Law of Mohr-Coulomb - Code_Aster · 2017. 12. 14. · Code_Aster Version 12 Titre : Loi de...

Code_Aster Version 12

Titre : Loi de Mohr-Coulomb Date : 23/09/2016 Page : 1/21Responsable : KHAM Marc Clé : R7.01.28 Révision :

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Law of Mohr-Coulomb

Summary:

This document presents the method of resolution of the law of Mohr-Coulomb in Code_Aster.

Warning : The translation process used on this website is a "Machine Translation". It may be imprecise and inaccurate in whole or in part and isprovided as a convenience.Copyright 2017 EDF R&D - Licensed under the terms of the GNU FDL (http://www.gnu.org/copyleft/fdl.html)



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Contents1 Notations.............................................................................................................................................. 3

1.1 General data.................................................................................................................................. 3

1.2 Convention on the tensorial notations............................................................................................3

2 Formulation in terms of principal constraints........................................................................................6

3 Local integration of the law of Mohr-Coulomb......................................................................................7

3.1 Case where only one mechanism is active....................................................................................7

3.2 Case where two mechanisms are active........................................................................................8

3.2.1 Formulation of the solution...................................................................................................8

3.2.2 Choice of the second mechanism.......................................................................................11

3.3 Case of projection at the top of the cone.....................................................................................12

4 Form of the consistent tangent matrix in the principal base...............................................................14

4.1 Case where only one mechanism is active..................................................................................14

4.2 Case where two mechanisms are active......................................................................................14

4.3 Case of projection at the top of the cone.....................................................................................14

5 Form of the consistent tangent matrix in the total base.....................................................................15

5.1 Some results on the isotropic symmetrical tensors of order two..................................................15

5.2 Derived from an isotropic tensorial function of order two............................................................15

5.2.1 Two-dimensional case of type forced plane (C_PLAN)......................................................16

5.2.2 Two-dimensional case of plane deformations type (D_PLAN) and axisymmetric (AXIS)...17

5.2.3 Three-dimensional case......................................................................................................17

5.2.4 Application to the case of Mohr-Coulomb...........................................................................19

6 Bibliography........................................................................................................................................ 21




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1 Notations

1.1 General data

1≥ 2≥ 3 Principal constraints (in this order)

E Young modulus

Poisson's ratio

K=E

3 1−2

Elastic module of compressibility

G=E

2 1

Elastic modulus of rigidity

ϕ Natural angle of repose of material

ψ Angle of dilatancy of material

c Cohesion of material

s=sin (ϕ )

t=sin (ψ )

p=I 1

3=trace (σ )

3

Average constraint

p0 Convention of sign for the constraint in compression

e Tensor of elastic prediction constraints

ε=εe+d ε p Tensors of the deflections total, elastic and increment

of plastic deformation

d εvp=trace (d ε p ) Increment of the voluminal plastic deformation

d e p=d ε p−d εv

p

31

Increment of the deviatoric plastic deformation

d e p=∥d e p∥=√ 32d e p .d e p

Increment of deviatoric plastic deformation, orincrement of equivalent deformation normalizes

1.2 Convention on the tensorial notations

Vectors of the strains and the stresses in the principal base d 1 ,d 2 ,d3 are noted:

ε={ε1ε2ε3} and σ={

σ1σ2σ3} (1)

The tensor of elasticity C allowing to connect and in the principal base, such as =C . is written:




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C=[K +

43G K−

23G K−

23G

K−23G K + 4

3G K− 2

3G

K−23G K−

23G K+

43G ] (2)

With K the elastic module of compressibility and G the elastic modulus of rigidity. The strains and thestresses are of the symmetrical tensors of order two. One generally exploits this symmetry (six independentcomponents) by representing them by vectors of dimension six resulting from the projection of these tensors insuitable bases.The strains and the stresses given as starter and produced at exit of the resolution of the law of behavior areexpressed in the orthonormal base of the symmetrical tensors of order two, noted b̄ :

b={ex⊗e xe y⊗e ye z⊗ez

e x⊗e ye y⊗ex2

e x⊗e zez⊗ex2

e y⊗e zez⊗e y2

(3)

Where ( ex , e y ,ez ) represent the unit vectors of the total Cartesian base orthonormal, presumedly fixed. The

condensed expression of the tensors of the strains and the stresses projected in the base b is written:

ε̄={εxxε yyε zz

√2 εxy√2 εyz√2εxz

} and σ̄={σ xxσ yyσ zz

√2σ xy√2σ yz√2σ xz

} (4)

This writing reveals a term in √2 in front of the cross components. It allows:• To express the tensor of elasticity of order four of 81 components by a tensor of order two of 36

components; • To symmetrize this tensor of elasticity.

