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Elasticity pp 227-235 | Cite as

Antiplane Shear

  • J. R. Barber
Chapter
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 172)

Abstract

In Chapters 3–14, we have considered two-dimensional states of stress involving the in-plane displacements ux, uy and stress components σxx, σxy, σyy. Another class of two-dimensional stress states that satisfy the elasticity equations exactly is that in which the in-plane displacements ux, uy are everywhere zero, whilst the out-of-plane displacement uz 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
$$e_{xx} = e_{yy} = e_{zz} = 0$$
(15.2)
and
$$e_{xy} = 0;e_{yz} = \frac{1}{2}\frac{{\partial f}}{{\partial y}};e_{zx} = \frac{1}{2}\frac{{\partial f}}{{\partial x}}.$$
(15.3)
It then follows from Hooke’s law (1.71) that
$$\sigma _{xx} = \sigma _{yy} = \sigma _{zz} = 0$$
(15.4)
and
$$\sigma _{xy} = 0;\sigma _{yz} = \mu \frac{{\partial f}}{{\partial y}};\sigma _{zx} = \mu \frac{{\partial f}}{{\partial x}}.$$
(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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Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of Mechanical Engineering and Applied MechanicsUniversity of MichiganAnn ArborUSA

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