Code_Aster ®
Version
7.2
Titrate:
Voluminal element of hull SHB8


Date:
04/08/04
Author (S):
J.M. PROIX, S. BAGUET, A. COMBESCURE Key
:
R3.07.07-A Page
: 1/18

Organization (S): EDF-R & D/AMA, INSA-LYON
Handbook of Référence
R3.07 booklet: Machine elements on average surface
Document: R3.07.07

Code_Aster: Voluminal element of hull SHB8

Summary:

We present in this document the theoretical formulation of element SHB8PS and its establishment
numerical for non-linear incremental analyzes implicit (great displacements, small rotations,
small deformations).

It is about a three-dimensional cubic element with 8 nodes with a called privileged direction thickness.
Thus, it can be used to represent mean structures while correctly taking into account them
phenomena through the thickness (inflection, elastoplasticity), thanks a numerical integration to 5 points of
Gauss in this privileged direction.

In order to reduce the calculating time considerably and to draw aside various likely blockings
to appear, this element under-is integrated. It requires consequently a mechanism of stabilization in order to
to control the modes of deformation to null energy (modes of Hourglass).

In addition to its cost of relatively weak calculation and its good performances in elastoplasticity, this element
have another advantage. Since it is based on a three-dimensional formulation and that it only has
degrees of freedom of translation, it is very easy to couple it with voluminal elements 3D, which is
very useful in systems where voluminal hulls and elements must cohabit.

Handbook of Référence
R3.07: Machine elements on average surface
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Code_Aster ®
Version
7.2
Titrate:
Voluminal element of hull SHB8


Date:
04/08/04
Author (S):
J.M. PROIX, S. BAGUET, A. COMBESCURE Key
:
R3.07.07-A Page
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Count

matters

1 Introduction ............................................................................................................................................ 3
2 Kinematics of the element ...................................................................................................................... 4
3 variational Formulation ...................................................................................................................... 4
4 Discretization ......................................................................................................................................... 5
4.1 Discretization of the field of displacement ....................................................................................... 5
4.2 Operator gradient discretized .......................................................................................................... 5
4.3 Stamp rigidity ............................................................................................................................ 7
4.4 Stamp geometrical rigidity K .................................................................................................. 8
4.5 Stamp pressure Kp ..................................................................................................................... 9
5 Stabilization of the element ..................................................................................................................... 10
5.1 Motivations ..................................................................................................................................... 10
5.2 Modes of “Hourglass” ................................................................................................................... 11
5.3 Stabilization of the type “Assumed Strain Method” ............................................................................ 11
6 Strategy for non-linear calculations ............................................................................................... 13
6.1 Geometrical non-linearities .......................................................................................................... 13
6.2 Small displacements ...................................................................................................................... 14
6.3 Forces of stabilization ................................................................................................................... 14
6.4 Plasticity ......................................................................................................................................... 15
7 Establishment of element SHB8 in Code_Aster ......................................................................... 15
7.1 Description ..................................................................................................................................... 15
7.2 Use ....................................................................................................................................... 15
7.2.1 Grid ................................................................................................................................ 15
7.2.2 Modeling ......................................................................................................................... 15
7.2.3 Material ................................................................................................................................ 16
7.2.4 Boundary conditions and loading .................................................................................. 16
7.2.5 Calculation in linear elasticity ................................................................................................... 16
7.2.6 Calculation in linear buckling .............................................................................................. 16
7.2.7 Calculation in geometrical nonlinear “elasticity” .................................................................. 16
7.2.8 Plastic nonlinear calculation ................................................................................................ 16
7.3 Establishment ................................................................................................................................... 17
7.4 Validation ....................................................................................................................................... 17
8 Bibliography ........................................................................................................................................ 18

Handbook of Référence
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Code_Aster ®
Version
7.2
Titrate:
Voluminal element of hull SHB8


Date:
04/08/04
Author (S):
J.M. PROIX, S. BAGUET, A. COMBESCURE Key
:
R3.07.07-A Page
: 3/18

1 Introduction

Many recent work proposed to use a voluminal formulation for the structures
thin. Two principal families of methods, which rest all on the introduction of a field of
postulated deformation (“assumed strain”), emerge. Methods of the first family
consist in using a conventional numerical integration with an adequate control of all them
modes of blocking and locking (volume, transverse shearing, membrane). Methods of
the second family consist in under-integrating the elements to remove blockings and controlling
the modes of Hourglass which rise from this under-integration (see [bib3] [bib4]). Both
approaches were studied in details in the case of an elastic behavior. On the other hand, very little
work treats elastoplastic case.

