# Simple shear

Simple shear

Simple shear is a deformation in which parallel planes in a material remain parallel and maintain a constant distance, while translating relative to each other.

## In fluid mechanics

In fluid mechanics, simple shear is a special case of deformation where only one component of velocity vectors has a non-zero value:

${\displaystyle V_{x}=f(x,y)}$
${\displaystyle V_{y}=V_{z}=0}$

And the gradient of velocity is constant and perpendicular to the velocity itself:

${\displaystyle {\frac {\partial V_{x}}{\partial y}}={\dot {\gamma }}}$ ,

where ${\displaystyle {\dot {\gamma }}}$  is the shear rate and:

${\displaystyle {\frac {\partial V_{x}}{\partial x}}={\frac {\partial V_{x}}{\partial z}}=0}$

The displacement gradient tensor Γ for this deformation has only one nonzero term:

${\displaystyle \Gamma ={\begin{bmatrix}0&{\dot {\gamma }}&0\\0&0&0\\0&0&0\end{bmatrix}}}$

Simple shear with the rate ${\displaystyle {\dot {\gamma }}}$  is the combination of pure shear strain with the rate of 1/2${\displaystyle {\dot {\gamma }}}$  and rotation with the rate of 1/2${\displaystyle {\dot {\gamma }}}$ :

${\displaystyle \Gamma ={\begin{matrix}\underbrace {\begin{bmatrix}0&{\dot {\gamma }}&0\\0&0&0\\0&0&0\end{bmatrix}} \\{\mbox{simple shear}}\end{matrix}}={\begin{matrix}\underbrace {\begin{bmatrix}0&{{\tfrac {1}{2}}{\dot {\gamma }}}&0\\{{\tfrac {1}{2}}{\dot {\gamma }}}&0&0\\0&0&0\end{bmatrix}} \\{\mbox{pure shear}}\end{matrix}}+{\begin{matrix}\underbrace {\begin{bmatrix}0&{{\tfrac {1}{2}}{\dot {\gamma }}}&0\\{-{{\tfrac {1}{2}}{\dot {\gamma }}}}&0&0\\0&0&0\end{bmatrix}} \\{\mbox{solid rotation}}\end{matrix}}}$

The mathematical model representing simple shear is a shear mapping restricted to the physical limits. It is an elementary linear transformation represented by a matrix. The model may represent laminar flow velocity at varying depths of a long channel with constant cross-section. Limited shear deformation is also used in vibration control, for instance base isolation of buildings for limiting earthquake damage.

## In solid mechanics

In solid mechanics, a simple shear deformation is defined as an isochoric plane deformation in which there are a set of line elements with a given reference orientation that do not change length and orientation during the deformation.[1] This deformation is differentiated from a pure shear by virtue of the presence of a rigid rotation of the material.[2][3] When rubber deforms under simple shear, its stress-strain behavior is approximately linear.[4] A rod under torsion is a practical example for a body under simple shear.[5]

If e1 is the fixed reference orientation in which line elements do not deform during the deformation and e1 − e2 is the plane of deformation, then the deformation gradient in simple shear can be expressed as

${\displaystyle {\boldsymbol {F}}={\begin{bmatrix}1&\gamma &0\\0&1&0\\0&0&1\end{bmatrix}}.}$

We can also write the deformation gradient as

${\displaystyle {\boldsymbol {F}}={\boldsymbol {\mathit {1}}}+\gamma \mathbf {e} _{1}\otimes \mathbf {e} _{2}.}$

### Simple shear stress–strain relation

In linear elasticity, shear stress, denoted ${\displaystyle \tau }$ , is related to shear strain, denoted ${\displaystyle \gamma }$ , by the following equation:[6]

${\displaystyle \tau =\gamma G\,}$

where ${\displaystyle G}$  is the shear modulus of the material, given by

${\displaystyle G={\frac {E}{2(1+\nu )}}}$

Here ${\displaystyle E}$  is Young's modulus and ${\displaystyle \nu }$  is Poisson's ratio. Combining gives

${\displaystyle \tau ={\frac {\gamma E}{2(1+\nu )}}}$