Plasticity Model for Bidirectional Fabric-Reinforced Composite Materials

The shear plasticity model of the matrix is formulated in the local directions of the bidirectional fabric-reinforced material and is assumed to be decoupled from the response along the local 1 and 2 fiber directions. The following elastic relation gives the in-plane shear stress in terms of the elastic shear strain, ${\epsilon }_{12}^{el}$ :

where $G_{12}$ is the shear modulus of the matrix, ${\epsilon }_{12}$ is the total shear strain, and ${\epsilon }_{12}^{pl}$ is the plastic shear strain.

The yield surface and flow rule are defined as:

where $\overline{\sigma }$ is the yield stress, ${\overline{\epsilon }}^{pl}$ is the equivalent plastic strain, and ${\dot{\overline{\epsilon }}}^{pl}$ is the equivalent plastic strain rate.

The hardening behavior and strain rate dependency are incorporated into the yield stress $\overline{\sigma }$ assuming this separable form:

where ${\sigma }^0$ is the static yield stress, and $R$ is a ratio, defined as $R=1.0$ at ${\dot{\overline{\epsilon }}}^{pl}=0.0$ .

The hardening behavior defines the evolution of the yield surface size, ${\sigma }^0$ , as a function of the equivalent plastic strain, ${\overline{\epsilon }}^{pl}$ . This evolution can be introduced by specifying ${\sigma }^0$ directly as a function of ${\overline{\epsilon }}^{pl}$ in tabular form or by using the Johnson-Cook law:

where ${\sigma |}_0$ is the yield stress at zero plastic strain and $C$ and $n$ are material parameters. A schematic figure of the hardening curve based on the Johnson-Cook law is shown in Figure 1.

Shear hardening curve of bidirectional fabric-reinforced composite materials.

You can specify rate-dependent effects using the Cowper-Symonds overstress power law form or the Johnson-Cook form. For more information on these methods, see Rate-Dependent Yield.