Elasticity pp 227-235 | Cite as

# Antiplane Shear

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## Abstract

In Chapters 3–14, we have considered two-dimensional states of stress involving the in-plane displacements u

_{x}, u_{y}and stress components σ_{xx}, σ_{xy}, σ_{yy}. Another class of two-dimensional stress states that satisfy the elasticity equations exactly is that in which the in-plane displacements u_{x}, u_{y}are everywhere zero, whilst the out-of-plane displacement u_{z}is independent of z — i.e.$$u_x = u_y = 0; \quad u_x = f(x,y).$$

(15.1)

Substituting these results into the strain-displacement relations (1.51) yields
and

$$e_{xx} = e_{yy} = e_{zz} = 0$$

(15.2)

$$e_{xy} = 0;e_{yz} = \frac{1}{2}\frac{{\partial f}}{{\partial y}};e_{zx} = \frac{1}{2}\frac{{\partial f}}{{\partial x}}.$$

(15.3)

It then follows from Hooke’s law (1.71) that
and

$$\sigma _{xx} = \sigma _{yy} = \sigma _{zz} = 0$$

(15.4)

$$\sigma _{xy} = 0;\sigma _{yz} = \mu \frac{{\partial f}}{{\partial y}};\sigma _{zx} = \mu \frac{{\partial f}}{{\partial x}}.$$

(15.5)

In other words, the only non-zero stress components are the two shear stresses σ_{ zx }, σ_{ zy } and these are functions of *x, y* only. Such a stress state is known as *antiplane shear* or *antiplane strain*.

## Keywords

Screw Dislocation Stress Concentration Factor Antiplane Shear Uniform Shear Rigid Support
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer Science+Business Media B.V. 2010