Project ForgeConsider two concentric circular cylinders of different materials M and N in contact with each other at r=b, as shown below. The interface at r=b is frictionless. The composite cylinder system is subjected to internal pressure P. Let (u_r^M,u_θ^M) and (σ_rr^M,σ_θθ^M) denote the radial and tangential displacement and stress components, respectively, in material M. Similarly, (u_r^N,u_θ^N) and (σ_rr^N,σ_θθ^N) denote the radial and tangential displacement and stress components, respectively, in material N. The boundary conditions that need to be satisfied at the frictionless interface between the two cylinders are:
- u_r^M=u_r^N and σ_rr^M=σ_rr^N only
- Bu_r^M=u_r^N and σ_rr^M=σ_rr^N and u_θ^M=u_θ^N and σ_θθ^M=σ_θθ^N
- Cu_θ^M=u_θ^N and σ_θθ^M=σ_θθ^N only
- Dσ_rr^M=σ_rr^N and σ_θθ^M=σ_θθ^N only
Explanation
At a bonded but frictionless (no-shear) interface, compatibility requires the radial displacements to match (no gap opens): u_r^M=u_r^N. Equilibrium (Newton's third law across the interface, in the radial direction only) requires the radial normal stress to be continuous: σ_rr^M=σ_rr^N. The hoop stress σ_θθ is NOT a traction on this interface (it acts parallel to it) and need not match, since the two materials are free to have different tangential stiffness/response.
Source: GATE ME 2019 Session 2 (Set 3) (2019)
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