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Deformation of a Body
When a deformable body is subjected to external forces, a body may translate, rotate and deform as well. Thus, after deformation the body occupies a new region. The initial region occupied by the body is called Reference Configuration and the new region occupied by the body after translation, rotation and deformation is called Deformed Configuration. Let us consider a point P in reference configuration. Its position with respect to origin of a reference axes system  is shown in Figure 2.3. The point P occupies a new position  and its position vector  is also given.
| Figure 2.3 Reference and deformed configurations |
The deformation map is defined as
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(2.9) |
Thus, deformation map is a vector valued function. Similarly, for deformation of a point Q to  , we can write
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(2.10) |
We can find the deformation  as
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(2.11) |
where  is called Deformation Gradient. In component form, one can write
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(2.12) |
Now, let us give the deformation map for the displacement of a point. Let us consider the point P in reference configuration again. It undergoes a deformation  and occupies a new position . Thus, we can write this deformation as follows
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(2.13) |
This gives us the deformation gradient as
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(2.14) |
or in component form
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(2.15) |
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