Simple shear

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{γ}

where $\dot{γ}$ 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:

Γ = [\begin{matrix} 0 & \dot{γ} & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]

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

Γ = \begin{matrix} \underset{⏟}{[\begin{matrix} 0 & \dot{γ} & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]} \\ simple shear \end{matrix} = \begin{matrix} \underset{⏟}{[\begin{matrix} 0 & \frac{1}{2} \dot{γ} & 0 \\ \frac{1}{2} \dot{γ} & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]} \\ pure shear \end{matrix} + \begin{matrix} \underset{⏟}{[\begin{matrix} 0 & \frac{1}{2} \dot{γ} & 0 \\ - \frac{1}{2} \dot{γ} & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]} \\ 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.

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In fluid mechanics

In solid mechanics

Simple shear stress–strain relation

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