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Thermo-mechanical analysis of functionally graded material beams using micropolar theory and higher-order unified formulation

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Abstract

In this work, thermo-mechanical analysis of functionally graded material beams using micropolar theory is studied. The analysis is based on a higher-order model in the framework of the Carrera unified formulation (CUF). Regarding use of CUF methodology, the three-dimensional (3D) displacement and micro-rotation fields are expressed as the approximation of the arbitrary order of the displacement and micro-rotation unknowns over the cross-section. Taylor expansion and Lagrange expansion in terms of the cross-section coordinates are considered in the model for approximating the displacement and micro-rotation fields. It is assumed that the mechanical and thermal properties of the beams are nonhomogeneous over the thickness; therefore, the nonlinear temperature rise profile is taken into account in the analysis. So, the effect of the steady-state heat conduction has been taken into account in the formulation. Here, uncoupled thermo-elastic problems are considered. The governing equations are written in terms of the so-called fundamental nuclei, which are independent of the order of expansion. Several numerical examples of free vibration analysis are presented and the effect of several parameters such as thermal loadings, power law indexes, orders of expansion and boundary conditions on the results are demonstrated. The equations can be used to analyze the beam structures in macro‐, micro‐, and nano‐scale through taking the micropolar couple stress and micro‐rotation effects into account.

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  1. Civil Engineering Department, Shahid Bahonar University of Kerman, Kerman, Iran

    Behnam Daraei, Saeed Shojaee & Saleh Hamzehei-Javaran

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  1. Behnam Daraei
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  2. Saeed Shojaee
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Appendix

Appendix

The variation of the internal strain energy \(\delta {L}_{\rm {int}}\) is as follows:

