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where the vectors 
x, y, z are unit vectors in the direction of axes x, y, z, respectively.
Symmetry relations apply, i.e., ? yx = ? xy, ? zy = ? yz, ? xz = ? zx
Using the differentiation shorthand (12), we write Eq. (14) in the form
In expanded form:
Dilatation, ?, is a measure of the volumetric strain, ? v, i.e., associated with an uniform expansion of the elastic medium. Several equivalent expressions of the dilatation are as follows.
Cartesian notations
Tensor notations
or
Vector notations
Using the expansion rule for the vector product we write
The following equivalent notations are used interchangeably:
and
where
The elasticity relation in compliance formulation can be expressed as
where E is Young elastic modulus, v is Poisson ratio, and G is the shear modulus given by
Inversion of Eq. (31) yields the stiffness formulation of the constitutive relation
Lame constants are defined in relation with Eq. (33) as:
Conversely, we can relate the engineering elasticity constants, E and G, to the Lame constants, i.e.,
Substitution of Eq. (34) into Eq.
