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:

:$\frac = \dot \gamma$,

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

:$\frac = \frac = 0$

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

:$\Gamma = \begin 0 & & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end$

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

:$\Gamma = \begin \underbrace \begin 0 & & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end \\ \mbox\end = \begin \underbrace \begin 0 & & 0 \\ & 0 & 0 \\ 0 & 0 & 0 \end \\ \mbox \end + \begin \underbrace \begin 0 & & 0 \\ & 0 & 0 \\ 0 & 0 & 0 \end \\ \mbox \end$

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
:$\boldsymbol = \begin 1 & \gamma & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end.$
We can also write the deformation gradient as
:$\boldsymbol = \boldsymbol + \gamma\mathbf_1\otimes\mathbf_2.$

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

$G = \frac$

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

$\tau = \frac$

See also

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

References

{{DEFAULTSORT:Simple Shear
Fluid mechanics
Continuum mechanics