The element presented here rests on an under-integrated formulation especially developed for
elastoplastic behavior of the structures in inflection. The basic idea consists first of all with
to make sure that there are sufficient points of Gauss in the thickness to represent it correctly
phenomenon of inflection, then to calculate rigidities of stabilization in an adaptive way according to the state
plastic of the element. That represents an unquestionable improvement compared to the formulations
traditional for the forces of stabilization, because these last rest on an elastic stabilization
who becomes too rigid when the effects of plasticity dominate the response of the structure.

Element SHB8 is a continuous three-dimensional cube with eight nodes, in which a direction
privileged, called thickness, was selected. It can thus be used to model the structures
thin and to take into account the phenomena which develop in the thickness within the framework
mechanics of the continuous mediums three-dimensional. Since this element is under
integrated, it exhibe of the modes of Hourglass which must be stabilized. We chose the method of
stabilization introduced by Belytschko, Bindeman and Flanagan [bib3] [bib4]. This element and this
method of stabilization were already implemented in an explicit formulation by Abed-Meraim
and Combescure [bib2]. This documentation describes the formulation of this element, its implementation
numerical for the prediction of elastic and elastoplastic structural instabilities, like sound
establishment in Code_Aster. For the non-linear problems, an incremental formulation
implicit of Newton-Raphson type is used [R5.03.01]. The equilibrium equations are solved by
method of Lagrangien updated. The control of the increments of load and displacement is
based on a method of control close to the algorithm to Riks [bib5]. Implementation the numerical
of this element within a non-linear framework was proposed by Legay and Combescure in [bib1].
Handbook of Référence
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Code_Aster ®
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Titrate:
Voluminal element of hull SHB8


Date:
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Author (S):
J.M. PROIX, S. BAGUET, A. COMBESCURE Key
:
R3.07.07-A Page
: 4/18

2
Kinematics of the element

Element SHB8 is a hexahedron with 8 nodes. The five points of integration are selected along
direction in the reference mark of the local co-ordinates. The shape of the element of reference as well as
points of integration are represented on [Figure 2-a].


7
8
6
5

4
5

3
2
3
1
4
2
1

Appear 2-a: Géométrie of the element of reference and points of integration

This element is isoparametric and has the same linear interpolation and same kinematics
that hexaèdraux elements with 8 standard nodes.

3 Formulation
variational

The formulation used for the construction of element SHB8PS differs from a formulation
traditional simply by the choice of a postulated deformation &, therefore of an operator gradient
discretized, allowing to avoid the induced modes parasitized by under integration.

Thus, the variational principle is written:

(v, &) = (&): FD -
ext.
U & F
= 0
V

where represents the total virtual power, the variation, the v field speed, U & speeds
nodal, & the rate of postulated deformation (assumed strain misses), the constraint of Cauchy, V it
updated volume and ext.
F
external forces.

The discretized equations thus require the only interpolation the speed v and rate of
deformation postulated & in the element. We now will build element SHB8PS to be left
of this equation. The complete developments and the demonstrations concerning this element are
explained in details in [bib2].
Handbook of Référence
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Code_Aster ®
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Titrate:
Voluminal element of hull SHB8


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:
R3.07.07-A Page
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4 Discretization

4.1
Discretization of the field of displacement

The space co-ordinates X of the element are connected to the nodal co-ordinates X by means of
I
II
isoparametric functions of forms NR by the formulas:
I

8
X = X NR (,) = NR (,) X
I
II
I
I
II
I 1
=

In the continuation, and except contrary mention, one will adopt the convention of summation for the indices
repeated. The indices in small letters I vary from one to three and represent the directions of
space co-ordinates. Those in capital letters I vary from one to eight and correspond to the nodes of
the element.

