Simple shear

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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:

$\mathrm{\Gamma }=\left[\begin{array}{ccc}0& \dot{\gamma }& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$

Simple shear with the rate $\dot{\gamma }$ is the combination of pure shear strain with the rate of Template:Sfrac$\dot{\gamma }$ and rotation with the rate of Template:Sfrac$\dot{\gamma }$:

$\mathrm{\Gamma }=\begin{array}{c}\underbrace{\left[\begin{array}{ccc}0& \dot{\gamma }& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]}\\ \text{simple shear}\end{array}=\begin{array}{c}\underbrace{\left[\begin{array}{ccc}0& \frac{1}{2}\dot{\gamma }& 0\\ \frac{1}{2}\dot{\gamma }& 0& 0\\ 0& 0& 0\end{array}\right]}\\ \text{pure shear}\end{array}+\begin{array}{c}\underbrace{\left[\begin{array}{ccc}0& \frac{1}{2}\dot{\gamma }& 0\\ -\frac{1}{2}\dot{\gamma }& 0& 0\\ 0& 0& 0\end{array}\right]}\\ \text{solid rotation}\end{array}$

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

Template:Main 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

$𝑭=\left[\begin{array}{ccc}1& \gamma & 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right].$

We can also write the deformation gradient as

$𝑭=1+\gamma {𝐞}_1\otimes {𝐞}_2.$

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