Shear

2010 Mathematics Subject Classification: Primary: 15-XX [MSN][ZBL]

An affine transformation in the plane under which each point is displaced in the direction of the $x$-axis by a distance proportional to its ordinate. In a Cartesian coordinate system a shear is defined by the relations

$$x'=x+ky,\quad y'=y,\quad k\ne 0.$$ Area and orientation are preserved under a shear.

A shear in space in the direction of the $x$-axis is defined by the relations

$$x'=x+ky,\quad y'=y,\quad z'=z,\quad k\ne 0.$$ Volume and orientation are preserved under a shear in space.

The terminology "shear" (instead of transvection) is especially used in continuum mechanics (deformation of an elastic body e.g.). If the deformation is given by $x_1 = p_1+\gamma p_2,\ x_2 = p_2,\ x_3 = p_3$, the coefficient $\gamma$ is called the shearing strain. This is a simple shear.