Large-amplitude vibrations of functionally graded shallow arches subjected to cooling shock

Abstract

In this paper, the nonlinear large-amplitude vibrations of shallow arch structures made of functionally graded materials (FGMs) under cooling shock have been investigated. It is considered that the FG shallow arch is made of low carbon steel $\left(\mathrm{AISI }1020\right)$ and stainless steel $(\mathrm{SUS }304)$, whose material properties change in the thickness direction. Using the kinematic assumptions that are modeled based on the first-order shear deformation theory (FSDT) and the von Kármán’s geometrical nonlinearity; along with the aid of Hamilton’s principle, the shallow arch motion equations are obtained. The material properties vary in the direction of arch’s thickness due to the temperature changes and material distribution. Based on the Voigt rule of mixture and power law distribution, the dependence of material properties on temperature and material distribution is defined. Assuming uncoupled theory of thermoelasticity, first, the one-dimensional heat conduction equation is solved along the thickness of the arch in order to obtain the temperature distribution. Afterward, the equations of motion are solved. For the numerical solution of the heat conduction equation and the nonlinear equations of motion, the iterative hybrid method of generalized differential quadrature and the Newmark time integration scheme has been used in an iterative Newton–Raphson loop. After validating the present formulation, a parametric scrutiny is conducted regarding the influence of various parameters, namely, thermal load rapidity time, FG-index, dimensional parameters on the mid-plane non-dimensional lateral deflection of the arch as well as the changes in stress and material properties.

Appendix 1

$${\varvec{\mathfrak{M}}}_{11} = - I_{0} {\mathbf{I}}_{{N_{\theta } \times N_{\theta } }}$$

$${\varvec{\mathfrak{M}}}_{12} = 0_{{N_{\theta } \times N_{\theta } }}$$

$${\varvec{\mathfrak{M}}}_{13} = - I_{1} {\mathbf{I}}_{{N_{{N_{\theta } }} \times N_{{N_{\theta } }} }}$$

$${\varvec{\mathfrak{M}}}_{21} = 0_{{N_{\theta } \times N_{\theta } }}$$

$${\varvec{\mathfrak{M}}}_{22} = - I_{0} {\mathbf{I}}_{{N_{\theta } \times N_{\theta } }}$$

$${\varvec{\mathfrak{M}}}_{23} = 0_{{N_{\theta } \times N_{\theta } }}$$

$${\varvec{\mathfrak{M}}}_{31} = - I_{1} {\mathbf{I}}_{{N_{\theta } \times N_{\theta } }}$$

$${\varvec{\mathfrak{M}}}_{32} = 0_{{N_{\theta } \times N_{\theta } }}$$

$${\varvec{\mathfrak{M}}}_{33} = - I_{2} {\mathbf{I}}_{{N_{\theta } \times N_{\theta } }}$$

$$\hbar_{11} = \frac{{A_{11} }}{{R^{2} }}{\mathbf{D}}_{\theta }^{2}$$

$$\hbar_{12} = \frac{{A_{11} }}{{R^{3} }}{\mathbf{D}}_{\theta }^{2} \left( {{\mathbf{I}}_{{N_{\theta } \times N_{\theta } }} {\mathbf{w}}_{{N_{\theta } \times 1}} } \right){\mathbf{D}}_{\theta }^{1} - \frac{{A_{11} }}{{R^{2} }}{\mathbf{D}}_{\theta }^{1}$$

$$\hbar_{13} = \frac{{B_{11} }}{{R^{2} }}{\mathbf{D}}_{\theta }^{2}$$

$$\hbar_{21} = \frac{{A_{11} }}{{R^{2} }}{\mathbf{D}}_{\theta }^{1} + \frac{{A_{11} }}{{R^{3} }}\left[ {{\mathbf{D}}_{{{\varvec{\uptheta}}}}^{1} \left( {{\mathbf{I}}_{{N_{\theta } \times N_{\theta } }} {\mathbf{w}}_{{N_{\theta } \times 1}} } \right){\mathbf{D}}_{\theta }^{2} + {\mathbf{D}}_{\theta }^{2} \left( {{\mathbf{I}}_{{N_{\theta } \times N_{\theta } }} {\mathbf{w}}_{{N_{\theta } \times 1}} } \right){\mathbf{D}}_{\theta }^{1} } \right]$$

