An adaptive XIGA with locally refined NURBS for modeling cracked composite FG Mindlin–Reissner plates

Abstract

This paper is devoted to numerical investigations on mechanical behavior of cracked composite functionally graded (FG) plates. We thus develop an efficient adaptive approach in terms of the extended isogeometric analysis (XIGA) enhanced by locally refined non-uniform rational B-spline (LR NURBS) for natural frequency and buckling analysis of cracked FG Mindlin–Reissner plates. In this setting, the crack geometries, which are described by the level sets, are independent of the computational mesh; and the LR NURBS basis functions, which have the abilities of local refinement and modeling the complex shape, are used as the shape functions of XIGA. According to the recovered stresses of the first buckling or free vibration mode, a posteriori error estimator for driving the adaptive process is defined. The accuracy can be effectively improved through adaptive local refinements, demonstrated through numerical experiments with complex geometries. Compared with uniform global refinement, XIGA based on adaptive local refinement has the characteristics of high precision at low cost and fast convergence rate. The effects of some factors such as crack length and location, plate thickness, gradient index, boundary condition, loading type, etc. on critical buckling loads and natural frequencies are investigated.

Appendices

Appendix 1

$$\begin{aligned}&{\varvec{N}}^u=\begin{bmatrix} 0 &{}R_i &{}0 \\ 0 &{} 0 &{} R_i\\ R_i &{}0 &{}0 \end{bmatrix} \end{aligned}$$

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$$\begin{aligned}&{\varvec{N}}^d=\begin{bmatrix} 0 &{}{\tilde{H}}R_i &{}0 \\ 0 &{} 0 &{}{\tilde{H}} R_i\\ {\tilde{H}} R_i &{}0 &{}0 \end{bmatrix} \end{aligned}$$

(31)

$$\begin{aligned}&{\varvec{N}}^{c\alpha }=\begin{bmatrix} 0 &{}\tilde{F_\alpha }R_i &{}0 \\ 0 &{} 0 &{}\tilde{F_\alpha } R_i\\ \tilde{G_\alpha } R_i &{}0 &{}0 \end{bmatrix}, ~~\alpha =1,\ldots ,4 \end{aligned}$$

(32)

$$\begin{aligned}&{\varvec{N}}^{c5 }=\begin{bmatrix} 0 &{}0 &{}0 \\ 0 &{} 0 &{}0\\ \tilde{G_5} R_i &{}0 &{}0 \end{bmatrix} \end{aligned}$$

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Appendix 2

$$\begin{aligned}&{\varvec{G}}_\mathrm{{b}}^u=\begin{bmatrix} R_{i,x} &{}0 &{}0 \\ R_{i,y} &{} 0 &{}0 \end{bmatrix} \end{aligned}$$

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$$\begin{aligned}&{\varvec{G}}_\mathrm{{b}}^d=\begin{bmatrix} {\tilde{H}}R_{i,x} &{}0 &{}0 \\ {\tilde{H}} R_{i,y} &{} 0 &{}0 \end{bmatrix} \end{aligned}$$

(35)

$$\begin{aligned}&{\varvec{G}}_\mathrm{{b}}^{c\alpha }=\begin{bmatrix} \tilde{G_\alpha }R_{i,x} &{}0 &{}0 \\ \tilde{G_\alpha } R_{i,y} &{} 0 &{}0 \end{bmatrix},~~\alpha =1,\ldots ,5 \end{aligned}$$

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Appendix 3

$$\begin{aligned}&{\varvec{G}}_{\mathrm{{s}}1}^u=\begin{bmatrix} 0 &{}R_{i,x} &{}0 \\ 0 &{} R_{i,y} &{}0 \end{bmatrix} \end{aligned}$$

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$$\begin{aligned}&{\varvec{G}}_{\mathrm{{s}}1}^d=\begin{bmatrix} 0 &{}{\tilde{H}}R_{i,x} &{}0 \\ 0 &{}{\tilde{H}} R_{i,y} &{}0 \end{bmatrix} \end{aligned}$$

(38)

$$\begin{aligned}&{\varvec{G}}_{\mathrm{{s}}1}^{c\alpha }=\begin{bmatrix} 0 &{}\tilde{F_\alpha }R_{i,x} &{}0 \\ 0 &{}\tilde{F_\alpha } R_{i,y} &{}0 \end{bmatrix},~~\alpha =1,\ldots ,4 \end{aligned}$$

(39)

$$\begin{aligned}&{\varvec{G}}_{\mathrm{{s}}1}^{c5 }=\begin{bmatrix} 0 &{}0 &{}0 \\ 0 &{}0 &{}0 \end{bmatrix} \end{aligned}$$

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Appendix 4

$$\begin{aligned}&{\varvec{G}}_{\mathrm{{s}}2}^u=\begin{bmatrix} 0 &{} 0 &{}R_{i,x} \\ 0 &{} 0 &{}R_{i,y} \end{bmatrix} \end{aligned}$$

(41)

$$\begin{aligned}&{\varvec{G}}_{\mathrm{{s}}2}^u=\begin{bmatrix} 0 &{} 0 &{}{\tilde{H}}R_{i,x} \\ 0 &{} 0 &{}{\tilde{H}}R_{i,y} \end{bmatrix} \end{aligned}$$

(42)

$$\begin{aligned}&{\varvec{G}}_{\mathrm{{s}}2}^u=\begin{bmatrix} 0 &{} 0 &{}\tilde{F_\alpha }R_{i,x} \\ 0 &{} 0 &{}\tilde{F_\alpha }R_{i,y} \end{bmatrix},~~\alpha =1,\ldots ,4 \end{aligned}$$

(43)

$$\begin{aligned}&{\varvec{G}}_{\mathrm{{s}}2}^{c5 }=\begin{bmatrix} 0 &{}0 &{}0 \\ 0 &{}0 &{}0 \end{bmatrix} \end{aligned}$$

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