Indeed, while noting σ ij=C ijklεkl , its form projected in the base b becomes σ̄ i=C̄ ij ε̄ j , where one has the

following expression for C :

C=[C xxxx C xxyy C xxzz 2C xxxy 2C xxxz 2C xxyz

C xxyy C yyyy C yyzz 2C yyxy 2C yyxz 2C yyyz

C zzxx C zzyy C zzzz 2C zzxy 2C zzxz 2C zzyz

2C xyxx 2C xyyy 2C xyzz 2C xyxy 2C xyxz 2C xyyz

2C xzxx 2C xzyy 2C xzzz 2C xzxy 2C xzxz 2C xzyz

2C yzxx 2C yzyy 2C yzzz 2C yzxy 2C yzxz 2C yzyz

] (5)




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The condensed form (5) is not convenient to use because of the need for handling the terms in √2 at the timeas of matric operations. One prefers to him another writing, based on projection in base known as of Voigt,noted b and having the following expression:

b̃={e x⊗e xe y⊗e ye z⊗e ze x⊗e ye x⊗e ze y⊗ez

(6)

The condensed expression of the tensors of the strains and the stresses projected in the base of Voigt b iswritten:

ε̃={εxxε yyεzzεxyεyzεxz} and σ̃={

σ xxσ yyσ zzσ xyσ yzσxz} (7)

This writing makes it possible to be freed from the terms in √2 in front of the crossed components, and ismore convenient to use during the digital resolution of the law of behavior.




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2 Formulation in terms of principal constraints

This formulation is valid only under the assumption of one isotropy of material [79, 82]. Indeed, this condition isnecessary to guarantee that the method of radial return preserves the principal directions. Its interest lies in thefact that it simplifies the writing of the equations and authorizes of this fact of the very powerful methods ofresolution (bus quasi-analytical).

The elastic behavior is purely linear.

The surface of charge is characterized by six plans within the space of principal constraints (σ1 ,σ2 ,σ3) . Each

one of these plans is characterized by an equation of the type:

F 13 (σ1 ,σ3)=σ1−σ3+ (σ1+σ3 ) sinϕ−2c cosϕ=0 (8)

Where ϕ and c are data material and respectively characterize the natural angle of repose and the cohesionof material.

The law is nonassociated and the plastic potential of flow G13 associated with the surface of load F 13 iswritten in the same way:

G13 (σ1 ,σ3 )=σ1−σ3+(σ1+σ3 )sinψ−2c cosψ (9)

Where ψ is a data material and characterizes the angle of dilatancy of material.

When ψ=ϕ , the plastic law of flow becomes associated.

A chart of the surface of load of Mohr-Coulomb within the space of principal constraints is on figure 2-1 and inthe plan on figure 2-2.

It is observed that the six plans intersect two to two following an angular edge, and meeting at the top of thecone characterized by the equation:

p=c cot (ϕ ) (10)

These edges, six, as well as the top of the cone, form singularities which pose problems of digital integration,because the derivative of the surface of load are not defined in these places. One will discuss more far fromthe methods allowing to solve this difficulty.

Figure 2-1: Representation of the surface of load of Mohr-Coulomb in the space three-dimensional of the principal

constraints

Figure 2-2: Representation of thesurface of load of Mohr-Coulomb in theplan π diverters of the constraints (any

vector represented in this plancorresponds to a deviatoric constraint).




