implicit numerical integration of the mohr-coulomb surface in principal stress space

SEECCM 2009 2nd South-East European Conference on Computational Mechanics

An IACM-ECCOMAS Special Interest Conference M. Papadrakakis, M. Kojic, V. Papadopoulos (eds.)

Rhodes, Greece, 22–24 June 2009

IMPLICIT NUMERICAL INTEGRATION OF THE MOHR-COULOMB SURFACE IN PRINCIPAL STRESS SPACE

Fotios E. Karaoulanis1 and Theodoros Chatzigogos2

1 Aristotle University of Thessaloniki Thessaloniki, GR-54124, Greece

e-mail: [email protected]

2 Aristotle University of Thessaloniki Thessaloniki, GR-54124, Greece e-mail: [email protected]

Keywords: Nonsmooth yield surfaces, return mapping, spectral representation, Mohr-Coulomb.

Abstract. The Mohr-Coulomb yield criterion is acknowledged as one of the first and most important criteria, widely used to describe the yield behavior of a wide range of materials. However singularities rising due to the nonsmoothness of the Mohr-Coulomb yield surface introduce severe complexities that lead to different numerical approaches, proposed algo-rithms and computer implementations. In this article an extremely simple, robust, efficient and general return mapping method for implicit integration of nonsmooth yield criteria is utilized. The algorithm is based on a spectral representation of stresses and strains and a re-turn mapping scheme in principal stress directions. It is shown that in the Mohr-Coulomb case the return mapping reduces to a one step closest point projection, minimizing the compu-tational cost and increasing the accuracy when compared to a typical return mapping scheme. The validation, verification and performance of the present return mapping algorithm is dem-onstrated by numerical examples.

Fotios E. Karaoulanis and Theodoros Chatzigogos

2

1 INTRODUCTION

The Mohr-Coulomb yield criterion is acknowledged as one of the first and most important criteria, widely used to describe the yield behavior of a wide range of materials. Named in honor of Charles-Augustin de Coulomb and Christian Otto Mohr, uses the Coulomb's friction hypothesis to determine the combination of shear and normal stress that will cause a fracture of a material and Mohr's circle to determine which principal stresses will produce this combi-nation of shear and normal stress as well as the angle of the plane in which this will occur. In three dimensions, the Mohr-Coulomb failure surface is a nonsmooth yield surface, rendered as a cone with a hexagonal cross section in the deviatoric stress space.

From a mathematical standpoint, the extension of classical plasticity models to accommo-date nonsmooth yield surface goes back to the fundamental work of Koiter [1]. Later formula-tions of plasticity employing convex analysis as in Moreau [2] encompass these classical treatments as a particular case. Modern formulations are usually based on the work of Simo [3], where the standard Kuhn-Tucker complementarity conditions are used to provide the characterization of plastic loading/unloading.

However singularities rising due to the nonsmoothness of the Mohr-Coulomb yield surface introduce severe complexities that lead to different numerical approaches, proposed algo-rithms and computer implementations. Hence, from a computational standpoint, effort has been initially directed towards replacing the yield surface in near singular areas (see e.g. [4] where the authors proposed the replacement of Mohr-Coulomb and Tresca surfaces by Drucker-Prager’s and Von-Mises’ ones respectively, leading inevitably to a gradient jump) or smoothing the yield surface in those areas (e.g. by modifying the yield surface in the vicinity of singularities as in [5]). In early 1990s and within the context of the return mapping algo-rithm [6], effort has been focused on the proper implementation of the plastic corrector near corner regions following specific surface dependent implementations.

Formulation of the return mapping algorithm in the principal stress space is not new. Pankaj and Bićanić [7] elaborate on the detection of the proper stress return in principal stress space for the Mohr-Coulomb surface, Perić and Neto [8] for the Tresca one and Borja et al. [9] for three invariant elastoplastic models. The main benefit of dealing with stress return in prin-cipal stress space is that differentiation of yield surfaces w.r.t. stresses (namely σ1, σ2 and σ3) in this space is usually much simpler, since almost all yield surfaces of interest are defined in this space, leading to more robust and efficient algorithms. Recall that within the context of the return mapping algorithm, the yield function must be at least twice differentiable w.r.t. stresses. However the above mentioned approaches, have been either focused on specific yield surfaces ([7]-[8]), or avoided nonsmoothness of the yield surface [9].

