# A finite strain theory for elastic-plastic deformation

## A finite strain theory for elastic-plastic deformation

Int. J. Non-Lmear Mechonrcs, Vol. 6, pp. 435-450 A FINITE STRAIN Pergamon Press 1971. Printed in Great Britain THEORY FOR ELASTIC-PLASTIC DEFORM...

Int. J. Non-Lmear

Mechonrcs,

Vol. 6, pp. 435-450

A FINITE

STRAIN

Pergamon Press 1971. Printed in Great Britain

THEORY FOR ELASTIC-PLASTIC DEFORMATION

J. B. HADDOW and T. M. HRUDEY University of Alberta, Edmonton, Canada

Abstract--An elastic-plastic theory that is applicable when the elastic part of the strain is finite is proposed. A flow rule for an incompressible solid is obtained from Drucker’s postulate 111.Isothermal simple shear of a material which is neo-Hookean both before yielding and during elastic unloading after yielding is considered as an application of the theory. The problem is solved for two yield conditions and associated flow rules.

1. INTRODUCTION CLASSICAL elastic-plastic

theory is largely based on the assumption that all deformations are small and that the strain is described by the infinitesimal approximation [2], although it has been applied to some problems that involve finite elastic-plastic deformation with the elastic part of the strain infinitesimal, for example the expansion of a spherical cavity in an infinite medium [3]. Clearly, application of the classical theory is restricted to materials for which the shear yield stress k is very small compared with the modulus of rigidity p. For most metals k/p is of the order of magnitude of 10e3, consequently the elastic shear strain during plastic flow is considered infinitesimal. However, if the negative hydrostatic part of the stress is not negligible compared with the bulk modulus, the dilatational elastic strain is finite since it has been found experimentally that hydrostatic stress does not produce plastic flow. Another situation that involves finite elastic strain during plastic flow, is the plastic deformation of a material that yields after finite elastic shear strain. Whiskers of certain non-metallic crystalline materials have been found to behave elastically at tensile strains up to 4 per cent, and whiskers of tin have been bent elastically to a strain of 2 per cent [4]. These elastic strains are too large for the infinitesimal approximation to be satisfactory, and classical elastic-plastic theory is not strictly applicable to consideration of any elastic-plastic flow that occurs between the elastic limit and fracture. Some plastics and elastomers undergo permanent set after finite elastic shear deformation, and a finite strain elastic-plastic theory may be applicable to such materials for certain ranges of temperature and deformation rate. The purpose of this paper is to present a theory for elastic-plastic flow when the elastic part of the strain is finite. The kinematic and thermoelastic theory is valid for finite elastic shear and dilatational strains, and for both perfectly plastic and work-hardening solids that may be plastically anisotropic but the yield conditions discussed and the problem which is solved are for an incompressible perfectly plastic solid. An extension of the theory to consider elastically anisotropic solids is possible but is not considered here. Lee [5] has proposed a finite deformation elastic-plastic theory which uses a different approach from the theory presented here although both theories are based on three 435

436

J. B. HADDOW and T. M. HRUDEY

configurations of the body. Green and Naghdi [6] have also proposed an elastic-plastic theory and this theory appears to be fundamentally different from that in this paper. Certain of the assumptions of classical elastic-plastic theory are retained in this paper, namely that there is no plastic volume change. and that the elastic constants are unchanged ‘myprevious plastic straining. 2. KINEMATICS

Three configurations of an initially homogeneous body. the initial unstrained and unstressed configuration at the uniform reference temperature, the unstressed plastically strained configuration at the reference temperature and finally the stressed elastic-plastic configuration. are henceforth called C.l. C.2, and C.3 respectively. C.2 differs from C.l by at most a rigid body displacement if no plastic deformation has occurred. The concept of the unstressed C.2 at uniform temperature was introduced by Lee [5] and the use of three configurations to consider plastic deformation has also been suggested by Backman [7] and Sedov [8]. If non-uniform elastic-plastic deformation has occurred, in general. residual stresses will remain after the body has been unloaded and returned to the reference temperature. C.2 is then obtained by imagining every infinitesimal element of the body to be isolated from the surrounding material when unloaded from the elastic-plastic state so that it is stress free. A distribution of residual plastic strain is then obtained which, in general, does not satisfy compatibility relations and the space corresponding to C.2 is non-Euclidean. Let the coordinates of a particle with respect to a fixed set of rectangular Cartesian axes be X1, xcr. xi for C.l. C.2, and C.3 respectively. Upper case Latin subscripts refer to Cl, lower case Greek subscripts to C.2 and lower case Latin subscripts to C.3. The deformation gradient for the elastic-plastic deformation of C.3 is