$$\begin{aligned}\delta {L}_{\rm {int}}=&{\int }_{\Omega }{\int }_{l}\left\{\left({F}_{s}{N}_{j}{\delta \omega }_{xsj}+{N}_{j}{\delta u}_{ysj}\frac{\partial {F}_{s}}{\partial z}\right)\left({C}_{66}^{MT}\left(-{F}_{\tau }{N}_{i}{\omega }_{x\tau i}+{N}_{i}{u}_{z\tau i}\frac{\partial {F}_{\tau }}{\partial y}\right)\right.\right.\\ &\left.+{C}_{66}^{M}\left({F}_{\tau }{N}_{i}{\omega }_{x\tau i}+{N}_{i}{u}_{y\tau i}\frac{\partial {F}_{\tau }}{\partial z}\right)\right)+\left(-{F}_{s}{N}_{j}{\delta \omega }_{xsj}+{N}_{j}{\delta u}_{zsj}{F}_{s,y}\right)\\ &\left({C}_{66}^{M}\left(-{F}_{\tau }{N}_{i}{\omega }_{x\tau i}+{N}_{i}{u}_{z\tau i}\frac{\partial {F}_{\tau }}{\partial y}\right)+{C}_{66}^{MT}\left({F}_{\tau }{N}_{i}{\omega }_{x\tau i}+{N}_{i}{u}_{y\tau i}\frac{\partial {F}_{\tau }}{\partial z}\right)\right)\\ &+{N}_{j}{\delta \omega }_{ysj}\frac{\partial {F}_{s}}{\partial z}\left({A}_{66}^{MT}{N}_{i}{\omega }_{z\tau i}\frac{\partial {F}_{\tau }}{\partial y}+{A}_{66}^{M}{N}_{i}{\omega }_{y\tau i}\frac{\partial {F}_{\tau }}{\partial z}\right)\\ &+{N}_{j}{\delta \omega }_{zsj}\frac{\partial {F}_{s}}{\partial y}\left({A}_{66}^{M}{N}_{i}{\omega }_{z\tau i}\frac{\partial {F}_{\tau }}{\partial y}+{A}_{66}^{MT}{N}_{i}{\omega }_{y\tau i}\frac{\partial {F}_{\tau }}{\partial z}\right)\\ &+{N}_{j}{\delta u}_{ysj}\frac{\partial {F}_{s}}{\partial y}\left({C}_{22}{N}_{i}{u}_{y\tau i}\frac{\partial {F}_{\tau }}{\partial y}+{C}_{23}{N}_{i}{u}_{z\tau i}\frac{\partial {F}_{\tau }}{\partial z}+{C}_{12}{F}_{\tau }{u}_{x\tau i}\frac{\partial {N}_{i}}{\partial x}\right)\\ &+{N}_{j}{\delta u}_{zsj}\frac{\partial {F}_{s}}{\partial z}\left({C}_{23}{N}_{i}{u}_{y\tau i}\frac{\partial {F}_{\tau }}{\partial y}+{C}_{33}{N}_{i}{u}_{z\tau i}\frac{\partial {F}_{\tau }}{\partial z}+{C}_{13}{F}_{\tau }{u}_{x\tau i}\frac{\partial {N}_{i}}{\partial x}\right)\\ &+{N}_{j}{\delta \omega }_{ysj}\frac{\partial {F}_{s}}{\partial y}\left({A}_{22}{N}_{i}{\omega }_{y\tau i}\frac{\partial {F}_{\tau }}{\partial y}+{A}_{23}{N}_{i}{\omega }_{z\tau i}\frac{\partial {F}_{\tau }}{\partial z}+{A}_{12}{F}_{\tau }{\omega }_{x\tau i}\frac{\partial {N}_{i}}{\partial x}\right)\\ &+{N}_{j}{\delta \omega }_{zsj}\frac{\partial {F}_{s}}{\partial z}\left({A}_{23}{N}_{i}{\omega }_{y\tau i}\frac{\partial {F}_{\tau }}{\partial y}+{A}_{33}{N}_{i}{\omega }_{z\tau i}\frac{\partial {F}_{\tau }}{\partial y}+{A}_{13}{F}_{\tau }{\omega }_{x\tau i}\frac{\partial {N}_{i}}{\partial x}\right)\\ &+{N}_{j}{\delta \omega }_{xsj}\frac{\partial {F}_{s}}{\partial y}\left({A}_{44}^{M}{N}_{i}{\omega }_{x\tau i}\frac{\partial {F}_{\tau }}{\partial y}+{A}_{44}^{MT}{F}_{\tau }{\omega }_{y\tau i}\frac{\partial {N}_{i}}{\partial x}\right)\\ &+{N}_{j}{\delta \omega }_{xsj}\frac{\partial {F}_{s}}{\partial