Interfacial Strength
The interfacial strength refers to the strength of the bond between the matrix phase and the dispersed phase. Usually interfacial strength is desired.
                                                  

Interfacial Strength in PMCs and MMCs
In polymeric matrix and metal matrix composites
high interfacial bonding is desirable so that the stress can be transmitted from the matrix phase to the dispersed phase in order to maximize the overall composite strength. (The dispersed phase is usually the stronger material.) If the bond between the matrix phase and the dispersed phase is not strong enough to transmit the stress, then the reinforcing phase slips out of the matrix and the strength of the fibers will not be transmitted to the matrix.
                                                   
Interfacial Strength in CMCs
A case where interfacial strength is not desirable is the case of ceramic matrix composites. In these composites failure originates in the matrix. In order to maximize the fracture toughness for these, it is desirable to have a relatively weak interfacial bond allowing the fibers to pull out.  As a result, a crack initiated in the matrix can be deflected along the fiber-matrix interface.  This improves fracture toughness.

In this picture we we see a matrix crack approaching a fiber in figure (a).
It is deflected along the fiber-matrix interface as shown in figure (b).
For the overall composite shown in figure (c), the increased crack path length due to fiber pullout significantly improves fracture toughness.

In figure (a) we see a matrix crack approaching a fiber. It is deflected along the fiber-matrix interface as shown in figure (b).  For the overall composite shown in figure (c), the increased crack path length due to fiber pullout significantly improves fracture toughness.

                 
Factors to consider for interfacial bonding:

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