Simple shear

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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[edit]

In fluid mechanics, simple shear is a special case of deformation where only one component of velocity vectors has a non-zero value:

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

,

where is the shear rate and:

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

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

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[edit]

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

We can also write the deformation gradient as

Simple shear stress–strain relation[edit]

Shear stress, denoted , is related to shear strain, denoted , by the following equation:[6]

where is the shear modulus of the material, given by

Here is Young's modulus and is Poisson's ratio. Combining gives

See also[edit]

References[edit]

  1. ^ Ogden, R. W. (1984). Non-Linear Elastic Deformations. Dover. ISBN 9780486696485.
  2. ^ "Where do the Pure and Shear come from in the Pure Shear test?" (PDF). Retrieved 12 April 2013.
  3. ^ "Comparing Simple Shear and Pure Shear" (PDF). Retrieved 12 April 2013.
  4. ^ Yeoh, O. H. (1990). "Characterization of elastic properties of carbon-black-filled rubber vulcanizates". Rubber chemistry and technology. 63 (5): 792–805. doi:10.5254/1.3538289.
  5. ^ Roylance, David. "SHEAR AND TORSION" (PDF). mit.edu. MIT. Retrieved 17 February 2018.
  6. ^ "Strength of Materials". Eformulae.com. Retrieved 24 December 2011.