Damage Initiation for Fiber-Reinforced Composites

Damage initiation refers to the onset of degradation at a material point. Abaqus supports two damage initiation criteria for unidirectional fiber-reinforced composites: the Hashin criterion and the LaRC05 criterion; and the ply fabric damage initiation criterion for bidirectional fabric-reinforced composites.

Damage is characterized by the degradation of material stiffness. It plays an important role in the analysis of fiber-reinforced composite materials.

Unidirectional fiber-reinforced composite materials exhibit elastic-brittle behavior; that is, damage in these materials is initiated without significant plastic deformation. Consequently, plasticity is often neglected when modeling the behavior of these materials. The fibers in the fiber-reinforced material are assumed to be parallel, as depicted in Figure 1.

Figure 1. Unidirectional lamina.

For unidirectional fiber-reinforced composite materials, you must specify material properties in a user-defined local coordinate system. The lamina is in the 1–2 plane, and the local 1-direction corresponds to the fiber direction. You must specify the undamaged material response using one of the methods for defining an orthotropic linear elastic material (see Linear Elastic Behavior); the most convenient of which is the method for defining an orthotropic material in plane stress (see Defining Orthotropic Elasticity in Plane Stress). However, the material response can also be defined in terms of the engineering constants or by specifying the elastic stiffness matrix directly.

For bidirectional fabric-reinforced composite materials, the shear response is dominated by the nonlinear behavior of the matrix, which includes both plasticity and stiffness degradation due to matrix microcracking. The fiber directions are assumed to be orthogonal. You must specify material properties in a user-defined local coordinate system, with the local 1-direction and 2-direction aligned with the fiber directions, as shown in Figure 2. The material response along the fiber directions is characterized with damaged elasticity, and the model differentiates between tensile and compressive fiber failure modes. The elastic undamaged response of the material must be defined using the bilamina elasticity model (see Defining Orthotropic Elasticity in Plane Stress with Different Moduli in Tension and Compression).

Figure 2. Bidirectional fabric.

The Abaqus anisotropic damage model for unidirectional fiber-reinforced composite is based on the work of Matzenmiller et al. (1995), Hashin and Rotem (1973), Hashin (1980), and Camanho and Davila (2002).

Four different modes of failure are considered:

• fiber rupture in tension;

• fiber buckling and kinking in compression;

• matrix cracking under transverse tension and shearing; and

• matrix crushing under transverse compression and shearing.

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=$\alpha$ 
$X^T$, $X^C$, $Y^T$, $Y^C$, $S^L$, $S^T$

The LaRC05 criterion (available only in Abaqus/Standard) for unidirectional fiber-reinforced composite is a three-dimensional model based on the theory of Pinho et al. (2012). It can be applied generally to polymer-matrix fiber-reinforced composites. It can also be used for fiber-reinforced composite lamina similar to the Hashin criterion. The criterion considers four different damage initiation mechanisms (matrix cracking, fiber kinking, fiber splitting, and fiber tension), as follows:

Matrix cracking:

where

The angle, $\alpha$ , represents the orientation of a critical plane perpendicular to the local 2–3 plane. The value of $\alpha$ is found numerically such that it maximizes $F_m^{crack}$ .

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

Fiber splitting $\left(-\frac{X_C}{2}\le {\sigma }_{11}\le 0\right)$ :

Fiber kinking $\left({\sigma }_{11}\le -\frac{X_C}{2}\right)$ :

In the fiber splitting and fiber kinking equations above, stresses in the fiber-misalignment frame are computed as:

where ${\sigma }_{ij}^{\psi }$ represents the stresses in the fiber-kinking plane:

For the three fiber-failure mechanisms above, Abaqus/Standard first checks the sign of ${\sigma }_{11}$ . If ${\sigma }_{11}$ is positive, the fiber tension criterion, $F_f^{tens}$ , is calculated; otherwise, for fiber splitting and kinking, stresses are rotated to the fiber-kinking plane, and then these obtained stresses are further rotated to the misalignment frame. During this process, the angles $\psi$ and $\phi$ are varied together until the maximum value of $F_f^{split}$ or $F_f^{kink}$ is found.

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 plane shear strength;

$S_T$

denotes the transverse shear strength;

${\alpha }_0$

denotes the fracture plane angle for pure compression (by default, ${\alpha }_0$ =53°);

${\eta }_L$

denotes the longitudinal shear friction coefficient; and

${\eta }_T$

denotes the transverse shear friction coefficient.

If ${\eta }_L$ and ${\eta }_T$ are not defined, they are calculated as:

An output variable is associated with each initiation criterion (matrix cracking, fiber kinking, fiber splitting, fiber tension) 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). The initiation criteria affect only output. Therefore, you can use these criteria to evaluate the propensity of the material to undergo damage without modeling the damage process.

