Code_Aster. Law of Mohr-Coulomb

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1 Titre : Loi de Mohr-Coulomb Date : 0/05/016 Page : 1/1 adaab1bf74 Law of Mohr-Coulomb Summary: This document presents the method of resolution of the law of Mohr-Coulomb in Code_Aster.

2 Titre : Loi de Mohr-Coulomb Date : 0/05/016 Page : /1 adaab1bf74 Contents 1 Notations General data Convention on the tensorial notations... Formulation in terms of principal constraints...6 Local integration of the law of Mohr-Coulomb Case where only one mechanism is active...7. Case where two mechanisms are active Formulation of the solution Choice of the second mechanism Case of projection at the top of the cone Form of the consistent tangent matrix in the principal base Case where only one mechanism is active Case where two mechanisms are active Case of projection at the top of the cone Form of the consistent tangent matrix in the total base Some results on the isotropic symmetrical tensors of order two Derived from an isotropic tensorial function of order two Two-dimensional case of type forced plane (C_PLAN) Two-dimensional case of plane deformations type (D_PLAN) and axisymmetric (AXIS) Three-dimensional case Application to the case of Mohr-Coulomb Bibliography... 1

3 Titre : Loi de Mohr-Coulomb Date : 0/05/016 Page : /1 adaab1bf74 1 Notations 1.1 General data 1 E E K = 1 G= E 1 ϕ ψ c Principal constraints (in this order) Young modulus Poisson's ratio Elastic module of compressibility Elastic modulus of rigidity Natural angle of repose of material Angle of dilatancy of material Cohesion of material s=sin (ϕ ) t=sin ( ψ ) p= I 1 = trace (σ ) p 0 e ε=ε e +d ε p d ε v p =trace (d ε p ) d e p =d ε p d ε p v 1 d e p = d e p = d e p.d e p Average constraint Convention of sign for the constraint in compression Tensor of elastic prediction constraints Tensors of the deflections total, elastic and increment of plastic deformation Increment of the voluminal plastic deformation Increment of the deviatoric plastic deformation Increment of deviatoric plastic deformation, or increment of equivalent deformation normalizes 1. Convention on the tensorial notations Vectors of the strains and the stresses in the principal base d 1, d, d are noted: ε= { ε 1 ε } { ε and σ= σ 1 σ } σ (1) The tensor of elasticity C allowing to connect and in the principal base, such as =C. is written:

4 Titre : Loi de Mohr-Coulomb Date : 0/05/016 Page : 4/1 adaab1bf74 =[ K C 4 + G K G K G ] K G K + 4 G K G K G K G K + 4 G () With K the elastic module of compressibility and G the elastic modulus of rigidity. The strains and the stresses are of the symmetrical tensors of order two. One generally exploits this symmetry (six independent components) by representing them by vectors of dimension six resulting from the projection of these tensors in suitable bases. The strains and the stresses given as starter and produced at exit of the resolution of the law of behavior are expressed in the orthonormal base of the symmetrical tensors of order two, noted b : e x e x e y e y e z e z e b={ x e y e y e x () e x e z e z e x e y e z e z e y Where ( e x, e y,e z ) 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: εxx ε yy ε zz ε ε={ } σ xx σ yy σ zz xy and σ ε yz ε xz σ={ xy (4) σ yz σ xz} This writing reveals a term in 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 6 components; To symmetrize this tensor of elasticity. Indeed, while noting =[ σ ij =C ijkl ε kl, its form projected in the base b becomes σ i = C ij ε j, where one has the following expression for C : C xxxx C xxyy C xxzz C xxxy C xxxz C xxyz C xxyy C yyyy C yyzz C yyxy C yyxz C yyyz C C zzxx C zzyy C zzzz C zzxy C zzxz C zzyz (5) C xyxx C xyyy C xyzz C xyxy C xyxz C xyyz C xzxx C xzyy C xzzz C xzxy C xzxz C xzyz C yzxx C yzyy C yzzz C yzxy C yzxz C yzyz ]

5 Titre : Loi de Mohr-Coulomb Date : 0/05/016 Page : 5/1 adaab1bf74 The condensed form (5) is not convenient to use because of the need for handling the terms in at the time of the matric operations. One prefers to him another writing, based on projection in base known as of Voigt, noted b and having the following expression: x e y y b={ex e e z z e x e y e x e z e y e z (6) The condensed expression of the tensors of the strains and the stresses projected in the base of Voigt b is written: xx ε yy σ yy ε ε={εxx zz ε and σ={σ σ zz xy ε yz ε xz} } σ (7) xy σ yz σ xz This writing makes it possible to be freed from the terms in in front of the crossed components, and is more convenient to use during the digital resolution of the law of behavior.

