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GATE 2019 ME (ME2) – Question 40

Mechanics of Materials · Stress, strain and elastic constants · 2 marks · Multiple choice

Two concentric cylinders M and N contact at r = b. The interface is frictionless and the composite is loaded by internal pressure P. Let $u_r,u_\theta$ and $\sigma_{rr},\sigma_{\theta\theta}$ denote radial/tangential displacements and normal stresses, with superscripts denoting material. Which boundary conditions hold at the interface?

Diagram for GATE 2019 ME (ME2) question 40
  1. $u_r^M=u_r^N,\ \sigma_{rr}^M=\sigma_{rr}^N$ only
  2. $u_r^M=u_r^N,\ \sigma_{rr}^M=\sigma_{rr}^N,\ u_\theta^M=u_\theta^N,\ \sigma_{\theta\theta}^M=\sigma_{\theta\theta}^N$
  3. $u_\theta^M=u_\theta^N,\ \sigma_{\theta\theta}^M=\sigma_{\theta\theta}^N$ only
  4. $\sigma_{rr}^M=\sigma_{rr}^N,\ \sigma_{\theta\theta}^M=\sigma_{\theta\theta}^N$ only

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Correct answer: (A) $u_r^M=u_r^N,\ \sigma_{rr}^M=\sigma_{rr}^N$ only

Explanation

Contact compatibility gives $u_r^M=u_r^N$ and radial traction equilibrium gives $\sigma_{rr}^M=\sigma_{rr}^N$. Frictionless contact permits tangential slip; neither tangential displacement nor hoop stress must match. **A**.