Almost all displacement fields induced by boundary loads, support movements, temperature, body forces, or other perturbations to the initial condition are, unfortunately, nonlinear; that is: u, v, and w are cross-products or power functions of x, y, z (and perhaps other variables).

**Deformation
(Relative Displacement)**

Almost all displacement
fields induced by boundary loads, support movements, temperature, body forces,
or other perturbations to the initial condition are, unfortunately, *nonlinear*;
that is: *u*, *v*, and *w* are cross-products or power functions
of *x*, *y*, *z* (and perhaps other variables). However, as
shown in Figure. 2.1,* the fundamental linear assumption of calculus allows us
to directly use the relations of finite linear transformation to depict immediately
the relative displacement or deformation *du*, *dv*, *dw* of a
differential element *dx*, *dy*, *dz*.

On a differential scale, as long as *u*,
*v*, and *w* are continuous, smooth, and small, straight lines remain
straight and parallel lines and planes remain par-allel. Thus the standard
definition of a total derivative:

is more than a mathematical statement
that differential base lengths obey the laws of linear transformation.** The
resulting deformation tensor, *E _{ij}*, also

called the relative displacement tensor,
is directly analogous to the linear displacement tensor, ?* _{ij}*,
of coming pages, which transformed finite base-lengths. The elements of

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Civil : Principles of Solid Mechanics : Strain and Stress : Deformation (Relative Displacement) |

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