référence
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:
R4.01.01-B Page
: 9/20

For a finite element of type facets planes, contained in the tangent plan (T), one defines the reference mark
orthonormé (V, V room with the element using the classification of the nodes. For example for
1
2)
triangle:

N3
N2
V1
V2
0
E ref.
(T)
N1

Appear 3.1-b: local Repère of the element (V, V
1
2)

The directed angle
= V,
E
allows to pass from the local reference mark to the element to the reference mark of
0
(1 ref.)
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. There is the practice
1
2)
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)

(
2
2
Aij
I, J m, S
, I
,

() {
} () {, 1} 2)

in a vectorial form with 6 vectors by taking account of symmetries [§2.2]:

ij
With
K
11

11
H


Aij
ij
= A
22
=
I
P (x3). Pj (x3). k22 dx

3
ij
- H



A12
K


12

K
11

0
K

where K = K
indicate the thermal vector conductivity built using the tensor

22
0



K
0 0 K
12

33
(cf [§2.1]),

and of P (X, polynomials of LAGRANGE in the thickness. One makes in the same way for Bij Cij
,
.
I
3)
Handbook of Référence
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:
R4.01.01-B Page
: 10/20

While being placed in the reference mark of the element (V, V, one uses the matrix of passage (m)
P of the tensor
1
2)
K
K
11
of conductivity K = K
of (V, V towards the reference mark associated with E [bib3]:
1
2)
22
ref.



k12

C2
S2
2CS

(
2
2

m)
C = cos
P
= S
C
- 2CS
where
(0)

K



S = sin (0)

- CS CS C2 - S2
It results from it that the matrix from passage P (m) - 1 of the tensor of conductivity of the reference mark associated with E

K
ref.
towards (V, V is given by:
1
2)

C2
S2
- 2CS

(
1

C = cos (0)
2
2

m)
P - = S
C
2CS
where

K





S = sin (0)
CS

- CS C2 - S2

3.2.2 Thermomechanics

One also represents in a vectorial form in the reference mark (V, V:
1
2)

· on the one hand, normal constraints,
, shearing in the plan and it
11 22
12
transverse shearing and:
13
23


11
13
=,

22
=



23
12

· in addition, corresponding deformations:


11
1
13
=,
2

22
=
12 =


2


12
23
12

who break up with the generalized deformations of membrane E and inflection K:

(X =

-
3)
(U) (x3)
HT (x3)
with (U) (X = E + K

3)
x3
HT (X = D

-
3)
(x3)
ref.



(
T X


3)
T
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:
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for an ordinate X] - H, H [
3
, and:

E
K
D
11
11
11
E = E
,
,

22
K = K22
D = d22

E11

K11

d11

where D is the vector associated with the dilation coefficients thermal.
The vector forced is obtained using the matrix of rigidity (3 X 3):
=

R ((
. U)
HT
-
) with R, matrix of flexibility reverses (see in [§4.3]).
While being placed in the reference mark of the element (V, V, one uses the matrix of passage ()
P m of the tensor
1
2)

11
deformations =
of (V, V towards the reference mark associated with E [bib3]:
1
2)
22
ref.

12

C2
S2
CS

()

C = cos (0)
2
2

P m =
S
C
- CS
where




S = sin (0)

- CS
CS C2 - S2
2
2



While being placed in the reference mark of the element (V, V, one uses the matrix of passage (m)
P of the tensor
1
2)
2
1
13
deformations
= of (V, V towards the reference mark associated with E:
1
2)
2

ref.
23

C
S
C = cos (0)
(

m)
P
=
where
2




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


11
constraints =
of (V, V towards the reference mark associated with E is worth:
1
2)
22
ref.

12

C2
S2
2CS

(

C = cos (0)
2
2

m)
P
where

= S
C
- 2CS




S = sin (0)

- CS CS C2 - S2

It results from it that the form of the matrix of passage of the reference mark associated with E
towards the reference mark of
ref.
- 1
the element (V, V for the constraints above is such as: (m)
(m)
P
= P (
T
-)
. This
0
= P (m)
1
2)


property will be particularly useful in the continuation of the talk.
Handbook of Référence
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:
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4
Hulls made up of homogeneous layers

4.1
Description of the layers

One considers the hull made up of a stacking of NR
layers (parallel with the tangent plan) in
couch
the thickness] - H, H [made up each one of one of the M
orthotropic homogeneous materials (hull
to subdue
laminated [Figure 4.1-a]).

x3
+ H
2h
E
- H
N
Appear 4.1-a

A layer N is defined by:

· its thickness E with the ordinates of the interfaces lower and higher:
N
N 1
-
xn-1 = - H + E; xn = xn-1 + E;
3
J
3
3
N
J =1
· the constitutive material m, and its physical characteristics,
· the angle of the first direction of orthotropism (noted L) in the tangent plan (
N
T) by
report/ratio with the direction of reference E
(see figure [Figure 4.1-b]).
ref.

