Stress Concentration Factor in Functionally Graded Plates with Circular Holes Subjected to Anti-plane Shear Loading

Abstract

Stress concentration factors due to the presence of geometrical discontinuities (circular holes) in functionally graded plates are derived. The material property inhomogeneity is assumed to be in the radial direction originating at the center of the plate. Variable separable closed-form solutions are obtained for the stresses and displacements in functionally graded plates (without and with holes) subjected to anti-plane shear loading. The stresses in functionally graded plates without a hole are not homogeneous as it is in the case of homogeneous plates. Either a stress concentration (more than the applied stress) or dilution (less than the applied stress) occurs depending on whether the modulus increases (hardening graded material) or decreases (softening graded material) away from the center of the graded plate without a hole. A novel definition of the stress concentration factor due to the geometrical discontinuity in functionally graded plates is derived. The effect of the circular hole in functionally graded plates is to magnify (compared to homogeneous plates) the stress concentration when the modulus decreases away from the center of the hole (softening material). Beneficial reduction of the stress concentration factor is achieved in hardening functionally graded materials.

Appendix: Stress Concentration Factor in a Homogeneous Material

In this Appendix we list the stresses and displacements in a homogeneous plate with a circular hole subjected to an anti-plane shear loading. Starting from the variable separable solution (Eq. (14)) and with the boundary conditions given by Eq. (22) we can write the displacements and stresses in a homogeneous material as

$$ \begin{aligned} &w( r,\theta) = \frac{\sigma_{\infty }}{\mu} r\biggl[ 1 + \biggl( \frac{a}{r} \biggr)^{2} \biggr]\sin( \theta), \\ &\frac{\sigma_{rz}( r,\theta )}{\sigma_{\infty }} = \biggl[ 1 - \biggl( \frac{a}{r} \biggr)^{2} \biggr]\sin( \theta), \\ &\frac{\sigma_{\theta z}( r,\theta )}{\sigma_{\infty }} = \biggl[ 1 + \biggl( \frac{a}{r} \biggr)^{2} \biggr]\cos( \theta). \end{aligned} $$

(31)

The stress-ratio and stress concentration factor due to the circular hole in a homogeneous plate can now be written as

$$ \begin{aligned} &\varSigma_{\theta z}^{\mathrm{homogeneous}}( r ) = \frac{\sigma_{\theta z}( r,\theta = 0 )}{\sigma_{\infty }} = \biggl[ 1 + \biggl( \frac{a}{r} \biggr)^{2} \biggr], \\ &\mathit{SCF}^{\mathrm{homogeneous}} = \frac{\sigma_{\theta z}( r = a,\theta = 0 )}{\sigma_{\infty }} = 2. \end{aligned} $$

(32)

The stress-ratio and stress concentration factor are depicted in Figs. 5 and 6 along with those for the radial functionally graded materials.