(1) If the space corresponding to C.2 is Euclidean the deformation C.2 and C.2 to C.3 are the plastic deformation gradient F(P) = % &I ?X, and the thermoelastic

deformation

relating

C.l to

(2)

Also F,, = Fiz’ F(“) ar

(4)

If the space corresponding to C.2 is non-Euclidean the displacement gradients Fi!,) and Fjz’ cannot be expressed as partial derivatives as in equations (2) and (3) and the Pfaffian forms F\$)dX, and Fi”,)dXmare not exact differentials, but equations (1) and (4) are still valid. To simplify the presentation, the remainder of this paper is restricted to consideration of uniform plastic deformation and temperature distribution. consequently the space corresponding to C.2 is Euclidean and C.2 and C.3 are uniform. If this restriction is removed and the space corresponding to C.2 is non-Euclidean, the results obtained in this section are

Strain theory for elastic-plastic

437

deformation

valid provided velocity and deformation gradients that involve partial derivatives. Since the material is assumed to be plastically incompressible,

C.2 are not expressed

(5)

[1*

d& e ?X,

as

=1

and

where pO is the density corresponding to C. 1 and C.2 and p is the density corresponding to C.3. The (Lagrangian) strain tensors. referred to C.l, for the deformations of C.2 and C.3 are the plastic strain

(6) and the total strain 1

(7)

E,,y

respectively. Material differentiation, of equation (7) gives

that is differentiation

with respect to time? with x1 held constant,

where

is the rate of deformation tensor and ui = .ti. A superimposed tiation. Material differentiation of equation (6) gives

dot denotes

material

differen-

where

d\$‘=;(\$+\$) is the plastic rate of deformation

(9)

tensor and v, = 2,. Since there is no plastic volume change d(p) aa = 0 .

7 Since elastic-plastic deformation is inviscid, monotically varyingparameterifinertiaeffectsare

material differentiation neglected.

may be with respect

to any sutable

438

J. B.

This relation can be obtained referred to C.2 is

from equation

and material

differentiation

and T. M.

and substitution

HRUDEY

(5). The elastic

of equation

(Lagrangian)

strain

(8) gives the elastic

tensor

strain

rate

(11) C.2 is defined

only to within a rigid body rotation,

consequently

the plastic spin

is not uniquely defined. Since I?\$ given by equation (11) depends on the plastic spin W\$ it is not a satisfactory elastic strain rate. For example if dij = 0 and d\$ = 0 but W\$ # 0 then I\$\$ # 0 although neither elastic nor plastic deformation is occurring. An elastic strain rate tensor which is independent of W\$ and vanishes if dij = 0 and dhP,’= 0 is obtained as follows. The plastic deformation gradient (2) may be expressed in the polar form.

where Rpd is a proper such that

orthogonal

tensor

and [email protected] IS . a symmetric

The components of elastic strain given by equation of C.2 but the elastic strain components

(10) depend

positive

definite

tensor

on the rigid body rotation

EgL = R\$ Rf” ,E\$

(12)

are unchanged by rigid body rotation of C.2. Material differentiation of (12) gives

where

is an antisymmetric

tensor and _Q\$ zz &\$ + [email protected]! zy E!\$ 91 W!\$ - W(P)

is the co-rotational

derivative

of E\$’ and is independent

of the spin of C.2.

(14)

Struin theory for elustic-plustic

Using equations

(8). (9) and (11) equation

439

deform&on

(14) becomes

(15) The elastic rate of deformation

tensor is defined

as (16)

andsubstituting(l5)gives d!” = d..1, _ d?. 11’ 1, where (17) The components of d,*ido not depend on the rotation of C.2, nor does dt”jdepend on the spin flow with no of either C.2 or C.3. Also d\$ = 0 if ,!?\$1 = 0 and d\$ = 0 for elastic-plastic plastic volume change. These properties indicate that d; is a suitable definition of plastic rate of deformation. However it is shown in the next section that if the solid is elastically anisotropic the definitions of elastic and plastic rates of deformation given by equations (16) and (17) are not suitable.

