Architecting materials for extremal stiffness yield and buckling strength

2025-04-24 0 0 6.91MB 20 页 10玖币

Architecting materials for extremal stiﬀness, yield and buckling strength

Fengwen Wang∗, Ole Sigmund

Department of Civil and Mechanical Engineering, Technical University of Denmark, Koppels All´e, Building 404, 2800 Kgs. Lyngby, Denmark

Abstract

This paper proposes a methodology for architecting microstructures with extremal stiﬀness, yield, and buckling

strength using topology optimization. The optimized microstructures reveal an interesting transition from simple

lattice like structures for yield-dominated situations to hierarchical lattice structures for buckling-dominated situa-

tions. The transition from simple to hierarchical is governed by the relative yield strength of the constituent base

material as well as the volume fraction. The overall performances of the optimized microstructures indicate that max-

imum strength is determined by the buckling strength at low volume fractions and yield strength at higher volume

fractions, regardless of the base material’s relative yield strength. The non-normalized properties of the optimized

microstructures show that higher base material Young’s modulus leads to both higher Young’s modulus and strength

of the architected microstructures. Furthermore, the polynomial order of the maximum strength lines with respect to

mass density obtained from the optimized microstructures reduces as base material relative yield strength decreases,

reducing from 2.3 for buckling dominated Thermoplastic Polyurethane to 1 for yield dominated steel microstructures.

Keywords: Architected material, Stiﬀness, Yield strength, Buckling strength, Topology optimization

1. Introduction

Exploring novel material architectures with extremal properties has been a constant quest in the ﬁeld of material

design and lightweight engineering. These developments have been further promoted by advances in additive man-

ufacturing facilitating fabrication of functional materials with unprecedented complexity [1, 2]. As substitutions to

trial and error and limited human intuition, topology optimization methods have been shown to be powerful tools in

designing novel materials in various applications [3, 4, 5]. Up to now, however, the ﬁeld of architected materials has

∗Corresponding author

Email address: fwan@dtu.dk (Fengwen Wang)

Preprint submitted to Journal name April 17, 2023

arXiv:2210.00003v2 [cs.CE] 14 Apr 2023

focused on individual material properties like stiﬀness, yield or buckling strength but less so on the intricate trade-oﬀ

between these properties.

Material stiﬀness and strength are fundamental properties for determining material load-bearing capacity as they

measure the material’s deformation resistance and ultimate load-carrying capacity, respectively. The quest for op-

timal stiﬀand strong materials depends on many aspects, including the loading conditions and constituent base

materials. Stiﬀness-optimal materials meeting theoretical upper bounds have been obtained via systematic design

approaches [6, 7, 8, 9, 10]. It has been shown that plate microstructures reach the Hashin-Shtrikman bounds [11]

in the low volume fraction limit and remain within 10% of the theoretical upper bounds at moderate volume frac-

tions. A few studies have focused on improving microstructure strength, including stress minimization and buckling

strength optimization. It has been shown that the maximum von Mises stress can be reduced when taking stress

into account during the microstructure optimization procedure either as constraints or objectives [12, 13, 14, 15].

The stress singularity issue, where the stress at a point approaches inﬁnity as the density at that point approaches

zero, was handled by stress relaxation methods [16, 17, 18, 19]. Large numbers of local stress constraints can be

aggregated to a global quantity using the p-norm, Kresselmeier–Steinhauser (KS) [20] methods or by the augmented

Lagrangian method [21]. Microstructure buckling strength optimization using topology optimization was ﬁrst studied

by [22], focusing on cell-periodic buckling modes. More recently, 2D and 3D material microstructures were system-

atically designed to enhance buckling strength based on linear material analysis. This approach evaluates eﬀective

material properties using the homogenization method [23] and material buckling strength under a given macro stress

using linear buckling analysis (LBA) with Bloch-Floquet boundary conditions to capture all the possible buckling

modes [24, 25, 26]. Both studies showed that the optimized materials possess several times higher buckling strength

than their references at the cost of some stiﬀness degradation. Subsequent 2D experimental veriﬁcations have further

veriﬁed the buckling superiority of the optimized microstructures and validated the linear material evaluation [27].

