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:

$V_{x}=f(x,y)$

$V_{y}=V_{z}=0$

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

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

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

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

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

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

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

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

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

Main article: Deformation (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. This deformation is differentiated from a pure shear by virtue of the presence of a rigid rotation of the material. When rubber deforms under simple shear, its stress-strain behavior is approximately linear. A rod under torsion is a practical example for a body under simple shear.

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

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

We can also write the deformation gradient as

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

Simple shear stress–strain relation

In linear elasticity, shear stress, denoted $\tau$, is related to shear strain, denoted $\gamma$, by the following equation:

$\tau =\gamma G\,$

where $G$ is the shear modulus of the material, given by

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

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

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

See also

• Deformation (mechanics)
• Infinitesimal strain theory
• Finite strain theory
• Pure shear