3. THERMOELASTIC

It is convenient given by

to introduce

CONSIDERATIONS

the Piola-Kirchhoff

where aEj is the true stress referred depend on the rigid body rotation

stress tensor

S,, referred

to C.2 and

to C.3. The components of stress given by equation of C.2, but the stress components s,,

= R\$jj \$2

s,,

are unchanged by rigid body rotation of C.2. It is assumed that the elastic constants and heat capacity are unchanged plastic flow, consequently there is an elastic free energy function unit mass, [email protected]) = [email protected]) (E&

T).

(18)

(19) by previous

(20)

that is independent of previous plastic straining and the free energy is the sum of two parts, A’“’ and a part which is a function of the plastic strain history and perhaps T but not Egi. For an elastically isotropic solid [email protected]) is a function of the invariants of EEL and T and it may then be represented by [email protected]) = ,J’“‘(Eb’B, 7) since Rzd is a proper

orthogonal

tensor.

J. B. HADDOW and T. M. HKUDEY

440

from an elastic-plastic

state is considered

s,,

it follows

from the theory

= P”;;

of

(21) KL

and

(22) where SC” is the elastic part of the entropy. may be replaced by

From

For an elastically

isotropic

solid equation

(21)

(20)

and substituting

equations

(21) and (22) gives [email protected]’ = S,,_&po

- Se’ i:

(23)

When the solid is elastically isotropic the tensors Ej\$ and S,, are coaxial and E\$ and S,, are coaxial consequently from (13). (14) and (19). s,, and equation

l!& = s,, E\$’ = s,, ;Ij\$’

(24)

(23) becomes [email protected]’ = s,, rj\$j /p, - Se’ ,i.

For isothermal

deformation

it follows from equation

(23) that

\$e’ = /+” where &’ is the rate of increase of elastic,strain this may be expressed as

= S

KL

\$e’

energy/unit

KLIPO

mass. If the solid is elastically

[email protected]’ = 0 ,J.d!“/ I, ! P by using equation

(16). (18) and (24). The total rate of stress work/unit

isotropic (25)

mass is

l+” .+ tip’ = aijdij/p, where tip’ = Oijd&‘p is the rate of plastic energy dissipation/unit

mass.

(26)

Strain theory for elstic-phstic

deform&on

If the solid is elastically anistropic the rate of increase of strain energy plastic energy dissipation are no longer given by equations (25) and (26).

4. YIELD

CONDITION

Perfectly plastic solids are considered tesimal. as in the classical elastic-plastic function

AND FLOW

441

and the rate of

RULE

in this section. If strains and rotations are infinitheory. the yield condition may be described by a f(0ij.T)

= 0

(27)

and the sign of f is chosen so that f < 0 represents the elastic domain. If the solid is plastically isotropic f is a function of the stress invariants. consequently it is independent of the rotation of the body and the yield condition may be represented by an equation of the form (27) when large strain and rotation occurs. However, if the solid is plastically anisotropic the form of the function f depends on the rigid body rotation of C.3, consequently equation (27) is not a satisfactory representation of the yield condition when deformation with finite rotation occurs. This difficulty may be avoided by referring the true stress to axes fixed in the material. The deformation gradient (1) may be expressed in the polar form 2.K -’ = Ri, u,,, ?XK where U,,

is a symmetric

positive

definite

tensor

such that

and Ri, is a proper orthogonal second order tensor which represents the rigid body rotation that follows the irrotational deformation described by ULK. The yield condition for an anisotropic solid may then be described by f

bKL?

T) = 0,

(28)

where gKL = RiKRjL~ij. and this form is independent of the rigid body rotation of C.3. This procedure was suggested by Green and Naghdi [6]. The components of the rate of deformation tensors may also be referred to axes fixed in the material,

and since Ri, is an orthogonal

tensor

o,,[d,,.

d;ZiL.dly”L] = oij[dij, d\$.-d\$)].