In this work, the treatment of multisurface plasticity as proposed by Simo [3] is reformu-lated and implemented in principal stress space. Contrary to previous mentioned works, a general algorithm is proposed for the case of nonsmooth multisurface plasticity in this space, which is applied to the Mohr-Coulomb yield surface.

2 THEORY

In what follows a review of basic notation and theory is provided, the case of multisurface plasticity is defined and the return mapping in principal stress space is addressed.


3

2.1 Summary of governing equations

Let ε∈ℜnε denote the total strain at a fixed point X∈B of a solid, X∈ℜndim, where

ndim∈{1,2,3}; typically X refers to a quadrature point in finite element discretization of the equations governing its mechanical equilibrium. In the infinitesimal case, the strains ε are simply identified as the symmetric part of the gradient of the displacement vector and are as-sumed additively decomposed, as

peεεε += , (1)

where εe and εp are referred to as the elastic and plastic strain parts. Then let σ∈ℜnσ and κ∈ℜnκ denote the stress and a set of nκ strain-like internal variables

characterizing the hardening/softening response of the material, respectively. Standard ther-modynamic arguments identify the following constitutive relation,

,),(

e

eW

ε

κεσ

∂

∂= (2)

for the stored energy function W in terms of the elastic strains εe and a general set of strain-like variables κ.

The definition of the elastoplastic problem is completed with the introduction of the evolu-tion equations for the plastic internal variables, namely the plastic strains εp and hardening variables κ, called flow rule and hardening/softening law, respectively:

).,(ˆ ,),(

λκκσ

λ σκσ

ε&&&& =

∂

∂=

gp (3)

The parameter 0≥λ& is a nonnegative scalar, called the consistency parameter, which is as-sumed to obey the following Kuhn-Tucker complimentarity conditions,

0),( and 0),( ,0 =≤≥ κσλκσλ ff && (4)

and the consistency requirement,

.0),( =κσλf& (5)

In classical literature, conditions (4) and (5) go by the names loading/unloading and con-sistency conditions. The functions g(σ,κ) and ),(ˆ λκ σ are prescribed functions that define the direction of the flow rule and the type of hardening.

Finally the function f(σ,κ): ℜnσ×ℜnκ→ℜ is usually known as the yield function and defines the so called elastic domain, i.e. the following convex set,

}0),(|),{( ≤ℜ×ℜ∈=Ε κσκσ fnn κσσ (6)

in which the admissible stresses are constrained to lie. Of special significance is the case where:

f ≡ g, (7)

which is called associative flow rule.


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2.2 Nonsmooth multisurface plasticity

The extension of the above elastoplastic problem to nonsmooth multisurface plasticity is

rather straight forward. The essential feature is the characterization of the elastic domain Eσ, which is still a convex subset of ℜnσ×ℜnκ, however now is defined as

]},,...2,1[ allfor ,0),(|),{( nfann ∈≤ℜ×ℜ∈=Ε ακσ

σ κσκσ (8)

where fα(σ,κ) are n≥1 functions intersecting nonsmoothly. With this definition of the elastic domain, the evolution of plastic strain is governed by the following flow rule [1] often re-ferred to as Koiter’s rule:

.),(

1∑= ∂

∂=

n

a

aap g

σ

κσε λ&& (9)

Similarly the complimentarity conditions (4) and the consistency condition (5) are now re-defined as:

.0),( and 0),( ,0),( ,0 ==≤≥ κσκσκσaaaaaa fff &&& λλλ (10)

2.3 Return mapping in principal stress space

Recall that the spectral decomposition is given as:

, , )()()(3

1

)()1(1

AAAAnn nnmmσσ ⊗== ∑

=Α+Α+ (11)

where σA and n(A) are the principal Cauchy stresses and principal directions, respectively, ⊗ denotes a juxtaposition, e.g., (a⊗b)ij=aibj, and subscript n stands for a converged step in the incremental solution scheme. The gradients of any scalar functions of stress invariants, such as the yield and plastic potential functions f and g, can now be evaluated as

,,3

1

)(3

1

)( ∑∑=Α=Α ∂

∂=

∂

∂

∂

∂=

∂

∂ A

A

A

A

ggffm

σσm

σσ (12)

with m(A) as defined above. If isotropy in the elastic response is assumed, then it can be easily shown that,

,3

1

)(1 ∑

=Α+ =∆+= Ae

Ann mεεεε (13)

with m(A) being a spectral direction of ε as well, i.e. the strain spectral directions coincide with stress spectral directions and therefore the return mapping algorithm may be conveniently formulated in principal stress space.

