Simple shear - Alchetron, The Free Social Encyclopedia

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:

∂ V x ∂ y = γ ˙ ,

where γ ˙ is the shear rate and:

∂ V x ∂ x = ∂ V x ∂ 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 ]

Simple shear with the rate γ ˙ is the combination of pure shear strain with the rate of 1/2 γ ˙ and rotation with the rate of 1/2 γ ˙ :

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

Important examples of simple shear include laminar flow through long channels of constant cross-section (Poiseuille flow), and elastomeric bearing pads in base isolation systems to allow critical buildings to survive earthquakes undamaged.

Simple shear 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.

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 ] .

We can also write the deformation gradient as

F = 1 + γ e 1 ⊗ e 2 .

References

Simple shear Wikipedia

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