The same functions of forms are used to define the field of displacement of the element U in
I
function of nodal displacements U:
II

U = U NR (
,)
I
II
I

Trilinear isoparametric functions of form are chosen:


1
NR (
,) = (1+) (1+) (1+)
I

8
I
I
I




, [1
-, 1],
I = 1,, 8

These functions of form transform a unit cube in space (
,) in a hexahedron
unspecified in space (X, X, X).
1
2
3

4.2
Operator discretized gradient

The gradient U
U of node I in
I, of the field of displacement is a function of displacement
J
II
direction I:
U = U NR
I, J
II
I, J

The linear tensor of deformation is given by the symmetrical part of the gradient of displacement:

1
= (U + U)
ij
2 I, J
J, I

Let us introduce the three vectors B, derived from the functions of form at the points of Gauss P:
I
K

NR

T
B (P) =

I
K
X
I =0, =0, =k
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Titrate:
Voluminal element of hull SHB8


Date:
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:
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Also let us introduce the following vectors:

T
S
= (1
1
1
1
1
1 1 1
)
T
H
= (1
1
- 1
1
-
1
-
- 1 1 1)
1
T
H
= (1
1
-
- 1 1
1
-
1 1 - 1)
2
T
H
= (1
1
-
1
1
-
1
1
-
1 - 1)
3

T
H
= (
1
-
1
- 1 1
1
1
-
1 - 1)
4
T
X
= (
1
-
1
1
1
-
1
-
1 1 - 1)
1
T
X
= (
1
-
1
-
1
1
1
-
- 1 1 1)
2
T
X
= (
1
-
1
-
- 1 - 1 1
1 1 1
)
3

The three vectors T
X represent the nodal co-ordinates of the eight nodes. The four vectors T
I

H
the functions H, H, H and H for each of the eight nodes represent respectively, which are
1
2
3
4
defined by:

H =
H =
H =
H =
1
2
3
4

Let us introduce finally the four following vectors:


3
= 1

H -

(T
h. X

J)

B
8
J
j=1




The gradient of the field of displacement can be now written in the form (without any
approximation [bib3]):


4

T
T
T
T
U = B + h. U = B + h. U
I J
J
J
I
J
J
,
,
,
I





1
=


Or, in the form of vector:


U


X, X




U


y, y


U



Z, Z


U =

S
U + U
X y
y X
,
,


U + U
X Z
Z X
,
,


U + U
y, Z
Z, y



with U nodal displacement in direction I. The symmetrical operator gradient (noted)
I
S
discretized connecting the tensor of deformation to the vector of nodal displacements

U =.
B U
S
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Titrate:
Voluminal element of hull SHB8


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:
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takes the matric form then:

T
T
B + H
0
0

X
X

,



0
T
T
B + H
0


y
, y




0
0
T
T
B + H

Z
, Z
B =


T
T
T
T
B +h
B +h
0


y
, y
X
, X

T
T
B + H
0
T
T
B + H


Z
, Z
X
, X



0
T
T
T
T
B + H
B +h

Z
Z
y
y
,
,



The detailed formulation was presented by Belytschko in [bib3].

4.3
Stamp rigidity

The matrix of rigidity of the element is given by:

T
K =
B CB D
E
E

The five points of integration considered are on the same vertical line. Their co-ordinates
are (
,) and their weights of integration are the roots of the polynomial of Gauss-Legendre:




P (1)
0 0
= 0 9
. 1
= 0 2
. 4
1
1
P (2)
0 0
= 0 5
. 4
= 0 4
. 8
2
2
P (3)
0 0
0
0.57
P (4)
0 0

-

2
2
P (5)
0 0

-

1
1
Thus, the expression of rigidity K is:
E

5
K = () J () T
B () CB ()
E
J
J
J
J
J 1
=

where J () is Jacobien, calculated at the point of Gauss J, of the transformation between the configuration
J
unit of reference and an arbitrary hexahedron. The elastic matrix of behavior C chosen with
following form:
+ 2µ

