Computers and Geotechnics 36 (2009) 911–913

Contents lists available at ScienceDirect

Computers and Geotechnics

journal homepage: www.elsevier .com/locate /compgeo

Comments on ‘‘Flow rule effects in the Tresca model” by H.A. Taiebat andJ.P. Carter [Computers and Geotechnics 35 (2008) 500–503]

Lars Andersen, Johan Clausen *

Department of Civil Engineering, Aalborg University, 9000 Aalborg, Denmark

The authors compare two different approaches of handling thediscontinuities of the Tresca criterion when performing three-dimensional elasto-plastic analyses. The discontinuities arisewhere the six yield planes intersect, as shown in Fig. 1. On theseintersections the derivatives of the yield criterion, oF=or, and theplastic potential, oG=or, are not uniquely defined. As stated thesederivatives are needed for the calculation of the elasto-plastic con-stitutive matrix, Dep. Both of the presented methods redefine theTresca criterion in an approximate manner with the purpose ofrendering these derivatives well-defined everywhere.

However, the discussers feel that the most important means ofdealing with the formation of the constitutive matrix at cornershas been left out and would like to add some comments regardingthis. As a side remark, the discussers would like to point out an er-ror in the definition of the rounded Tresca criterion in Eq. (5) whereit should be RðhÞ ¼ 1=ðA� B sin 3hÞ instead ofRðhÞ ¼ 1=ðAþ B sin 3hÞ, according to Abbo and Sloan [1]. Further-more, the constant B should be multiplied by �1 if the current Lodeangle, h, is negative.

1. Elasto-plastic constitutive model at intersecting yieldsurfaces

As an alternative to carrying out different smoothing proce-dures to the Tresca yield surface and plastic potential, Dep can bedetermined directly at a corner where two surfaces intersect. Thisis an attractive approach, as the smoothing of yield surfaces inev-itably will introduce an error in the computed response and bear-ing capacity, albeit this error will probably be of no importance inpractical applications.

In the references [2–4] the elasto-plastic constitutive matrix hasbeen calculated at corners without smoothing of the intersectingyield surfaces, and examples of two-dimensional calculations aregiven for a Mohr–Coulomb material. An example of the expressionfor Dep at a corner for a Tresca material is given in the book byCrisfield [5]:

Dep ¼ De I� a22

qaaTDe þ a12

qabTDe þ a12

qbaTDe � a11

qaaTDe

� �ð8Þ

0266-352X/$ - see front matter � 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.compgeo.2008.12.004

DOI of original article: 10.1016/j.compgeo.2008.12.002* Corresponding author. Tel.: +45 9940 7234.

E-mail address: jc@civil.aau.dk (J. Clausen).

where De is the elastic constitutive matrix, I is the identity matrix,a11 ¼ aTDea, a22 ¼ bTDeb, a12 ¼ a21 ¼ aTDeb and q ¼ a11a22 � a12a21.The directions a and b are the partial derivatives of the intersectingyield functions with respect to the stress vector, i.e. the plane nor-mals, a ¼ oF1=or ¼ G1=or and b ¼ oF2=or ¼ G2=or. Eq. (8) is a ratherlengthy expression as the calculation of a and b requires the deriv-atives of the stress invariants, as can be seen from Eq. (1).

A much simpler expression for Dep can be obtained if it is calcu-lated in the principal stress space. In the following a hat over a ma-trix or a vector means that it is expressed in the principal co-ordinate system, e.g. bDep and a. Similarly, an overbar indicates thatthe matrix or vector only contains elements related to the normaldirections, and that they are expressed in the principal co-ordi-nates, e.g. Dep and �a. As an example this allows us to partitionthe elastic and the elasto-plastic constitutive matrices as

De ¼ bDe ¼ De

Ge

" #and bDep ¼ Dep

Ge

" #ð9Þ

where Ge contains the elastic shear stiffness. The fact that the shearpartition of bDep in Eq. (9b) reduces to Ge can be seen from Eq. (2)when it is recalled that the last three components in the yield planenormal vanish when they are expressed in the principal stressspace,

oFor¼ a ¼ ½a1 a2 a3 0 0 0�T ð10Þ

In the principal stress space the Tresca criterion simplifies fromthe form given in Eq. (1) into

F ¼ r1 � r3 � 2c ¼ 0 ð11Þ

under the assumption r1 P r2 P r3. This gives the Tresca planenormal

�a ¼10�1

8><>:9>=>; ð12Þ

The straight lines that arise at the intersections of the yield planes(see Fig. 1) are parallel to the hydrostatic axis and therefore havethe direction vector

�‘ ¼111

8><>:9>=>; ð13Þ

a b c

σ1 = σ2σ2 = σ1

σ3

ε3ε3εerr

1

ε1 = ε2ε2 = ε1

Fig. 6. A unit cube of Tresca material in a state of triaxial stress. (a) Undeformed configuration; (b) erroneous dilatative behaviour and (c) correct dilatative behaviour.

N

912 L. Andersen, J. Clausen / Computers and Geotechnics 36 (2009) 911–913

With the definitions given above, Clausen et al. [6] show that theexpression for the elasto-plastic constitutive matrix in the principalstress space can be simplified into

Dep ¼�‘�‘T

�‘TðDeÞ�1�‘¼ E

3ð1� 2mÞ

1 1 11 1 11 1 1

264375 ð14Þ

which combined with Eq. (9b) gives the elasto-plastic constitutivematrix on a line in principal stress space. When it is transformedinto the general stress space by standard co-ordinate transforma-tion it yields the exact same result as Eq. 8.

