Transport Theorems and Jump Condition

A volume element t/F in the undeformed body S2o is related to the volume element J V in the deformed body through the determinant, 7, of the deformation gradient 

Under this relation the material time derivative of the integral of an arbitrary function d can be calculated as 

Next we give the transport theorem for a surface integral The determinant of a (3 X 3)-matrix A is given by 

A surface element dS — dX aBX consists of line elements dX and 8X in the undeformed body and the corresponding surface element ds = dx aSx consists of line elements dx and Sx in the deformed body 

Thus the transport theorem for the surface integral of a scalar-valued function is given by 

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