z}\left({A}_{55}^{M}{N}_{i}{\omega }_{x\tau i}\frac{\partial {F}_{\tau }}{\partial z}+{A}_{55}^{MT}{F}_{\tau }{\omega }_{z\tau i}\frac{\partial {N}_{i}}{\partial x}\right)+\left({F}_{s}{N}_{j}{\delta \omega }_{zsj}+{N}_{j}{\delta u}_{xsj}\frac{\partial {F}_{s}}{\partial y}\right)\\ &\left({C}_{44}^{M}\left({F}_{\tau }{N}_{i}{\omega }_{z\tau i}+{N}_{i}{u}_{x\tau i}\frac{\partial {F}_{\tau }}{\partial y}\right)+{C}_{44}^{MT}\left(-{F}_{\tau }{N}_{i}{\omega }_{z\tau i}+{F}_{\tau }{u}_{y\tau i}\frac{\partial {N}_{i}}{\partial x}\right)\right)\\ &+\left(-{F}_{s}{N}_{j}{\delta \omega }_{ysj}+{N}_{j}{\delta u}_{xsj}\frac{\partial {F}_{s}}{\partial z}\right)\\ &\left({C}_{55}^{M}\left(-{F}_{\tau }{N}_{i}{\omega }_{y\tau i}+{N}_{i}{u}_{x\tau i}\frac{\partial {F}_{\tau }}{\partial z}\right)+{C}_{55}^{MT}\left({F}_{\tau }{N}_{i}{\omega }_{y\tau i}+{F}_{\tau }{u}_{z\tau i}\frac{\partial {N}_{i}}{\partial x}\right)\right)\\ &+{F}_{s}{\delta u}_{xsj}\left({C}_{12}{N}_{i}{u}_{y\tau i}\frac{\partial {F}_{\tau }}{\partial y}+{C}_{13}{N}_{i}{u}_{z\tau i}\frac{\partial {F}_{\tau }}{\partial y}+{C}_{11}{F}_{\tau }{u}_{x\tau i}\frac{\partial {N}_{i}}{\partial x}\right)\frac{\partial {N}_{j}}{\partial x}\\ &+{F}_{s}{\delta \omega }_{xsj}\left({A}_{12}{N}_{i}{\omega }_{y\tau i}\frac{\partial {F}_{\tau }}{\partial y}+{A}_{13}{N}_{i}{\omega }_{z\tau i}\frac{\partial {F}_{\tau }}{\partial z}+{A}_{11}{F}_{\tau }{\omega }_{x\tau i}\frac{\partial {N}_{i}}{\partial x}\right)\frac{\partial {N}_{j}}{\partial x}\\ &+{F}_{s}{\delta \omega }_{ysj}\left({A}_{44}^{MT}{N}_{i}{\omega }_{x\tau i}\frac{\partial {F}_{\tau }}{\partial y}+{A}_{44}^{M}{F}_{\tau }{\omega }_{y\tau i}\frac{\partial {N}_{i}}{\partial x}\right)\frac{\partial {N}_{j}}{\partial x}\\ &+{F}_{s}{\delta \omega }_{zsj}\left({A}_{55}^{MT}{N}_{i}{\omega }_{x\tau i}\frac{\partial {F}_{\tau }}{\partial z}+{A}_{55}^{M}{F}_{\tau }{\omega }_{z\tau i}\frac{\partial {N}_{i}}{\partial x}\right)\frac{\partial {N}_{j}}{\partial x}\\ &+\left({C}_{44}^{MT}\left({F}_{\tau }{N}_{i}{\omega }_{z\tau i}+{N}_{i}\frac{\partial {F}_{\tau }}{\partial y}\right)+{C}_{44}^{M}\left(-{F}_{\tau }{N}_{i}{\omega }_{z\tau i}+{F}_{\tau }{u}_{y\tau i}\frac{\partial {N}_{i}}{\partial x}\right)\right)\\ &\left(-{F}_{s}{N}_{j}{\delta \omega }_{zsj}+{F}_{s}{\delta u}_{ysj}\frac{\partial {N}_{j}}{\partial x}\right)\\ &+\left({C}_{55}^{MT}\left(-{F}_{\tau }{N}_{i}{\omega }_{y\tau i}+{N}_{i}{u}_{x\tau i}\frac{\partial {F}_{\tau }}{\partial z}\right)+{C}_{55}^{M}\left({F}_{\tau }{N}_{i}{\omega }_{y\tau i}+{F}_{\tau }{u}_{z\tau i}\frac{\partial {N}_{i}}{\partial x}\right)\right)\\ &\left.\left({F}_{s}{N}_{j}{\delta \omega }_{ysj}+{F}_{s}{\delta u}_{zsj}\frac{\partial {N}_{j}}{\partial x}\right)\right\}dxd\Omega \end{aligned}$$

