JP LIMIT STATE CRITERION FOR BRITTLE MATERIALS

 

 

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)

 

where:                  

   - 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.