0
0
0
0



+ 2µ 0 0 0 0


0
0
E 0
0
0
C =

0
0
0 µ 0 0


0
0
0
0 µ 0



0
0
0
0
0 µ

E
where E is the modulus Young, the Poisson's ratio, µ = (
the modulus of rigidity and
2 1+)

= E
the coefficient of modified Lamé. This law is specific to element SHB8. It resembles
2
1
with that which one would have in the case of the assumption of the plane constraints, put aside the term (3,3). One
can note that this choice involves an artificial anisotropic behavior.
This choice makes it possible to satisfy all the tests without introducing blocking.
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Voluminal element of hull SHB8


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:
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4.4
Stamp geometrical rigidity K

By introducing the quadratic deformation Q
E:



Q
1
E

=
U U


2
K, I K, J



K 1 3

=,


one can define this matrix of geometrical rigidity by:

T
Q
the U.K.U =
: E (,
U U) =
: (
T
D


U .u) D




0
0

In order to express this matrix in discretized space, let us introduce the operators gradient
quadratic discretized Q
B such as:


Q

T
Q
Q
T
Q

Q
Q
E
E
E

U .B ()
T
.U U .B () .U U .B () .U
xx
yy
zz
xx
J


yy
J
zz
J











Q
Q
Q
Q
E
E
E +e
T
U. Q
B ()
T
.U U. Q
B ()
T
.U U. Q
B () .U

yy
zz
xy
yx
yy
J
zz
J
xy
J












T
Q
T
Q
T

Q


Q
Q
E
E + Q
Q
E
E + Q
E
U .B () .U U .B () .U U .B () .U

zz
xy
yx
xz
zx
zz
J
xy
J
xz
J


Q
E (U (), U ())

=
=

J
J





Q
Q
Q
Q
Q
Q
T
Q
T
Q
T
Q
E +e
E +e
E +e
U .B () .U U .B () .U U .B () .U
xy
yx
xz
zx
yz
zy
xy
J
xz
J
yz
J














Q
Q
Q
Q

E +e
E +e
T
U. Q
B ()
T
.U U. Q
B () .U
xz
zx
yz
zy

xz
J
yz
J














T
Q
Q
E + Q
E

U .B () .U

yz
zy


yz
J


The various terms Q
B are given by the following equations:
ij

B. T
B
0
0


X
X

Q
B ()
=
B. T
B

xx

0
0
J
X
X




0
0
T
B .b


X
X
B. T
B
0
0


y
y

Q
B ()
=
B. T
B

yy

0
0
J
y
y




0
0
T
B .b


y
y
B. T
B
0
0


Z
Z

Q
B ()
=
B. T
B

zz

0
0
J
Z
Z




0
0
T
B .b


Z
Z
B. T
B + B. T
B
0
0



X
y
y
X

Q
B ()
=
B. T
B + B. T
B

xy

0
0

J
X
y
y
X




0
0
T
T
B .b + B .b


X
y
y
X
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Voluminal element of hull SHB8


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B. T
B + B. T
B
0
0



X
Z
Z
X

Q
B ()
=
B. T
B + B. T
B

xz

0
0

J
X
Z
Z
X




0
0
T
T
B .b + B .b


X
Z
Z
X
B. T
B + B. T
B
0
0



y
Z
Z
y

Q
B ()
=
B. T
B + B. T
B

yz

0
0

J
y
Z
Z
y




0
0
T
T
B .b + B .b


y
Z
Z
y

With these notations, the geometrical matrix of rigidity K

at the point of Gauss
is given by:
J

K () = (). Q
B () + (). Q
B () + (). Q
B ()


J
xx
J
xx
J
yy
J
yy
J
zz
J
zz
J

+
(). Q
B () + (). Q
B () + (). Q
B ()
xy
J
xy
J
xz
J
xz
J
yz
J
yz
J

and geometrical rigidity of the element stamps it is given by:

5
K = () J () K ()


J
J

J
J 1
=

4.5
Stamp Kp pressure

The following compressive forces are present in the tangent matrix via the matrix K, because them
P
following external forces depend on displacement. The following compressive forces are written:

T
1
p.
N U dS =
p of [
T F (U)]
T
N .F
-
U
dS = p F - p K.


