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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
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{{TEX|done}}{{MSC|15}}
  
<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>
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An
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[[Affine transformation|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
  
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$$x'=x+ky,\quad y'=y,\quad k\ne 0.$$
 
Area and orientation are preserved under a shear.
 
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
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A shear in space in the direction of the $x$-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>
 
  
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$$x'=x+ky,\quad y'=y,\quad z'=z,\quad k\ne 0.$$
 
Volume and orientation are preserved under a shear in space.
 
Volume and orientation are preserved under a shear in space.
  
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====Comments====
 
====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.
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For shears in an arbitrary direction in a linear space, see
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[[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.
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The terminology  "shear"  (instead of transvection) is especially used
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in continuum mechanics (deformation of an elastic body e.g.). If the
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deformation is given by $x_1 = p_1+\gamma p_2,\ x_2 = p_2,\ x_3 =
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p_3$, the coefficient $\gamma$ is called the shearing strain. This is a simple shear.
  
 
====References====
 
====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>
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|valign="top"|{{Ref|Gu}}||valign="top"| M.E. Gurtin,  "An introduction to continuum mechanics", Acad. Press  (1981)  pp. Chapt. IX, §26 {{MR|0636255}}  {{ZBL|0559.73001}}
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Latest revision as of 18:28, 21 December 2016

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
How to Cite This Entry:
Shear. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Shear&oldid=16755
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article