$$\hbar_{22} = 1.5 \times \frac{{A_{11} }}{{R^{4} }}{\mathbf{D}}_{\theta }^{2} \left( {{\mathbf{I}}_{{N_{\theta } \times N_{\theta } }} {\mathbf{w}}_{{N_{\theta } \times 1}} } \right){\mathbf{D}}_{\theta }^{1} \left( {{\mathbf{I}}_{{N_{\theta } \times N_{\theta } }} {\mathbf{w}}_{{N_{\theta } \times 1}} } \right){\mathbf{D}}_{\theta }^{1}$$

$$- \frac{{A_{11} }}{{R^{3} }}\left[ {0.5 \times {\mathbf{D}}_{\theta }^{1} \left( {{\mathbf{I}}_{{N_{\theta } \times N_{\theta } }} {\mathbf{w}}_{{N_{\theta } \times 1}} } \right){\mathbf{D}}_{\theta }^{1} + \left( {{\mathbf{I}}_{{N_{\theta } \times N_{\theta } }} {\mathbf{w}}_{{N_{\theta } \times 1}} } \right){\mathbf{D}}_{\theta }^{2} } \right] + \frac{{A_{55} }}{{R^{2} }}{\mathbf{D}}_{\theta }^{2} - \frac{{A_{11} }}{{R^{2} }}{\mathbf{I}}_{{N_{\theta } \times N_{\theta } }}$$

$$\hbar_{23} = \frac{{B_{11} }}{{R^{2} }}{\mathbf{D}}_{\theta }^{1} + \frac{{B_{11} }}{{R^{3} }}\left[ {{\mathbf{D}}_{{{\varvec{\uptheta}}}}^{1} \left( {{\mathbf{I}}_{{N_{\theta } \times N_{\theta } }} {\mathbf{w}}_{{N_{\theta } \times 1}} } \right){\mathbf{D}}_{\theta }^{2} + {\mathbf{D}}_{\theta }^{2} \left( {{\mathbf{I}}_{{N_{\theta } \times N_{\theta } }} {\mathbf{w}}_{{N_{\theta } \times 1}} } \right){\mathbf{D}}_{\theta }^{1} } \right] + \frac{{A_{55} }}{R}{\mathbf{D}}_{\theta }^{1}$$

$$\hbar_{31} = \frac{{B_{11} }}{{R^{2} }}{\mathbf{D}}_{\theta }^{2}$$

$$\hbar_{32} = \frac{{B_{11} }}{{R^{3} }}{\mathbf{D}}_{\theta }^{2} \left( {{\mathbf{I}}_{{N_{\theta } \times N_{\theta } }} {\mathbf{w}}_{{N_{\theta } \times 1}} } \right){\mathbf{D}}_{\theta }^{1} - \frac{{B_{11} }}{{R^{2} }}{\mathbf{D}}_{\theta }^{1} - \frac{{A_{55} }}{R}{\mathbf{D}}_{\theta }^{1}$$

$$\hbar_{{33}} = \frac{{\varvec{D}_{{11}} }}{{R^{2} }}{\varvec{D}}_{\theta }^{2} - A_{{55}} {\varvec{I}}_{{N_{\theta } \times N_{\theta } }}$$

Herein, ${\mathbf{I}}$ is the $N_{\theta } \times N_{\theta }$ identity matrix, ${\mathbf{D}}_{\theta }^{{\mathbf{m}}}$ represents the derivative matrix of order $m$, and ${\mathbf{w}}$ is the vector of nodal lateral deflections.