Multiphysical time-dependent creep response of FGMEE hollow cylinder in thermal and humid environment

Abstract

In this paper, we investigate the history of radial displacement, stresses, electric potential, and magnetic potential of a functionally graded magneto-electro-elastic (FGMEE) hollow cylinder subjected to an axisymmetric hygro-thermo-magneto-electro-mechanical loading for the plane strain condition. The material properties are taken as a power-law function of radius. Using stress-displacement relations, equations of equilibrium, electrostatic and magnetostatic equations, we find a differential equation including creep strains. Initially, eliminating creep strains, we obtain an analytical solution for the primitive stresses and electric and magnetic potential. In the next step, considering creep strains, we find the creep stress rates by applying the Norton law and Prandtl–Reuss equations for steady-state hygrothermal boundary condition. Finally, using an iterative method, we find the time-dependent creep stresses, radial displacement, and magnetic and potential field redistributions at any time. In numerical section, are comprehensively investigate the effects of grading index, hygrothermal environmental conditions, rotating speed, and temperature- and moisture-dependency of elastic constant of FGMEE.

Appendix

The components of matrices $H$ and $E$ for the primitive state can be expressed as

$$\begin{aligned} H =& \left [ \textstyle\begin{array}{c@{\quad }c@{\quad }c@{\quad }c@{\quad }c@{\quad }c} a^{\beta + m_{1} - 1}V_{1} & a^{\beta + m_{2} - 1}V_{2} & V_{3}/a^{2} & V_{6}/a^{2} & 0& 0\\ b^{\beta + m_{1} - 1}V_{1} & b^{\beta + m_{2} - 1}V_{2} & V_{3}/b^{2} & V_{6}/b^{2} & 0& 0\\ a^{m_{1}}E_{1} &a^{m_{2}}E_{2} & a^{ - \beta } V_{4}&a^{ - \beta } V_{7} &1 &0\\ b^{m_{1}}E_{1} &b^{m_{2}}E_{2}&b^{ - \beta } V_{4}&b^{ - \beta } V_{7}&1&0\\ a^{m_{1}}N_{1} &a^{m_{2}}N_{2}&a^{ - \beta } V_{5} &a^{ - \beta } V_{8}&0&1\\ b^{m_{1}}N_{1} &b^{m_{2}}N_{2}&b^{ - \beta } V_{5} &b^{ - \beta } V_{8} &0&1\\ \end{array}\displaystyle \right ], \\ V_{i} =& ( C_{1} m_{i} + C_{2} ), \qquad E_{i} =\biggl( \frac{L_{2}}{m_{i}} + L_{1} \biggr), \qquad N_{i} =\biggl( P_{1} + \frac{P_{2}}{m_{i}} \biggr), \quad i= 1, \! 2, \\ V_{3} = & C_{4} - C_{1} \beta B_{8} + C_{2} B_{8}, \qquad V_{4} =\biggl( \frac{L_{5}}{\beta } - \frac{L_{2} B_{8}}{\beta } + L_{1} B_{8} \biggr), \\ V_{5} =& \biggl( P_{1} B_{8} - \frac{P_{2} B_{8}}{\beta } - \frac{P_{3}}{\beta } \biggr), \\ V_{6} =& C_{3} - C_{1} \beta B_{7} + C_{2} B_{7}, \qquad V_{7} = \biggl( - \frac{L_{3}}{\beta } - \frac{L_{2} B_{7}}{\beta } + L_{1} B_{7} \biggr), \\ V_{8} = & \biggl( B_{7} P_{1} - \frac{P_{2} B_{7}}{\beta } + \frac{P_{5}}{\beta } \biggr), \\ E =& \left [ \textstyle\begin{array}{c} P_{a} - S_{1}(a) \\ P_{b} - S_{1}(b) \\ \phi _{a} - S_{2}(a) \\ \phi _{b} - S_{2}(b) \\ \psi _{a} - S_{3}(a) \\ \psi _{b} - S_{3}(b) \end{array}\displaystyle \right ], \\ S_{1} = & C_{1}\bigl(B_{3}r^{\beta } + B_{4}(\beta + 1)r^{2\beta } + B_{5}(1 - \beta ) + 3B_{6}r^{2 + \beta } \bigr) \\ &{} + C_{2}\bigl(B_{3}r^{\beta } + B_{4}r^{2\beta } + B_{5} + B_{6}r^{2 + \beta } \bigr) \\ &{} + \bigl( C_{5}r^{\beta } - \dot{\lambda }_{1}r^{2\beta } \bigr) \bigl( W_{1}r^{ - \beta } + W_{2} \bigr) + \bigl( C_{6}r^{\beta } - \bar{ \zeta }_{1}r^{2\beta } \bigr) \bigl( S_{1}r^{ - \beta } + S_{2} \bigr) , \\ S_{2} =& L_{1} \bigl( B_{3} r+ B_{4} r^{\beta +1} + B_{5} r^{1-\beta } + B_{6} r^{3} \bigr) \\ &{} + L_{2} \biggl( B_{3} r+ \frac{B_{4}}{\beta +1} r^{\beta +1} + \frac{B_{5}}{1-\beta } r^{1-\beta } + \frac{B_{6}}{3} r^{3} \biggr) \\ &{} + L_{4} \biggl( \frac{W_{1}}{-\beta +1} r^{-\beta +1} + W_{2} r \biggr) + L_{6} \biggl( \frac{S_{1}}{-\beta +1} r^{-\beta +1} + S_{2} r \biggr), \\ S_{3} =& P_{1} \bigl( B_{3} r+ B_{4} r^{\beta +1} + B_{5} r^{1-\beta } + B_{6} r^{3} \bigr) \\ &{} + P_{2} \biggl( B_{3} r+ \frac{B_{4}}{\beta +1} r^{\beta +1} + \frac{B_{5}}{1-\beta } r^{1-\beta } + \frac{B_{6}}{3} r^{3} \biggr) \\ &{} + P_{4} \biggl( \frac{W_{1}}{-\beta +1} r^{-\beta +1} + W_{2} r \biggr) + P_{6} \biggl( \frac{S_{1}}{-\beta +1} r^{-\beta +1} + S_{2} r \biggr). \end{aligned}$$

