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Organization (S): EDF/AMA
Handbook of Validation
V6.04 booklet: Nonlinear statics of the voluminal structures
Document: V6.04.156
SSNV156 - Colonne under voluminal loading.
Elastoplastic law has gradient

Summary:

In addition to the elastoplastic law with gradient which it tests only in its version with linear work hardening, this test has especially
for object to validate the algorithm of integration of the laws of behavior to gradient of internal variables. In
effect, the problem suggested, the setting in traction of a column under voluminal forces, led to a solution
nonhomogeneous which activates the various components of algorithm (Newton, linear search, BFGS) and
for which one can obtain an analytical expression. The results obtained are in agreement with this one, and
it with a high degree of accuracy.

Handbook of Validation
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HT-66/02/001/A

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1
Problem of reference

1.1 Geometry

The structure is a cylinder, with the most general direction, height L = 2 mm and of which the form of
section does not influence the solution. So one will adopt a square section for modelings
in plane deformations and 3D (side has =
mm

1
,
0
) as well as a circular section for
axisymmetric modeling (radius R =
mm

1
,
0
).

1.2
Properties of material

The material follows an elastoplastic law of behavior to gradient whose work hardening is isotropic and
linear. Characteristics of material, respectively the Young modulus E, the coefficient of
NAKED Poisson, elastic limit SY, the slope of work hardening D_SIGM_EPSI and the length
characteristic LONG_CARA, are equal to:

E = 100
MPa

000

=,
0 3
y =
MPa

100

T
E = 10
MPa

000

L


B =
,
0 982707 mm


1.3
Boundary conditions and loading

Vertical displacements are blocked on its higher face, horizontal displacements are
blocked on the side faces and the lower face is free of any kinematic condition. By
elsewhere, the structure is subjected to a voluminal force vertical and directed to the bottom of intensity
increasing F T
() = F T where
3
F =
1 NR/mm and T is a parameter of loading (without unit).
0
0
Handbook of Validation
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2
Reference solution

2.1
Method of calculation of the reference solution

This problem admits an analytical solution. Sought sizes, namely the constraints,
deformations, plastic deformations p
and cumulated plastic deformation p,
depend that level on loading T and dimension Z of the section considered, where Z = 0
indicate the lower face (free) cylinder and Z = L its face higher (blocked).
Because of the blocking of the side faces, the sections do not become deformed horizontally, so that
the fields of displacements and deformations are written:

U (Z, T) = U (Z, T) E
(Z, T)
Z
= (Z, T) ez E
with
U (L, T)
Z
= 0 éq 2.1-1

As for the tensor of constraints, it is diagonal. Its vertical component is fixed by the equation
of balance while its horizontal components, identical in the two directions, depend
law of behavior (effect of fastening):

(Z, T) = S (Z, T) (ex ex + ey ey) + (Z, T) ez E
with
Z
(Z, T) = Z F (T) éq
2.1-2

One can notice that the evolution of the diverter of the constraints is radial:

D
1
2

= eq - (ex ex + ey ey) + ez E
and
Z
= - S

eq

éq 2.1-3
3
3


Having proposed a field of displacements kinematically acceptable and a stress field
statically acceptable, it any more but does not remain to show than they are bound by the law of behavior. Well
that it is about a nonlocal model, the law of flow preserves its usual form:

D
p
3
p
1
& = p&
= p - (E
E
E
E
E
E éq
2.1-4
X
X +
y
y)


+ Z Z
2 eq
2


The same applies to the relation stress-strain, where and µ is the coefficients of Lamé which
result from the Young modulus and the Poisson's ratio:

= (tr) Id + µ
2 (
p
-) éq
2.1-5
While deferring [éq 2.1-1], [éq 2.1-2] and [éq 2.1-4] in [éq 2.1-5], one deduces the expression from it from
constraints and of the deformations according to the only cumulated plastic deformation:

1 E


2 2 Z F


S =

p + Z F
= 1
+ 1
p






éq 2.1-6
1 - 2

1 - E



1 -
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The condition of coherence then makes it possible to determine p. Dans la mesure où the evolution of the diverter
constraints is radial and monotonous, one can directly determine the current state without having with
to integrate the rate of plastic deformation. Indeed, one can distinguish two zones in the structure:
one, 0 Z B, in which the plastic deformation is null and where the threshold of plasticity is not
reached, and the other, B Z L, where the plastic deformation is nonnull and the threshold reached. The border B
between these two zones is a new unknown factor of the problem. As follows:

0 Z B p = 0


éq
2.1-7
B Z L F = 0
According to [bib1], the threshold of plasticity has as an expression:
T
2
(
E E
H L
F p)
y
4
,
B
= eq - H p - + C p

where
H =
and
C =
éq
2.1-8
T
E - E
13
Moreover, always according to [bib1], the field of cumulated plastic deformation p is 1
C and of derivative
null at the edge of the structure. That implies in particular:

p (b) = 0
p (b) = 0
p (L) = 0
éq
2.1-9

Taking into account [éq 2.1-7] and [éq 2.1-8], p checks a linear differential equation of the second
command on the field B Z L:

E

1 - 2
y
C p (Z) +
+
()
éq
2.1-10
2 (1 -) H p Z

=
Z F -

1 -

The data of the 3 boundary conditions [éq 2.1-9] then makes it possible to determine p completely like
the position B of the free border.
To reduce the expressions, the following notations are introduced:

E
C
4 H
1 - 2
H = (
H
L
L
F
2 1 -) +
C =
= B
=
H
13 H
H (1 -)

éq 2.1-11
y
Z - L
Z - L
(Z) = Z -
C (Z) = cosh
S (Z) =




sinh


H
L
L
C


C

Then, after some calculations, one obtains the following expression for p:

L
S (b)
B
()
C
-
With =
p (Z) = (Z) + AC (Z) + BS (Z)
where
C (b)

éq 2.1-12

B = - LLC
As for the free border, it is given according to the level of loading by the equation
following (or conversely, the level of loading corresponding to a certain position of
free border):

y

S (b)
=
éq
2.1-13
H bS (b) + L
-
C (1
C (b))
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2.2
Results of reference

One examines the cumulated plastic deformation p, the total deflection, the equivalent constraint
and the horizontal constraint S at the point Z = L for various levels of loading which
eq
correspond to various positions of the free border.

B L
F (
3
NR/mm)
p



(
S MPa)
eq (
MPa)
0, 75
104, 811 963
1, 165.975 E-4 1, 623.833 E-3 111, 456 702
98, 167 224
0, 50
146, 159 407
6, 125.415 E-4 2, 521.534 E-3 123, 286 355
169,032 459
0, 25
250, 078 993
1, 905.213 E-3 4, 804.152 E-3 149, 717 896
350, 440 090
0, 00
875, 079 453
9, 693.407 E-3 1, 854.027 E-2 307, 704.531 1.442, 454 356

2.3
Uncertainties on the solution

It is about an analytical solution.

2.4 References
bibliographical

[1]
Lorentz E., Andrieux S.: With variational formulation for nonlocal ramming models. Int. J. Plas.,
15, pp. 119-138 (1999)
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3 Modeling
With

3.1
Characteristics of modeling

It is about a modeling 3D. The section of the cylinder is square. Linear work hardening is characterized
initially by a curve of work hardening (VMIS_ISOT_TRAC) then by its limit
of elasticity and its module (VMIS_ISOT_LINE), which makes it possible to test the two establishments.

3.2
Characteristics of the grid

The grid is carried out by GMSH. The meshs are of smaller size in the zone where one examines
the results (higher face). On the whole, 1749 TETRA10 are counted.

3.3 Functionalities
tested

Commands
DEFI_MATERIAU
NON_LOCAL
LONG_CARA
AFFE_MODELE
AFFE
MODELISATION = “3d_GRADIENT”
STAT_NON_LINE
LAGR_NON_LOCAL

4
Results of modeling A

4.1 Values
tested

The various values are tested with the dimension Z = L, as presented to [§2.2]. It is about
cumulated plastic deformation (component “V1” of CHAM_NO “VARI_NOEU_ELGA” and component
“VANL” of CHAM_NO “DEPL”), vertical deformation (component “EPZZ” of the CHAM_NO
“EPSI_NOEU_DEPL”), of the constraint of von Mises (component “VMIS” of the CHAM_NO
“EQUI_NOEU_SIGM”) and finally of horizontal constraint (component “SIXX” of the CHAM_NO
“SIEF_NOEU_DEPL”). For recall, the analytical values are as follows.

F (
3
NR/mm)
p



(
S MPa)
eq (
MPa)
104, 811 963
1, 165.975 E-4
1, 623.833 E-3
111, 456 702
98, 167 224
146, 159 407
6, 125.415 E-4
2, 521.534 E-3
123, 286 355
169,032 459
250, 078 993
1, 905.213 E-3
4, 804.152 E-3
149, 717 896
350, 440 090
875, 079 453
9, 693.407 E-3
1, 854.027 E-2
307, 704 531
1.442, 454 356

The results obtained differ extremely little from the analytical solution (lower relative difference
with the precision owing to lack of 0, 1%).

