A refined quasi-3D isogeometric nonlinear model of functionally graded triply periodic minimal surface plates

Abstract

In recent years, research and applications of bioinspired structures in advanced engineering fields have gained increased attention from the research community, thanks to their fascinating properties. In this study, we address an efficient computational approach for performing nonlinear static and dynamic analyses of functionally graded plates based on triply periodic minimal surface architectures for the first time. We named a functionally graded triply periodic minimal surface (FG-TPMS) plate. A key idea of modelling FG-TPMS plates is to rely on the four-unknown refined quasi-3D plate theory, von-Kármán assumptions, and NURBS-based isogeometric analysis. The nonlinear behavior of three TPMS structures including Primitive (P), Gyroid (G), and I-graph and Wrapped Package-graph (IWP) under various conditions are intensively studied in this work. To estimate the effective mechanical features of the TPMS architectures, we utilize a two-phase fitting model with respect to the relative density. The influence of several parameters of TPMS structures on the nonlinear static and dynamic characteristics is evaluated. In addition, four types of dynamic loads including rectangular, triangular, half-sine, and explosive blast are also considered here. The key contribution of this study is the development of an efficient and powerful nonlinear numerical model to explore the static and dynamic behavior of TPMS architectures-based FG plates. The present method not only effectively accounts for the thickness stretching effect but also includes the consideration of structural damping, thereby facilitating a more accurate solution to engineering problems under real-world conditions. Furthermore, the current results indicate that FG-TPMS plates exhibit a superior energy absorption capacity compared to isotropic ones of the same weight under geometric nonlinearity conditions. Finally, the findings obtained from this study enhance our understanding of nonlinear behavior as well as provide valuable design strategies for future advanced engineering structures based on TPMS architectures.

Appendix

1.1 A static bending analysis

For static bending problems, we can obtain the global nonlinear equation system of FG-TPMS plates subjected to transverse distributed load by removing the inertia term from Eq. (23), as presented below [51]

$$\begin{aligned} \left[ {{\textbf{K}}_{L}}+{{\textbf{K}_{NL}(\textbf{q})}} \right] \textbf{q}=\textbf{F}. \end{aligned}$$

(34)

To solve the above nonlinear equation, the Newton–Raphson iterative scheme is implemented in this work. First, at the $j^{th}$ load increment step, the vector of the residual force ${\varvec{\mathcal {R}}}(\textbf{q}_i)$ for the $i^{th}$ iteration is calculated as follows

$$\begin{aligned}{} & {} {\varvec{\mathcal {R}}}\left( {{\textbf{q}}_i} \right) \nonumber \\{} & {} \quad =\left[ {{\textbf{K}}_{L}}+{{\textbf{K}}_{NL}}\left( {{\textbf{q}}_i} \right) \right] {{\textbf{q}}_i}-{{\textbf{F}}^{j}}, \end{aligned}$$

(35)

Then, the updated displacement vector for the $(i+1)^{th}$ iteration is obtained and expressed as follows [58]

$$\begin{aligned} {{\textbf{q}}_{i+1}}={{\textbf{q}}_i}+\delta {{\textbf{q}}_{i+1}}, \end{aligned}$$

(36)

herein, the vector of the increment displacement $\delta {{\textbf{q}}_{i+1}}$ can be defined as follows

(37)

in which

(38)

The iteration process continues until the specified convergence criterion in terms of displacement, as stated below, is met

$$\begin{aligned} \frac{\left\| {{\textbf{q}}_{i+1}}-{{\textbf{q}}_i} \right\| }{\left\| {{\textbf{q}}_i} \right\| }<\epsilon , \end{aligned}$$

(39)

where $\epsilon =10^{-4}$ is the prescribed tolerance in this work.

