#### 5.3.4.5. Puck Failure Criteria

There are several different Puck Failure criteria, the following sections describe them.

##### 5.3.4.5.1. Simple and Modified Puck Criterion

The two oldest Puck failure criterion formulations are simple Puck and modified Puck. Both criteria consider failure due to longitudinal loads and matrix failure mode due to transverse and shear loads separately ([ 27 ] and [ 28 ]).

For both the simple and modified Puck criteria, failure in fiber direction is calculated the same way as in the maximum stress criterion:

| (5–60)

Matrix failure is calculated differently for each formulation as illustrated in Equation 5–61 for simple Puck. Equation 5–62 demonstrates how tensile or compressive failure stresses are used depending on the stress state.

| (5–61)

where:

| (5–62)

The modified Puck criterion differs from the simple criterion only in the formulation for matrix failure:

| (5–63)

As in Hashin Failure Criterion, the failure occurs when either or reaches one, so the failure criterion function is:

| (5–64)

Despite being called simple in the failure criteria configuration in the Failure Criteria Definition dialog the Puck modified version is actually implemented. The name is referring to the simplicity of that criterion in comparison to Puck's Action Plane Strength Criterion.

##### 5.3.4.5.2. Puck's Action Plane Strength Criterion

The following sections describe the different failure modes for Puck’s action plane strength criterion.

###### 5.3.4.5.2.1. Fiber Failure (FF)

As in the simple Puck criterion, one option for evaluating fiber failure is to use the maximum stress criterion for that case ([ 29 ], [ 30 ], and [ 31 ]):

| (5–65)

and similarly a maximum strain criterion:

| (5–66)

A more complicated version for FF criterion was presented by Puck for the World Wide Failure Exercise, but the maximum stress criterion is considered sufficient for the case of FF.

###### 5.3.4.5.2.2. Interfiber Failure (IFF)

Interfiber failure is formulated differently depending on the model type.

Plane stress-state

Interfiber failure, or interfiber fracture ([ 29 ] and [ 30 ]) can be explained in the cutting plane for which the principal stress of a UD layer is zero in the case of plane stress.

Figure : Fracture Curve in σ2, τ21 Space for σ1 = 0

The curve consists of two ellipses (modes and ) and one parabola (mode ). Generally Puck's action plane strength criterion is formed utilizing the following 7 parameters, , where stands for fracture resistances and for slope parameters of the fracture curves. The symbols and denote the reference to direction parallel to the fibers and transverse (perpendicular) to the fibers. The values for and define the intersections of the curve with -axis, as well as for the intersection with -axis. The slope parameters and are the inclinations in the latter intersections.

The failure conditions for IFF are:

| (5–67)

The superscript denotes that the fracture resistance belongs to the action plane.

| (5–68)

The assumption is valid here and leads to:

| (5–69)

Equation 5–70 is also valid.

| (5–70)

As the failure criterion functions and the functions for their corresponding stress exposure factors are the same, they can be written as follows (given Equation 5–69 and Equation 5–70):

| (5–71)

3-D Stress State

While the latter formulations have been a reduced case working in ()-stress space, the 3-D stress state can be described with Equation 5–72:

| (5–72)

where:

From the above equations, the failure criterion function is formulated in the fracture (action) plane using the corresponding stresses and strains. The formulations for the stresses , , and in an arbitrary plane with the inclination angle are:

| (5–73)

To find the stress exposure factor the angle is iterated to find the global maximum, as the failure will occur for that angle. An analytical solution for the fracture angle is only available for plane stress-state by assuming:

| (5–74)

which leads to formulations for the exposure factor:

| (5–75)

Puck illustrated in [ 29 ] that the latter criterion can be used as a criterion to determine delamination, if an additional weakening factor for the interface is applied, finally resulting in:

| (5–76)

The active failure mode depends on the fraction angle and the sign of . Delamination can occur if is positive and is 90 degree. The failure modes and happen with negative .

Puck Constants

Different default values for the coefficients are set for carbon and glass fiber plies to:

Carbon:
Glass:

Those values are compliant with recommendations given in [ 32 ].

Influence of fiber parallel stresses on inter-fiber failure

To take into account that some fibers might break already under uniaxial loads much lower than loads which cause ultimate failure (which can be seen as a kind of degradation), weakening factors can be introduced for the strength parameters. Puck formulated a power law relation in [ 29 ]:

| (5–77)

where and and can be can be experimentally determined.

Different approaches exist to handle this problem numerically. The function given in Equation 5–77 can be replaced by an elliptic function:

| (5–78)

where:

and are degradation parameters.

In ACP, the stress exposure factor is calculated by intersecting the weakening factor ellipse with a straight line defined by the stress vector using the parameters:

Otherwise the fiber failure criterion determines the stress exposure factor .

Default values for the degradation parameters are and .

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