For the plane stress-state the Tsai-Wu criterion coefficients have the values

(2–9)

Thus, the criterion can be written as:

(2–10)

Note that Equation 2–10 has been updated in the documentation of Release 2025 R1 with negative stress limits for the compression.

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:

(2–11)

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 2–12. Similarly , can be dealt with using the corresponding values and , which leads to the Tsai-Wu 3D expression:

(2–12)

Note that Equation 2–12 has been updated in the documentation of Release 2025 R1 with negative stress limits for the compression.

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:

(2–13)

where the values of and are:

(2–14)

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

(2–15)

For woven plies, the function becomes

(2–16)

where .

For the full 3D case, the following formulation can be used, as in [13]:

(2–17)

where

(2–18)

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

(2–19)

The Hoffman failure criterion for a 3D stress state can be written as:

(2–20)

The biaxial coefficient for the plane stress state reduces to:

(2–21)

The entire Hoffman criterion in the plane stress case reduces to:

(2–22)

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 ([23] and [24]).

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

(2–29)

Matrix failure is calculated differently for each formulation as illustrated in Equation 2–30 for simple Puck. Equation 2–31 demonstrates how tensile or compressive failure stresses are used depending on the stress state.

(2–30)

where:

(2–31)

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

(2–32)

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

(2–33)

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.

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

 

2.5.5.2.1. 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 ([25], [26], and [27]):

(2–34)

and similarly a maximum strain criterion:

(2–35)

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.

 

2.5.5.2.2. Interfiber Failure (IFF)

Interfiber failure is formulated differently depending on the model type.

Plane stress-state

Interfiber failure, or interfiber fracture ([ 25 ] and [ 26 ]) 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 2.1: 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:

(2–36)

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

(2–37)

The assumption is valid here and leads to:

(2–38)

Equation 2–39 is also valid.

(2–39)

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 2–38 and Equation 2–39):

(2–40)

3D Stress State

While the latter formulations have been a reduced case working in ()-stress space, the 3D stress state can be described with Equation 2–41:

(2–41)

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:

(2–42)

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:

(2–43)

which leads to formulations for the exposure factor:

(2–44)

Puck illustrated in [ 25 ] 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:

(2–45)

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 [28].

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 [ 25 ]:

(2–46)

where and and can be can be experimentally determined.

Different approaches exist to handle this problem numerically. The function given in Equation 2–46 can be replaced by an elliptic function:

(2–47)

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 .

The required unidirectional properties for the criteria are:

, , , , , , , , , , and .

where is the longitudinal shear strength and and are the fracture toughness for mode I and II.

The following LaRC Constants are required for ACP:

The fracture angle can be determined in tests or taken to be which has proven to have good results for carbon/epoxy and glass/epoxy laminates [ 26 ]. The Thin Ply Thickness Limit is the only default value set for the LaRC parameters. The following reference values are drawn from [33]:

Parameter	Typical Values (Carbon/epoxy)
Elastic Modulus, (GPa)	128
Elastic Modulus, , (GPa)	7.63
Fracture Angle (deg)	53
Fracture Toughness Mode 1, (N/mm)	0.28
Fracture Toughness Mode 2, (N/mm)	0.79
Fracture Toughness Ratio, g	0.35
Thin Ply Thickness Limit (mm)	0.7

Several failure functions involve the friction coefficients, in-situ strengths, and fiber misalignment. These values are described in the following sections.

Friction Coefficients

Laminates tend not too fail in the plane of maximum shear stress. This is attributed to internal friction and considered in the LaRC failure criteria with two friction coefficients:

Transverse Friction Coefficient:

Longitudinal Friction Coefficient:

In-Situ Ply Strength

The in-situ transverse direct strength and longitudinal shear strength for a thin ply are:

(2–48)

where:

= thickness of an embedded ply

For a thick ply, the in-situ strengths are not a function of the ply thickness:

(2–49)

Fiber compression, where the plies fail due to fiber kinking, is handled separately for transverse tension and transverse compression. In the model, imperfections in the fiber alignment are represented by regions of waviness, where transformed stresses can be calculated using a misalignment frame transforming the "original stresses". There are two different misalignment frames for LaRC03 (2D) and LaRC04 (3D).

LaRC03

For LaRC03, the stresses in the misaligned frame are computed as follows:

(2–50)

The misalignment angle for pure compression can be derived to 114 using and in the equations above as well as the stresses and the quadratic interaction criterion presented in Equation 2–62 for matrix compression.