6 Titre : Loi de Mohr-Coulomb Date : 0/05/016 Page : 6/1 adaab1bf74 Formulation in terms of principal constraints This formulation is valid only under the assumption of one isotropy of material [79, 8]. Indeed, this condition is necessary to guarantee that the method of radial return preserves the principal directions. Its interest lies in the fact that it simplifies the writing of the equations and authorizes of this fact of the very powerful methods of resolution (bus quasi-analytical). The elastic behavior is purely linear. The surface of charge is characterized by six plans within the space of principal constraints (σ 1,σ, σ ). Each one of these plans is characterized by an equation of the type: F 1 (σ 1, σ )=σ 1 σ + ( σ 1 +σ )sin ϕ c cos ϕ=0 (8) Where ϕ and c are data material and respectively characterize the natural angle of repose and the cohesion of material. The law is nonassociated and the plastic potential of flow G 1 associated with the surface of load F 1 is written in the same way: G 1 (σ 1,σ )=σ 1 σ + ( σ 1 +σ )sin ψ c cos ψ (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 -1 and in the plan on figure -. It is observed that the six plans intersect two to two following an angular edge, and meeting at the top of the cone 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 from the methods allowing to solve this difficulty. Figure -1: Representation of the surface of load of Mohr- Coulomb in the space three-dimensional of the principal constraints Figure -: Representation of the surface of load of Mohr-Coulomb in the plan π diverters of the constraints (any vector represented in this plan corresponds to a deviatoric constraint).

7 Titre : Loi de Mohr-Coulomb Date : 0/05/016 Page : 7/1 adaab1bf74 Local integration of the law of Mohr-Coulomb The rate of plastic deformation is given using the formula of Koiter: m d ε p = j=1 m G d λ j j σ = d λ j n G j (11) j=1 Where m characterize the number of active mechanisms, equal to one, two or six following following situations: 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= ; 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 σ e and of a correction σ c =C. d ε p so that: m σ + =σ e d σ c =σ e C.d ε p =σ e d λ j C. n G j (1) Plastic multipliers d j are calculated by injecting the equation (1) in the equation (8), which gives: m d j [ C. n G j 1 C.n G j C.n G j 1 C. n G j s ]= e 1 e e 1 e s ccos (1) j =1 In what follows, one details the expressions corresponding to the various situations mentioned above. The procedure of resolution is recalled in the synoptic one figure Case where only one mechanism is active j=1 One detects that the prediction activates a criterion when: There is the following expression for the normal: F 1 (σ e )=σ 1 e σ e +(σ 1 e +σ e ) s c cos ϕ 0 (14) And thus: n G = t +1 0 t 1 (15) C. n G = ( K + G ) t+g ( K G ) t ( K + G ) t G (16) That one introduces into the equation (1), which becomes: 4 d λ[ G+ ( K + G ) ] t s =σ e 1 σ e +(σ e 1 +σ e ) s c cos ϕ (17) From where the plastic multiplier is deduced: d = F 1 e + 4[ G K G t s ] (18) Where + indicate the positive part of a size. One obtains finally:

8 Titre : Loi de Mohr-Coulomb Date : 0/05/016 Page : 8/1 adaab1bf74 + e {σ1 + e σ σ +}={σ1 σ σ e} F 1 (σ e ) + G+ ( K + G ) t s { (K G + ) t+g t G} ( K G ) t (19) ( K +G ) If one breaks up the rate of plastic deformation into a deviatoric part d ε v p =trace (d ε p ) and a spherical part d e p =d ε p d ε p v 1, one a: {d εv p =t d λ d e p =d λ t +1 t t 1 That is to say the following expression of the increment of equivalent deformation: (0). Case where two mechanisms are active..1 Formulation of the solution d e p =d λ t + (1) It is checked that + obtained starting from the preceding stage (19) always check: σ 1 + σ + σ + If it is not the case, then it is advisable to activate two mechanisms in the following way: () { σ + σ σ σ + 1 σ + + σ F 1 et F actifs [ LEFT ] F 1 et F 1 actifs [ RIGHT ] The definition of the mechanisms LEFT and RIGHT is purely conventional, and obeys the geometrical logic represented in figure () σ 1 σ σ σ 1 RIGHT σ 1 σ σ LEFT σ σ 1 σ σ σ Figure..1-1: Definition of the mechanisms LEFT and RIGHT