Hence, starting from the flow rule (the eigenbases m(A) are dropped for clarity’s shake),

∑= ∂

∂=

mp g

1α

ααλσ

ε && , (14)


5

where m is the number of active yield surfaces and applying a backward Euler return, yields

11

1

+=+ ∑ ∂

∂∆+=

n

m

a

pn

pn

g

σεε

ααλ . (15)

Considering that ∆εn+1 = εn+1 – εn and σn+1= c(εn+1 – εpn+1), the above equation may be written

as

0:11

11

1 =∂

∂∆−∆−∆

+=+

−+ ∑

n

m

ann

g

σσcε

ααλ , (16)

or in residual form

11

11

1 :+=

+−

+ ∑ ∂

∂∆−∆−∆=

n

m

annn

g

σσcεr

αασ λ . (17)

The tensor c-1 is the elastic compliance matrix in principal stress/strain space, which in the case of isotropic elasticity is given as:

−−

−−

−−

=−

1

1

111

νννν

νν

Ec (18)

and depends on the Young’s modulus E and Poisson’s ratio ν. Linearization of the above equation yields:

)(

)(

11

)()(

11

)()(

)(

11

2

2)(1)(

1 : k

k

n

m

a

kk

n

m

a

kk

k

n

m

a

kkn

gggκ

κσσσ

σ

cr δλδλδλα

αα

αα

ασ

+=+=+

=

−+ ∑∑∑ ∂∂

∂∆+

∂∂

+

∂

∂∆+= (19)

Similarly linearization of the hardening rule is given as (I denoting the identity matrix):

nnnk

nk

k

n

k

k

n

kn

kκκσκrκIσ

σr +−=−

∂

∂+

∂

∂= ++

++

+ 11)()(

)(

1

)(

)(

1

)(1 ),(ˆ where,

ˆˆλδδλ

λκ

δκ κ

κ . (20)

Finally linearization of the yield function 0=af yields

. where, 1)(

)(

1

)(

)(

1

)(1

an

fak

k

n

ak

k

n

ak

nfa f

ff−=

∂

∂+

∂

∂= +

++

+ rκκ

σσ

r δδ (21)

Combining the above linearized equations (19)-(21) the following system is defined:


6

)(

1

1

1)(

)(

311

11

11

)(

1111

1

ˆˆ

k

n

nfa

nk

a

k

nm

nnn

m

n

aT

n

a

n

m

i

k

n

m

an

m

a

aa

ff

ggg

=

−∂

∂

∂

∂

∂

∂

∂

∂

∂∂

∂∆

∂

∂

∂

∂∆+

+

+

+

++++

++

+=+=+=

− ∑∑∑

r

r

r

κ

σ

Iσσ

0σ

κσσσc

κ

σ

κ

κ

αα

α

δδλ

δ

λκ

δκ

λ

λλ

(22)

for all active (m≥1) yield surfaces.

The stress tensor is then updated in terms of principal stresses as

∑+=+)()(

1k

nk

n σσσ δ , (23)

the hardening parameter as

∑+=+)()(

1k

nk

n κκκ δ , (24)

and the parameters ∆λα as

∑=∆k

ka )(δλλα . (25)

The active set of surfaces, defined as actJ :{α ∈ {1,2,...,m}| f αtrial,n+1>0} is updated during

the iteration procedure, so as the admissibility constraint ∆λα remains nonnegative for all

α∈ actJ . I.e. ∆λα is updated and if negative, the α surface is removed from the active set of

surfaces.

3 IMPLEMENTATION

The above described return mapping algorithm is applied in this section to the Mohr-Coulomb perfectly-plastic yield criterion, where the algorithm’s simplicity, efficiency and robustness is demonstrated.

3.1 The Mohr-Coulomb yield criterion

The Mohr-Coulomb yield function is usually defined in terms of the difference between the maximum and minimum principal stresses, as:

( ) ( ) )cos(2)sin(minmaxminmax φφσσσσ cf −++−= , (26)

where σmax and σmin are the maximum and the minimum principal stress, φ is the friction angel and c is the cohesion. It is subdifferentiable in stress space and for a fixed σy defines an elas-tic region given as:

{ }0| <= fE σσ , (27)


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which is plotted in Fig.1.

Figure 1: The Mohr-Coulomb criterion in principal stress space.

