Damage initiation for fiber-reinforced composites

Damage initiation refers to the onset of degradation at a material point. In Abaqus the damage initiation criteria for fiber-reinforced composites are based on Hashin's theory (see Hashin and Rotem, 1973, and Hashin, 1980). These criteria consider four different damage initiation mechanisms: fiber tension, fiber compression, matrix tension, and matrix compression.

The initiation criteria have the following general forms:

Fiber tension $\left({\widehat{\sigma }}_{11}\ge 0\right)$:

Fiber compression $\left({\widehat{\sigma }}_{11}<0\right)$:

Matrix tension $\left({\widehat{\sigma }}_{22}\ge 0\right)$:

Matrix compression $\left(\widehat{{\sigma }_{22}}<0\right)$:

In the above equations

$X^T$

denotes the longitudinal tensile strength;

$X^C$

denotes the longitudinal compressive strength;

$Y^T$

denotes the transverse tensile strength;

$Y^C$

denotes the transverse compressive strength;

$S^L$

denotes the longitudinal shear strength;

$S^T$

denotes the transverse shear strength;

$\alpha$

is a coefficient that determines the contribution of the shear stress to the fiber tensile initiation criterion; and

${\widehat{\sigma }}_{11},{\widehat{\sigma }}_{22},{\widehat{\tau }}_{12}$

are components of the effective stress tensor, $\widehat{\sigma }$, that is used to evaluate the initiation criteria and which is computed from:

$$\widehat{\sigma }=\mathbf{M}\sigma ,$$

where $\sigma$ is the true stress and $\mathbf{M}$ is the damage operator:

$$\mathbf{M}=\left[\begin{array}{ccc}\frac{1}{\left(1-d_f\right)}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \frac{1}{\left(1-d_m\right)}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \frac{1}{\left(1-d_s\right)}\hfill \end{array}\right].$$

$d_f$, $d_m$, and $d_s$ are internal (damage) variables that characterize fiber, matrix, and shear damage, which are derived from damage variables $d_f^t$, $d_f^c$, $d_m^t$, and $d_m^c$, corresponding to the four modes previously discussed, as follows:

$$\begin{array}{cc}\hfill d_f& =\{\begin{array}{cc}\hfill d_f^t& \mathrm{if} {\widehat{\sigma }}_{11}\ge 0,\hfill \\ \hfill d_f^c& \mathrm{if} {\widehat{\sigma }}_{11}<0,\hfill \end{array}\hfill \\ \hfill d_m& =\{\begin{array}{cc}\hfill d_m^t& \mathrm{if} {\widehat{\sigma }}_{22}\ge 0,\hfill \\ \hfill d_m^c& \mathrm{if} {\widehat{\sigma }}_{22}<0,\hfill \end{array}\hfill \\ \hfill d_s& =1-\left(1-d_f^t\right)\left(1-d_f^c\right)\left(1-d_m^t\right)\left(1-d_m^c\right).\hfill \end{array}$$

Prior to any damage initiation and evolution the damage operator, $\mathbf{M}$, is equal to the identity matrix, so $\widehat{\sigma }=\sigma$. Once damage initiation and evolution has occurred for at least one mode, the damage operator becomes significant in the criteria for damage initiation of other modes (see Damage evolution and element removal for fiber-reinforced composites for discussion of damage evolution). The effective stress, $\widehat{\sigma }$, is intended to represent the stress acting over the damaged area that effectively resists the internal forces.

The initiation criteria presented above can be specialized to obtain the model proposed in Hashin and Rotem (1973) by setting $\alpha =0.0$ and $S^T=Y^C/2$ or the model proposed in Hashin (1980) by setting $\alpha =1.0$.

An output variable is associated with each initiation criterion (fiber tension, fiber compression, matrix tension, matrix compression) to indicate whether the criterion has been met. A value of 1.0 or higher indicates that the initiation criterion has been met (see Output for further details). If you define a damage initiation model without defining an associated evolution law, the initiation criteria will affect only output. Thus, you can use these criteria to evaluate the propensity of the material to undergo damage without modeling the damage process.

DAMAGE INITIATION, CRITERION=HASHIN, ALPHA=$\alpha$ 
$X^T$, $X^C$, $Y^T$, $Y^C$, $S^L$, $S^T$