In quadratic criteria all the stress or strain components are combined into one expression. Many commonly used criteria for fiber-reinforced composites belong to a subset of fully interactive criteria called quadratic criteria. The general form of quadratic criteria can be expressed as a second-order polynomial.
 | (5–38) |
In plane stress-state (
), the polynomial
reduces to the simpler form
 | (5–39) |
The quadratic failure criteria in ACP (Tsai-Wu, Tsai-Hill, and Hoffman) differ
in how the coefficients 
and 
are defined.
Generally, the coefficients 
and 
are determined
so that the value of the failure criterion function corresponds to
the material strength when a unidirectional stress state is present.
However, not all coefficients can be determined in this way.
For the plane stress-state the Tsai-Wu criterion coefficients 
have the
values
 | (5–40) |
Thus, the criterion can be written as:
 | (5–41) |
The coefficient 
cannot be obtained directly from
the failure stresses of uniaxial load cases. For accurate results
it should be determined through biaxial load tests. In practice, it
is often given in the form of a non-dimensional interaction coefficient:
 | (5–42) |
To insure that the criterion represents a closed conical failure surface, the value of
must be within the range 
. However, the value range for physically meaningful material behavior is
more limited. The often used value 
corresponds to a "generalized Von Mises criterion". The final Tsai-Wu
constant 
becomes -1 as used in Equation 5–43. Similarly 
,
can be dealt with using the corresponding values 
and 
, which leads to the Tsai-Wu 3D expression:
 | (5–43) |
In ACP the Tsai-Wu constants are:
 , default -1 |
 , default -1 |
 , default -1 |
In the Tsai-Hill criterion, either tensile or compressive strengths are used for
determining the coefficients 
depending on the loading condition. The coefficients are:
 | (5–44) |
where the values of 
and 
are:
 | (5–45) |
The Tsai-Hill failure criterion differentiates between UD and woven plies. The Tsai-Hill failure criterion function for UD plies can be written in the form
 | (5–46) |
For woven plies, the function becomes
 | (5–47) |
where 
.
For the full 3D case, the following formulation can be used, as in [25]:
 | (5–48) |
where
 | (5–49) |
The Hoffman criterion defines the biaxial coefficients 
, 
, and 
with the following material strength expressions for the 3D stress
state:
 | (5–50) |
The Hoffman failure criterion for a 3D stress state can be written as:
 | (5–51) |
The biaxial coefficient 
for the plane stress state reduces
to:
 | (5–52) |
The entire Hoffman criterion in the plane stress case reduces to:
 | (5–53) |