U
0
()
0
0

0
P
F (U) = 1+ U

by using the notations:

·
T
N = (N, N, N), normal on the surface external of the element in the configuration of
0
X
y
Z
reference
·
B %, vector of dimension 4, drift of the functions of form to the 4 nodes of the face of
I
the element charged in pressure
·
S surface of the face charged in pressure
0
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Voluminal element of hull SHB8


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:
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The preceding formulation leads to a not-symmetrical matrix. It is known that one can nevertheless
to use a symmetrical formulation if the external forces due to the pressure derive from a potential.
It is the case if the compressive forces do not work on the border of the modelled field. One
thus consider that the symmetrical part of the matrix is enough. The symmetrized matrix takes the form
following:


0
T
T
T
T
B %

y N-B
% N
N -
N
X
X
y
B % Z X B % X Z



0
T
T
T
T
B %

y N-B
% N
N -
N
X
X
y
B % Z X B % X Z



0
T
T
T
T
B % y N-B % N
N -
N
X
X
y
B %

Z
X B
% X Z


0
T
T
T
T

B % y N-B % N
N -
N
X
X
y
B % Z X B % X Z
T
T
N -

%
% N
0
T
T
B
B
B % N-B %
X
y
Z
y N
y
X
y
Z
T
T

N -
N
0
T
T
B %
B %
B % N-B %
X
y
Z
y N
y
X
y
Z
K = S
P
0

T
T
N -
N
0
T
T
B %
B %
B % N-B %
X
y
Z
y N
y
X
y
Z


Tn
T
T
T
B %
-
N
0
B %
B % N - %
y
Z
by N
X
y
X
y
Z
T
T
T
T
N
N
N
N
0

-
-
B %
%
%
X
Z
bz X
by Z B % Z y


T
T

T
T
B % N
B % N
N
N
0

-
-
X
Z
Z
X
B % y Z B % Z y


T
T
T

T
B % N
B % N
N
N
0

-
-
X
Z
Z
X
B % y Z B % Z y


T
T
T

T
B % N
B % N
N
N
0

-
-
X
Z
Z
X
B % y Z B % Z y



It is a matrix (12×12), which it is necessary to multiply by displacements of the 4 nodes of the face on
which one applies a pressure.

5
Stabilization of the element

5.1 Motivations

The under-integration of element SHB8 (5 points of Gauss only) aims at reducing considerably
calculating time (gradient displacement, law of behavior,…). It also makes it possible to draw aside
the various blockings met in implementation the numerical of the finite elements.

However, this under-integration does not have only advantages: it introduces unfortunately
parasitic modes associated a null energy (mode of Hourglass or sand glass). In statics, that
can lead to a singularity of the matrix of total stiffness for certain boundary conditions.
In transitory dynamics, on the other hand, that led to modes of sand glass which will deform it
unrealistic grid of way and which ends up exploding the solution. This deficiency of the matrix
of stiffness, due to under-integration, must thus be compensated while adding to elementary rigidity
a matrix of stabilization. The core of the new rigidity, thus obtained, must be reduced to only
modes corresponding to the rigid movements of solids.
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Titrate:
Voluminal element of hull SHB8


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:
R3.07.07-A Page
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5.2
Modes of “Hourglass”

Since the points of integration are on the same vertical line (privileged direction),
the derivative of the functions H and H are cancelled in these points. The operator discretized gradient is thus
3
4
tiny room to:


2

T
T
B + H
0
0

X
X

,



1
=



2




0
T
T
B + H
0


y
, y


1
=




2




0
0
T
T
B + H
Z
, Z



1
=

B =



2
2



T
T
T
T
B +
H
B +
H
0



y
, y
X
, X



1
=
1
=




2
2

T
T

B + H
0
T
T
B + H
Z
, Z
X
, X

1
=
1
=




2
2




0
T
T
T
T
B + H
B +
H
Z
, Z

y
, y



1
=

1
=



The modes of Hourglass are modes of displacement to null energy, i.e they check Bu = 0.
Six modes, others that rigid modes of solids, which check this equation are:


H
0
0
H
0
0
3
4




0
H
0 0
H
0


3

4




0
0

H
0
0

H


3
4

5.3
Stabilization of the type “Assumed Strain Method”

In this approach, inspired of work of Belytschko, Bindeman and Flanagan [bib3] [bib4], them
derived B from the functions of form are not calculated at the points of Gauss but are realized on
I
the element:
T
1
^ =
(,) D, I = 1, 2,3
I
NR
B
, I
O C

Thus, the new operator discretized gradient can be written:

^
^
B = B + B

stab
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:
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The expression of ^
B
is given by:
stab
4


H ^T

,
0
0
X

=3


4


0
H ^T

,
0
y

=3
^


B
=

stab

0
0
H ^T
,
4 Z
4


0
0
0



0
0
0



0
0
0

and that of the vectors ^
by:
3
1

^ = H - (T
h. X) ^

B
8


J
J

J 1
=


The new matrix of rigidity becomes:

T
T

^
K =
B CBd +
B C B D





stab
E
E


^
T
^
T
^
+
B
CBd +
B
CB D


stab




stab
stab
E
E

144424443

stab
K

The last term of the preceding equation (
stab
K
) is enough to stabilize the element. One can thus reduce
the matrix of rigidity stabilized with:

=
+ stab
K K
K

E

stab
^
T
^
K
=
B
CB D



stab
stab
E

The many cases which were studied showed that it is enough to calculate the diagonal terms of
stamp stabilization
stab
K
, I = 1, 2,3, which is given by:
II
T
stab
1
K
=
(+ 2) [
T
H
µ +]
11
11
$ $
$ $
3
3
4
4
3

T
stab
1
K
=
(+ 2) [
T
H

µ +]
22
22
$ $
$ $
3
3
4
4
3

stab
H33
T
K
=
E

33
$ $
4
4
3

The Hii coefficients themselves are given by the following equation, in which there is not
summation on the repeated indices:
1 T

T

X X
X X
J
J
K
K
H



=

II
3

T

X X
I
I


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6
Strategy for non-linear calculations

6.1 Non-linearities
geometrical

One treats here the case of great displacements, but weak rotations (see further) and small
deformations. One adopts for that an updated Lagrangian formulation.

Into nonlinear we seek to write balance between internal forces and force external at the end of
the increment of load (located by index 2):

int
extr
F
= F

2
2

The expression of the internal forces is written:

int
T
F =
FD
2
B

2
2
2

In the preceding equation the B2 operator is the operator allowing to pass from displacement to
linear deformation calculated on the geometry at the end of the step, the constraint is the constraint of
2
Cauchy at the end of the step and integration is made on the volume deformed at the end of the step.
2

We chose this updated Lagrangienne formulation.

The element available to date in Aster is programmed in small rotations. Indeed the increment of
deformation is calculated by using only the linear deformation:

1
E
= (1 (U) + T1 (U))
2

The operator gradient is calculated on the geometry of beginning of step. This writing of the deformation is
limited to small rotations (<5 degrees).

One can without difficulty of extending the formulation to great rotations by including in the deformation them
terms of second command:

1
T
T
E =
U + U + U U
2 (
.
1 (
) 1 () 1 () 1 ())

In elasticity, the law of behavior is written:

= It E

where C is the matrix of Hooke. Let us notice that for the SHB8 this matrix is a matrix
orthotropic transverse which is written in the axes of the lamina:

+ 2µ
µ
0
0
0
0
µ
2µ 0 0 0 0
+


[C] 0
0
E 0 0
0
'=

0
0
0 µ 0 0


0
0
0
0 µ 0


0
0
0
0 0 µ
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The formula allowing to calculate the constraint of Cauchy starting from the constraint of
2
Piola Kirchoff II is:
2


= +

2
1



1
T


=
F F
2
det

(F)
2

F = I +
1


U

The combination of the four last equations with the expression of the internal forces gives
formulation of the element in great deformations in updated Lagrangien.

