Simple shear

right|frame|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:

:<math>V_x=f(x,y)</math>

:<math>V_y=V_z=0</math>

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

:<math>\frac {\partial V_x} {\partial y} = \dot \gamma </math>,

where <math>\dot \gamma </math> is the shear rate and:

:<math>\frac {\partial V_x} {\partial x} = \frac {\partial V_x} {\partial z} = 0 </math>

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

:<math>\Gamma = \begin{bmatrix} 0 & {\dot \gamma} & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}</math>

Simple shear with the rate <math>\dot \gamma</math> is the combination of pure shear strain with the rate of <math>\dot \gamma</math> and rotation with the rate of <math>\dot \gamma</math>:

:<math>\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} </math>

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

:<math> \boldsymbol{F} = \begin{bmatrix} 1 & \gamma & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}. </math>

We can also write the deformation gradient as

:<math> \boldsymbol{F} = \boldsymbol{\mathit{1 + \gamma\mathbf{e}_1\otimes\mathbf{e}_2. </math>

Simple shear stress–strain relation

In linear elasticity, shear stress, denoted <math>\tau</math>, is related to shear strain, denoted <math>\gamma</math>, by the following equation:

<math>\tau = \gamma G\,</math>

where <math>G</math> is the shear modulus of the material, given by

Here <math>E</math> is Young's modulus and <math>\nu</math> is Poisson's ratio. Combining gives

<math>\tau = \frac{\gamma E}{2(1+\nu)}</math>

In fluid mechanics

In solid mechanics

Simple shear stress–strain relation

See also

References