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, 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 ) {\displaystyle V_{x}=f(x,y)}

V y = V z = 0 {\displaystyle V_{y}=V_{z}=0}

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

∂ V x ∂ y = γ ˙ {\displaystyle {\frac {\partial V_{x}}{\partial y}}={\dot {\gamma }}} ,

where γ ˙ {\displaystyle {\dot {\gamma }}} is the shear rate and:

∂ V x ∂ x = ∂ V x ∂ z = 0 {\displaystyle {\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 = [ 0 γ ˙ 0 0 0 0 0 0 0 ] {\displaystyle \Gamma ={\begin{bmatrix}0&{\dot {\gamma }}&0\\0&0&0\\0&0&0\end{bmatrix}}}

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

Γ = [ 0 γ ˙ 0 0 0 0 0 0 0 ] ⏟ simple shear = [ 0 1 2 γ ˙ 0 1 2 γ ˙ 0 0 0 0 0 ] ⏟ pure shear + [ 0 1 2 γ ˙ 0 − 1 2 γ ˙ 0 0 0 0 0 ] ⏟ solid rotation {\displaystyle \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, 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.^[1] This deformation is differentiated from a pure shear by virtue of the presence of a rigid rotation of the material.^[2][3] When rubber deforms under simple shear, its stress-strain behavior is approximately linear.^[4] A rod under torsion is a practical example for a body under simple shear.^[5]

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 = [ 1 γ 0 0 1 0 0 0 1 ] . {\displaystyle {\boldsymbol {F}}={\begin{bmatrix}1&\gamma &0\\0&1&0\\0&0&1\end{bmatrix}}.}

We can also write the deformation gradient as

F = 1 + γ e 1 ⊗ e 2 . {\displaystyle {\boldsymbol {F}}={\boldsymbol {\mathit {1}}}+\gamma \mathbf {e} _{1}\otimes \mathbf {e} _{2}.}

In linear elasticity, shear stress, denoted τ {\displaystyle \tau } , is related to shear strain, denoted γ {\displaystyle \gamma } , by the following equation:^[6]

τ = γ G {\displaystyle \tau =\gamma G\,}

where G {\displaystyle G} is the shear modulus of the material, given by

G = E 2 ( 1 + ν ) {\displaystyle G={\frac {E}{2(1+\nu )}}}

Here E {\displaystyle E} is Young's modulus and ν {\displaystyle \nu } is Poisson's ratio. Combining gives

τ = γ E 2 ( 1 + ν ) {\displaystyle \tau ={\frac {\gamma E}{2(1+\nu )}}}

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