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3 Local integration of the law of Mohr-Coulomb

The rate of plastic deformation is given using the formula of Koiter:

d ε p=∑j=1

m

d λ j

∂G j

∂σ=∑

j=1

m

d λ j nG j (11)

Where m characterize the number of active mechanisms, equal to one, two or six following followingsituations:

• the final constraint is inside the surface of load, the point is regular and m=1 ;• the final constraint is on an edge of the cone, the point is singular and m=2 ;• the final constraint is neither inside the surface of load nor on an edge. It is then projected at the top of

the cone, the point is singular and m=6 ;The final constraint σ

+ is calculated starting from a noted elastic prediction σe and of a correction

σc=C .d ε p so that:

σ+=σ

e−d σ c=σ e−C .d ε p=σ e−∑

j=1

m

d λ jC .nG j (12)

Plastic multipliers d j are calculated by injecting the equation (12) in the equation (8), which gives:

∑j=1

m

d j [ C . nG j 1−C .nG j 3 C .nG j 1C . nG j 3 s ]= 1e−3

e 1

e3

e s−2ccos (13)

In what follows, one details the expressions corresponding to the various situations mentioned above. Theprocedure of resolution is recalled in the synoptic one figure 3.3-1.

3.1 Case where only one mechanism is active

One detects that the prediction activates a criterion when:

F 13(σ e )=σ1

e−σ3e+(σ1

e+σ3e ) s−2c cosϕ≥0 (14)

There is the following expression for the normal:

nG=⟨ t+1 0 t−1⟩ (15)

And thus:

C .nG=2 ⟨(K+G3 )t+G (K−23G)t (K +G3 )t−G ⟩ (16)

That one introduces into the equation (13), which becomes:

4 d λ [G+(K+G3 )t s ]=σ1e−σ3

e+(σ1

e+σ3

e ) s−2c cosϕ (17)

From where the plastic multiplier is deduced:

d =⟨F 13

e ⟩+

4 [GKG3 t s ] (18)

Where ⟨ ⟩+ indicate the positive part of a size. One obtains finally:




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{σ1

+

σ2+

σ3+}={

σ1e

σ2e

σ3e}− ⟨F 13

(σ e ) ⟩+2G+2(K +G3 )t s {

(K+G3 )t+G

(K−23G )t

(K+G3 )t−G} (19)

If one breaks up the rate of plastic deformation into a deviatoric part d εvp=trace (d ε p ) and a spherical part

d e p=d ε p−d εv

p

31 , one a:

{d εv

p=2td λ

d e p=d λ ⟨ t3 +1 −2t3

t3−1⟩ (20)

That is to say the following expression of the increment of equivalent deformation:

d e p=d λ √t 2+3 (21)

3.2 Case where two mechanisms are active3.2.1 Formulation of the solution

It is checked that + obtained starting from the preceding stage (19) always check:

σ1+≥σ2

+≥σ3

+ (22)

If it is not the case, then it is advisable to activate two mechanisms in the following way:

{σ2+≥σ1

+≥σ3

+⇒ F 13et F 23 actifs [ LEFT ]

σ1+≥σ3

+≥σ2

+⇒ F 13et F 12 actifs [ RIGHT ]

(23)

The definition of the mechanisms LEFT and RIGHT is purely conventional, and obeys the geometrical logicrepresented in figure 3.2.1-1.

Figure 3.2.1-1: Definition of the mechanisms LEFTand RIGHT


RIGHT σ

1 ≥σ

3 ≥σ

2

LEFT σ

2 ≥σ

1 ≥σ

3

σ 1 ≥σ

2 ≥σ

3

σ 1

σ 2 σ

3



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There are the following expressions for the normals:

nG13=⟨ t+1 0 t−1 ⟩ nG

12=⟨ t+1 t−1 0 ⟩ nG

23=⟨0 t+1 t−1 ⟩ (24)

Thus:

{C .nG

13=2⟨(K+G3 )t+G (K− 2

3G)t (K +G3 )t−G ⟩

C .nG12=2⟨(K+G3 )t+G (K+G3 )t−G (K−2

3G )t ⟩

C .nG23=2 ⟨(K−2

3G) t (K +G3 )t+G (K +G3 )t−G ⟩

(25)

One introduces these expressions into the equation (13). For the mechanism LEFT, one obtains the followingsystem to solve:

{4d λ13[G+(K+G3 )t s ]+2d λ23[G (1−t−s )+(2K−

G3 )t s ]=F 13

(σe )

2d λ13[G (1−t−s )+(2K−G3 )t s]+4d λ23[G+(K+G3 )t s ]=F 23

(σ e ) (26)

For the mechanism RIGHT, one obtains:

{4d λ13[G+(K +G3 )t s ]+2d λ12[G (1+t+ s )+(2K−

G3 )t s]=F 13

(σe )

2d λ13[G (1+ t+ s )+(2K−G3 )t s]+4d λ12[G+(K+G3 )t s]=F 12

(σ e )

(27)

For implementation the algorithmic, it is more pleasant to rewrite (26) and (27) as follows:

{Ad λ1+Bside d λ2

=F 1(σ e )

Bsided λ1+Ad λ2

=F 2(σe )

(28)

With the following plastic multipliers:

d λ1=d λ13 and d λ

2={d λ

23 [ LEFT ]d λ12 [ RIGHT ] (29)

For surfaces of load:

F 1=F 13 and F 2={F 23 [ LEFT ]F 12 [ RIGHT ] (30)

The expression of A :

A=4 [G+(K +G3 )t s] (31)

And the expression of Bside :




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Bside={2[G (1−t−s )+(2K−G3 )t s ] [ LEFT ]

2[G (1+t+s )+(2K−G3 )t s ] [ RIGHT ]

(32)

The solution of the system of equations (32) exist if and only if its determinant is nonnull, that is to say:

det=∣A Bside

Bside A ∣=A2−(Bside )

2≠0 (33)

One can show that never arrives for “physical” values of the parameters K , G , ϕ and ψ . One has then assolutions:

{d λ1=A F 1

(σe )−BsideF 2

(σe )

det

d λ2=A F 2

(σ e )−BsideF 1(σ e )

det

(34)

For the mechanism LEFT, one obtains finally:

{σ1

+

σ2+

σ3+}={

σ1e

σ2e

σ3e}−2d λ13 {

(K+G3 )t+G

(K−23G )t

(K+G3 )t−G}−2 d λ23 {(K−2

3G)t

(K +G3 )t+G

(K +G3 )t−G} (35)

And for the mechanism RIGHT :

{σ1

+

σ2+

σ3+}={

σ1e

σ2e

σ3e}−2d λ13{

(K+G3 )t+G

(K−23G )t

(K+G3 )t−G }−2 d λ12{(K+G3 )t+G

(K+G3 )t−G

(K−23G)t } (36)

Or, by simplifying the writing:

σ+=σ

e−2d λ1v t

1−2d λ2 v t

2 (37)

With the following plastic multipliers:

d λ1=d λ13 and d λ

2={d λ

23 [ LEFT ]d λ12 [ RIGHT ] (38)

And vectors:




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v t1={(K +G3 )t+G

(K− 23G)t

(K+G3 )t−G} and v t

2={((K−

23G )t (K +G3 )t+G (K +G3 )t−G) [ LEFT ]

((K+G3 )t+G (K +G3 )t−G (K−23G )t) [RIGHT ]

(39)

In the same way, the rate of plastic deformation is written:

{d εv

p=2t (d λ1

+d λ2 )

d e p=d λ1( t3+1 −2t3

t3−1)+d λ2 {(

−2t3

t3+1

t3−1) [ LEFT ]

( t3+1t3−1

−2t3 ) [ RIGHT ]

(40)

That is to say the following expression of the increment of equivalent deformation:

d e p=√((d λ1 )2+ (d λ2 )

2) (t 2+3)+(−t2+sign×6 t+3)d λ1d λ2 (41)

With sign={−1 [ LEFT ]+1 [RIGHT ]

.

