Tangent mapping


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.

Background

Grade school children (or college freshman in some countries) learn that any nonlinear function y=f(x) has a local tangent mapping at any point of interest, x_0 , which is simply the straight-line tangent to the nonlinear curve at that point.  The equation of the tangent mapping in this case is

y-y_0 = m (x-x_0) , where m=dy/dx evaluated at x=x_0 and y_0=f(x_0) .

Analogously, consider the nonlinear “material mapping” transformation commonly studied in continuum mechanics:

Input:  Initial position vector of a point \mathbf{X}  in the body

Output: Deformed position vector \mathbf{x}  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 y=f(x) has a local straight-line tangent line, this vector-to-vector transform also has a tangent mapping.  The tangent mapping at a point \mathbf{X}_0 in the initial configuration of the body (red dot in the animations) is defined by

\mathbf{x}-\mathbf{x}_0 = \mathbf{F}\cdot(\mathbf{X}-\mathbf{X}_0) ,

where

\mathbf{F} is the deformation gradient tensor having components F_{ij}=\partial{x_i}/\partial{X_j}

\mathbf{x}_0 is the deformed location of the point initially located at \mathbf{X}_0 (blue dot in the animations)

Evidently, the deformation gradient tensor plays a role similar to the role played by slope m=dy/dx in the scalar case of y=f(x) .

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