You can basically classify composite materials into two categories based on : (1) constituents materials i.e fibre- reinforced composite or particle-reinforced composite and (2) based on material structure i.e laminates or sandwich panels
You can find a good explanation for this question in "Mechanics of Composite Materials" by Autar Kaw.
There are many factors such as constituent materials nature, type, construction and etc.., will influence in the classification of composite materials. You can find the detailed information regarding this in the Mechanics of materials by Autar k. kaw book. Any how i list out few
1. Based on the nature
a. Natural Composites, b. man-made composites
2. Based on the nature of reinforcement
a. Continuous fiber reinforced composite, b. Short fiber reinforced composite, c. Particle reinforced composite
3. based on the type of matrix
a. i. Thermoset polymer based, a. ii. Thermoplastic polymer based, b. Metal Matrix based, c. Ceramic matrix based
4. based on the construction
a. laminated composite, b. Sandwich composite, c. Hybrid composite
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