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.

L
V1


N
0
T
L
V
E
2
ref.
(T)

Appear 4.1-b: Sur sleep orthotropic
Handbook of Référence
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:
R4.01.01-B Page
: 13/20

4.2 Thermics

2
The expression of the vectors Aij (I, J) {m, S, I}, I J) defined in [§3.2.1] is obtained from

conductivities km of the material m constituting layers N.

In the cases of orthotropism (L, T) of the material m, the coefficients of conductivity are:

K
L

K (
= K
L, T)
T

0

In the case of a transverse isotropic material the coefficient K is equal to K.
33
T

To have the expression of Aij in the reference mark of the element (V, V one must apply rotation
1
2)
following, of the reference mark of orthotropism towards the reference mark of the element, as clarified with [§3]:

K C2
S2
with
C
=
(
cos

+

I
0)
11

K
()

K m
L
= K
2
2

S
=
(
sin

+

I
0)
22 = S
C
K




T (L, T)
K12 CS - CS

The Aij vectors can then be expressed by integration in the thickness of the contributions of
sleep:

Ncouch
N
ij
WITH = X3 P
.
X P
.
X .k
dx
.
éq
4.2-1
n-1
I (3) J (3) ()
X
m
3
n=1
3

The ij terms ((I, J) {
,
2}
3 2
B
, I J) are:

Ncouch
N
X


3
P X
P X
ij
I (3)
J (3)
B =
.
.k
dx
.

n-1
(
33
)
X
m
3
X
X
3
n=1
3
3

In the same way for Cij:

Ncouch
N
ij
C = x3 P X .P X .C dx
.

n-1
I (3)
J (3)
()
X
m
3
n=1
3
Handbook of Référence
R4.01 booklet: Composite materials
HT-66/02/004/A

Code_Aster ®
Version
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Titrate:
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Date:
13/09/02
Author (S):
P. MASSIN, F. NAGOT, F. VOLDOIRE, J.M. PROIX Key
:
R4.01.01-B Page
: 14/20

4.3 Thermomechanics

4.3.1 Relation of behavior

In the case of the laminated hulls, it is shown that the relation between the strains and the stress
in layer “N” depends on the constants of orthotropic material” m “:

That is to say:

(m) (m) (m) (m) (m) (m)

ELL,

T
E T, T
L, G T
L, GLZ, T
G Z
elastic coefficients


(

m)
(m)

dilation coefficients
DLL, dTT


In the axes of orthotropism (L, T) of the material m, the matrix of flexibility S is expressed by:

1
LT


-
0
E
L
T
E T

TL
1

S (m) (L T,) =
0


T
E T
T
E T


1
0
0

G

T
L


(m)

with

TL
LT =

ELL
ETT

Rigidity (m)
1
-
=S (being:
m)


E

L
TL .ELL

0
-
1
.
-
1
.

T L

T
L
T L
T
L


LT. ETT
ETT

(m) (L T,) =
0
-
1 H T.
-

T
L
1 H T. T
L


0
0
G

T
L





(m)

Rigidity in transverse shearing is expressed for its part in the following way:

G
LZ
0
(m) (L T,) =


0
T
G Z (m)
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:
R4.01.01-B Page
: 15/20

While being placed in the reference mark of the element (V, V, one uses the matrix of passage ()
P m of the tensor
1
2)
deformations defined in [§3] of (V, V towards the reference mark of orthotropism:
1
2)

C2
S2
2CS

()

C = cos (+
I
0)
2
2

P m =
S
C
- 2CS
where




S = sin (+
I
0)

- CS CS C2 - S2

In the same way the vector dilation is expressed in the reference mark (V, V:
1
2)

D
D
C2
S2
11
L





D
(m)
D
= D
1
2
2

22 = Pm-
L L
D
= S
C


TT
D






T T (L, T)
D
0
2
2
12

(
CS - CS
L, T)

One thus has in layer N (material: m), in X:
3

(m) - 1
(m)

HT
T
m
m
HT
HT
= P
.
. P
.
U -
= P.
. P
.
U -
=
U -

N

L, T
(()) ()
()
L, T
(())
m (()
)
()
()
()
()

with:

E
K
D
11
11
11
(U) E
and
.

