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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:
And the gradient of velocity is constant and perpendicular to the velocity itself:
where γ ˙ {\displaystyle {\dot {\gamma }}} is the shear rate and:
The displacement gradient tensor Γ for this deformation has only one nonzero term:
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 }}} :
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
We can also write the deformation gradient as
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 )}}}