DAMAGE INITIATION, CRITERION=LARC05 
$X_T$, $X_C$, $Y_T$, $Y_C$, $S_L$, ${\alpha }_0$, ${\phi }_0$, $S_T$
${\eta }_L$, ${\eta }_T$

The ply fabric damage initiation criterion for bidirectional fiber-reinforced composite (available only in Abaqus/Explicit) is an anisotropic damage model based on the work of Johnson, 2001 and Sokolinsky et al., 2011. The material response along the fiber directions is characterized with damaged elasticity. The model incorporates different initial (undamaged) stiffness in tension and compression (see Defining Orthotropic Elasticity in Plane Stress with Different Moduli in Tension and Compression), and differentiates between tensile and compressive fiber failure modes by activating the corresponding damage variable depending on the stress state in the fiber directions. The shear response is dominated by the nonlinear behavior of the matrix, which includes both plasticity (see Plasticity Model for Bidirectional Fabric-Reinforced Composite Materials) and stiffness degradation due to matrix microcracking.

Four different modes of damage initiation for the fibers and one for shear response of the matrix are considered:

• Fiber rupture in tension in the local 1-direction, with a damage variable $d_{1+}$ ;

• Fiber buckling and kinking in compression in the local 1-direction, with a damage variable $d_{1-}$ ;

• Fiber rupture in tension in the local 2-direction, with a damage variable $d_{2+}$ ;

• Fiber buckling and kinking in compression in the local 2-direction, with a damage variable $d_{2-}$ ; and

• Matrix microcracking due to shear deformation, with a damage variable $d_{12}$ .

The four fiber damage initiation criteria have the following general form:

In the above equation, an index $\alpha$ is used to simplify notation and is used in subsequent discussions, such that it takes the values $1+,1-,2+,2-$ depending on the sign of the corresponding stresses.

In the above equation:

${\varphi }_{\alpha }$

are functions that provide initiation criteria for fiber damage and are assumed to take the form:

$${\varphi }_{\alpha }=\frac{{\widehat{\sigma }}_{\alpha }}{X_{\alpha }},$$

where $X_{\alpha }$ are the tensile/compressive strengths for uniaxial loading along the fiber directions, and ${\widehat{\sigma }}_{\alpha }$ are effective stresses defined as:

$${\widehat{\sigma }}_{1+}=\frac{<{\sigma }_{11}>}{(1-d_{1+})},{\widehat{\sigma }}_{1-}=\frac{<-{\sigma }_{11}>}{(1-d_{1-})},$$

$${\widehat{\sigma }}_{2+}=\frac{<{\sigma }_{22}>}{(1-d_{2+})},{\widehat{\sigma }}_{2-}=\frac{<-{\sigma }_{22}>}{(1-d_{2-})}\mathrm{.}$$

The symbol $⟨⟩$ in the equations above represents the Macaulay bracket operator, which is defined for every $x\in \mathrm{\Re }$ as $⟨x⟩=\left(x+\left|x\right|\right)/2$ .

$r_{\alpha }$

are damage thresholds that are initially set to one. After damage activation ( ${\varphi }_{\alpha }=1$ ), they increase with increasing damage according to:

$$r_{\alpha }\left(t\right)=\underset{\tau \le t}{\max }{\varphi }_{\alpha }\left(t\right)\mathrm{.}$$

The definition ensures that the damage thresholds are nondecreasing quantities.

The damage initiation criterion for matrix shear failure has the following form:

In the above equation:

${\varphi }_{12}$

is the matrix shear failure criterion and is assumed to take the form:

$${\varphi }_{12}=\max (\frac{{\widehat{\sigma }}_{12}}{S},\frac{{\overline{\epsilon }}^{pl}}{{\overline{\epsilon }}_{\max }^{pl}}),$$

where $S$ is the shear stress for initial matrix damage, ${\widehat{\sigma }}_{12}$ is the effective shear stress defined as ${\widehat{\sigma }}_{12}={\sigma }_{12}/(1-d_{12})$ , ${\overline{\epsilon }}^{pl}$ is the equivalent plastic strain due to shear deformation, and ${\overline{\epsilon }}_{\max }^{pl}$ is a specified maximum value. The damage shear stress-strain relations are computed based as discussed in Plasticity Model for Bidirectional Fabric-Reinforced Composite Materials.

$r_{12}$

is the shear damage threshold that is initially set to one. After damage activation ( ${\varphi }_{12}=1$ ), it increases with increasing damage according to:

$$r_{12}\left(t\right)=\underset{\tau \le t}{\max }{\varphi }_{12}\left(t\right)\mathrm{.}$$

Once a particular damage initiation criterion is satisfied, the material stiffness is degraded according to the specified damage evolution law for that criterion. For a discussion of damage evolution, see Damage Evolution and Element Removal for Fiber-Reinforced Composites.

An output variable is associated with each initiation criterion (fiber tension in the local 1-direction, fiber compression in the local 1-direction, fiber tension in the local 2-direction, fiber compression in the local 2-direction, and matrix shear) to indicate whether the criterion has been met. A value of 1.0 or higher indicates that the initiation criterion has been met (for more details, see Output). If you define a damage initiation model without defining an associated evolution law, the initiation criteria affects only output. Therefore, you can use these criteria to evaluate the propensity of the material to undergo damage without modeling the damage process.

DAMAGE INITIATION, CRITERION=PLY FABRIC
$X_{1+}$, $X_{1-}$, $X_{2+}$, $X_{2-}$, $S$