3.2.2 Choice of the second mechanism

A simple criterion is proposed and purely geometrical to undoubtedly determine if the second mechanism, if itwould be activated, is on the left (LEFT) or on the right (RIGHT). The vector is defined tG perpendicular withthe direction of flow as represented in figure 3.2.2-1, in the following way:

{t−1−2t+1 }=tG⊥nG=nG13

={t+10t−1} (42)

Line passing by 0 and parallel with nG , represented in dotted lines, characterizes the states of stresses such

as σ . tG=0 . One thus has the following choice:

• If . tG0 and that a second mechanism must be active, this mechanism is necessarily located onthe right, that is to say the mechanism RIGHT;

• If . tG0 and that a second mechanism must be active, this mechanism is necessarily located onthe left, that is to say the mechanism LEFT;




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Figure 3.2.2-1: Selection criteria of the mechanisms LEFT and RIGHT

3.3 Case of projection at the top of the cone

It is checked that + obtained starting from the preceding stage (35) or (36) always check:

σ1+≥σ2

+≥σ3

+ (43)

If it is not the case, then it is appropriate to carry out a projection at the top of the cone of equation:

pc=c cotϕ (44)

One thus imposes:

{p+= pe−K d εv

p :=c cotϕ

σ + := p+ 1 (45)

In the same way, the rate of plastic deformation is written:

d εP=d λ1nG

13+d λ2nG

12+d λ3nG

23 (46)

That is to say:

{d εv

p=2t (d λ1

+d λ2+d λ3 )=

1K

( pe−c cotϕ )

d e p=d λ1( t3+1−2t

3t3−1)+d λ2( t3+1

t3−1

−2t3 )+d λ3(−2t

3t3+1

t3−1)

(47)




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Figure 3.3-1: Synoptic of resolution of the law of Mohr-Coulomb




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4 Form of the consistent tangent matrix in the principal base4.1 Case where only one mechanism is active

The coherent tangent matrix T in the principal base is obtained by deriving the equation (19), which gives:

d σ +=C .d ε−

2d F 13(σ

e )

A {(K+G3 )t+G

(K−23G )t

(K+G3 )t−G}=C .d ε−

2d F 13(σ

e )

Av t (48)

Knowing that:

d F 13(σ

e )=d σ1e−d σ3

e+(d σ1

e+d σ3

e ) s=2 {(K+G3 )s+G

(K− 23G)s

(K +G3 ) s−G}.d ε=2 v s .d ε (49)

One obtains:

d σ +=(C−

4Av t⊗v s⏟

D)

⏟T

.d ε (50)

Where A is given by the equation (22).

4.2 Case where two mechanisms are active

In the same way, the coherent tangent matrix T is obtained by deriving the equations (34) and (38), whichgives:

d σ +=(C−

4det (v t1⊗v s1+v t2⊗vs2−

Bside

A(v t

1⊗v s

2+v t

2⊗v s

1))⏟D

)⏟

T

.d ε (51)

Where A is given by the equation (31), Bside by (32), det by (33) and v by (39).

4.3 Case of projection at the top of the cone

According to the equation (45), one has crudely T=0 .




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5 Form of the consistent tangent matrix in the total base

The paragraph §4 allows to build the consistent tangent matrix in the principal base, noted T . It is advisablefrom now on to bring back this matrix in the total base (Cartesian), that one will note T̄ .

Notice important:

It should be noted that the construction of this consistent tangent matrix is a crucial stage at the same time forthe robustness and the performance of the algorithm:

• Firstly, it is perfectly known that such a matrix allows a quadratic rate of convergence for the process ofNewton;

• Secondly, this matrix gives an account of the rotation of the principal directions during an increment.Without it, the formulation of the law of Mohr-Coulomb in terms of principal constraints described in theparagraph §2 would not be complete, since principal constraints, maintained fixed during the localintegration of the law (§3), could not turn on the total level of the structure.

In this paragraph, one describes in detail the method allowing to build T from T .

5.1 Some results on the isotropic symmetrical tensors of order two

One defines by S3 the space of symmetrical tensors of order two in the vector space of dimension n=3 , and

tensors Y∈S3 and X ∈S3 such as:

Y (X ) :S3→S 3 (52)

The tensorial function Y ( X ) is known as isotropic if:

R .Y (X ) .R t=Y (R . X .R t ) (53)

Whatever the rotation R . The assumption of isotropy implies that Y and X are coaxial, i.e. qu‘they have

the same principal directions d =1,2,3 . One notes:

X=∑α=1

3

xα (d α⊗dα )⏟Eα

=∑α=1

3

xα Eα

Y (X )=∑α=1

3

yα (dα⊗d α )=∑α=1

3

yα Eα

(54)

Where yα= yα ( x1 , x2 , x3) and xα they represent eigenvalues of Y and X , respectively.

