JP LIMIT STATE CRITERION FOR BRITTLE MATERIALS

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

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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 (*I*_{1}, *J*_{2}, *J*_{3})

§
well
known Drucker-Prager criterion, (*I*_{1}, *J*_{2})

§
classical
Huber-Mises criterion (*J*_{2 })

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 t

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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:

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*f*_{c} - failure stress in uniaxial compression,

§
* f*_{t} - failure stress in uniaxial tension,

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*f*_{cc} - failure stress in biaxial compression at s_{1}/s_{2} = 1,

§
*f*_{0c} - failure stress in biaxial compression at s_{1}/s_{2} = 2,

§
*f*_{v} - failure stress in triaxial tension at s_{1}/s_{2}/s_{3} = 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:

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*f*_{cc}=1.1
*f*_{c} , *f*_{0c}=1.25 *f*_{c}
, *f*_{v}= *f*_{t}.

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

_{}. (2)_{}

Two material constants *C*_{0}
and *C*_{1} can be evaluated on the basis of uniaxial test results
like

*f*_{t} and *f*_{c} .

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

_{}. (3)_{}

Material constant *C*_{0 },
in this analysis, is evaluated with uniaxial tension failure stress *f*_{t}.

Fig. 1. Limit curves in biaxial state of stress

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Fig.
2. ** PJ** and Drucker-Prager limit surface cross section by t

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

[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.*