The above yield surface may be defined in terms of principal stresses using the following

six linear equations:

( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) )cos(2)sin(

)cos(2)sin(

)cos(2)sin(

)cos(2)sin(

)cos(2)sin(

)cos(2)sin(

12126

23235

13134

21213

32322

31311

φφσσσσ

φφσσσσ

φφσσσσ

φφσσσσ

φφσσσσ

φφσσσσ

cf

cf

cf

cf

cf

cf

−++−=

−++−=

−++−=

−++−=

−++−=

−++−=

. (28)

3.2 Return mapping scheme for the Mohr-Coulomb yield criterion

Without any loss of generality, one may assume that the principal stresses are ordered in

descending order, i.e. σ1≥σ2≥σ3 and hence only three of the above constraints can be used, namely f1, f2 and f3 defining the sextant shown in Fig.2.

The normal vectors w.r.t. to the principal stress tensor can now be easily calculated as:

+−

++

=

+−

++=

+−

++

=

0

)sin(1

)sin(1

,

)sin(1

)sin(1

0

,

)sin(1

0

)sin(1321 φ

φ

φφ

φ

φ

σσσ d

df

d

df

d

df. (29)

while the second derivatives are all zero vectors, i.e. 0σ

=2

2

d

fd i , and the above described algo-

rithm is applied with no modifications.


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Figure 2. The Mohr-Coulomb criterion in π-plane.

It should be clear that due to the linearity of the gradients the return map degenerates to one step closest point projection, dramatically increasing the accuracy and robustness of the algo-rithm, while minimizing the computation cost, when compared with typical implementations of the return mapping schemes, found in classical textbooks of finite element literature (see e.g. [10]).

4 NUMERICAL EXAMPLES

The Mohr-Coulomb criterion is extensively used in Soil Mechanics, hence typical exam-ples referring to soil plasticity are presented in this section. In the beginning, a series of soil tests is presented and then the problem of vertically unsupported excavation is stated and solved. All examples have been solved using nemesis [11], which is an experimental finite element code, implemented by the first author.

4.1 Triaxial test

The triaxial test [12] is the most widely laboratory test used to evaluate the mechanical properties of soils. It is performed in cylindrical soil specimens immersed in a water cell, which provides the radial pressure σ3, while vertical pressure σ1 is applied mechanically on the bottom and top sides. Assuming a consolidated drained compression test, the loading is applied in two stages:

1. The soil is consolidating isotropically to a desired effective stress level by pressur-

izing the water in the cell. 2. The water pressure in the cell is kept constant and additional axial displacements in

a very slow rate are added until the soil fails.


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Figure 3: Idealized triaxial test and corresponding finite element model.

The finite element model consists of one quadrilateral, displacement based, axisymmetric element as shown in Fig.3. Two load cases are defined:

1. P1 = P3 = 1 using load control and thus simulating the isotropic consolidation. 2. P3 = 1, P1 = λ, where λ is found using displacement control such as an imposed dis-

placement of 0.005 inwards (compression, see Fig.4(a)) or outwards (tension, see Fig.4(b)).

(a)

(b)

Figure 4: Numerical and analytical results for the triaxial test, in case of compression (a) and tension (b).

In both cases the results are identical to the analytical solution, namely P1=44.884 for com-pression and P1=–20.516 for tension respectively.

4.2 Direct shear test

The direct shear test [12] is used to find the shear strength parameters of soil. A soil speci-men is sheared under a constant vertical force and fails across a predefined zone, constrained by the test apparatus. From the recorded applied forces and the resulting displacements the


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peak and residual shear strength, the friction and the dilation angle and as well as the cohesion can be determined.

Figure 5: Idealized shear test and corresponding finite element model.

In Fig.5 an idealized shear test is shown, which will be used to verify the accuracy of the implemented model, concerning a non-associative flow rule, which in the case of the Mohr-Coulomb yield criterion is defined using the dilation angle α instead of the friction angle φ, when enforcing the normality hypothesis (see Eq. 9 and 28). The finite element model con-sists of one quadrilateral, displacement based, plane strain element as shown in Fig.5. Assum-ing for the material E=1000., ν=0.2, c=15, φ=200 and α=100, two load cases are defined:

1. P1 = 1 using load control. 2. P1 = 1, P2 = λ, where λ is found using displacement control such as an imposed dis-

placement of 0.01 is applied.