9 Titre : Loi de Mohr-Coulomb Date : 0/05/016 Page : 9/1 adaab1bf74 There are the following expressions for the normals: Thus: n G 1 = t+1 0 t 1 n G 1 = t+1 t 1 0 n G = 0 t+1 t 1 (4) {C. ng = ( 1 K + G ) t+g ( K G ) t ( K + G ) t G C. n 1 G = ( K + G ) t+g ( K + G ) t G ( K G ) t C. n G = ( K G ) t ( K + G ) t+g ( K + G ) t G (5) One introduces these expressions into the equation (1). For the mechanism LEFT, one obtains the following system to solve: {4 d λ 1[ G+ ( K +G ) t s ] +d λ [ G (1 t s )+ ( K G ) t s ] =F 1 ( σ e ) d λ 1[ G (1 t s)+ ( K G ) t s ] +4d λ [ G+ ( K +G ) t s ] =F ( σ e ) (6) For the mechanism RIGHT, one obtains: {4 d λ 1[ G+ ( K + G ) t s ] +d λ1[ G (1+t+ s )+ ( K G ) t s ] =F 1 ( σ e ) d λ 1[ G (1+t+ s )+ ( K G ) t s ] +4 d λ1[ G+ ( K + G ) t s ] =F 1 ( σ e ) For implementation the algorithmic, it is more pleasant to rewrite (6) and (7) as follows: With the following plastic multipliers: { A d λ 1 + B side d λ =F 1 (σ e ) B side d λ 1 + A d λ =F (σ e ) (7) (8) d λ 1 =d λ 1 and d λ ={ d λ [ LEFT ] d λ 1 [ RIGHT ] (9) For surfaces of load: F 1 =F 1 and F = { F [ LEFT ] F 1 [ RIGHT ] (0) The expression of A : A=4[ G+ ( K + G ) t s ] (1) And the expression of B side :

10 Titre : Loi de Mohr-Coulomb Date : 0/05/016 Page : 10/1 adaab1bf74 B ={[ G (1 t s )+ ( K G side ) ] t s [ LEFT ] [ G (1+t +s )+ ( K G ) ] t s [ RIGHT ] () The solution of the system of equations () exist if and only if its determinant is nonnull, that is to say: det= A Bside B side A = A (B side ) 0 () One can show that never arrives for physical values of the parameters K, G, ϕ and ψ. One has then as solutions: For the mechanism LEFT, one obtains finally: {d λ 1 = A F ( 1 σ e ) B side F (σ e ) det d λ = A F ( σ e ) B side F 1 (σ e ) det (4) + e {σ1 + e σ σ +}={σ1 σ σ e} d λ And for the mechanism RIGHT : + e {σ1 + e σ σ +}={σ1 σ Or, by simplifying the writing: σ e} d λ G 1{ (K + ) t+g ( K G ) t ( K +G ) t G} d λ G 1{ (K+ ) t+g ( K G t G} ) t d λ ( K +G ) { (K G ) t ( K +G ) t+g ( K +G ) t G} (5) G 1{ (K + ) t +G ( K +G ) t G ( K G ) t } (6) With the following plastic multipliers: σ + =σ e d λ 1 v t 1 d λ v t (7) d λ 1 =d λ 1 and d λ ={ d λ [ LEFT ] d λ 1 [ RIGHT ] (8) And vectors:

11 Titre : Loi de Mohr-Coulomb Date : 0/05/016 Page : 11/1 ={ adaab1bf74 v 1 t ( K G ) ((K t G ) t ( K +G ) t+g ( K +G ) ) t G [ LEFT ] (9) (( K +G ) t+g ( K +G ) t G ( K G ) ) t [RIGHT ] G (K + ) t+g t G} ={ and vt ( K + G ) In the same way, the rate of plastic deformation is written: { {d εv p =t (d λ 1 +d λ ) d e p =d λ ( 1 t +1 t t 1 ) ( t +d λ ( t +1 t t +1 t 1 ) [ LEFT ] That is to say the following expression of the increment of equivalent deformation: With sign= { 1 [ LEFT ]. +1 [RIGHT ] 1 t ) [ RIGHT ] (40) d e p = ((d λ 1 ) + (d λ ) ) (t +)+( t +sign 6t +)d λ 1 d λ (41).. Choice of the second mechanism A simple criterion is proposed and purely geometrical to undoubtedly determine if the second mechanism, if it would be activated, is on the left (LEFT) or on the right (RIGHT). The vector is defined t G perpendicular with the direction of flow as represented in figure..-1, in the following way: { t 1 } ={ t+1 =t n =n 1 G G G 0 (4) t+1 t 1} Line passing by 0 and parallel with n G, represented in dotted lines, characterizes the states of stresses such as σ.t G =0. One thus has the following choice: If. t G 0 and that a second mechanism must be active, this mechanism is necessarily located on the right, that is to say the mechanism RIGHT; If. t G 0 and that a second mechanism must be active, this mechanism is necessarily located on the left, that is to say the mechanism LEFT;

12 Titre : Loi de Mohr-Coulomb Date : 0/05/016 Page : 1/1 adaab1bf74 Figure..-1: Selection criteria of the mechanisms LEFT and RIGHT. Case of projection at the top of the cone It is checked that + obtained starting from the preceding stage (5) or (6) always check: σ 1 + σ + σ + If it is not the case, then it is appropriate to carry out a projection at the top of the cone of equation: (4) One thus imposes: { p+ p c =c cot ϕ (44) = p e K d ε p v :=c cot ϕ σ + := p + 1 In the same way, the rate of plastic deformation is written: (45) That is to say: d ε P =d λ 1 n G 1 +d λ n G 1 +d λ n G (46) {d εv p =t (d λ 1 +d λ +d λ )= 1 K ( pe c cot ϕ) d e p =d λ 1 ( t +1 t t 1 ) +d λ ( t +1 t t 1 ) +d λ ( t t +1 t 1 ) (47)

13 Titre : Loi de Mohr-Coulomb Date : 0/05/016 Page : 1/1 adaab1bf74 Figure.-1: Synoptic of resolution of the law of Mohr-Coulomb

14 Titre : Loi de Mohr-Coulomb Date : 0/05/016 Page : 14/1 adaab1bf74 4 Form of the consistent tangent matrix in the principal base 4.1 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: Knowing that: One obtains: d σ + =C.d ε d F 1 ( σ e ) A { d F 1 (σ e )=d σ 1 e d σ e +( d σ 1 e +d σ e ) s= { Where A is given by the equation (). G (K + ) t+g ( K G t G}=C ) t d F. d ε 1 (σ e ) v A t (48) ( K + G ) d σ + = (C 4 A v t v s D ) T G (K + ) s+g ( K G s G}.d ) s ε= v s. d ε (49) ( K + G ).d ε (50) 4. Case where two mechanisms are active In the same way, the coherent tangent matrix T is obtained by deriving the equations (4) and (8), which gives: =( d σ + C 4 det ( v 1 t v 1 s +v t v s Bside ).d ε D T A ( v t 1 v s +v t v s1 ) ) Where A is given by the equation (1), B side by (), det by () and v by (9). (51) 4. Case of projection at the top of the cone According to the equation (45), one has crudely T =0.

15 Titre : Loi de Mohr-Coulomb Date : 0/05/016 Page : 15/1 adaab1bf74 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 advisable from now on to bring back this matrix in the total base (Cartesian), that one will note T. Notice important: It should be noted that the construction of this consistent tangent matrix is a crucial stage at the same time for the 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 of Newton; 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 the paragraph would not be complete, since principal constraints, maintained fixed during the local integration of the law ( ), 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 S tensors Y S and X S such as: the space of symmetrical tensors of order two in the vector space of dimension n=, and The tensorial function Y ( X ) is known as isotropic if: Y ( X ) : S S (5) R. Y ( X ). R t =Y ( R. X. R t ) (5) 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,,. One notes: X = α=1 Y ( X )= α=1 x α (d α d α ) = α=1 E α x α E α y α (d α d α )= α=1 y α E α (54) Where y α = y α ( x 1, x, x ) and x α they represent eigenvalues of Y and X, respectively. 5. 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 Applied to the equation (41), the following expression is obtained: (55) D ( X )= α=1 ( E α d y α d X + y d E α α d X ) = α=1 ( y d E α α d X + β=1 y α x β E α d x β d X ) (56)

