For simplicity, consider a rectangular block of material with dimensions a0, b0, and c0. Its volume V0 is given by,
When the block is loaded by stress, its volume will change since each dimension now includes a direct strain measure. To calculate the volume when loaded Vf, we multiply the new dimensions of the block,
Products of strain measures will be much smaller than individual strain measures when the overall strain in the block is small (i.e. linear strain theory). Therefore, we were able to drop the strain products in the equation above.
The relative change in volume is found by dividing the volume difference by the initial volume,
Hence, the relative volume change (for small strains) is equal to the sum of the 3 direct strains.