View source for Shear

An [[Affine transformation|affine transformation]] in the plane under which each point is displaced in the direction of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084850/s0848501.png" />-axis by a distance proportional to its ordinate. In a Cartesian coordinate system a shear is defined by the relations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084850/s0848502.png" /></td> </tr></table>

Area and orientation are preserved under a shear.

A shear in space in the direction of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084850/s0848503.png" />-axis is defined by the relations

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084850/s0848504.png" /></td> </tr></table>

Volume and orientation are preserved under a shear in space.



====Comments====
For shears in an arbitrary direction in a linear space, see [[Transvection|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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084850/s0848505.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084850/s0848506.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084850/s0848507.png" />, the coefficient <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s084/s084850/s0848508.png" /> is called the shearing strain. This is a simple shear.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.E. Gurtin,   "An introduction to continuum mechanics" , Acad. Press  (1981)  pp. Chapt. IX, §26</TD></TR></table>