16 Titre : Loi de Mohr-Coulomb Date : 0/05/016 Page : 16/1 adaab1bf Two-dimensional case of type forced plane (C_PLAN) In dimension two (cases C_PLAN), the characteristic equation det ( X x α I )=0 give a quadratic equation of the eigenvalues x α=1, of X following type: With: x α I 1 x α + I =0 with α=1, (57) { I 1 =trace ( X )= X 11 + X (58) I =det ( X )= X 11 X X 1 X 1 The resolution of the spectral problem easily gives the following solutions for the eigenvalues: {x1= I 1 + I 1 4 I x = I 1 I 1 4 I And clean vectors, taking account of the multiplicity of the eigenvalues: (59) X + ( x α I 1 ) I In particular, Carlson and Hoger show that if x 1 x, one a: {E α = si x x α I 1 x 1 (60) E 1 =I si x 1 =x d x α d X =E α (61) By using the equations (59), (60) and (61) in (56), the expression of the derivative is obtained D ( X ), taking account of the multiplicity )={ of the eigenvalues: y 1 y y [ I x D ( X 1 x S E 1 E 1 E E ]+ α E α=1 β=1 x α E β si x 1 x ( β y 1 y 1 x 1 x ) I S+ y (6) 1 I I si x x 1 = x With the matrix identity I : I ijkl = ik jl (6) The matrix of transposition I t ijkl = il jk and symmetrization stamps it I S, such as: ( I S ) ijkl = 1 ( I + I t )= 1 ( δ ik δ jl +δ il δ jk ) (64) Note: It is noticed that the term express the rotation of the principal directions in the plan. y 1 y x 1 x [ I S E 1 E 1 E E ] in the derivative D ( X ) first equation of (6)

17 Titre : Loi de Mohr-Coulomb Date : 0/05/016 Page : 17/1 adaab1bf Two-dimensional case of plane deformations type (D_PLAN) and axisymmetric (AXIS) The direction out-plan α= being fixed, the expression of the derivative D ( X ) is obtained starting from the preceding case. Indeed, by isolating the term α= in the equation (56), there is the following expression: D ( X )= ( y d E α α α=1 d X + y α E β=1 x α d x β y β + α E α=1 x α d x d X + y E β=1 x d x β d X β (65) d X ) D D ( X ) Where D D ( X ) is given by the equation (56). The complementary term D ( X ) is written, by taking account of the multiplicity of the eigenvalues, in the following way: α=1 D ( X )={ D ( X ) ( y α x E α E + y E x E α α) + y E x E si x 1 x y 1 x I p E + y x 1 E I p + y x E E si x 1 = x Where I p is the matrix of the orthogonal projection of I S in the plan e x, e y : (66) I p ={ 1 ( δ ik δ jl +δ il δ jk ) si i, j, k,l {1,} 0 sinon (67) 5.. Three-dimensional case In dimension three, the characteristic equation det ( X x α I )=0 give a cubic equation of the eigenvalues x α=1,, of X following type: With: x α I 1 x α + I x α I =0 with α=1,, (68) {I 1=trace ( X ) I =det ( X ) I = 1 [ trace ( X ) trace ( X ) ] The resolution of the spectral problem easily gives the following solutions for the eigenvalues: {x1= Q cos( θ ) + I 1 x = Q cos ( θ π x = Q cos ( θ+π ) + I 1 ) + I 1 (69) (70) Where Q and θ are given by:

18 Titre : Loi de Mohr-Coulomb Date : 0/05/016 Page : 18/1 adaab1bf74 With: Q= I 1 I 9 1( R and θ=cos Q ) (71) R= I 1+9 I 1 I 7 I 54 And clean vectors, by taking account of the multiplicity of the eigenvalues: (7) x α X +( x α I 1 ) X + I x α I ] si x 1 x x {E α= x α I 1 x α + I [ E β = I E α si x α x β E 1 =I si x 1 = x = x (7) In the second equation of (7), 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 the eigenvalues, is written finally: α=1 )={ ( x α x β x γ ) E α E α ( x β x γ ) ( E β E β E γ E γ )] D ( X y α ( x α x β ) ( x α x γ ) [ d X d X ( x β +x γ ) I S + a=1 b=1 y a x b E a E b si x 1 x x d X s 1 d X s I S s X X +s 4 X I +s 5 I X s 6 I I si x α x β =x γ ( y 1 y 1 x 1 x ) I S+ y 1 I I x si x 1 =x =x (74) Where (α,β, γ ) corresponds to a cyclic permutation of (1,,). I and I S are given by the equations (6) 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 X ( d X d X )ijkl d X, which gives the following form: = d ( X i m X mj ) = 1 d X kl ( δ ik δ lm +δ il δ km ) X mj + X i m ( δ mk δ jl +δ ml δ kj ) = 1 ( δ ik X lj +δ il X kj +δ jl X ik +δ kj X il ) (75) Lastly, expressions of s i=1,6 are the following ones:

19 Titre : Loi de Mohr-Coulomb Date : 0/05/016 Page : 19/1 adaab1bf74 s 1 = y α y γ ( x α x γ ) + 1 x α x γ ( y γ y γ x β x γ ) s = x γ y α y γ ( x α x γ ) + x α+ x γ x α x γ ( y γ y γ x β x γ ) y α + y γ y α y γ x γ x α x α x γ ) + x γ ( x α x γ ) ( s = y α y γ ( x α x γ ) + 1 ( x α x γ ) ( y s 4 = x α y γ γ ( x α x γ ) + 1 x α x γ ( y α y γ x γ x β ) y s 5 = x α y γ γ ( x α x γ ) + 1 x α x γ ( y γ y γ x α x β ) + x γ ( x α x γ ) ( y s 6 = x α y γ γ ( x α x γ ) + x α x γ ( x α x γ ) ( y α + y γ x γ x α ) x γ ( x α x γ ) ( y α + y γ x α x γ ) Where,, corresponds to a cyclic permutation of 1,,. y α + y γ y α y γ x γ x α x α x γ ) y α + y γ y α y γ x γ x α x α x γ ) x α+ x γ y γ x α x γ x β (76) Note: It is noticed that the following term (all but commonplace): α=1 y α ( x α x β ) ( x α x γ ) [ d X d X ( x β + x γ ) I S ( x α x β x γ ) E α E α ( x β x γ ) ( E β E β E γ E γ )] Who appears in the derivative D X first equation of (74) express the rotation of the principal directions in space three-dimensional. (77) 5..4 Application to the case of Mohr-Coulomb The transposition of the preceding formulas to the numeric work implementation deserves some precise details. There are first of all the following correspondences: X = ε pred and x α =ε α pred ; Y = σ + and y α =σ α + ; E α = d α pred d α pred ; (T ) α β = y α is the consistent tangent matrix in the principal base calculated in the paragraph 4 ; x β 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 pred base of Voigt. It will be noted that the predicted principal directions d 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 deduced the consistent tangent matrix T = D expressed in the base of projection b defined in the paragraph 1. : The equation (6) 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 (D);

20 Titre : Loi de Mohr-Coulomb Date : 0/05/016 Page : 0/1 adaab1bf74 The second important information relates to the convention of writing of the various tensors. Indeed, by preoccupation with general information, a notation used for the tensors in all the paragraph 5 is the classical notation, revealing of the tensors until the order four. This writing is unsuitable with the digital resolution, where one prefers to use condensed notations made possible by the fact that one works with tensors symmetrical of order two (constraints and deformations are it always). One distinguishes two forms from notations condensed correspondent at two bases of projection (see 1.): The orthonormal base b symmetrical tensors of order two. It is in this base that the constraints and the 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 at the time of the matric operations; The diagram of resolution is summarized in figure Figure : Process of resolution of the law of Mohr-Coulomb. Description of the writing in the various bases of projection.

21 Titre : Loi de Mohr-Coulomb Date : 0/05/016 Page : 1/1 adaab1bf74 6 Bibliography [78] Jiang Hua, 11th Y. with note one the Mohr-Coulomb and Drucker-Prager strenght criteria, Mechanics Research Communications, 8, pp, 09-14, 011 [79] Fotios E. Karaoulanis Nonsmooth multisurface plasticity in principal stress space, 6 HT International GRACM Congress one Computational Mechanics, Thessaloniki, 008 [80] Pankaj, Bićanić NR. multiple Detection of activates yield conditions for Mohr-Coulomb elasto- plasticity, Computers and Structures, 6, 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,, pp, , 1986 [8] Ronaldo I. Borja, Sama K.M., Sanz P.F. One the numerical integration of three-invariant elastoplastic constitutive models, Comput. Methods Appl. Mech. Engrg., 19, pp, [8] X. Wang, Wang L.B., Xu L.M. Formulation of the return mapping algorithm for elastoplastic soil models, Computers and Geotechnics, 1, pp, 15-8, 004