5.2 Derived from an isotropic tensorial function of order two

It is supposed that the isotropic tensorial function Y (X ) is differentiable compared to X , and his derivative

is defined D such as:

D ( X )≝d Y ( X )

d X (55)

Applied to the equation (41), the following expression is obtained:

D ( X )=∑α=1

3

(Eα⊗d yαd X

+ yαd Eα

d X )=∑α=1

3

( yα d Eα

d X+∑β=1

3∂ yα∂ xβ

Eα⊗d xβd X ) (56)




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5.2.1 Two-dimensional case of type forced plane (C_PLAN)

In dimension two (cases C_PLAN), the characteristic equation det (X−xα I )=0 give a quadratic equation of

the eigenvalues xα=1,2 of X following type:

xα2− I 1 xα+ I 2=0 with α=1,2 (57)

With:

{I 1=trace (X )=X 11+X 22

I 2=det (X )=X 11 X 22−X 12 X 21

(58)

The resolution of the spectral problem easily gives the following solutions for the eigenvalues:

{x1=I 1+√ I 1

2−4 I 2

2

x2=I1−√ I 1

2−4 I 2

2

(59)

And clean vectors, taking account of the multiplicity of the eigenvalues:

{Eα=X + ( xα− I 1) I

2 xα−I 1

si x1≠x2

E1=I si x1=x2

(60)

In particular, Carlson and Hoger show that if x1≠x2 , one a:

d xαd X

=Eα (61)

By using the equations (59), (60) and (61) in (56), the expression of the derivative is obtained D ( X ) , takingaccount of the multiplicity of the eigenvalues:

D (X )={y1−y2

x1−x2[ I S−E1⊗E 1−E2⊗E 2 ]+∑

α=1

2

∑β=1

2∂ yα∂ xβ

Eα⊗Eβ si x1≠x2

(∂ y1

∂ x1

−∂ y1

∂ x2)I S+∂ y1

∂ x2

I⊗ I si x1=x2

(62)

With the matrix identity I :

I ijkl= ik jl (63)

The matrix of transposition I t ijkl=il jk and symmetrization stamps it I S , such as:

( I S )ijkl=12( I+ I t )=

12 (δik δ jl+δil δ jk ) (64)

Note:

It is noticed that the term y1− y2

x1−x2[ I S−E 1⊗E1−E2⊗E 2 ] in the derivative D ( X ) first equation of (62)

express the rotation of the principal directions in the plan.




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5.2.2 Two-dimensional case of plane deformations type (D_PLAN) and axisymmetric(AXIS)

The direction out-plan α=3 being fixed, the expression of the derivative D ( X ) is obtained starting from the

preceding case. Indeed, by isolating the term α=3 in the equation (56), there is the following expression:

D ( X )=∑α=1

2

( yα d Eα

d X+∑β=1

2∂ yα∂ xβ

Eα⊗d xβd X )⏟

D2D (X )

+∑α=1

2∂ yα∂ x3

Eα⊗d x3

d X+∑β=1

3 ∂ y3

∂ xβE3⊗

d xβd X⏟

D3 (X )

(65)

Where D2D ( X ) is given by the equation (56). The complementary term D3 ( X ) is written, by taking accountof the multiplicity of the eigenvalues, in the following way:

D3 ( X )={∑α=1

2

(∂ yα∂ x3

Eα⊗E3+∂ y3

∂ xαE3⊗Eα)+∂ y3

∂ x3

E 3⊗E3 si x1≠x2

∂ y1

∂ x3

I p⊗E3+∂ y3

∂ x1

E3⊗I p+∂ y3

∂ x3

E3⊗E3 si x1=x2

(66)

Where I p is the matrix of the orthogonal projection of I S in the plan e x ,e y :

I p={12

(δik δ jl+δilδ jk ) si i , j , k , l∈{1,2}

0 sinon (67)