Besides, for the state of creep progress, the components of $H$ are as before, and the components of $E$ can be rewritten as

$$\begin{aligned} E = & - \left [ \textstyle\begin{array}{c} S_{1}(a) \\ S_{1}(b) \\ S_{2}(a) \\ S_{2}(b) \\ S_{3}(a) \\ S_{3}(b) \end{array}\displaystyle \right ], \\ S_{1} = & C_{1} r^{\beta } \biggl( \frac{\partial G_{11}}{\partial r} r^{m_{1}} + G_{11} m_{1} r^{m_{1} -1} + \frac{\partial G_{21}}{\partial r} r^{m_{2}} + G_{21} m_{2} r^{m_{2} -1} \biggr) \\ &{} + C_{2} r^{\beta -1} \bigl( G_{11} r^{m_{1}} + G_{21} r^{m_{2}} \bigr) + ( C_{1} - C_{2} ) \frac{\sqrt{3}}{2} b_{0} r^{b_{1} +\beta } \sigma _{e}^{n_{0}}, \\ S_{2} =& + \int \biggl( L_{1} \biggl( \frac{\partial G_{11}}{\partial r} r^{m_{1}} + G_{11} m_{1} r^{m_{1} -1} + \frac{\partial G_{21}}{\partial r} r^{m_{2}} + G_{21} m_{2} r^{m_{2} -1} \biggr) \\ &{} + L_{2} \bigl(+ G_{11} r^{m_{1} -1} + G_{21} r^{m_{2} -1} \bigr)+( L_{1} - L_{2} ) \frac{\sqrt{3}}{2} b_{0} r^{b_{1}} \sigma _{e}^{n_{0}} \biggr)\,dr, \\ S_{3} = &+ \int \biggl( P_{1} \biggl( \frac{\partial G_{11}}{\partial r} r^{m_{1}} + G_{11} m_{1} r^{m_{1} -1} + \frac{\partial G_{21}}{\partial r} r^{m_{2}} + G_{21} m_{2} r^{m_{2} -1} \biggr) \\ & + P_{2} \bigl(+ G_{11} r^{m_{1} -1} + G_{21} r^{m_{2} -1} \bigr)+( P_{1} - P_{2} ) \frac{\sqrt{3}}{2} b_{0} r^{b_{1}} \sigma _{e}^{n_{0}} \biggr)\,dr. \end{aligned}$$