#### 5.3.4.6. LaRC Failure Criterion

LaRC03 (2-D) and LaRC04 (3-D) are two sets of failure criteria for laminated fiber-reinforced composites. They are based on physical models for each failure mode and distinguish between fiber and matrix failure for different transverse fiber and matrix tension and compression modes. The LaRC criteria take into account that the apparent (in-situ) strength of an embedded ply, constrained by plies of different fiber orientations, is different compared to the same ply embedded in a UD laminate. Specifically, moderate transverse compression increases the apparent shear strength of a ply. Similarly in-plane shear significantly reduces the compressive strength of a ply. The evaluation of the in-situ strength also makes a distinction between thin and thick plies. The definition for a thick ply is a ply in which the slit crack is much smaller than the ply thickness. For epoxy E-glass and epoxy carbon laminates, the suggested threshold between thin and thick plies is 0.7 mm ([ 21 ] and [ 26 ]).

The implemented LaRC04 (3-D) failure criterion ACP assumes linear shear behavior and small angle deflection. The abbreviation LaRC stands for Langley Research Center.

##### 5.3.4.6.1. LaRC03/LaRC04 Constants

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 postprocessing in ACP:

- Fracture Toughness Ratio: (Dimensionless)

- Fracture Toughness Mode I: (Force / Length)

- Fracture Toughness Mode II: (Force / Length)

- Fracture Angle under Compression: (Degrees)

- Thin Ply Thickness Limit (Length)

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 [ 30 ]. The Thin Ply Thickness Limit is the only default value set for the LaRC parameters. The following reference values are drawn from [ 37 ]:

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

##### 5.3.4.6.2. General Expressions

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:

| (5–79)

where:

= thickness of an embedded ply

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

| (5–80)

##### 5.3.4.6.3. Fiber Misalignment Frame

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 (2-D) and LaRC04 (3-D).

LaRC03

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

| (5–81)

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 5–93 for matrix compression.

| (5–82)

The total misalignment angle is calculated from:

| (5–83)

LaRC04

The 2-D misalignment model assumes that the kinking occurs in the plane of the lamina. LaRC04 incorporates a more complex 3-D 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 thus given by:

| (5–84)

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:

| (5–85)

where:

= the misalignment angle for pure compression.

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

##### 5.3.4.6.4. LaRC03 (2-D)

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

###### Fiber Failure

For fiber tension a simple maximum strain approach is applied:

| (5–86)

Fiber compression failure for matrix compression is calculated as follows:

| (5–87)

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

| (5–88)

###### 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.

| (5–89)

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

| (5–90)

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.

| (5–91)

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

| (5–92)

The failure function for the second case is:

| (5–93)

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

##### 5.3.4.6.5. LaRC04 (3-D)

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

###### Fiber Failure

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

| (5–94)

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

| (5–95)

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

| (5–96)

###### Matrix Failure

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

| (5–97)

Matrix compressive failure is given by:

| (5–98)

where:

Matrix compressive failure with transverse tension is given by:

| (5–99)

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