The plastic potential flow rule of classical plasticity may be obtained from Drucker’s postulate [l] which states that if a body is in an equilibrium state under an arbitrary set of body forces and surface tractions, and an external agency applies and removes an additional load under isothermal conditions, the work done by the external agency is positive during the loading and non-negative during the complete cycle of application and removal of the additional load. A plastic flow rule can be obtained from this postulate, for an incompressible elastically isotropic solid that undergoes finite elastic deformation, by modifying slightly the reasoning used to obtain the classical flow rule. Consider the body to be at

J. B. HADDOW and T. M. HRUDEY

442

temperature T and stress G& which is just within the yield surface in stress space, so that when the stress point moves to a stress G;(~ which is on the yield surface, as shown diagrammatically in Fig. 1. the additional elastic strain increment is infinitesimal. Let the external agency be such as to move the stress point isothermally from C& at time 0 to aZL at time t and then along the yield surface as shown in Fig. 1 to oLL at time t + 6t. Removal

FIG. 1. Diagrammatic representation of isothermal yield surface in stress space.

of the external agency returns the stress point to a& at time t*. The cycle ‘of loading and unloading of the external agency is such as to produce infinitesimal increments of strain so that changes in the configuration of the body can be neglected during the cycle of loading and unloading by the external agency. The total work done/unit volume by both the external agency and the original loading during the cycle is W =

i a,,d,,dz

+ ‘;f

o,,d,,dz

+

[

tibt 1. i

o,,d,,dr

Then. since d KL = d\$\$ + d& and d&, # 0 only from t to t + 6t.

f+dt

= where the symbol \$ denotes integration ments during the cycle are infinitesimal

\$a,,dj\$.dr

+

;c o,,d&dr

1 ,

(29)

around the complete cycle. Since the strain increthe elastic term in (29) is zero and

A part

of W is due to the original loading. From Drucker’s postulate w-w;30

&ruin theory for elustic-pbstic

deformution

443

consequently

and lim

w -

__~_

df-0

W”

=

6t

(4~

-

&)d&,

3 0.

According to equation (30) the yield surface must be convex and the vector in stress space that represents the diL must be parallel to the normal to the yield surface at the corresponding stress point. Thus

where 3, is a non-negative factor of proportionality. rule may be expressed in the form

When

d?F = & II

aqj

the solid is isotropic

the flow

(31)

The condition d& = 0 and the flow rule imply that the yield condition does not involve the hydrostatic part of the stress tensor. Two yield conditions are now considered for isothermal deformation of a material whose elastic behaviour is neo-Hookean. The strain energy function? for such a material is W = 2CEk’,

(32)

where C is an elastic constant and 2C is equivalent to the classical shear modulus for small deformation. If yielding occurs when the true octahedral shearing stress reaches a critical value the von Mises yield condition +aijgij

= k2,

Or

f = Oijaij - 2k2 = 0,

(33)

where alj = gij - (crkk/3)hij is the deviatoric part of the stress and k is the shearing stress which acting alone would produce yielding, is valid as in classical plasticity. In classical plasticity another interpretation of the Mises yield condition is that it implies that yielding begins when the shear strain energy reaches a critical value. If yielding of a neo-Hookean solid begins when the shear strain energy/unit volume reaches a critical value W* then the yield condition may be expressed in the form w = w*,

(34)

where the strain energy must be expressed as function of cij. It is more convenient to express this yield condition in a different form. Since the neo-Hookean solid is incompressible and t For the incompressible rather

than per unit mass.

neo-Hookean

solid the symbol

Wis used here to denote strain energy per unit volume

J. B. HADDOW and T. M. HRUDEY

444

isotropic. the maximum shear strain stress deviator. The relation

energy yield condition

involves

the invariants

of the

where J, = aij&/(8C2),

J, = &,~?,~5~~/(24C~)

and Icp) = 2E”’ aa + 3 * is easily obtained for a neo-Hookean solid. Using this relation and equations (32) and (34) the maximum shear strain energy yield condition may be expressed in the form

(35) It is easily verified that f < 0 if W < W*. For a solid that yields after infinitesimal strain

elastic

W N aijoij/(8C) and W* < C. Equation

(35) then becomes 35, - 3 7

and becomes

equivalent

+ 0( W*2/C2)

to the Mises yield condition

5. SIMPLE

SHEAR

= 0

when terms 0( W*2/C2) are neglected.

PROBLEM

Simple shear of a cuboid is considered. The sides in the undeformed C.l are parallel to the axes of a fixed rectangular Cartesian coordinate system. After deformation the position of a particle originally at X, is given by x1

=x, + KX,.

x2

=x2,

x3 =x3.

