Elasticity pp 227-235 | Cite as

Antiplane Shear

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


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.
$$u_x = u_y = 0; \quad u_x = f(x,y).$$
Substituting these results into the strain-displacement relations (1.51) yields
$$e_{xx} = e_{yy} = e_{zz} = 0$$
$$e_{xy} = 0;e_{yz} = \frac{1}{2}\frac{{\partial f}}{{\partial y}};e_{zx} = \frac{1}{2}\frac{{\partial f}}{{\partial x}}.$$
It then follows from Hooke’s law (1.71) that
$$\sigma _{xx} = \sigma _{yy} = \sigma _{zz} = 0$$
$$\sigma _{xy} = 0;\sigma _{yz} = \mu \frac{{\partial f}}{{\partial y}};\sigma _{zx} = \mu \frac{{\partial f}}{{\partial x}}.$$

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.


Screw Dislocation Stress Concentration Factor Antiplane Shear Uniform Shear Rigid Support 
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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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