Simple shear
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Simple shear is a
deformation Deformation can refer to: * Deformation (engineering), changes in an object's shape or form due to the application of a force or forces. ** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies. * Defor ...
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 bio ...
, simple shear is a special case of
deformation Deformation can refer to: * Deformation (engineering), changes in an object's shape or form due to the application of a force or forces. ** Deformation (physics), such changes considered and analyzed as displacements of continuum bodies. * Defor ...
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 is a ...
vectors has a non-zero value: :V_x=f(x,y) :V_y=V_z=0 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 gradi ...
of velocity is constant and perpendicular to the velocity itself: :\frac = \dot \gamma , where \dot \gamma is the
shear rate In physics, shear rate is the rate at which a progressive shearing deformation is applied to some material. Simple shear The shear rate for a fluid flowing between two parallel plates, one moving at a constant speed and the other one stationary ...
and: :\frac = \frac = 0 The displacement gradient tensor Γ for this deformation has only one nonzero term: :\Gamma = \begin 0 & & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end Simple shear with the rate \dot \gamma is the combination of pure shear strain with the rate of \dot \gamma 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 \dot \gamma: :\Gamma = \begin \underbrace \begin 0 & & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end \\ \mbox\end = \begin \underbrace \begin 0 & & 0 \\ & 0 & 0 \\ 0 & 0 & 0 \end \\ \mbox \end + \begin \underbrace \begin 0 & & 0 \\ & 0 & 0 \\ 0 & 0 & 0 \end \\ \mbox \end The mathematical model representing simple shear is a
shear mapping In plane geometry, a shear mapping is a linear map that displaces each point in a fixed direction, by an amount proportional to its signed distance from the line that is parallel to that direction and goes through the origin. This type of mappi ...
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'' (franchis ...
. 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 In earthquake engineering, vibration control is a set of technical means aimed to mitigate seismic impacts in building and non-building structures. All seismic vibration control devices may be classified as ''passive'', ''active'' or ''hybrid'' ...
, 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 decoupl ...
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 In mechanics and geology, pure shear is a three-dimensional homogeneous flattening of a body. It is an example of irrotational strain in which body is elongated in one direction while being shortened perpendicularly. For soft materials, such as ru ...
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 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 : \boldsymbol = \begin 1 & \gamma & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end. We can also write the deformation gradient as : \boldsymbol = \boldsymbol + \gamma\mathbf_1\otimes\mathbf_2.


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 the ot ...
, denoted \tau, is related to
shear strain In physics, deformation is the continuum mechanics transformation of a body from a ''reference'' configuration to a ''current'' configuration. A configuration is a set containing the positions of all particles of the body. A deformation can ...
, denoted \gamma, by the following equation: \tau = \gamma G\, where G is the
shear modulus In materials science, shear modulus or modulus of rigidity, denoted by ''G'', or sometimes ''S'' or ''μ'', is a measure of the elastic shear stiffness of a material and is defined as the ratio of shear stress to the shear strain: :G \ \stackrel ...
of the material, given by G = \frac Here E 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 leng ...
and \nu is
Poisson's ratio In materials science and solid mechanics, Poisson's ratio \nu ( nu) is a measure of the Poisson effect, the deformation (expansion or contraction) of a material in directions perpendicular to the specific direction of loading. The value of Pois ...
. Combining gives \tau = \frac


See also

*
Deformation (mechanics) In physics, deformation is the continuum mechanics transformation of a body from a ''reference'' configuration to a ''current'' configuration. A configuration is a set containing the positions of all particles of the body. A deformation can ...
*
Infinitesimal strain theory In continuum mechanics, the infinitesimal strain theory is a mathematical approach to the description of the deformation of a solid body in which the displacements of the material particles are assumed to be much smaller (indeed, infinitesimally ...
*
Finite strain theory In continuum mechanics, the finite strain theory—also called large strain theory, or large deformation theory—deals with deformations in which strains and/or rotations are large enough to invalidate assumptions inherent in infinitesimal strai ...
*
Pure shear In mechanics and geology, pure shear is a three-dimensional homogeneous flattening of a body. It is an example of irrotational strain in which body is elongated in one direction while being shortened perpendicularly. For soft materials, such as ru ...


References

{{DEFAULTSORT:Simple Shear Fluid mechanics Continuum mechanics