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- To see how this formula is derived, we start with the oriented area elements in the reference and current configurations:
The reference and current volumes of an element are
where .
Therefore,
or,
so,
So we get
or,
Q.E.D. (en)
- A measure of deformation is the difference between the squares of the differential line element , in the undeformed configuration, and , in the deformed configuration . Deformation has occurred if the difference is non zero, otherwise a rigid-body displacement has occurred. Thus we have,
In the Lagrangian description, using the material coordinates as the frame of reference, the linear transformation between the differential lines is
Then we have,
where are the components of the right Cauchy–Green deformation tensor, . Then, replacing this equation into the first equation we have,
or
where , are the components of a second-order tensor called the Green – St-Venant strain tensor or the Lagrangian finite strain tensor,
In the Eulerian description, using the spatial coordinates as the frame of reference, the linear transformation between the differential lines is
where are the components of the spatial deformation gradient tensor, . Thus we have
where the second order tensor is called Cauchy's deformation tensor, . Then we have,
or
where , are the components of a second-order tensor called the Eulerian-Almansi finite strain tensor,
Both the Lagrangian and Eulerian finite strain tensors can be conveniently expressed in terms of the displacement gradient tensor. For the Lagrangian strain tensor, first we differentiate the displacement vector with respect to the material coordinates to obtain the material displacement gradient tensor,
Replacing this equation into the expression for the Lagrangian finite strain tensor we have
or
Similarly, the Eulerian-Almansi finite strain tensor can be expressed as (en)
- The stretch ratio for the differential element in the direction of the unit vector at the material point , in the undeformed configuration, is defined as
where is the deformed magnitude of the differential element .
Similarly, the stretch ratio for the differential element , in the direction of the unit vector at the material point , in the deformed configuration, is defined as
The square of the stretch ratio is defined as
Knowing that
we have
where and are unit vectors.
The normal strain or engineering strain in any direction can be expressed as a function of the stretch ratio,
Thus, the normal strain in the direction at the material point may be expressed in terms of the stretch ratio as
solving for we have
The shear strain, or change in angle between two line elements and initially perpendicular, and oriented in the principal directions and , respectively, can also be expressed as a function of the stretch ratio. From the dot product between the deformed lines and we have
where is the angle between the lines and in the deformed configuration. Defining as the shear strain or reduction in the angle between two line elements that were originally perpendicular, we have
thus,
then
or (en)
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