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0)( =++

p

vb

dt

ddiv (2)

Where:

: is partial constraint tensor for the phase .

p : is the contribution of quantity of movement to the

phase, must have some 0ppp =++ aws

b : is the strength of volume of the phase.

Contributions of quantity of movement

p are governed by

the law of diffusion:

awpg

s,,)( =

+=

xvv

kp (3)

Where:

k > 0: represents the permeability.

g : The acceleration of the gravity.

p : is the intrinsic pressure.

3. ELASTOPLASTICS MISCELLANIES

The solid distortion rate s and rates of content in fluid

volume wv and av are decomposed in an elastic part and a

plastic part:

wavvv pe

s

p

s

e

s

,, =+=

+=

(4)

The function of load (f) and the potential plastic (g)dependent of the efficient constraint of Bishop (1959) that

holds amount of the suction in pores (difference of pressure

of air and the one of water), and of one whole of variable

intern, the suction (s) and (X) variable of ecrouissage:

II )( waa PPP += (6)

Where:

aP : Pressure of air,

wP : Pressure of water,

: Total constraint content.

Laws of outflows are defined with the help of a rule of

normality generalized applied to triplets ),,( wa PP and

),,( was

vv under the shape:

waP

gv

gp

s

p,; =

=

=

(07)

Where: 0 : Plastic multiplier.

The law of evolution of ecrouissage variable is applied for

under the shape:(s)0 == ;X (08)

Equations of behaviours under shape incrementales write

themselves:

++=

++=

=

aaawaws

aa

awawwws

ww

aepaw

epw

s

PaPav

PaPav

PP

:

:

:

I

I

A

(09)

==

= wa

K

K

s

s

s

DS

,)1(;)(

(10)

waK

CdsdfaDS

,,;;),,( ===

(11)

Where:

DSK : Module of compressibility of the skeleton,

sK : Module of compressibility of the constituent solid,

C : Material parameters.

With:

DSDSDS fg

HEEEA ::

1

= (12)

s

fg

H

DSw

epw

=

:

1EI (13)

s

fg

H

DSa

epa

=

:

1EI (14)

The H module is supposed positive in order to exclude the

case of materials bloquantses.

=>

+

+=

X

Ef

hgf

hHsDS

;0:: (15)

Where:H: modulates ecrouissage.

In what follows, the surface of load and the potential plasticare supposed of Cam-Clay modified model type. This surface

is elliptic in the plan of the efficient constraints P and

constraints deviatoricq defined by:

))((),,( 02

2

PPPPM

qsf s += (16)

sPP

PM

qsg ++

=2

2

),,( (17)

21):3

2(;

3

1 devdevqtrP == (18)

Where: 0,, PPP s are respectively middle pressures, pressure of

cohesion according to the suction, pressure of pre

consolidation for a value of suction data, Mslope of the state

criticizes (CCM model). Let's put:

=

fQ : The normal to the surface of load.

=

gP : Normal to the potential, here P = Q (associated

plasticity).

Journal of Advanced Research in Mechanical Engineering (Vol.1-2010/Iss.2)Abderahim / On the Instability by Localization of Deformation in / pp. 120-123

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4. VISCOPLASTICS MISCELLANIES

We use here the model of Duvaut and Lions (1971) adapted

by Loret and Prevost to the porous middles, and extent byHarireche et al(1991) in the analysis of the phenomenon of

localization of the distortion in the submissive samples to

impacteses of solid amplitude.

[ ] [ ]),,()(:1 1 XE sfTrDSvps

H=

(19)

)(1

XXX =

(20)

Where:

),,( Xsf : represents the answer of the solid elastoplastic ofreference.

Can be seen in particularTr

as the projection of the

constrainton the surface of load. The parameter

represents the time of relaxation and the H symbol

designates the function of Heaviside.