1.2 A.2 Transient analysis

In the current research, we utilize a combination of Newmark’s integration procedure, the Picard method and the average acceleration scheme [59] to achieve the nonlinear transient responses of the FG-TPMS plates. It is assumed that at the first time step, namely $t = 0$, the displacement $\textbf{q}$, the velocity $\mathbf{\dot{q}}$ and the acceleration $\mathbf{\ddot{q}}$ are initially assigned a value of 0. The acceleration and velocity fields at the $t=(n+1)\Delta t$ time step are then updated from the obtained displacement field, respectively, as follows [22, 59]

$$\begin{aligned} {\mathbf{\ddot{q}}}^{{n + 1}} = & \frac{1}{{\beta \Delta t^{2} }}\left( {{\mathbf{q}}^{{n + 1}} - {\mathbf{q}}^{n} } \right) \\ - \frac{1}{{\beta \Delta t}}{\mathbf{\dot{q}}}^{n} - \left( {\frac{1}{{2\beta }} - 1} \right){\mathbf{\ddot{q}}}^{n} , \\ \end{aligned}$$

(40a)

$$\begin{aligned} {\mathbf{\dot{q}}}^{{n + 1}} = & {\mathbf{\dot{q}}}^{n} + \left( {1 - \gamma } \right)\Delta t{\mathbf{\ddot{q}}}^{n} \\ + \gamma \Delta t{\mathbf{\ddot{q}}}^{{n + 1}} , \\ \end{aligned}$$

(40b)

in which $\alpha =0.5$ and $\beta =0.25$ are the constants from the average acceleration scheme. By substituting Eq. (40) into Eq. (27), the global nonlinear equation system is briefly written as follows

$$\begin{aligned} {{\hat{\textbf{K}}}^{n+1}}{{\textbf{q}}^{n+1}}={{{\hat{\textbf{F}}}}^{n+1}}, \end{aligned}$$

(41)

where the matrix of stiffness and the vector of force at the time step $(n+1)\Delta t$ are given as follows [22, 59]

$$\begin{aligned} \widehat{{\mathbf{K}}}^{{n + 1}} = & {\mathbf{K}}^{{n + 1}} + \frac{1}{{\beta \Delta t^{2} }}{\mathbf{M}} \\ + \frac{\gamma }{{\beta \Delta t}}{\mathbf{C}}, \\ \end{aligned}$$

(42a)

$$\begin{aligned} \widehat{{\mathbf{F}}}^{{n + 1}} = & {\mathbf{F}}^{{n + 1}} + \left[ {\frac{1}{{\beta \Delta t^{2} }}{\mathbf{q}}^{n} } \right. \\ \left. { + \frac{1}{{\beta \Delta t}}{\mathbf{\dot{q}}}^{n} + \left( {\frac{1}{{2\beta }} - 1} \right){\mathbf{\ddot{q}}}^{n} } \right]{\mathbf{M}} + \\ \left[ {\frac{\gamma }{{\beta \Delta t}}{\mathbf{q}}^{n} + \left( {\frac{\gamma }{\beta } - 1} \right)} \right. \\ \left. {{\mathbf{\dot{q}}}^{n} + \frac{{\Delta t}}{2}\left( {\frac{\gamma }{\beta } - 2} \right){\mathbf{\ddot{q}}}^{n} } \right]{\mathbf{C}}. \\ \end{aligned}$$

(42b)

It is known that the matrix of ${{\hat{\textbf{K}}}}^{n+1}$ represents nonlinearly dependent on the displacements at the current time step, i.e. $t=(n+1)\Delta t$. Hence, the Picard approach is applied in this study and then Eq. (41) is rewritten as follows [22]

$$\widehat{{\mathbf{K}}}\left( {{\mathbf{q}}_{i}^{{n + 1}} } \right)\;{\mathbf{q}}_{{i + 1}}^{{n + 1}} = \widehat{{\mathbf{F}}}^{{n + 1}} ,$$

(43)

in which i refers to the iteration number. It is worth noting that the final nonlinear equation, as presented in Eq. (43), is iteratively solved using the Newton–Raphson scheme, as presented in the preceding subsection.