Central to the computation of stresses and strains is the Jacobian matrix which relates the
initial and deformed configuration:
(1)
(2)
The transformation is fully described by the elements of the Jacobian matrix
:
(3)
So that
式 1 can be written in matrix
notation:
(4)
The Jacobian, or determinant of the Jacobian matrix, measures the relation between the initial
volume
and the volume in the initial configuration containing the same
points:
(5)
Physically, the value of the Jacobian cannot take the zero value without cancelling the volume
of a set of material points. So the Jacobian must not become negative whatever the final
configuration. This property insures the existence and uniqueness of the inverse
transformation:
(6)
At a regular point whereby definition of the field
is differentiable, the vicinity transformation is defined
by:
(7)
or in matrix form:
(8)
So, the Jacobian matrix
can be obtained from the matrix of gradients of
displacements:
(9)