Consider 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 $\left( {u_r^M,u_\theta ^M} \right)and\;\left( {\sigma _{rr}^M,\sigma _{\theta \theta }^M} \right)$ denote the radial and tangential displacement and stress components, respectively material M. Similarly, $\left( {u_r^N,u_\theta ^N} \right)\;and\;\left( {\sigma _{rr}^N,\sigma _{\theta \theta }^N} \right)$ 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:

Question

Consider 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 $\left( {u_r^M,u_\theta ^M} \right)and\;\left( {\sigma _{rr}^M,\sigma _{\theta \theta }^M} \right)$ denote the radial and tangential displacement and stress components, respectively material M. Similarly, $\left( {u_r^N,u_\theta ^N} \right)\;and\;\left( {\sigma _{rr}^N,\sigma _{\theta \theta }^N} \right)$ 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:

TuteeHUB · Accepted Answer

$u_r^M = u_r^N\;and\;\sigma _{rr}^M = \sigma _{rr}^N\;and\;u_\theta ^M = u_\theta ^Nand\;\sigma _{\theta \theta }^M = \sigma _{\theta \theta }^N$

TuteeHUB · Answer

$u_r^M = u_r^N\;and\;\sigma _{rr}^M = \sigma _{rr}^N\;only$

TuteeHUB · Answer

$u_\theta ^M = u_\theta ^N\;and\;\sigma _{\theta \theta }^M = \sigma _{\theta \theta }^N\;only$

TuteeHUB · Answer

$u_{rr}^M = u_{rr}^N\;and\;\sigma _{\theta \theta }^M = \sigma _{\theta \theta }^N\;only$

1.	Consider 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 \(\left( {u_r^M,u_\theta ^M} \right)and\;\left( {\sigma _{rr}^M,\sigma _{\theta \theta }^M} \right)\) denote the radial and tangential displacement and stress components, respectively material M. Similarly, \(\left( {u_r^N,u_\theta ^N} \right)\;and\;\left( {\sigma _{rr}^N,\sigma _{\theta \theta }^N} \right)\) 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:
A.	\(u_r^M = u_r^N\;and\;\sigma _{rr}^M = \sigma _{rr}^N\;only\)
B.	\(u_r^M = u_r^N\;and\;\sigma _{rr}^M = \sigma _{rr}^N\;and\;u_\theta ^M = u_\theta ^Nand\;\sigma _{\theta \theta }^M = \sigma _{\theta \theta }^N\)
C.	\(u_\theta ^M = u_\theta ^N\;and\;\sigma _{\theta \theta }^M = \sigma _{\theta \theta }^N\;only\)
D.	\(u_{rr}^M = u_{rr}^N\;and\;\sigma _{\theta \theta }^M = \sigma _{\theta \theta }^N\;only\)
Answer» B. \(u_r^M = u_r^N\;and\;\sigma _{rr}^M = \sigma _{rr}^N\;and\;u_\theta ^M = u_\theta ^Nand\;\sigma _{\theta \theta }^M = \sigma _{\theta \theta }^N\)

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