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HT-66/02/001/A

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5 Modeling
B

5.1
Characteristics of modeling

It is about a modeling 3D. The section of the cylinder is square. Linear work hardening is characterized
initially by a curve of work hardening (VMIS_ISOT_TRAC) then by its limit
of elasticity and its module (VMIS_ISOT_LINE), which makes it possible to test the two establishments.

5.2
Characteristics of the grid

The grid is carried out by GMSH. The meshs are of smaller size in the zone where one examines
the results (higher face). On the whole, 169 TRIA6 are counted.

5.3 Functionalities
tested

Commands
DEFI_MATERIAU
NON_LOCAL
LONG_CARA
AFFE_MODELE
AFFE
MODELING = “D_PLAN_GRADIENT”
STAT_NON_LINE
LAGR_NON_LOCAL

6
Results of modeling B

6.1 Values
tested

The various values are tested with the dimension Z = L, as presented to [§2.2]. It is about
cumulated plastic deformation (component “V1” of CHAM_NO “VARI_NOEU_ELGA” and component
“VANL” of CHAM_NO “DEPL”), vertical deformation (component “EPYY” of the CHAM_NO
“EPSI_NOEU_DEPL”), of the constraint of von Mises (component “VMIS” of the CHAM_NO
“EQUI_NOEU_SIGM”) and finally of horizontal constraint (component “SIXX” of the CHAM_NO
“SIEF_NOEU_DEPL”). For recall, the analytical values are as follows.

F (
3
NR/mm)
p



(
S MPa)
eq (
MPa)
104, 811 963
1, 165.975 E-4
1, 623.833 E-3
111, 456 702
98, 167 224
146, 159 407
6, 125.415 E-4
2, 521.534 E-3
123, 286 355
169,032 459
250, 078 993
1, 905.213 E-3
4, 804.152 E-3
149, 717 896
350, 440 090
875, 079 453
9, 693.407 E-3
1, 854.027 E-2
307, 704 531
1.442, 454 356

The results obtained differ extremely little from the analytical solution (lower relative difference
with the precision owing to lack of 0, 1%).

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HT-66/02/001/A

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7 Modeling
C

7.1
Characteristics of modeling

It is about an axisymmetric modeling 2D. The section of the cylinder is necessarily circular.
Linear work hardening is characterized initially by a curve of work hardening
(VMIS_ISOT_TRAC) then by its elastic limit and its module (VMIS_ISOT_LINE), which allows
to test the two establishments.

7.2
Characteristics of the grid

The grid is carried out by GMSH. With the difference in two preceding modelings, it is about one
regulated grid. The meshs are smaller sizes in the zone where one examines the results (face
higher). On the whole, 50 QUAD8 are counted.

7.3 Functionalities
tested

Commands
DEFI_MATERIAU
NON_LOCAL
LONG_CARA
AFFE_MODELE
AFFE
MODELING = “AXIS_GRADIENT”
STAT_NON_LINE
LAGR_NON_LOCAL

8
Results of modeling C

8.1 Values
tested

The various values are tested with the dimension Z = L, as presented to [§2.2]. It is about
cumulated plastic deformation (component “V1” of CHAM_NO “VARI_NOEU_ELGA” and component
“VANL” of CHAM_NO “DEPL”), vertical deformation (component “EPYY” of the CHAM_NO
“EPSI_NOEU_DEPL”), of the constraint of von Mises (component “VMIS” of the CHAM_NO
“EQUI_NOEU_SIGM”) and finally of horizontal constraint (component “SIXX” of the CHAM_NO
“SIEF_NOEU_DEPL”). For recall, the analytical values are as follows.

F (
3
NR/mm)
p



(
S MPa)
eq (
MPa)
104, 811 963
1, 165.975 E-4
1, 623.833 E-3
111, 456 702
98, 167 224
146, 159 407
6, 125.415 E-4
2, 521.534 E-3
123, 286 355
169,032 459
250, 078 993
1, 905.213 E-3
4, 804.152 E-3
149, 717 896
350, 440 090
875, 079 453
9, 693.407 E-3
1, 854.027 E-2
307, 704 531
1.442, 454 356

The results obtained differ extremely little from the analytical solution (lower relative difference
with the precision owing to lack of 0, 1%).

9
Summary of the results

Because of the positive character of work hardening, the problem exhibe not of instabilities, which confirms
good convergence of calculations. The validation relates thus to the law of behavior itself and
on the algorithm of integration of the nonlocal laws, from which the various components are activated
(primal and dual iterations observed). The remarkable agreement enters the values of reference
and those calculated shows the capacities of the algorithm when a fine precision is necessary (criteria
of convergence severe) while the size of the dealt with problems, particularly in 3D, seems
to prove its robustness.
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