The explicit form of the components of \({\mathbf{k}}_{uu}^{ij\tau s}\) is given as follows:

$${k}_{uuxx}^{ij\tau s}={\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }{C}_{44}^{M}\frac{{\partial F}_{\tau }}{\partial y}\frac{{\partial F}_{s}}{\partial y}dydz+{\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }{C}_{55}^{M}\frac{{\partial F}_{\tau }}{\partial z}\frac{{\partial F}_{s}}{\partial z}dydz+{\int }_{l}\frac{{\partial N}_{i}}{\partial x}\frac{{\partial N}_{j}}{\partial x}dx{\int }_{\Omega }{C}_{11}{F}_{\tau }{F}_{s}dydz$$
$${k}_{uuxy}^{ij\tau s}={\int }_{l}\frac{{\partial N}_{i}}{\partial x}{N}_{j}dx{\int }_{\Omega }{C}_{12}{F}_{\tau }\frac{{\partial F}_{s}}{\partial y}dydz+{\int }_{l}{N}_{i}\frac{{\partial N}_{j}}{\partial x}dx{\int }_{\Omega }{C}_{44}^{MT}\frac{{\partial F}_{\tau }}{\partial y}{F}_{s}dydz$$
$${k}_{uuxz}^{ij\tau s}={\int }_{l}\frac{{\partial N}_{i}}{\partial x}{N}_{j}dx{\int }_{\Omega }{C}_{13}{F}_{\tau }\frac{{\partial F}_{s}}{\partial z}dydz+{\int }_{l}{N}_{i}\frac{{\partial N}_{j}}{\partial x}dx{\int }_{\Omega }{C}_{55}^{MT}\frac{{\partial F}_{\tau }}{\partial z}{F}_{s}dydz$$
$${k}_{uuyx}^{ij\tau s}={\int }_{l}\frac{{\partial N}_{i}}{\partial x}{N}_{j}dx{\int }_{\Omega }{C}_{44}^{MT}{F}_{\tau }\frac{{\partial F}_{s}}{\partial y}dydz+{\int }_{l}{N}_{i}\frac{{\partial N}_{j}}{\partial x}dx{\int }_{\Omega }{C}_{12}\frac{{\partial F}_{\tau }}{\partial y}{F}_{s}dydz$$
$${k}_{uuyy}^{ij\tau s}={\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }{C}_{66}^{M}\frac{{\partial F}_{\tau }}{\partial z}\frac{{\partial F}_{s}}{\partial z}dydz+{\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }{C}_{22}\frac{{\partial F}_{\tau }}{\partial y}\frac{{\partial F}_{s}}{\partial y}dydz+{\int }_{l}\frac{{\partial N}_{i}}{\partial x}\frac{{\partial N}_{j}}{\partial x}dx{\int }_{\Omega }{C}_{44}^{M}{F}_{\tau }{F}_{s}dydz$$
$${k}_{uuyz}^{ij\tau s}={\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }{C}_{66}^{MT}\frac{{\partial F}_{\tau }}{\partial z}\frac{{\partial F}_{s}}{\partial y}dydz+{\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }{C}_{23}\frac{{\partial F}_{\tau }}{\partial y}\frac{{\partial F}_{s}}{\partial z}dydz$$
$${k}_{uuzx}^{ij\tau s}={\int }_{l}\frac{{\partial N}_{i}}{\partial x}{N}_{j}dx{\int }_{\Omega }{C}_{55}^{MT}{F}_{\tau }\frac{{\partial F}_{s}}{\partial z}dydz+{\int }_{l}{N}_{i}\frac{{\partial N}_{j}}{\partial x}dx{\int }_{\Omega }{C}_{13}\frac{{\partial F}_{\tau }}{\partial z}{F}_{s}dydz$$
$${k}_{uuzy}^{ij\tau s}={\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }{C}_{66}^{MT}\frac{{\partial F}_{\tau }}{\partial y}\frac{{\partial F}_{s}}{\partial z}dydz+{\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }{C}_{23}\frac{{\partial F}_{\tau }}{\partial z}\frac{{\partial F}_{s}}{\partial y}dydz$$
$${k}_{uuzz}^{ij\tau s}={\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }{C}_{66}^{M}\frac{{\partial F}_{\tau }}{\partial y}\frac{{\partial F}_{s}}{\partial y}dydz+{\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }{C}_{33}\frac{{\partial F}_{\tau }}{\partial z}\frac{{\partial F}_{s}}{\partial z}dydz+{\int }_{l}\frac{{\partial N}_{i}}{\partial x}\frac{{\partial N}_{j}}{\partial x}dx{\int }_{\Omega }{C}_{55}^{M}{F}_{\tau }{F}_{s}dydz$$

The fundamental nuclei for \({k}_{\omega \omega }^{ij\tau s}\) are given as follows:

$${k}_{\omega \omega xx}^{ij\tau s}={\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }2\left({C}_{66}^{M}-{C}_{66}^{MT}\right){F}_{\tau }{F}_{s}dydz+{\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }{A}_{44}^{M}\frac{{\partial F}_{\tau }}{\partial y}\frac{{\partial F}_{s}}{\partial y}dydz+{\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }{A}_{55}^{M}\frac{{\partial F}_{\tau }}{\partial z}\frac{{\partial F}_{s}}{\partial z}dydz{\int }_{l}\frac{{\partial N}_{i}}{\partial x}\frac{{\partial N}_{j}}{\partial x}dx{\int }_{\Omega }{A}_{11}^{M}{F}_{\tau }{F}_{s}dydz$$
$${k}_{\omega \omega xy}^{ij\tau s}={\int }_{l}\frac{{\partial N}_{i}}{\partial x}{N}_{j}dx{\int }_{\Omega }{A}_{12}{F}_{\tau }\frac{{\partial F}_{s}}{\partial y}dydz+{\int }_{l}{N}_{i}\frac{{\partial N}_{j}}{\partial x}dx{\int }_{\Omega }{A}_{44}^{MT}\frac{{\partial F}_{\tau }}{\partial y}{F}_{s}dydz$$
$${k}_{\omega \omega xz}^{ij\tau s}={\int }_{l}\frac{{\partial N}_{i}}{\partial x}{N}_{j}dx{\int }_{\Omega }{A}_{13}{F}_{\tau }\frac{{\partial F}_{s}}{\partial z}dydz+{\int }_{l}{N}_{i}\frac{{\partial N}_{j}}{\partial x}dx{\int }_{\Omega }{A}_{55}^{MT}\frac{{\partial F}_{\tau }}{\partial z}{F}_{s}dydz$$
$${k}_{\omega \omega yx}^{ij\tau s}={\int }_{l}\frac{{\partial N}_{i}}{\partial x}{N}_{j}dx{\int }_{\Omega }{A}_{44}^{MT}{F}_{\tau }\frac{{\partial F}_{s}}{\partial y}dydz+{\int }_{l}{N}_{i}\frac{{\partial N}_{j}}{\partial x}dx{\int }_{\Omega }{A}_{12}\frac{{\partial F}_{\tau }}{\partial y}{F}_{s}dydz$$
$${k}_{\omega \omega yy}^{ij\tau s}={\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }2\left({C}_{55}^{M}-{C}_{55}^{MT}\right){F}_{\tau }{F}_{s}dydz+{\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }{A}_{66}^{M}\frac{{\partial F}_{\tau }}{\partial z}\frac{{\partial F}_{s}}{\partial z}dydz+{\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }{A}_{22}\frac{{\partial F}_{\tau }}{\partial y}\frac{{\partial F}_{s}}{\partial y}dydz{\int }_{l}\frac{{\partial N}_{i}}{\partial x}\frac{{\partial N}_{j}}{\partial x}dx{\int }_{\Omega }{A}_{44}^{M}{F}_{\tau }{F}_{s}dydz$$
$${k}_{\omega \omega yz}^{ij\tau s}={\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }{A}_{66}^{MT}\frac{{\partial F}_{\tau }}{\partial z}\frac{{\partial F}_{s}}{\partial y}dydz+{\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }{A}_{23}\frac{{\partial F}_{\tau }}{\partial y}\frac{{\partial F}_{s}}{\partial z}dydz$$
$${k}_{\omega \omega zx}^{ij\tau s}={\int }_{l}\frac{{\partial N}_{i}}{\partial x}{N}_{j}dx{\int }_{\Omega }{A}_{55}^{MT}{F}_{\tau }\frac{{\partial F}_{s}}{\partial z}dydz+{\int }_{l}{N}_{i}\frac{{\partial N}_{j}}{\partial x}dx{\int }_{\Omega }{A}_{13}\frac{{\partial F}_{\tau }}{\partial z}{F}_{s}dydz$$
$${k}_{\omega \omega zy}^{ij\tau s}={\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }{A}_{66}^{MT}\frac{{\partial F}_{\tau }}{\partial y}\frac{{\partial F}_{s}}{\partial z}dydz+{\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }{A}_{23}\frac{{\partial F}_{\tau }}{\partial z}\frac{{\partial F}_{s}}{\partial y}dydz$$
$${k}_{\omega \omega zz}^{ij\tau s}={\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }2\left({C}_{44}^{M}-{C}_{44}^{MT}\right){F}_{\tau }{F}_{s}dydz+{\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }{A}_{66}^{M}\frac{{\partial F}_{\tau }}{\partial y}\frac{{\partial F}_{s}}{\partial y}dydz+{\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }{A}_{33}\frac{{\partial F}_{\tau }}{\partial z}\frac{{\partial F}_{s}}{\partial z}dydz{\int }_{l}\frac{{\partial N}_{i}}{\partial x}\frac{{\partial N}_{j}}{\partial x}dx{\int }_{\Omega }{A}_{55}^{M}{F}_{\tau }{F}_{s}dydz$$

