These images show the initial configuration of a body (square) and a nonlinear deformation of that body into a curvy shape (to the right of the square). Overlaid on the actual deformed shape is the so-called tangent mapping at the indicated point. It coincides with the nonlinear mapping to first-order accuracy.
Grade school children (or college freshman in some countries) learn that any nonlinear function has a local tangent mapping at any point of interest, , which is simply the straight-line tangent to the nonlinear curve at that point. The equation of the tangent mapping in this case is
, where evaluated at and .
Analogously, consider the nonlinear “material mapping” transformation commonly studied in continuum mechanics:
Input: Initial position vector of a point in the body
Output: Deformed position vector of the same point.
The “output” (deformed location of a point) is generally nonlinearly related to the “input” (initial location of a point), which is why the initial square in the animations deforms to a curvy shape. However, just as the nonlinear function has a local straight-line tangent line, this vector-to-vector transform also has a tangent mapping. The tangent mapping at a point in the initial configuration of the body (red dot in the animations) is defined by
is the deformation gradient tensor having components
is the deformed location of the point initially located at (blue dot in the animations)
Evidently, the deformation gradient tensor plays a role similar to the role played by slope ““ in the scalar case of .