Deformation Mappings and Strain

Here we have introduced a tensor F known as the deformation gradient tensor whose components reflect the various gradients in the deformation mapping and are given by 

Use of the fact that the deformation mapping may be written as x(t) = X + u(X, t) reveals that the deformation gradient may be written alternatively as 

Once we have determined F, we may query a particular state of deformation as to the disposition of the vector dX as a result of the deformation. For example, the simplest question one might ask about the vector between two neighboring material particles is how its length changes under deformation. To compute the length change, we compute the difference 

The strain tensor provides a more suitable geometric measure of relative displacements, and in the present context illustrates that the length change between neighboring material points is given by 

In addition to the use of the strain to deduce the length change as a result of deformation, it is possible to determine the ways in which angles change as well. The interested reader is invited either to deduce such results for him or herself or to consult any of the pertinent references at the end of the chapter. 

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