The fundamental nuclei for \({k}_{u\omega }^{ij\tau s}\) are given as:

$${k}_{u\omega xy}^{ij\tau s}={\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }\left({C}_{55}^{MT}-{C}_{55}^{M}\right)\frac{{\partial F}_{\tau }}{\partial z}{F}_{s}dydz$$
$${k}_{u\omega xz}^{ij\tau s}={\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }\left({C}_{44}^{M}-{C}_{44}^{MT}\right)\frac{{\partial F}_{\tau }}{\partial y}{F}_{s}dydz$$
$${k}_{u\omega yx}^{ij\tau s}={\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }\left({C}_{66}^{M}-{C}_{66}^{MT}\right)\frac{{\partial F}_{\tau }}{\partial z}{F}_{s}dydz$$
$${k}_{u\omega yz}^{ij\tau s}={\int }_{l}\frac{{\partial N}_{i}}{\partial x}{N}_{j}dx{\int }_{\Omega }\left({C}_{44}^{MT}-{C}_{44}^{M}\right){F}_{\tau }{F}_{s}dydz$$
$${k}_{u\omega zx}^{ij\tau s}={\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }\left({C}_{66}^{MT}-{C}_{66}^{M}\right)\frac{{\partial F}_{\tau }}{\partial y}{F}_{s}dydz$$
$${k}_{u\omega zy}^{ij\tau s}={\int }_{l}\frac{{\partial N}_{i}}{\partial x}{N}_{j}dx{\int }_{\Omega }\left({C}_{55}^{M}-{C}_{55}^{MT}\right){F}_{\tau }{F}_{s}dydz$$
$${k}_{u\omega xx}^{ij\tau s}={k}_{u\omega yy}^{ij\tau s}={k}_{u\omega zz}^{ij\tau s}=0$$

The fundamental nuclei for \({k}_{\omega u}^{ij\tau s}\) are:

$${k}_{\omega uxy}^{ij\tau s}={\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }\left({C}_{66}^{M}-{C}_{66}^{MT}\right){F}_{\tau }\frac{{\partial F}_{s}}{\partial z}dydz$$
$${k}_{\omega uxz}^{ij\tau s}={\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }\left({C}_{66}^{MT}-{C}_{66}^{M}\right){F}_{\tau }\frac{{\partial F}_{s}}{\partial y}dydz$$
$${k}_{\omega uyx}^{ij\tau s}={\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }\left({C}_{55}^{MT}-{C}_{55}^{M}\right){F}_{\tau }\frac{{\partial F}_{s}}{\partial z}dydz$$
$${k}_{\omega uyz}^{ij\tau s}={\int }_{l}{N}_{i}\frac{{\partial N}_{j}}{\partial x}dx{\int }_{\Omega }\left({C}_{55}^{M}-{C}_{55}^{MT}\right){F}_{\tau }{F}_{s}dydz$$
$${k}_{\omega uzx}^{ij\tau s}={\int }_{l}{N}_{i}{N}_{j}dx{\int }_{\Omega }\left({C}_{44}^{M}-{C}_{44}^{MT}\right){F}_{\tau }\frac{{\partial F}_{s}}{\partial y}dydz$$
$${k}_{\omega uzy}^{ij\tau s}={\int }_{l}{N}_{i}\frac{{\partial N}_{j}}{\partial x}dx{\int }_{\Omega }\left({C}_{44}^{MT}-{C}_{44}^{M}\right){F}_{\tau }{F}_{s}dydz$$
$${k}_{\omega uxx}^{ij\tau s}={k}_{\omega uyy}^{ij\tau s}={k}_{\omega uzz}^{ij\tau s}=0$$

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Daraei, B., Shojaee, S. & Hamzehei-Javaran, S. Thermo-mechanical analysis of functionally graded material beams using micropolar theory and higher-order unified formulation. Arch Appl Mech 93, 109–128 (2023). https://doi.org/10.1007/s00419-022-02143-z

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