Comments on ‘‘Flow rule effects in the Tresca model” by H.A. Taiebat and J.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 the discontinuities of the Tresca criterion when performing three- dimensional elasto-plastic analyses. The discontinuities arise where the six yield planes intersect, as shown in Fig. 1. On these intersections the derivatives of the yield criterion, oF =or, and the plastic potential, oG=or, are not uniquely defined. As stated these derivatives are needed for the calculation of the elasto-plastic con- stitutive matrix, D ep . Both of the presented methods redefine the Tresca criterion in an approximate manner with the purpose of rendering these derivatives well-defined everywhere. However, the discussers feel that the most important means of dealing with the formation of the constitutive matrix at corners has been left out and would like to add some comments regarding this. 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) where it should be RðhÞ¼ 1=ðA  B sin 3hÞ instead of RðhÞ¼ 1=ðA þ B sin 3hÞ, according to Abbo and Sloan [1]. Further- more, the constant B should be multiplied by 1 if the current Lode angle, h, is negative. 1. Elasto-plastic constitutive model at intersecting yield surfaces As an alternative to carrying out different smoothing proce- dures to the Tresca yield surface and plastic potential, D ep can be determined directly at a corner where two surfaces intersect. This is 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 in practical applications. In the references [2–4] the elasto-plastic constitutive matrix has been calculated at corners without smoothing of the intersecting yield surfaces, and examples of two-dimensional calculations are given for a Mohr–Coulomb material. An example of the expression for D ep at a corner for a Tresca material is given in the book by Crisfield [5]: D ep ¼ D e I  a 22 q aa T D e þ a 12 q ab T D e þ a 12 q ba T D e  a 11 q aa T D e   ð8Þ where D e is the elastic constitutive matrix, I is the identity matrix, a 11 ¼ a T D e a, a 22 ¼ b T D e b, a 12 ¼ a 21 ¼ a T D e b and q ¼ a 11 a 22  a 12 a 21 . The directions a and b are the partial derivatives of the intersecting yield functions with respect to the stress vector, i.e. the plane nor- mals, a ¼ oF 1 =or ¼ G 1 =or and b ¼ oF 2 =or ¼ G 2 =or. Eq. (8) is a rather lengthy 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 D ep 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. b D ep and ^ a. Similarly, an overbar indicates that the matrix or vector only contains elements related to the normal directions, and that they are expressed in the principal co-ordi- nates, e.g. D ep and  a. As an example this allows us to partition the elastic and the elasto-plastic constitutive matrices as D e ¼ b D e ¼ D e G e " # and b D ep ¼ D ep G e " # ð9Þ where G e contains the elastic shear stiffness. The fact that the shear partition of b D ep in Eq. (9b) reduces to G e can be seen from Eq. (2) when it is recalled that the last three components in the yield plane normal vanish when they are expressed in the principal stress space, oF o^ r ¼ ^ a ¼½ ^ a 1 ^ a 2 ^ a 3 000 T ð10Þ In the principal stress space the Tresca criterion simplifies from the form given in Eq. (1) into F ¼ r 1  r 3  2c ¼ 0 ð11Þ under the assumption r 1 P r 2 P r 3 . This gives the Tresca plane normal  a ¼ 1 0 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 have the direction vector  ‘ ¼ 1 1 1 8 > < > : 9 > = > ; ð13Þ 0266-352X/$ - see front matter Ó 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). Computers and Geotechnics 36 (2009) 911–913 Contents lists available at ScienceDirect Computers and Geotechnics journal homepage: www.elsevier.com/locate/compgeo