5.2.3 Three-dimensional case

In dimension three, the characteristic equation det (X−xα I )=0 give a cubic equation of the eigenvalues

xα=1,2 ,3 of X following type:

xα3− I 1 xα

2+ I 2 xα−I 3=0 with α=1,2 ,3 (68)

With:

{I 1=trace (X )

I 2=12[ trace (X )2−trace (X 2 ) ]

I 3=det ( X )

(69)

The resolution of the spectral problem easily gives the following solutions for the eigenvalues:

{x1=−2√Q cos (θ3 )+

I 1

3

x2=−2√Q cos(θ+2π3 )+

I 1

3

x3=−2√Q cos(θ−2 π3 )+

I 1

3

(70)

Where Q and θ are given by:




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Q=I 1

2−3 I 2

9 and θ=cos−1( R√Q3 ) (71)

With:

R=−2 I 1

3+9 I 1 I 2−27 I 3

54 (72)

And clean vectors, by taking account of the multiplicity of the eigenvalues:

{Eα=xα

2 xα3−I 1 xα

2+ I 3

[X 2+( xα−I 1 )X +I 3

xαI ] si x1≠x2≠x3

Eβ= I−Eα si xα≠xβE1=I si x1=x2=x3

(73)

In the second equation of (73), E is calculated with the assistance the first equation. Without giving the

intermediate stages of calculation, the derivative D ( X ) , by taking account of the multiplicity of theeigenvalues, is written finally:

D ( X )={∑α=1

3 yα( xα−xβ) ( xα−xγ ) [

d X 2

d X−( xβ+xγ ) I S

−(2 xα−xβ−x γ )Eα⊗Eα−( xβ−xγ ) (Eβ⊗Eβ−Eγ⊗Eγ ) ]

+∑a=1

3

∑b=1

3 ∂ ya∂ xb

Ea⊗Eb

si x1≠x2≠x3

s1d X 2

d X−s2 I S−s3 X⊗X + s4 X⊗ I+ s5 I⊗X−s6 I⊗I si xα≠xβ=xγ

(∂ y1

∂ x1

−∂ y1

∂ x2) I S+∂ y1

∂ x2

I⊗I si x1=x2=x3

(74)

Where (α ,β ,γ ) corresponds to a cyclic permutation of (1 ,2 ,3) . I and I S are given by the equations

(63) and (64), respectively. By noticing that X is a tensor symmetrical, care should be taken to apply

itsymmetrical operator of derivation for the evaluation of d X2

d X , which gives the following form:

( d X2

d X )ijkl

=d (X i m X mj )d X kl

=12 (δik δlm+δil δkm ) X mj+

X im

2 (δmk δ jl+δml δkj )

=12(δik X lj+δil X kj+δ jl X ik+δkj X il )

(75)

Lastly, expressions of si=1,6 are the following ones:




b6e63cc29956

s1=yα−yγ

( xα−xγ )2+

1xα−x γ (

∂ yγ∂ xβ

−∂ yγ∂ xγ )

s2=2 xγyα−yγ

( xα−xγ )2+xα+x γxα−x γ (

∂ y γ∂ xβ

−∂ y γ∂ xγ )

s3=2yα−y γ

( xα−xγ )3+

1

( xα−x γ )2 (∂ yα∂ xγ

+∂ yγ∂ xα

−∂ yα∂ xα

−∂ yγ∂ xγ )

s4=2 xγyα−y γ

( xα−xγ )3+

1xα−xγ (

∂ yα∂ xγ

−∂ yγ∂ xβ )+

x γ

( xα−xγ )2 (∂ yα∂ x γ

+∂ yγ∂ xα

−∂ yα∂ xα

−∂ y γ∂ xγ )

s5=2 xγyα− yγ

( xα−xγ )3+

1xα−x γ (

∂ y γ∂ xα

−∂ yγ∂ xβ )+

x γ

( xα−x γ )2 (∂ yα∂ xγ

+∂ yγ∂ xα

−∂ yα∂ xα

−∂ yγ∂ xγ )

s6=2 xγ2 yα− yγ

( xα−xγ )3+

xα xγ

( xα−xγ )2 (∂ yα∂ xγ

+∂ yγ∂ xα )−

xγ2

( xα−xγ )2 (∂ yα∂ xα

+∂ y γ∂ xγ )−

xα+ xγxα−xγ

∂ yγ∂ xβ

(76)

Where , , corresponds to a cyclic permutation of 1 ,2 ,3 .

