Shear

YES 2020 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.

Comments

For shears in an arbitrary direction in a linear space, see Transvection. From a projective point of view these are (projective) transvections (central collineations with incident centre and axis) with centre at infinity and an affine hyperplane as axis.

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.

References

[Gu]	M.E. Gurtin, "An introduction to continuum mechanics", Acad. Press (1981) pp. Chapt. IX, §26 MR0636255 Zbl 0559.73001