This is a homogeneous isochoric deformation and the parameter K = tan 0, where 8 is the angle of shear as shown in Fig. 2, is used as a time scale. The deformation gradient and its material derivative with respect to K are

Strain theory for ehstic-plustic

It is assumed that the deformation Hookean elastic behavior so that

is isothermal

aij = 2C[;\$’

deform&on

and

the solid

445

is isotropic

with neo-

\F\$) - 3 (\[email protected] FQ)&j].

(36)

The problem is solved for both the Mises yield condition and the maximum shear strain energy condition, and the associated plastic flow rules are obtained from equation (31).

FIG. 2. Simple shear ofcuboid.

A significant simplification in the analysis tensor is expressed in the polar form

results

if the elastic

deformation

F+” = l,‘/!:)R!“’ 111 LJ P’

(37)

where P\$’ is a proper orthogonal tensor which represents the rigid body rotation relative to C.2 and Vi;’ is a symmetric positive definite tensor such that

of C.3

l\$’ @? = F!“’ F(e), la ,a I

Substituting

(37) in (36) gives oij = 2C{ V\$’ V8) - 4 (YYi Vi5\$Jj>

which shows that the principal

axes of Oij and Vjy’ coincide.

Substituting

(38) (37) in (17) gives

d; = f[ l\$’ d& Vyj - 1 + vj, d&j vi”, - ‘1,

(39)

where [email protected] =

RIP,' R'f; d\$/PB.

(40)

For the isotropic yield conditions considered, flow rule (31) indicates that the principal axes of d\$ and ~ij coincide, consequently for the special case when the solid is both elastically and plastically isotropic, the principal axes of d,*jand ViJ’ coincide and it can then be deduced from (39) that diy’ also has the same principal axes and that d?. = d!P’ EJ 13. The

(41)

equation [d\$‘]

= +[V(“‘- l+)

+ (V (e) - \$7(@)f _ V”’ - 1&F - lV(‘4 _ (V”’ - 1i;F - lV(e))f] ,

(42)

J. B. HADDOW and T. M. HRUDEY

446

which is expressed in matrix notation, with a prime denoting from equations (39), (40). (41) and the relations

the transpose,

can be obtained

and

Matrix notation is used henceforth in this section. (31) and equations (38) and (42) give V(e)- +e)

+ (V”‘- IV’@‘) = V&‘- 1kF- Ijib’) + (V”‘-

For the Mises yield condition, ‘i;F-

flow rule

IV”‘)

- 4XC[V’“‘2 - f fr(VCeJ2)I], matrix and tr denotes

where I is the identity

(43)

the trace of the matrix argument.

1.0 Y

0.8

z ;

0.6

5

FJ0.4 z E 0.2 z 0

-0.2

L

0

0.1

0.2

0.3

0.5

0.4

0.6

0.7

0.8

09

10

K FIG. 3. Stress deviator-K

Expressing gives

the yield condition

curves.

in terms

Mises yield condition

and k:C = 0.1.

of V”’ and differentiating

ry(V(@3V(C))_ ftr(V(e)‘)tr(V(r)\i(~))

= 0.

with respect

to K (44)

When K < K,, where K, = { - 3 + \$9

+ 3(k2/C2)]+}+

is the elastic limit value of K for the Mises yield condition, the stress deviator is obtained from equation (36). When K > K,, the system of non-linear ordinary differential equations (43) and (44), must be solved for I/@) and x as functions of K using the known value of I’@) at the elastic limit K, as an initial condition. The stress deviator as a function of K is then obtained from equation (38).

Struin theory for elustic-plustic

deformution

447

1.0

0.8 2 --g

2 a

0.6

0.4

3 E 0.2 2 z z

0

- 0.2

-0.4

0

0.1

0.2

0.4

0.3

0.5

K FIG. 4. Mises yield condition

For the maximum

shear strain

energy

cdg’] = i, [lz and substituting v”‘-

[email protected])

+

equations

(V(e)-

1+9)’

0.6

0.7

0.9

1.0

and k/C = 0.5.

yield condition

(w* c

0.8

the plastic flow rule (31) gives

+

(38) and (42) gives _

V”‘-

‘kF-

IV(e)

_

(V”‘-

1

@F-

‘V’“‘)

0.8

= E a

0.6

0.4

> ;

0.2

2 ii

0

z -0.2

0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

K FIG. 5. Mises yield condition

and k/C = 0.1.