22 + x3 K22
HT = d22 (T (x3) - Tréf
=
)







E12
K12
d12

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. The 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
traditional 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
D

T

L


is written:
T
(m)
HT (N) = - P. (.d T
.
L, T)
TT


0

4.3.2 Shearing
transverse

Rigidity in transverse shearing of each layer is written in the reference mark (V, V of same
1
2)
way that dilation:

T
(m)
(m)
(
= P.
. P

m) (V, V
2
2
1
2)


(m)
C
S
with (m)
P
=
V, V towards the reference mark of orthotropism.
2

stamp passage vectorial from (1
2)
- C M
Handbook of Référence
R4.01 booklet: Composite materials
HT-66/02/004/A

Code_Aster ®
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Titrate:
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Date:
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Author (S):
P. MASSIN, F. NAGOT, F. VOLDOIRE, J.M. PROIX Key
:
R4.01.01-B Page
: 16/20

Rigidity in transverse shearing total of the hull [R calculated so as to be equal to
C]
that given by the law of three-dimensional elasticity [bib2], the matrix [R is defined so that
C]
surface density of transverse energy of shearing U2 obtained for a distribution
three-dimensional of the constraints and is identical to that associated the model of plate of
13
23
Noted REISSNER-MINDLIN U.
2

1 H
- 1
U =


=
1
2 -
[m
H
] {}
()
d3
13
23

1
- 1
1
H
- 1
H

U =
V R
V =

H

2
[C]
{} D



- H
3
[C]
{} D



- H
3
2
2




x3

= -
D
,
+
13

- H (11 1
12,2) 3
with the equilibrium equations:
X

3
= -
D
,
+
23


- H (12 1
22,2) 3

and conditions: 0 = = for X = ± h.
13
23
3

Plane constraints,
,
express themselves according to the resulting efforts while making
11
22 12
the assumption of pure inflection and absence of coupling membrane/inflection. It results from it that:

(X
1
=
-
. () P.M and A (X
1
=
-
()

3)
X P
3)
X
X
3
(m)
3
(m)
3

where P is the matrix of rigidity of inflection 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)} = D (X) V + D (X

3
1
3
2
3) {}

with V = M
;
11,1 + M
M
12,2
12,1 + M22,2


= M
;
;
;
11,1 - M
M
12,2
12,1 - M
M
M
22,2
22,1
11,2

H
Z A + A
WITH + A
11
33
13
32
and D = -
dz
1


- x3
2 A + A
WITH + A
31
23
22
33


H
Z A - With
WITH - WITH
2 A
2 A
D
11
33
13
32
12
31
= -
dz
2


- X
WITH - WITH
With -
3
2
With
2 A
2 A
31
23
33
22
32
21

1
C
C V
U
11
12
is thus written: U =
V

1
1

T
2

C
C

12
22

Handbook of Référence
R4.01 booklet: Composite materials
HT-66/02/004/A

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
: 17/20


1
H
-
with
T
11
C = 1
D ()
1
D d3
-
2 2
H
m
×
1
H
-
T
12
C =
1
D () D 2D3
-
2 4
H
m
×
1
H
-
T
C22 =
D
2 () D 2d3
-
4 4
H
m
×

1
C

- H -
C
C
V
from where U1 =U 2 V
11
12

=0V, {}
C T
C


12
22



one thus proposes the solution H = C - 1.
C
11

The coefficients of transverse correction of shearing correspond to the report/ratio of the terms of Hc
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
N
1
E
-
N =
N
3
X
- 3
X
):

M11


Ncouch
N
M=
X
M
3
22 = .x dx
3.
3 =
.
1
X dx
L
N () 3 3


X
N
n=
3
M12
1

N11


Ncouch
N
N=
X
NR
3
22 = dx
. 3 =
.
1
dx
L
N () 3


X
N
n=
3
N12
1

If one expresses like previously (with m material of layer N):

N = Mr. (E+x3.K-d m (
ref.
T X3 - T)
()
()
() ()


one can note the efforts generalized in the form: (cf [§1.3])

M-Mth =P K
. +Q E
.
N-Nth =Q K
. +R E
.

with P, Q, R of the matrices 3 X 3 being expressed by:

Ncouch
N
Ncouch
3
1
3
3
P =
X
2
N
N 1

(m) 1 X dx
.
. X
X
N
3
3
= (m) (3) - (-
3
)
x3
3
n=1
n=


1
Ncouch
N
Ncouch
3
1
2
2
Q =
X
N
N 1

(m) 1 X dx
.
. X
X
N
3
3
= (m) (3) - (-
3
)
x3
2
n=1
n=


1
Ncouch
Ncouch
R =
N
N
(
1
m) (
. x3 - x3) = (m) e.n
n=1
n=1
Handbook of Référence
R4.01 booklet: Composite materials
HT-66/02/004/A

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
: 18/20

the shearing action V is obtained by derivation of the moment [§4.3.2].

