3D architected isotropic materials with tunable stiffness and buckling strength

Highlights

• 3D isotropic material class with tunable properties is designed using optimization.

• Material buckling strength is significantly enhanced by optimized microstructures.

• Proposed material class improves the uniaxial buckling strength from 180% to 767%.

• Proposed material class owns decreased Young’s modulus range from 79% to 58%.

Abstract

This paper presents a class of 3D single-scale isotropic materials with tunable stiffness and buckling strength obtained via topology optimization and subsequent shape optimization. Compared to stiffness-optimal closed-cell plate material, the material class reduces the Young’s modulus to a range from 79% to 58%, but improves the uniaxial buckling strength to a range from 180% to 767%. Based on small deformation theory, material stiffness is evaluated using the homogenization method. Buckling strength under a given macroscopic stress state is estimated using linear buckling analysis with Block–Floquet boundary conditions to capture both short and long wavelength buckling modes. The 3D isotropic single-scale materials with tunable properties are designed using topology optimization, and are then further simplified using shape optimization. Both topology and shape optimized results demonstrate that material buckling strength can be significantly enhanced by hybrids between truss and variable thickness plate structures.

Introduction

Materials with extreme mechanical properties are highly attractive for many applications. Among them, stiffness and strength are fundamental for determining material load-bearing capability. Stiffness accounts for the ability to resist deformation while strength measures the ultimate load-carrying capability.

Many studies have been devoted to exploring materials with optimal stiffness (Hashin, 1962, Francfort and Murat, 1986, Milton, 1986, Sigmund, 1994, Sigmund, 2000, Berger et al., 2017, Tancogne-Dejean et al., 2018, Wang et al., 2019). It has been shown that isotropic stiffness-optimal plate materials can achieve the Hashin–Shtrikman upper bounds on isotropic elastic stiffness and show up to three times the stiffness of the isotropic stiffness-optimal truss materials in the low volume fraction limit (Berger et al., 2017, Tancogne-Dejean et al., 2018, Christensen, 1986, Sigmund et al., 2016). The stiffness advantage is attributed to the multiaxial stiffness offered by the constituent plates while bars in the truss materials only offer axial stiffness. However, a recent study has shown that isotropic stiffness-optimal truss material is superior from a bucking strength point of view compared to the isotropic plate counterparts for the same volume fraction when the volume fraction is below 31% (Andersen et al., 2021a) because of higher bending stiffness associated with the bars than the constituent plates. It is for example shown that the isotropic stiffness-optimal truss material possesses 48% higher uniaxial buckling strength and 52% lower Young’s modulus than the isotropic plate material for a volume fraction of 20%.

Material geometry strongly dictates material properties. Novel materials with exotic properties have been achieved through careful tailoring of material geometries via different design approaches. Among them, topology optimization methods (Bendsøe and Sigmund, 2003) have been proven powerful tools in designing novel materials ranging from mechanical properties, such as optimal stiffness (Sigmund, 1994, Andreassen et al., 2014), auxetic behavior (Sigmund, 1994, Andreassen et al., 2014, Wang, 2018), to acoustic and optical properties (Christiansen and Sigmund, 2016, Wang et al., 2018). Regarding elastic stability, material buckling failure may develop at different scales spanning from highly localized short wavelength modes to long wavelength modes. Previous numerical studies have employed homogenization methods assuming separation of scales (Guedes and Kikuchi, 1990) and Bloch–Floquet theory for detecting short and long wavelength buckling (Geymonat et al., 1993). A general methodology for characterizing material strength due to bifurcation failure was proposed in Triantafyllidis and Schnaidt (1993). Assuming small strains and ignoring material and geometric nonlinearities, topology optimization of material strength was first studied in Neves et al. (2002a), where only cell-periodic buckling modes were taken into account. Later, this work was extended to cover both local and global modes via the Bloch–Floquet theory (Neves et al., 2002b). More recently, 2D materials with enhanced buckling strength (Thomsen et al., 2018) have been systematically designed using topology optimization methods for different macro-level stress situations based on the homogenization theory and linear buckling analysis (LBA) with Bloch–Floquet boundary conditions. It was shown that optimized first-order hierarchical materials outperform their non-hierarchical counterparts in terms of buckling strength at the cost of slightly decreased stiffness. Further material evaluations considering both geometrical and material nonlinearity have proven that the superior buckling strength of the optimized hierarchical materials compared to reference materials also hold for finite structures (Wang and Sigmund, 2020) and geometrically nonlinear modeling (Bluhm et al., 2020). Hence, a buckling optimization approach assuming small strains has been demonstrated to be efficient and practically relevant, even for nonlinear structures.

This study extends the work in 2D material design with enhanced buckling strength in Thomsen et al. (2018) to 3D material design with tunable stiffness and buckling response utilizing a flexible framework for large scale topology optimization based on the Portable, Extensible Toolkit for Scientific Computation (PETSc) (Balay et al., 2016, Aage et al., 2015, Wang, 2018) and the Scalable Library for Eigenvalue Problem Computations (SLEPc) (Hernandez et al., 2005). Considered materials are constrained to be cubic symmetric and elastically isotropic. The homogenization method is employed to characterize the effective material properties, and LBA, together with Bloch–Floquet boundary conditions evaluated over the boundaries of the irreducible Brillouin zone (IBZ) (Brillouin, 1953), is employed to evaluate material buckling strength. The optimization problem for designing materials with tunable stiffness and buckling strength is formulated to minimize the weighted stiffness and buckling response. 3D materials are designed to achieve tunable stiffness and buckling response by assigning different weight factors for stiffness and buckling strength under uniaxial compression. Moreover, inspired by the topology optimized material configurations, a subsequent feature-based shape optimization approach is employed to simplify material geometries. In the feature-based approach, material architectures are parameterized using several hollow and one solid super-ellipsoids (Wang, 2018, Wein et al., 2020).

The paper is organized as follows. Section 2 presents finite element formulations to evaluate stiffness and buckling strength, and formulates the optimization problem for designing materials with tunable properties. Section 3 first validates the proposed optimization formulation by optimizing a material microstructure for maximum buckling strength under hydrostatic compression. Topology optimized single-length scale material microstructures with enhanced stiffness and strength under uniaxial compression are then systematically designed. Inspired by the optimized microstructure configurations, a shape optimization scheme is proposed to simplify the optimized microstructure geometries further. The paper ends with the conclusions in Section 4.