Antiplane shear or antiplane strainW. S. Slaughter, 2002, ''The Linearized Theory of Elasticity'', Birkhauser is a special state of

strain Strain may refer to: Science and technology * Strain (biology), variants of plants, viruses or bacteria; or an inbred animal used for experimental purposes * Strain (chemistry), a chemical stress of a molecule * Strain (injury), an injury to a mu ...

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 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 ...

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 Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...

, linear elastic material, the

stress Stress may refer to: Science and medicine * Stress (biology), an organism's response to a stressor such as an environmental condition * Stress (linguistics), relative emphasis or prominence given to a syllable in a word, or to a word in a phrase ...

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 In materials science, a dislocation or Taylor's dislocation is a linear crystallographic defect or irregularity within a crystal structure that contains an abrupt change in the arrangement of atoms. The movement of dislocations allow atoms to sl ...

Displacements

Stresses

Equilibrium equation for antiplane shear

Applications

References

See also