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
Fluid mechanics is the branch of physics concerned with the mechanics of fluids ( liquids, gases, and plasmas) and the forces on them.
It has applications in a wide range of disciplines, including mechanical, aerospace, civil, chemical and ...
, simple shear is a special case of
deformation where only one component of
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
vectors has a non-zero value:
:
:
And the
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gr ...
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
and
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
with the rate of
:
:
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
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
. The model may represent
laminar flow
In fluid dynamics, laminar flow is characterized by fluid particles following smooth paths in layers, with each layer moving smoothly past the adjacent layers with little or no mixing. At low velocities, the fluid tends to flow without lateral mi ...
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
Seismic base isolation, also known as base isolation, or base isolation system, is one of the most popular means of protecting a structure against earthquake forces. It is a collection of structural elements which should substantially decoup ...
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
:
We can also write the deformation gradient as
:
Simple shear stress–strain relation
In linear elasticity,
shear stress
Shear stress, often denoted by ( Greek: tau), is the component of stress coplanar with a material cross section. It arises from the shear force, the component of force vector parallel to the material cross section. '' Normal stress'', on ...
, denoted
, is related to
shear strain, denoted
, by the following equation:
where
is the
shear modulus of the material, given by
Here
is
Young's modulus
Young's modulus E, the Young modulus, or the modulus of elasticity in tension or compression (i.e., negative tension), is a mechanical property that measures the tensile or compressive stiffness of a solid material when the force is applied ...
and
is
Poisson's ratio. Combining gives
See also
*
Deformation (mechanics)
*
Infinitesimal strain theory
*
Finite strain theory
*
Pure shear
References
{{DEFAULTSORT:Simple Shear
Fluid mechanics
Continuum mechanics