(2–51)

The total misalignment angle is calculated from:

(2–52)

LaRC04

The 2D misalignment model assumes that the kinking occurs in the plane of the lamina. LaRC04 incorporates a more complex 3D model for the kink band formation. The kink plane is at an angle to the plane of the lamina. It is assumed to lie at an angle so that and is therefore given by:

(2–53)

and the stresses rotated in this plane are:

Following the definition of a kink plane, the stresses are rotated into a misaligned frame. This frame defined by evaluating the initial and the misalignment angles for pure compression as well as the shear strain under the assumption of linear shear behavior and small angle approximation:

(2–54)

where:

= the misalignment angle for pure compression.

Following this, the stresses can be rotated into the misaligned coordinate system:

The following sections describe the failure modes for LaRC03 (2D).

Fiber Failure

For fiber tension a simple maximum strain approach is applied:

(2–55)

Fiber compression failure for matrix compression is calculated as follows:

(2–56)

For fiber compression failure with matrix tension, the following quadratic equation has to be solved:

(2–57)

Matrix Failure

The formulation for matrix tensile failure is similar to that of fiber compressive failure under transverse compression. The difference is that the stress terms are not in the misaligned frame.

(2–58)

Matrix compression failure is divided into two separate cases depending on the longitudal loading. The failure function for the first case is:

(2–59)

where the effective shear stresses for matrix compression are based on the Mohr-Coulomb criterion which relates the effective shear stresses with the stresses of the fracture plane in Mohr's circle.

(2–60)

The transverse shear strength in terms of the transverse compressive strength and the fracture angle can be written as:

(2–61)

The failure function for the second case is:

(2–62)

where the effective shear stresses are rotated into the misaligned frame:

The following sections describe the failure modes for LaRC04 (3D).

Fiber Failure

The LaRC04 fiber tensile failure criteria is simply a maximum allowable stress criterion with no interaction of other components:

(2–63)

Fiber compressive failure is divided into two components depending of the direction of the transverse stress. For transverse compression it is:

(2–64)

The failure function for fiber compression and matrix tension is based on the Ansys Combined Stresses and Strains formulation for the LaRC criteria.

(2–65)

Matrix Failure

The failure function for matrix tension is based on the Ansys Combined Stresses and Strains formulation for the LaRC criteria.

(2–66)

Matrix compressive failure is given by:

(2–67)

where:

Matrix compressive failure with transverse tension is given by:

(2–68)

Replacing the invariants in Cuntze's invariant formulated SFCs [4] [5] with the associated stress states (factoring in IFF3 not the full stress state components but just the mode failure driving stress) and resolving for the simplified material stressing efforts yields the following:

(2–74)

where the superscripts and stand for tension and compression, respectively.

The friction parameter can be computed from the available friction coefficients of the UD material, derived in [ 5 ]

(2–75)

with typical ranges being:

If measurement data of the fracture angle is given, the friction coefficient is determined as:

(2–76)

or the directly related parameter as:

(2–77)

where .

If the IFF2 and IFF3 curves are based on enough test data, the 2 friction parameters can be determined using the following formulas [ 5 ]:

(2–78)

An interaction of the failure modes occurs due to the fact that the full (global) failure surface consists of five parts. Cuntze models these interactions by a simple, probability-based series spring model [ 3 ]. This model describes the lamina failure system as a series failure system which fails whenever any of its elements fail. Each mode is one element of this failure system and is treated as independent of the others. By this method, the interaction between FF and IFF modes as well as between the various IFF modes acts as a rounding-off procedure, enabling the determination of the final or inverse reserve factor (IRF).

(2–79)

In other words, the interaction equation includes all mode stressing efforts, and each of them represents a portion of the material's load-carrying capacity. In 2D practice at maximum 3 of the 5 modes will interact.

The modes' interaction exponent is obtained by curve-fitting of test data in the interaction zones. Experimental data showed that (for CFRP) .

The exponent is high in case of low scatter and low in the case of high scatter, hence chosing a low value for the interaction exponent is conservative. As an engineering assumption, is always given by the same value, regardless of the distinct mode interaction domain.

For pre-design, Cuntze recommends and

Cuntze's approach delivers the following simplified material stressing efforts for the five failure modes:

(2–80)