Imperfect Interfaces

Both NairnMPM and NairnFEA can model imperfect interfaces between phases in composite structures.

Imperfect Interface Theory

Modeling interfaces, which are often finite-thickness interphases, in composite materials is difficult. In numerical modeling, it would seemingly be straight-forward to discretize interphase zones and thereby explicitly model all effects. This approach has two problems. First, interphase zones may be much smaller than the bulk materials. Resolving both bulk materials and a thin interphase would require a highly refined model, which may exceed computational capacity. Second, interphase properties may be unknown and/or may vary within a transition zone from one material to another. Alternative methods for interphase modeling are needed to overcome these challenges. This need is especially important in nanocomposites because the amount of interphase per unit volume of reinforcement greatly exceeds the amount of interphase in composites with mi cron or larger reinforcement phases. As a consequence, interphases are expected to play a larger role, good or bad, in nanocomposite properties.

One way to model interphases is to abandon attempts for explicit modeling and instead replace 3D interphases with 2D interfaces The interphase effects are reduced to modeling the response of 2D interfaces due to tractions normal and tangential to the interfacial surface, which can be modeled by interface traction laws. Elimination of 3D interphases removes the resolution problem. The use of interface traction laws always replaces numerous unknown and potentially unmeasurable interphase properties with a much smaller number of interface parameters. If interface traction laws can be determined, one can potentially model interphase effects well. Example of using this for analytical modeling of interface effects in composite materials are in Hashin (1991) and Nairn and Liu (1997) Implementation of imperfect interfaces in FEA and in MPM is described in Nairn (2007). An alternative method for modeling imperfect interfaces in MPM using multimaterial MPM is described in Nairn (2013).

Imperfect Interface Traction Laws

To model 3D interphase effects using imperfect interfaces, the 2D interface is allowed to develop displacement discontinuities. For an isotropic interphase, it suffices to resolve interfacial traction into normal and tangential tractions (T_n and T_t) and assume they are functions of normal and tangential displacement discontinuities ([u_n] and [u_n]) at the interface:

      [math]\displaystyle{ T_n = f_n([u_n]) \qquad {\rm and } \qquad T_t = f_t([u_t]) }[/math]

Interfaces in composite materials may develop potential energy that is needed for effective property analysis. For an elastic interface, interfacial potential energy is:

      [math]\displaystyle{ \phi_i = \int_{S_i}\left(\int \vec T\cdot[d\vec u]\right) dS }[/math]

where S_i is interfacial area. The simplest assumption for traction laws is that they are linear and elastic. The tractions become:

      [math]\displaystyle{ T_n = D_n[u_n] \qquad {\rm and } \qquad T_t = D_t[u_t] }[/math]

and interfacial energy becomes

      [math]\displaystyle{ \phi_i = {1\over2} \int_{S_i} \left( D_n[u_n]^2 + D_t[u_t]^2\right) dS }[/math]

Notice that all 3D interphase properties have been reduced to just two interface parameters, D_n and D_t. These interface stiffnesses range from zero, for a debonded interface with no tractions, to infinity, for a perfect interface with no displacement discontinuity.