Antiplane shear or antiplane strain
[W. S. Slaughter, 2002, ''The Linearized Theory of Elasticity'', Birkhauser] is a special state of
strain in a body. This state of strain is achieved when the
displacement
Displacement may refer to:
Physical sciences
Mathematics and physics
*Displacement (geometry), is the difference between the final and initial position of a point trajectory (for instance, the center of mass of a moving object). The actual path ...
s in the body are zero in the plane of interest but nonzero in the direction perpendicular to the plane. For small strains, the
strain tensor
In mechanics, strain is defined as relative deformation, compared to a position configuration. Different equivalent choices may be made for the expression of a strain field depending on whether it is defined with respect to the initial or the ...
under antiplane shear can be written as
:
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
isotropic
In physics and geometry, isotropy () is uniformity in all orientations. Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence '' anisotropy''. ''Anisotropy'' is also ...
,
linear elastic material, the
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
See also
*
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, infinitesimal ...
*
Deformation (mechanics)
In physics and continuum mechanics, deformation is the change in the shape (geometry), shape or size of an object. It has dimension (physics), dimension of length with SI unit of metre (m). It is quantified as the residual displacement (geometr ...
Elasticity (physics)
Solid mechanics