The results are plotted in Fig.6, where the horizontal vs. the vertical displacements are plotted;

it can be verified that the resulted dilation angle, i.e. a =

∆

∆−

xy

y

γ

εarctan =

010arctan =

∆

∆−

x

y

u

u, equals to the given one.

Figure 6: Horizontal vs. vertical displacements in the numerical shear test. The dilation angle α is also shown.


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4.3 Unsupported excavation

Coulomb (see e.g [14]) proposed that a condition of limit equilibrium exists through which a soil mass behind a vertical retaining wall will slip along a plane inclined an angle

42

πφθ += to the horizontal (Fig.7(a)). Therefore, the critical height Hcr can be found, defined

as the maximum height for an unsupported excavation (Pa=0) and equals to:

γ

πφ 1

42tan4

+= cH cr (30)

A finite element analysis was performed for a soil described by E=40000kPa, ν=0.2, γ=20kN/m3, c=13.23771kPa and φ=230 which yield according to (34) a Hcr=4.0m. Two load cases are defined:

1. An initial stress field is applied for K0=1 – sin(φ) . 2. The excavation load case which is performed in one stage.

In Fig.7(b) the excavation, the slip line and the plastification zone are shown when a fail-

ure mechanism is deployed and the soil body collapses for an excavation depth of 4.0m, in perfectly agreement with the analytical solution.

(a)

(b)

Figure 7: Coulomb failure wedge (a) and numerical results for an unsupported excavation (b).

5 CONCLUSIONS

The Mohr-Coulomb yield criterion is acknowledged as one of the first and most important criteria, widely used to describe the yield behavior of a wide range of materials. However sin-gularities rising due to the nonsmoothness of the Mohr-Coulomb yield surface introduce se-vere complexities that lead to different numerical approaches, proposed algorithms and computer implementations. In this article, a robust and efficient return mapping method for the implicit integration of nonsmooth yield criteria, based on a spectral representation of stresses and strains and a return mapping scheme in principal stress directions is presented


12

and applied to the Mohr-Coulomb criterion. It is shown that in this case the return mapping reduces to a one step closest point projection, minimizing the computational cost and increas-ing the accuracy when compared with a typical return mapping scheme. Test cases of standard soil tests and an example of unsupported excavation failure are provided, with the results ob-tained to be in excellent agreement with the corresponding analytical solutions.

AKNOWLEDGEMENTS

The author gratefully acknowledge financial support from the Greek State Institute of Scholarships (I.K.Y.); contract/grant number: 4506/05.

REFERENCES

[1] W.T. Koiter, General Theorems for Elastic-plastic Solids, in Progress in Solid Mechan-ics 6, 167-221, eds. I.N. Sneddon and R. Hill, North-Holland Publishing Company, Amsterdam, 1960,

[2] J.J. Moreau, Application of Convex Analysis to the Treatment of Elastoplastic Systems, in Applications of Methods of Functional Analysis to Problems of Mechanics, eds. P. Germain and B. Nayroles, Springer-Verlag, Berlin, 1976.

[3] J.C. Simo and T.J.R. Hughes, Computational Inelasticity. Springer-Verlag, 1998.

[4] D. R. J. Owen, E. Hinton, Finite Elements in Plasticity: Theory and Practice. Pineridge Press, Swansea, U.K., 1980.

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

[6] J.C. Simo, R.L. Taylor, Return Mapping Algorithm for Plane Stress Elastoplasticity. In-ternational Journal for Numerical Methods in Engineering, 22, 649-670, 1986.

[7] Pankaj, N. Bićanić, Detection of multiple active yield conditions for Mohr–Coulomb elastoplasticity. Computers and Structures, 62(1), 51-61, 1997.

[8] D. Perić, E.A. de Souza Neto, A new computational model for Tresca plasticity at finite strains with an optimal parametrization in the principal space. Computer Methods in Applied Engineering, 71, 463-489, 1999.

[9] R.I.Borja, K.M. Sama, P.F. Sanz, On the numerical integration of three-invariant elas-toplastic constitutive models. Computer Methods in Applied Mechanics and Engineer-ing. 192, 1227-1258, 2003.

[10] M.A. Crisfield, Non–linear Finite Element Analysis of Solids and Structures, Vols. 1 & 2, John Wiley & Sons, 1991.

[11] nemesis, an experimental finite element code, http://www.nemesis-project.org. Re-trieved on February 2009.

[12] B. Muni, Soil Mechanics & Foundations, John Wiley & Sons, 2000.

implicit numerical integration of the mohr-coulomb surface in principal stress space

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