Assuming a uniform deformation, a point located at
will be shifted to
, which leads to the displacement
defined as
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(3.2) |
Further generalization leads to a description for non-uniform deformation, introducing a position dependent vector function
,
![]() |
(3.3) |
Restricting to small displacements from , the displacement function
can be developed into a Taylor series and truncated after the linear term at
, leading to a relation between the local displacement tensor and the displacement function,
![]() |
Therefore, the displacement can be expressed as
.
Frequently, the displacements
are expressed via the strain tensor
to describe the deformation of a body in three dimensions. In the limit of small deformations, the strain tensor is known as the Green tensor or Cauchy's infinitesimal strain tensor,
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(3.4) |
The relative length change in the
direction is described by the diagonal coefficients
, while the off-diagonal elements
denote the angular distortions by shear strains.
Also very common are the engineering strains , which are linked with the strain tensor as follows:
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(3.5) |
![]() |
(3.6) |