5. ALGORITHM OF RESOLUTION

We present below the procedure of regularization viscoplastic

introduced in a code of count by finished element and used inthe application that we will present in the following section:

)()1(;)exp(;; 121

tzzt

zknk

n

Tr=

===

):()1( 211k

n

DSkn

Trk

n

k

nzzz PE++=

)()1( 211k

nkn

kn

kn zzz ++=

6. NUMERIC RESOLUTIONS

We present in this section an example of application in order

to validate the method numeric clarification. The experienceconsists in submitting an oblong sample of dimension

L1=2.5m and L2=3.5m to an axial velocity of compression

on the two parallel faces. The speed is maintained constant

during all the interval of time in order to simulate an impact.

Strips take birth to the level of corners and propagate itself

until separation of the sample in four blocks. The point ofreference (A and A') represents the evolution of velocity of

displacement and pressures inside during the time of the

shearing strip, on the other hand the point (B and B ')

represents the evolution outside of this strip. Results of this

count are represented by the following figures, the difference

of the variation of pressure velocity (water, air) to the level ofpoints (A and B), represent the emigration of the fluid (water,

air) toward strips of shearing where appear a big dilatance,

one also notes an important difference between movements

of the solid and the fluid. The localized distortions are moreintense in the case of the material of which the constituent

solid is least compressible.

B

A

Fig. 1. Deformed mesh (compressible solid)

B

A

Fig. 2. Deformed mesh (incompressible solid)

0

0,5

1

1,5

2

2,5

3

3,5

4

100 200 300 400 500 600 700 800 900 1000

Time (s)

Velocity(m/s)

A

B

A'

B'

Fig. 3. Variation of the velocity of displacement of the solid

0

0,5

1

1,5

2

2,5

3

3,5

4

100 200 300 400 500 600 700 800 900 1000

Time (s)

Velocity(Kpa/s)

A

B

A'

B'

Fig. 4.Variation of the water velocity


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-5

-4

-3

-2

-1

0

1

2

3

4

100 200 300 400 500 600 700 800 900 1000

Time (s)

Velocity(Kpa/s)

AB

A'

B'

Fig. 5.Variation of the air velocity

7. CONCLUSION

Numeric experience results done on an oblong sample ofpartially saturated porous material showed that for two

porous materials no saturated correspondent to the even solid,distortions taking place in strips that matched to the same

time for the two materials, are more intense in the material ofwhich the constituent solid is least compressible. It is the

same observations observed in the works of Harireche et al(2000) done on the saturated porous materials.

REFERENCES

Gheris, A. (2002). Application du concept dtat critique la

modlisation des sols non satures par la mthode des

lments finis, Sminaire Nationale de Gnie Civil, CU,

TEBESSA.

Laffifi, B. (2001), Modlisation de la rponse transitoire dessols non saturs, Mmoire de magistre en gnie civil,

Centre Universitaire de TEBESSA

Harireche, O. Nouaouria, S. (2000). Sur linstabilit par

localisation de la dformation dans les gomatriaux

poreux satures, 1er Sminaire National de Mcanique

des Geomateriaux et Structures, U, GUELMA, pp 39-44Loret, B., Nasser, K. (2000). A three-phase model for

unsaturated soils, International Journal for Numerical

and Analytical Methods in Geomechanics, Vol 24 issue

11, pp 893-927

Harireche, O., Loret, B. (1992). 3D dynamic strain-

localization: shear-band pattern transition in solids,

European Journal of Mechanics: A/Solids, vol 11, N6,pp 733-751

AUTHORS PROFILE

Gheris Abderahim:

Msc in Mechanics of Soils and Structures, University of Tebessa,

Algeria 2002.

Bsc in Civil Engineering, University of Guelma, Algeria 1998

Teaching Areas

lecturer at University of Souk ahras, Algeria in: Soil Mechanics

Research Interests

member of Geotechnical Engineering and Hydraulics Laboratory,University of Souk ahras, Algeria

deep foundation and soil improviser

modeling the behavior of contaminated soils.


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JARME-4-2-2010