The Bresler-Pister yield criterion is a function that was originally devised to predict the strength of concrete under multiaxial stress states. This yield criterion is an extension of the Drucker-Prager yield criterion and can be expressed on terms of the stress invariants as
J
2
=
A
+
B
I
1
+
C
I
1
2
where
I
1
is the first invariant of the Cauchy stress,
J
2
is the second invariant of the deviatoric part of the Cauchy stress, and
A
,
B
,
C
are material constants.
Yield criteria of this form have also been used for polypropylene and polymeric foams.
The parameters
A
,
B
,
C
have to be chosen with care for reasonably shaped yield surfaces. If
σ
c
is the yield stress in uniaxial compression,
σ
t
is the yield stress in uniaxial tension, and
σ
b
is the yield stress in biaxial compression, the parameters can be expressed as
B
=
(
σ
t
−
σ
c
3
(
σ
t
+
σ
c
)
)
(
4
σ
b
2
−
σ
b
(
σ
c
+
σ
t
)
+
σ
c
σ
t
4
σ
b
2
+
2
σ
b
(
σ
t
−
σ
c
)
−
σ
c
σ
t
)
C
=
(
1
3
(
σ
t
+
σ
c
)
)
(
σ
b
(
3
σ
t
−
σ
c
)
−
2
σ
c
σ
t
4
σ
b
2
+
2
σ
b
(
σ
t
−
σ
c
)
−
σ
c
σ
t
)
A
=
σ
c
3
+
c
1
σ
c
−
c
2
σ
c
2
In terms of the equivalent stress (
σ
e
) and the mean stress (
σ
m
), the Bresler-Pister yield criterion can be written as
σ
e
=
a
+
b
σ
m
+
c
σ
m
2
;
σ
e
=
3
J
2
,
σ
m
=
I
1
/
3
.
The Etse-Willam form of the Bresler-Pister yield criterion for concrete can be expressed as
J
2
=
1
3
I
1
−
1
2
3
(
σ
t
σ
c
2
−
σ
t
2
)
I
1
2
where
σ
c
is the yield stress in uniaxial compression and
σ
t
is the yield stress in uniaxial tension.
The GAZT yield criterion for plastic collapse of foams also has a form similar to the Bresler-Pister yield criterion and can be expressed as
J
2
=
{
1
3
σ
t
−
0.03
3
ρ
ρ
m
σ
t
I
1
2
−
1
3
σ
c
+
0.03
3
ρ
ρ
m
σ
c
I
1
2
where
ρ
is the density of the foam and
ρ
m
is the density of the matrix material.
