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, 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, 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.²³ 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

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

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

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