Let us notice that this Lagrangienne formulation updated is completely equivalent to
total Lagrangienne formulation for which the internal forces are written:

int
T
NL
F =
B + B (U) FD
2
(
) 2
0
0

In this case all integrations are made on the initial geometry the constraint used is
0
2
the constraint of Piola Kirchoff II. This last method is probably preferable when it
grid becomes deformed significantly and thus makes it possible to deal with the problems into large
deformations but requires the development of operator NL
B (U).

The increment of deformation in total Lagrangien is expressed on the initial geometry of the structure.

1
T
T
E =
U + U + U U
2 (
.
0 (
) 0 () 0 () 0 ())

The combination of the two preceding equations gives the formulation of the element into large
deformations in linear behavior material.

6.2 Small
displacements

In the case of small displacements one confuses geometry in beginning and end of step, constraint of
Cauchy and of Piola Kirchoff II, moreover one uses the linear expression of the deformations.

6.3
Forces of stabilization

The forces of stabilization make it possible to avoid the modes of sand glass and are added in the calculation of
residues to balance the contribution of the matrix of stiffness of stabilization to the first member.
forces of stabilization stab
F
, to add to the forces intern int
F, are written:
2

stab = stab
F
K
U
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For reasons of effectiveness, one chooses not to assemble again
stab
K
to calculate stab
F
in
end of step, but rather to build stab
F
starting from ^
B
that one calculated previously. One must
stab
for that to place itself in the reference frame corotationnel of medium of step suggested in [bib3]. For this
reason, one does not obtain an exact expression of stab
F
, and some additional iterations are
generally necessary to converge. These some iterations are however unimportant by
report/ratio at the cost of calculation saved while not assembling
stab
K
.

6.4 Plasticity

The elastoplastic behavior of Von Mises, with isotropic work hardening, is calculated of each one of
5 points of integration. One thus uses quite simply the formulas and the usual programming of
plasticity with a three-dimensional state of stresses, but the linear matrix of behavior It is
orthotropic. We must quite simply slightly modify the usual algorithm of flow
elastoplastic three-dimensional by replacing the usual matrix of Hooke C by the matrix of
orthotropic behavior transverse It.

We must find the constraint at the end of the step which checks balance. In great displacements it
problem is written:

= +

2
1
(
p
It
-
)

This equation is solved as soon as the increment of plastic deformation is known. This deformation
is determined while forcing the final constraint to be plastically acceptable. This method is
completely similar to the usual three-dimensional method except that there is no explicit solution
with this problem if one uses for example the effective approximation of the radial return to calculate
solution. We chose to solve this nonlinear problem by a method of Newton.

7
Establishment of element SHB8 in Code_Aster

7.1 Description

This element is pressed on the voluminal meshs 3D HEXA8.

7.2 Use

This element is used in the following way:

7.2.1 Grid

Check the good orientation of the faces of the indicated elements (compatibility with the direction
privileged) by using ORIE_SHB8 of operator MODI_MAILLAGE.

7.2.2 Modeling

To assign modeling SHB8 to indicated meshs HEXA8.
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7.2.3 Material

For a homogeneous isotropic elastic behavior in the thickness one uses key word ELAS in
DEFI_MATERIAU where one defines the coefficients E, modulus Young and NAKED, Poisson's ratio.

To define a plastic behavior one uses key word TRACTION in DEFI_MATERIAU where one
the name of a traction diagram defines. Only this type of definition is available for the moment.

7.2.4 Boundary conditions and loading

DDL of volume 3D in the total reference mark. One imposes the boundary conditions on the ddl of volume
3D (AFFE_CHAR_MECA/DDL_IMPO), and efforts in the total reference mark (FORCE_NODALE).

They are the efforts in the total reference mark.

One defines the efforts of pressure distributed on the faces of the element (under key word PRES_REP). One
will have taken care as a preliminary to define meshs of skin QUAD4 and to suitably direct them
outgoing normals with these meshs of skin using the command
MODI_MAILLAGE key word ORIE_PEAU_3D

7.2.5 Calculation in linear elasticity

Order MECA_STATIQUE
The options of postprocessing available are SIEF_ELNO_ELGA and EQUI_ELNO_SIGM.

