Physical Interpretation of The Finite Strain Tensor
The diagonal components of the Lagrangian finite strain tensor are related to the normal strain, e.g.
where is the normal strain or engineering strain in the direction .
The off-diagonal components of the Lagrangian finite strain tensor are related to shear strain, e.g.
where is the change in the angle between two line elements that were originally perpendicular with directions and, respectively.
Under certain circumstances, i.e. small displacements and small displacement rates, the components of the Lagrangian finite strain tensor may be approximated by the components of the infinitesimal strain tensor
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Derivation of the physical interpretation of the Lagrangian and Eulerian finite strain tensors The stretch ratio for the differential element (Figure) 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 (Figure), 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, respectivelly, 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
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