#### 5.3.4.3. Quadratic Failure Criteria

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.

##### 5.3.4.3.1. Tsai-Wu Failure Criterion

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 3-D expression:

| (5–43)

In ACP the Tsai-Wu constants are:

, default -1
, default -1
, default -1

##### 5.3.4.3.2. Tsai-Hill Failure Criterion

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 3-D case another formulation can be used, as in [ 25 ]:

| (5–48)

where

| (5–49)

##### 5.3.4.3.3. Hoffman Failure Criterion

The Hoffman criterion defines the biaxial coefficients , , and with the following material strength expressions for the 3-D stress state:

| (5–50)

The Hoffman failure criterion for a 3-D 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)

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