title: Simple shear
text: # 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${\dot {\gamma }}$:
  
      $\Gamma ={\begin{matrix}\underbrace {\begin{bmatrix}0&{\dot {\gamma }}&0\\0&0&0\\0&0&0\end{bmatrix}} \\{\mbox{simple shear}}\end{matrix}}={\begin{matrix}\underbrace {\begin{bmatrix}0&{{\tfrac {1}{2}}{\dot {\gamma }}}&0\\{{\tfrac {1}{2}}{\dot {\gamma }}}&0&0\\0&0&0\end{bmatrix}} \\{\mbox{pure shear}}\end{matrix}}+{\begin{matrix}\underbrace {\begin{bmatrix}0&{{\tfrac {1}{2}}{\dot {\gamma }}}&0\\{-{{\tfrac {1}{2}}{\dot {\gamma }}}}&0&0\\0&0&0\end{bmatrix}} \\{\mbox{solid rotation}}\end{matrix}}$
  
  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 $e_1$ is the fixed reference orientation in which line elements do not deform during the deformation and $e_1$ − $e_2$ 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
  
  $\tau ={\frac {\gamma E}{2(1+\nu )}}$
  
  