Exact Solution for Classic Coupled Thermoelasticity in Cylindrical Coordinates

Overview

In this entry, the classic coupled thermoelasticity model of hollow and solid cylinders under radial-symmetric loading condition (r, t) is considered. A full analytical method is used, and an exact unique solution of the classic coupled equations is presented. The thermal and mechanical boundary conditions, the body force, and the heat source are considered in the most general forms, where no limiting assumption is used.

Introduction

The coupled thermoelasticity of structural problems is frequently referred to in literature, where the assumption is used in the advanced engineering design problems for the structures under thermal shock loads. A thick circular cylindrical pressure vessel, as a structural member, may be subjected to mechanical and thermal shock loads. A pressure vessel under thermal shock at refineries and power plants, when the period of thermal shock is of the same order of magnitude as the period of lowest natural frequency of the vessel, may experience the...

Appendices

Appendix 1 Mechanical Boundary Conditions

For mechanical boundary conditions, four options are available:known radial displacement, known radial stress, and a combination of them.

• 1. Radial displacements at inner and outer surfaces are known as

  $$ \eqalign{ u({r_i},t)={f_1}(t)\cr u({r_{\circ }},t)={f_2}(t)\cr } $$

  In this case, we have C ₁₁ = 1, C ₁₂ = 0, C ₂₁ = 1, and C ₂₂ = 0.

  For fixed surfaces, it is enough to consider f ₁(t) and f ₂(t) equal to zero.

• 2. When radial stress at inner and outer surfaces are known, by the help of (1) and (2), we can write

  $$ \eqalign{ {\sigma_{\it rr }}\left( {{r_i},t} \right)= \frac{\it E}{{\left( {1+\nu } \right)\left( {1-2\nu } \right)}}\left[ {\left( {1-\nu } \right){u_{,r }}+\nu \frac{1}{{{r_i}}}u} \right]\cr -\frac{{\it E\alpha }}{{\left( {1-2\nu } \right)}}T\left( {{r_i},t} \right)={f_1}(t)\cr {\sigma_{{\theta \theta }}}\left( {{r_o},t} \right)= \frac{\it E}{{\left( {1+\nu } \right)\left( {1-2\upsilon } \right)}}\left[ {\left( {1-\nu } \right){u_{,r }}+\nu \frac{1}{{{r_o}}}u} \right]\cr -\frac{{\it E\alpha }}{{\left( {1-2\nu } \right)}}T\left( {{r_o},t} \right)={f_2}(t)\cr } $$

In this case, we have

$$ \eqalign{ {C_{11 }}=\frac{E}{{\left( {1+\nu } \right)\left( {1-2\nu } \right)}},{C_{11 }}=\frac{{\rm E\left( {1-\nu } \right)}}{{\left( {1+\nu } \right)\left( {1-2\nu } \right)}} \cr {C_{21 }}=\frac{\rm E}{{\left( {1+\nu } \right)\left( {1-2\nu } \right)}},\ {C_{22 }}=\frac{{\rm E\left( {1-\nu } \right)}}{{\left( {1+\nu } \right)\left( {1-2\nu } \right)}}\cr } $$

For traction free, it is enough to consider ${f_1}(t)$ and ${f_2}(t)$ equal to zero.

The third and fourth mechanical boundary conditions are the combination of above mentioned first and second boundary conditions.

Appendix 2 Thermal Boundary Conditions

For thermal boundary conditions, six options are available:specified temperature, heat flux, and convection.

• 1. Temperatures at inner and outer surfaces are known as

  $$ \eqalign{ T({r_i},t)={f_1}(t)\cr T({r_{\circ }},t)={f_2}(t)\cr } $$

  In this case, we have C ₁₁ = 1, C ₁₂ = 0, C ₂₁ = 1, and C ₂₂ = 0.

• 2. Heat fluxes at inner and outer surfaces are known as

  $$ \eqalign{ T{,_r}({r_i},t)={f_1}(t)\cr T{,_r}({r_{\circ }},t)={f_2}(t)\cr } $$

  In this case, we have C ₃₁ = k, C ₃₂ = 0, C ₄₁ = k, and C ₄₂ = 0, where k is the thermal conduction coefficient.

  For an insulated surface, it is enough to consider ${f_1}(t)$ and ${f_2}(t)$ equal to zero.

• 3. Convections at inner and outer surfaces are known as

  $$ \eqalign{ {h_i}T({r_i},t)+kT{,_r}({r_i},t)={f_3}(t)\cr {h_i}T({r_o},t)+kT{,_r}({r_o},t)={f_4}(t)\cr } $$

In this case, we have C ₃₁ = h _i, C ₃₂ = k, C ₄₁ = h _o, and C ₄₂ = k, where h _i and h _o are the thermal convection coefficients at inner and outer surfaces of the cylinder, respectively. The fourth to sixth cases for thermal boundary conditions are combinations of the above-mentioned first to third boundary conditions.