### 4.1.2. Post-Failure Response

When an orthotropic equation of state is used in conjunction with material stress, strain or stress/strain failure criteria an option to use isotropic or orthotropic post failure response is included in the failure data input. Subsequent to failure initiation, the failed cell stiffness and strength properties are modified depending on the failure initiation modes described in the subsections below.

#### 4.1.2.1. Isotropic Post-Failure

The failure models outlined in Failure Initiation allow different tensile and shear failure stresses and/or strains for each of the “principal” directions. If the isotropic post-failure option is selected failed cells can only carry bulk compressive stresses. Therefore after failure is initiated in a cell the following occurs:

- The principal (material) stress in the direction of failure is set to zero.

- All shear moduli are set to zero.

- All shear stresses are set to zero.

- The average stress (in other words, pressure) is recomputed, using the normal calculation:

For the orthotropic equation of state, post-failure behavior is modelled as detailed below. This post-failure model is in effect an isotropic post-failure response.

- The orthotropic elastic incremental stress-strain relations are applied.

- The average stress (in other words, pressure) is recomputed, using the calculation above.

- If the cell is in compression,

  The principal stresses are set equal to the average stress (in other words, pressure):

- If the cell is in tension,

  All principal stresses, and therefore the average stress (in other words, pressure), are set to zero:

#### 4.1.2.2. Orthotropic Post-Failure Response

The post-failure behavior of laminated composite materials is better represented through the orthotropic post failure option. This option was developed specifically for simulating the performance and failure, including delamination, of fiber reinforced composite materials.

If the orthotropic option is selected, the user is prompted to define the post-failure parameters. This includes the post failure response mode for failure in each material direction and the failed material residual shear strength (see Table 4.1: Orthotropic Post-Failure Options).

Table 4.1: Orthotropic Post-Failure Options

Failed Direction | Post Failure Option | Post Failure Response
--- | --- | ---
FAIL 11 | 11 only Bulk | Zero tensile stress in 11 direction. Zero tensile stress in all directions.
FAIL 22 | 22 only Bulk | Zero tensile stress in 22 direction. Zero tensile stress in all directions.
FAIL 33 | 33 only Bulk | Zero tensile stress in 33 direction. Zero tensile stress in all directions.
FAIL 12 | 12 & 11 only 12 & 22 only Bulk | Zero tensile stress in 11 direction. Zero tensile stress in 22 direction. Zero tensile stress in all directions.
Residual Shear Stiffness Fracture | 0.0 to 1.0 | Residual shear modulus is set to the specified value times the intact shear modulus. Default value is 0.2.
Maximum Residual Shear Stress | 0.0 to 1.0e20 | Maximum shear stress allowed in a failed cell. A value equal to, or less than, the failure shear stress is recommended.

For all failure modes, on failure initiation, the stress in the failed material directions are set to zero. In addition the stresses in material directions orthogonal to the failed direction are reduced to account for the loss in the Poisson effect from strain in the failed direction. In subsequent cycles, cell tensile stresses are only allowed in non-failed directions.

If a cell fails in two or more directions, bulk failure is assumed.

Subsequent to failure initiation, the failed cell stiffness and strength properties are modified depending on the failure initiation modes described below.

Delamination

  In an axisymmetric simulation, the 11-direction is assumed to be through the thickness of the laminate and the 33-direction is the hoop direction. Delamination can result from excessive through thickness tensile stresses and/or strains or from excessive shear stress and or strain in the 12 plane. If failure is initiated in either of these two modes, the stress in the 11-direction is instantaneously set to zero and the strain in the 11-direction at failure is stored. Subsequently, if the tensile material strain in the 11-direction exceeds the failure strain, the material stiffness matrix is modified as

  | (4–1)

  This stiffness modification does not allow tensile through thickness stresses while tensile in-plane stresses are maintained.

  Also note that the delamination will in practice be associated with a reduction in shear stiffness. Often, in the absence of appropriate material data, a nominal value of 20% is typically used for the residual shear stiffness α.

In-plane Failure

  In an axisymmetric simulation, the 22- and 33-directions are assumed to be in the plane of the composite (in other words, in the fiber directions). If failure is initiated in these two modes, the stress in the failed direction is instantaneously set to zero and the strain in the failed direction at failure is stored. Subsequently, if the tensile material strain in the failed direction exceeds the failure strain, the material stiffness matrix is modified as

  22-failure:

    | (4–2)

  33-failure:

    | (4–3)

  This stiffness modification does not allow tensile stresses in the failed directions. Also note that these failure modes will in practice be associated with a reduction in shear stiffness. Often, in the absence of appropriate material data, a nominal value of 20% is typically used for the residual shear stiffness α.

Combined Failure

  The combined effect of failure in all three material directions is represented by a change in the material stiffness and strength to isotropic with no stress deviators and no tensile material stresses.

Melting and Decomposition

  Melting/vaporisation of epoxy and decomposition of the fiber material has been observed in composites with an epoxy matrix, for example Kevlar-epoxy [1]. This occurred in a finite region directly under the impact point under uniaxial strain conditions and velocities above or around 1000m/s.

  To represent this phenomena in an approximate way in a numerical model, the following features are available. An epoxy melting temperature can be specified. It is assumed that this has a very similar effect to delamination in the laminate. The procedure outlined in Section 4.1.2.2.1 is followed for this failure initiation mode.

  Additionally, a decomposition temperature for the fiber can be specified. Decomposition may lead to an inhomogeneous material of unknown properties. The model therefore assumes that the decomposed material has properties of the intact material under bulk compression. In bulk tension, pressure, deviatoric and tensile stresses are set to zero.

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