1.6

1.8

2.0

J. B. HADDOW and T. M. HKUDEY

448

2 E z

0.2

0 -0.2

I I

0

‘_f

b,z/k=.?,,,/k

01

02

03

04

05

06

07

0.8

yield

condition

K FIG.

6.

Maximum

shear strain energy k/C = 0.1.

Expressing the maximum shear strain energy yield condition tiating with respect to K gives tr(VW3+e))

_

~t,(vce)2)tr(vcp’irce,)

W* c + 3 - ‘{ 3tr(V(45V(9

(

in terms of I/@) and differen-

+ [+trqV’y

-

,~v(e’“)ltr(v(c)irc~))

1

-

2tr(Vce)2)tr(Vce)3~ce))j = 0.

(46)

Equations (45) and (46) must be solved simultaneously to determine I/@) as a function of K after yielding, and the stress deviator is then obtained from equation (38). Numerical solutions were obtained for both yield conditions, and the results are shown graphically in Figs. 3, 4, 5, 6, 7, and 8. The stress, k, used to non-dimensionalize the stress deviator for the maximum shear strain energy yield condition material, is the shear yield stress of the Mises material with same elastic constant C and the same uniaxial yield stress. 1.0

1

0.8

.

W./C = 0.0566

DC

0

0.6

2 5

w 0

0.4

2 0.2 kY zl 0 02 t 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

K FIG. 7. Maximum

shear strain

energy

yield condition

k/C

= 0.5.

Struin theory

for elustic-plastic

deformation

449

The numerical results indicate that as k/C becomes smaller the normal stresses become smaller and become negligible for k/C -g 1. Also for k/C G 1 the results approach those obtained from the classical theory.

0

0.2

0.4

FIG. 8. Maximum

0.6

0.8

1.0

1.2

1.4

K shear strain energy yield condition

1.6 k/C

1.8

2.0

= 1.0.

For the Mises yield condition the deviator component 5 33 approaches zero for values of K large compared with K, and if gZ3 = 0 then @I 1 = - oz2 = O11 for large K. The maximum shear strain energy yield condition results in quite different normal stress effects during elastic-plastic flow. The relation 5 22 = a,, which holds for elastic deformation before yielding also holds for the elastic-plastic deformation for all K and if CJ~~= 0, oZ2 = Oanda,, = - 3~7~~.

REFERENCES approach to plastic stress-strain relations. Proc. First U.S. Nutionul Ul D. C. DRUCKER, A more fundamental Congress of Applied Mechunics, 487 (1952). I21 W. T. KOITER, General theorems for elstic-plastic solids, Progress in Solid Mechunics, ed. R. HILL and I. N. SNEDDON, Vol. 1. North Holland, Amsterdam (1960). Theory of Plasticity, Oxford University Press (1950). [31 R. HILL, Muthemuticul [41J. E. GORDON, The New Science of Strong Muteriuls, Penguin Books (1968). deformation at finite strains, J. uppl. Mech. 36, 1 (1969). I51E. H. LEE, Elastic-plastic continuum, Archs rution. Mech. b1 A. E. GREEN and P. M. NAGHDI, A general theory of an elastic-plastic Anulysis 18. 251 (1965). [71M. E. BACKMAN, Form for the relation between stress and finite elastic and plastic strains under impulsive loading, J. uppl. Phys. 35,2524(1964). (1955). PI L. I. SEDOV, Introduction to the Mechanics of u Continuous Medium, Addison-Wesley (Received 24 Februury 1970 ; revised 1 June 1970)

R&urn&On propose une theorie elasto-plastique qui est applicable lorsque la partie Clastique de la deformation est finie. On obtient une loi d’ecoulement pour un solide incompressible a partir du postulat de Drucker [I]. On examine, comme application de la theorie, le cisaillement simple isotherme d’un materiau qui est neoHookeen a la fois avant la limite tlastique et durant la decharge Clastique apres la deformation plastique. Le probleme est resolu pour deux conditions de limite Clastique et les lois d’koulement assocites.

450

J.B.