where the plane is the plane of interest and the direction is perpendicular to that plane.
Displacements
The displacement field that leads to a state of antiplane shear is (in rectangular Cartesian coordinates)
where are the displacements in the directions.
Stresses
For an
stress
tensor that results from a state of antiplane shear can be expressed as
where is the shear modulus of the material.
Equilibrium equation for antiplane shear
The conservation of linear momentum in the absence of inertial forces takes the form of the equilibrium equation. For general states of stress there are three equilibrium equations. However, for antiplane shear, with the assumption that body forces in the 1 and 2 directions are 0, these reduce to one equilibrium equation which is expressed as
where is the body force in the direction and . Note that this equation is valid only for infinitesimal strains.
Applications
The antiplane shear assumption is used to determine the stresses and displacements due to a
screw dislocation
.
References
^W. S. Slaughter, 2002, The Linearized Theory of Elasticity, Birkhauser