The generalized efforts of thermal origin are calculated directly:

Ncouch
xn
Mth
3
ref.
=. 1.
.
.
-
3
3
-
N
X
T X
T
dx
m
X
(()
)
()
D (m)
3
N =1
3

Ncouch
xn
Nth
3
ref.
=. 1
.
.
-
3
-
N
T X
T
dx
m
X
(()
)
()
D (m)
3
N =1
3

4.3.4 Localization of the constraints (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 = 1, NR
in each
N (
couch)
()
lay down element.

It is necessary to calculate in each layer (N), the matrix (and the terms
m)
(T (X -
. D (cf [§3.2]) (m = chechmate represents the characteristics material of the layer
3)
T ref.) (m)
N
N).

Constraints
N 1
N
with an ordinate X
-,
in layer (N) are then:
3
] X X
3
3 [


X =
. E + X. K - D
T X - T

N (3)
m
[3
m (
(3) ref.)]
()
()
()

and transverse shearing:


X = D X. V + D X.
éq
4.3.4-1
N (3)
1 (3)
2 (3)
()


Note:

In the code postprocessings of the elements of plates are generally defined in
locate associated with ANGL_REP. Constraints in the reference mark in

trinsèque of the element are thus
brought back in the reference mark of the variety. One a:

2
2

C
S
+ 2CS

where C = cos (0)
11
11








2
2

= S
C
- 2CS

S = sin (
cf § 4.1
0) (
[])
22
22


2
2

- CS + CS C - S


12
12
where is the angle between V and E
eref
N
0
1
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):

X
L

feel



in

traction

in

limit
:

orthotropi

direction

(1ère
fibers)



feel
:

E
X
compressio

in

limit
:

L

feel



in

N
orthotropi

direction

(1ère
fibers)



feel
:

E
chechmate
Y
T

feel



in

traction

in

limit
:

orthogonal

direction

(2nd
er

1



with

E
E)
N

Y
compressio

in

limit
:

T

feel



in

N
orthogonal

direction

(2nd
er

1



with

E
E)
S
cisailleme

in

limit
:

LT

feel



in

NT
Handbook of Référence
R4.01 booklet: Composite materials
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Code_Aster ®
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Date:
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:
R4.01.01-B Page
: 19/20

It is necessary to calculate the constraints in the reference mark of the layer (defined by the axes
of orthotropism) starting from the constraints in the reference mark of the element:

the angle between V and E
is, and that between E and the reference mark of orthotropism is:
1
ref.
0
ref.
N


C2
S2
+ 2CS

where
C = cos (+
N
0)
L




11


= S2
C2
- 2CS

S = sin (+ cf §4.1
N
0) (
[])
T
22



2
2


- CS + CS C - S 12
L T

N
N

Maximum criterion of constraint:

The 5 following criteria are calculated by layer: (N = 1, NR - couch)



(
L N) (
L

if L >0
if

< 0

N
)
(N)
(Ln)
X (chechmate
X
N)
()
(matn)
()


(
T N) (
T
if
T > 0

if

< 0


N
) (N) (Tn)
(
Y chechmate
Y
N)
()
(matn)
()
(T
L N)
S (matn)

Criterion of TSAI-HILL:

This criterion is written in each layer in the following way:

2
L (N) L (N) .T (N)
2
T (N)
2
LT (N)
CTH =
-
+
+

X (2mat
X
Y
S
N)
(2matn)
(2matn)
(2matn)

The material is broken when C
1.
TH

Values X and Y are replaced by X and Y when constraints (
,

L N
T N)
()
()
corresponding are negative.
Handbook of Référence
R4.01 booklet: Composite materials
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Code_Aster ®
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Titrate:
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Date:
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:
R4.01.01-B Page
: 20/20

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” - Vol 2 Poutres and
plates - HERMES 1990.
[3]
J.R. TO BORE: “Elasticity”. Kluwer academic publishers.

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R4.01 booklet: Composite materials
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