Volume Friction is not one of specific terminology for friction stir welding or processing. It might mean the ratio of the volume of base metal stirred by the pin of tool vs volume of base metal stirred by the shoulder of the tool.

Volume fraction refers to the volume fraction of the reinforcement in a metal matrix composite (MMC) irrespective of the manufacturing method. The volume fraction of MMC fabricated through friction stir processing can be computed using the following expression for continuous grooves of various cross sections.

Volume fraction = (Cross sectional area of the groove/Cross sectional area of the stir zone) * 100

Cross sectional area of the groove = (Width x Depth) for square or rectangular grooves

Cross sectional area of the groove = (Width x Depth) /2 for V shaped grooves

It is assumed that the particles are packed to full density in the groove. However, manual packing does not reach 100% packing. Compacted reinforcements using any pressing methods of powder metallurgy may give closer to 100% packing.

The above equation is not applicable to drilled hole method of packing the particles where holes are separated by some distance. This will reduce the overall volume fraction. Therefore the following expression should be used for such method of packing.

Volume fraction = (Volume of a single hole x total number of holes)/(Cross sectional area of the stir zone x total length of FSP seam) * 100

Volume of a single hole = (22/7) x diameter of the hole2/4 x depth of the hole

Total length of FSP seam = Plate length - Tool Shoulder diameter

Volume fraction in solid materials science often refers to concentration of a given phase, that is, the ratio of volume of the given phase to total volume of the sample.

Thus, the volume fraction is a percentage(%) and dimensionless.

If someone wants to focus on weight fraction, any particular formula like you mentioned for volume fraction!! @Isaac Dinaharan

Volume fraction in FSP can be easily calculated by taking the volume of the sheet on which FSP is to be performed (without holes) as the total volume. Then, the volume fraction of reinforcement can be calculated as the sum of the volume of holes divided by total volume.

Volume fraction as Dr. Isaac said is approximate by calculating the volume fraction of the groove, I think the more accurate way is to analyze several SEM images of the composite surface using image J software

The fractional vol% was calculated using the expression given below:

Fractional vol% (theoretical)= (groove area/tool pin area) x 100

Groove area = width of groove x depth of groove

Area of tool pin (projected) = diameter of tool pin x  tool pin length.

Fractional vol% (actual) = (groove area/area of SC) x 100.

Ref: Investigating Effects of Groove Dimensions on Microstructure and Mechanical Properties of AA6063/SiC Surface Composites Produced by Friction Stir Processing.