Antiplane Shear
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Antiplane shear or antiplane strainW. 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 : \boldsymbol = \begin 0 & 0 & \epsilon_ \\ 0 & 0 & \epsilon_\\ \epsilon_ & \epsilon_ & 0\end where the 12\, plane is the plane of interest and the 3\, direction is perpendicular to that plane.


Displacements

The displacement field that leads to a state of antiplane shear is (in rectangular Cartesian coordinates) : u_1 = u_2 = 0 ~;~~ u_3 = \hat_3(x_1, x_2) where u_i,~ i=1,2,3 are the displacements in the x_1, x_2, x_3\, 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 : \boldsymbol \equiv \begin \sigma_ & \sigma_ & \sigma_ \\ \sigma_ & \sigma_ & \sigma_ \\ \sigma_ & \sigma_ & \sigma_ \end = \begin 0 & 0 & \mu~\cfrac \\ 0 & 0 & \mu~\cfrac \\ \mu~\cfrac & \mu~\cfrac & 0 \end where \mu\, 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 : \mu~\nabla^2 u_3 + b_3(x_1, x_2) = 0 where b_3 is the body force in the x_3 direction and \nabla^2 u_3 = \cfrac + \cfrac{\partial x_2^2}. 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

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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