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 & {\tfrac12 \dot \gamma} & 0 \\ {\tfrac12 \dot \gamma} & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \\ \mbox{pure shear} \end{matrix} + \begin{matrix} \underbrace \begin{bmatrix} 0 & {\tfrac12 \dot \gamma} & 0 \\ {- { \tfrac12 \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. 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

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