It should be noted that Dep is doubly singular at the lines de-fined by the intersections of the yield planes, i.e. Depa ¼ 0 andDepb ¼ 0 along these lines.

2. Issues in the application to three-dimensional problems

The doubly singular Dep shown in Eqs. 8 and 14 works well intwo-dimensional calculations. But when it is applied to three-dimensional calculations, problems arise as the double singularitycauses the direction of the plastic strain increment to be undefined.An example of an erroneous strain prediction can be seen in Fig. 6in which a cube of Tresca material is subjected to a forced verticaldisplacement on one of the horizontal surfaces. The calculationsare carried out in ABAQUS. It is seen in Fig. 6b that all the plasticstrain is directed in the e1 direction, as opposed to the correctstrain distribution seen in Fig. 6c.

A simple method of mending this problem is to add a littleamount of stiffness in the plane in which Dep is doubly singular,

ZT

R

X

Y

Z

Fig. 7. ABAQUS model with 1571 nodes. The boundary conditions are indicatedalong the edges of the models.

but in such a fashion that it is still singular in the plastic straindirection given by

�ep ¼ De� ��1�rB � �rC� �

ð15Þ

where �rB is the predictor stress in the return-mapping scheme and�rC is the corrected stress on the Tresca line. The details of this pro-cedure can be found in Ref. [7].

3. Numerical example

To indicate the performance of the proposed method and quan-tify potential inaccuracies of the rounded or modified Tresca mod-els, a comparison is made in which a circular footing with adiameter of D ¼ 2 m is pushed 0.2 m downwards into a Trescamaterial. Young’s modulus and Poisson’s ratio of the material are2 GPa and 0.25, respectively, and the cohesion is 20 MPa. Further,the interface between the footing and the Tresca material issmooth.

Ideally, the problem is axisymmetric. However, to study theperformance of the different implementations of the Tresca modelin three dimensions, a wedge with an internal angle of 15 has beenmodelled. The distance to the bottom of the computational mesh is5 m and the distance from the perimeter of the footing to theboundary of the model is 4 m, i.e. 1/24 of a cylinder with a radiusof 5 m is considered.

0 0.02 0.04 0.06 0.08 0.10

1

2

3

4

5

6

7

0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

5.8

5.9

6.0

6.1

c

δv/ D

TrescaRounded Tresca, θT = 29.5ºRounded Tresca, θT = 25º

Tresca-Mises

Fig. 8. Load–deflection curves obtained from the different material models.

L. Andersen, J. Clausen / Computers and Geotechnics 36 (2009) 911–913 913

ABAQUS [8] is utilised for the analysis, employing tetrahedralelements with quadratic shape functions and full integration. Themesh has 790 elements with a total of 1571 nodes and can be seenin Fig. 7.

The results for the different Tresca models are plotted in Fig. 8.The computation for the rounded Tresca model has been carriedout for two different values of the transition angle, hT, see Eq. (6).More specifically, the values hT ¼ 25� and hT ¼ 29:5� have been ap-plied. The exact solution, Nc ¼ 5:69, from Cox et al. [9], has beenplotted with a dotted line. In the figure it is seen that the curveof the true Tresca model almost coincides with the results fromthe rounded Tresca model with hT ¼ 29:5�, as would be expected.

Further comparisons of different means of handling the yieldfunction corners can be found in the recent paper by Huang andGriffiths, [10].

References

[1] Abbo AJ, Sloan SW. A smooth hyperbolic approximation to the Mohr–Coulombyield criterion. Comput Struct 1995;54(3):427–41.

[2] Lade PV, Nelson RB. Incrementalization procedure for elasto-plasticconstitutive model with multiple, intersecting yield surfaces. Int J NumerAnal Methods Geomech 1984;8:311–23.

[3] de Borst R. Integration of plasticity equations for singular yield functions.Comput Struct 1987;26(5):823–9.

[4] Larsson R, Runesson K. Implicit integration and consistent linearization foryield criteria of the Mohr–Coulomb type. Mech Cohesive-Friction Mater1996;1:367–83.

[5] Crisfield MA. Non-linear finite element analysis of solids and structures.Advanced topics, vol. 2. John Wiley & Sons; 1997.

[6] Clausen J, Damkilde L, Andersen L. Efficient return algorithms for associatedplasticity with multiple yield planes. Int J Numer Methods Eng2006;6:1036–59.

[7] Clausen J. Efficient Non-linear finite element implementation of elasto-plasticity for geotechnical problems. PhD thesis, Esbjerg Institute ofTechnology, Aalborg University, 2007. <http://vbn.aau.dk/fbspretrieve/14058639/JCthesis.pdf>.

[8] Simulia. Abaqus version 6.7 documentation.[9] Cox AD, Eason G, Hopkins HG. Axially symmetric plastic deformations in soils.

Philos Trans Roy Soc London, Ser. A: Math Phys Sci 1961;254(1036):1–45.[10] Huang J, Griffiths DV. Observations on return mapping algorithms for

piecewise linear yield criteria. Int J Geomech 2008;8(4):253–65.