Note:

It is noticed that the following term (all but commonplace):

∑α=1

3 yα( xα−xβ) ( xα−xγ ) [

d X 2

d X−( xβ+xγ ) I S−(2 xα−xβ−xγ )Eα⊗Eα

−( xβ−xγ ) (Eβ⊗Eβ−Eγ⊗E γ ) ] (77)

Who appears in the derivative D X first equation of (74) express the rotation of the principal directions inspace three-dimensional.

5.2.4 Application to the case of Mohr-Coulomb

The transposition of the preceding formulas to the numeric work implementation deserves some precisedetails. There are first of all the following correspondences:

• X=ε̃ pred and xα=εαpred ;

• Y=σ̃ + and yα=σα+ ;

• Eα=d̃ αpred⊗d̃ α

pred ;

• (T )αβ=∂ yα∂ xβ

is the consistent tangent matrix in the principal base calculated in the paragraph §4 ;

The notation pred indicate that one works with “predicted” sizes given as starter by the process of Newton, the

notation + with sizes resulting from the local resolution of the law of behavior, and the notation with the

base of Voigt. It will be noted that the predicted principal directions d

pred are fixed during the local resolution,

which is coherent with the assumption of isotropy adopted (see the explanations of the paragraph §5.1).

Having all this information at the conclusion of the local resolution of the law of behavior, one from of deducedthe consistent tangent matrix T= D expressed in the base of projection b defined in the paragraph §1.2 :

• The equation (62) in the two-dimensional case in plane constraints (C_PLAN);• The equation (66) in the two-dimensional case in deformation planes (D_PLAN) or axisymmetric

(AXIS);• The equation (74) in the three-dimensional case (3D);




b6e63cc29956

The second important information relates to the convention of writing of the various tensors. Indeed, bypreoccupation with general information, a notation used for the tensors in all the paragraph §5 is the classicalnotation, revealing of the tensors until the order four. This writing is unsuitable with the digital resolution, whereone prefers to use condensed notations made possible by the fact that one works with tensors symmetrical oforder two (constraints and deformations are it always). One distinguishes two forms from notations condensedcorrespondent at two bases of projection (see §1.2):

• The orthonormal base b symmetrical tensors of order two. It is in this base that the constraints andthe deformations are given to the entry and the exit of the local resolution of the behavior;

• The base known as of Voigt b , much more convenient to use at the time of the local digital resolution

of the behavior because it avoids having to handle coefficients in √2 at the time as of matricoperations;

The diagram of resolution is summarized in figure 5.2.4-1.

Figure 5.2.4-1: Process of resolution of the lawof Mohr-Coulomb. Description of the writing in

the various bases of projection.




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6 Bibliography

[78] Jiang Hua, 11th Y. “with note one the Mohr-Coulomb and Drucker-Prager strenght criteria”, Mechanics Research Communications, 38, pp, 309-314, 2011

[79] Fotios E. Karaoulanis “Nonsmooth multisurface plasticity in principal stress space”, 6HT International GRACM Congress one Computational Mechanics, Thessaloniki, 2008

[80] Pankaj, Bićanić NR. “multiple Detection of activates yield conditions for Mohr-Coulombelasto- plasticity”, Computers and Structures, 62, pp, 51-61, 1997

[81] S.W. Sloan, Booker J.R. “Removal of singularities in Tresca and Mohr-Coulomb yield functions”, Communications in Applied Numerical Methods, 2, pp, 173-179, 1986

[82] Ronaldo I. Borja, Sama K.M., Sanz P.F. “One the numerical integration of three-invariant elastoplastic constitutive models”, Comput. Methods Appl. Mech. Engrg., 192, pp, 1227-1258.2003

[83] X. Wang, Wang L.B., Xu L.M. “Formulation of the return mapping algorithm for elastoplasticsoil models”, Computers and Geotechnics, 31, pp, 315-338, 2004


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