J.Podgórski, Technical University of Lublin, Faculty of Civil Engineering, Poland




1.     Limit state conditions

The three failure criteria (Fig. 1) had been considered to analysis:

§           author’s (PJ) criterion, proposed in 1986 [1], which limit state depends on three tensor invariants (I1, J2, J3)

§           well known Drucker-Prager criterion, (I1, J2)

§           classical Huber-Mises criterion (J2 )


Limit curves described by eqs. (1), (2), (3) in biaxial stress state are shown in Fig. 1. Fig. 2 shows “tension meridian” and “compression meridian” of the PJ and Drucker-Prager limit surface in t0s0 plane and Fig. 3 shows isometric view of this surfaces.



1.1         PJ criterion

The PJ criterion was proposed by one of the author’s (J.P.) in 1986 [1] in the form:


,                         (1)



   - function describing the shape of limit surface in deviatoric plane,

                     - mean stress,

                 - octahedral shear stress,

                                             - first invariant of the stress tensor,

                         - second and third invariant of the stress deviator,

                - alternative invariant of the stress deviator,

       - material constants.


Classical failure criteria, like Huber-Mises, Tresca, Drucker-Prager, Coulomb-Mohr as well as some new ones proposed by Lade, Matsuoka Ottosen, are particular cases [cf. 1,2] of the general form (1) PJ criterion.

Material constants can be evaluated on the basis of some simple material test results like:

§         fc     - failure stress in uniaxial compression,

§          ft    - failure stress in uniaxial tension,

§         fcc    - failure stress in biaxial compression at s1/s2 = 1,

§         f0c    - failure stress in biaxial compression at s1/s2 = 2,

§         fv     - failure stress in triaxial tension at s1/s2/s3 = 1/1/1,


For concrete or rock-like materials some simplifications can be taken on the basis of test results in biaxial stress state and R. M. Haythornthwaite “tension cutoff” hypotesis:


fcc=1.1 fc ,             f0c=1.25 fc ,           fv= ft.


1.2         Drucker – Prager criterion

With notation used in eq. (1) well-known Drucker–Prager criterion can be written:


                                      .                                      (2)


Two material constants C0 and C1 can be evaluated on the basis of uniaxial test results like

ft  and  fc .


1.3         Huber – Mises criterion

Classical criterion proposed by T. Huber and  R. von Mises can be obtained by simplification of the general form (1):


                                      .                                                     (3)


Material constant  C0 , in this analysis,  is evaluated with uniaxial tension failure stress ft.





Fig. 1. Limit curves in biaxial state of stress





Fig. 2. PJ and Drucker-Prager limit surface cross section by t0s0 plane.


Fig. 3. PJ and Drucker-Prager limit surface – isometric view.



2.     References

[1]     J. Podgórski (1985). General Failure Criterion for Isotropic Media. Journal of Engineering Mechanics ASCE, 111, 2, 188-201.

[2]     Podgórski J.: Limit state condition and the dissipation function for isotropic materials.

Archives of Mechanics, 36(1984) 3, 323-342.