7.2.6 Calculation in linear buckling

Option RIGI_MECA_GE being activated in the catalog of the element, it is possible to carry out one
traditional calculation of buckling after assembly of the matrices of elastic and geometrical rigidity.

7.2.7 Calculation in geometrical nonlinear “elasticity”

One chooses behavior ELAS under key word COMP_INCR of STAT_NON_LINE, into small
deformations (“PETIT”) or in great deformations “GREEN” under key word DEFORMATION.

Strategy used being based on the use of a matrix of tangent rigidity during
iterations (reactualization at the beginning of step only), one will take care not to use another option
that that which is activated by defect, namely REAC_ITER = 0 pennies NEWTON.

Numerical integration in the thickness is carried out with 5 points of Gauss, just like in not
linear hardware.

7.2.8 Plastic nonlinear calculation

Only the criterion of Von Mises is available to date (RELATION = “VMIS_ISOT_TRAC” under
COMP_INCR). One defines the mode of calculation of the deformations as in the case of elasticity not
linear (DEFORMATION = “GREEN” or “PETIT”).

Strategy used being based on the use of a matrix of tangent rigidity during
iterations (reactualization at the beginning of step only), one will take care not to use another option
that that which is activated by defect, namely REAC_ITER = 0 pennies NEWTON.
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7.3 Establishment

Options RIGI_MECA, RIGI_MECA_GE, FORC_NODA, FULL_MECA, RIGI_MECA_TANG,
RAPH_MECA,
SIEF_ELGA_DEPL,
SIEF_ELNO_ELGA were activated in the catalog
gener_shb 3d_3.cata. They direct all calculation towards te0520.f, then towards shb8.f.

The forces of stabilization of the element require the storage of a vector of size 12 for each
not Gauss. We chose to store these terms like additional components of
stress field.

No development was necessary for the compressive forces distributed and the forces of
following pressures. Indeed, these loadings are pressed on meshs of skin identical to those
voluminal elements 3D.

7.4 Validation

The tests validating this element are, in version 7.2 of Code_Aster:

· SSLS108 C and D: beam bored in inflection, test allowing to check the absence of blocking
[V3.03.108],
· SSLS105 C: hemisphere doubly pinch [V3.03.105] traditional test to check
convergence of the element,
· SSLS123 a: sphere under external pressure [V3.03.123] to validate the loadings of
pressure and the orthotropic behavior particular to this element,
· SSLS124 A and thin b: section in inflection with various twinges, to delimit the field
of use of the element [V3.03.124]. The results are correct (less than 1% with the solution
analytical) for reports/ratios of twinge (thickness/width) going from 1 to 5 10­3,
· SSLS125 a: buckling (modes of Euler) of a free cylinder under external pressure [V3.03.125]
this test makes it possible to validate the geometrical nature of rigidity,
· SSNS101 A, B and C: breakdown of a cylindrical roof [V6.03.101]. This test makes it possible to validate it
geometrical nonlinear calculation and elastoplasticity,
· SSNS102 a: buckling of a hull with stiffeners in great displacements and pressure
following [V6.03.102].
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8 Bibliography

[1]
LEGAY A. and COMBESCURE A.: “Elastoplastic stability analysis off shells using the
physically stabilized finite element SHB8PS ", International Journal for Numerical Methods and
Engineering, vol. 57, 1299-1322, 2003
[2]
ABED-MERAIM F. and COMBESCURE A.: “SHB8PS has new adaptive assumed strain
continuum mechanics Shell element for impact analysis “, Computers and Structures, Vol. 80,
791-803, 2002
[3]
BELYTSCHKO T. and BINDEMAN L.P.: “Assumed strain stabilization off the eight node
hexahedral elements ", Computer Methods in Applied Mechanics and Engineering, Vol. 105,
225-260, 1993
[4]
FLANAGAN D.P. and BELYTSCHKO T.: “A uniform strain hexahedron and equilateral with
orthogonal hourglass control ", International Journal for Numerical Methods and Engineering,
Vol. 17, 679-706, 1981
[5]
RIKS E.: “Incremental Year approach to the solution off snapping and buckling problems”,
International Journal off Solids and Structures, Vol. 15, 524-551, 1979

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