So far, optimized microstructures have been designed considering material buckling or yield failure separately.

However, yield-optimized microstructures tend to be vulnerable to buckling failure. On the other hand, buckling-

optimized microstructures assume constituent base materials with relatively high yield strength (yield strength to

Young’s modulus ratio, σ1/E1), e.g., elastomers. For other base materials with low relative yield strengths, e.g., steel,

the buckling-optimized microstructures fail due to localized yield. Hence the optimized strength superiority in both

cases may signiﬁcantly degrade in real applications. Furthermore, a previous numerical study showed that for a given

microstructure topology, the failure mechanism switches from buckling- to yield-dominated failure as the volume

fraction increases [28]. Hence, it is crucial to consider both failure mechanisms in the design procedure consider-

ing diﬀerent volume fractions. Furthermore, it is essential to provide microstructure candidates with programmable

properties working for various base materials and volume fractions ﬁtting various applications.

In this study, we extend the stiﬀness/buckling studies from [25] and [26] to also include yield strength. Yield

strength and Young’s modulus share a monotonic relation in the optimized designs, as shown in the result section.

Hence microstructure will be designed systematically by maximizing buckling strength with diﬀerent yield strength

bounds and considering diﬀerent volume fractions. The optimized microstructure are further evaluated for diﬀerent

practical base materials ranging from low relative yield strength steel to high relative yield strength thermoplastic

polyurethane (TPU).

The paper is organized as follows. Section 2 summarizes basic formulations to evaluate microstructure stiﬀness

and strength, and formulates the optimization problem for designing 2D microstructures with extremal properties.

Section 3 validates the proposed approach and presents topology-optimized microstructure sets for diﬀerent volume

fractions and corresponding performances considering diﬀerent base materials. Finally, section 4 concludes the study.

2. Optimization problem for 2D architected materials with enhanced stiﬀness and strength

This section summarizes the basic formulations for designing 2D architected materials with enhanced stiﬀness and

strength using topology optimization. The ﬁnite element method is combined with homogenization theory and LBA

to evaluate material properties [29]. To accurately represent the stress situation, we employ the incompatible elements

from [30] and [31], i.e., the so-called Q6element. Two additional so-called incompatible modes are considered

to represent bending deformations accurately. The reader is referred to the work by [30] and [31] for additional

formulations for the Q6elements.

2.1. Material stiﬀness and strength evaluations

Under small strain assumptions, the material stiﬀness and strength are evaluated using the homogenization ap-

proach and LBA together with Bloch-Floquet theory to account for buckling modes at diﬀerent wavelengths based on

the periodic microstructure. Fig. 1 summarizes the material stiﬀness and strength evaluation procedure.

The microstructure is assumed of unit size here. The symmetry properties of the eﬀective elasticity matrix are

exploited to represent the equations in a more compact form using the abbreviation kl →α: 11 →1, 22 →2,

(12,21) →3. The eﬀective elasticity matrix is calculated by an equivalent energy-based homogenization formula-

Uniaxial Compression: σ0=[−1; 0; 0]

[0, π]

Linear buckling analysis

Dαβ

RVE

(b)

(a) (c)

Homogenization

[0,0]

[0, π][π, π]

σc/E1=min(λ)=0.00060

ΓX

max 



ϕ1



min 



ϕ1



σc/E1

Figure 1: Flowchart for material stiﬀness and strength evaluations based a stiﬀness-optimal orthotropic material with a volume

fraction of 0.2. (a) Homogenization of a periodic material using a representative volume element (RVE), i.e., microstructure. (b)

Calculation of yield strength and illustration of the corresponding irreducible Brillouin zone (IBZ) using uniaxial compression. (c)

Bucking band structure, buckling strength and mode.

Here |Y|denotes the volume of the microstructure, Prepresents a ﬁnite element assembly operation over all Nele-

ments, the superscript ()Tdenotes the transpose, ˆ

Bewith a size of 6 ×24 is the condensed strain-displacement matrix

of element ein the Q6element formulation, Deis the elasticity matrix of the material in element e,˜

εα=hδαβidenotes

the 3 independent unit strain ﬁelds, fαis the condensed equivalent load vector induced by the αth unit strain and K0

is the global condensed elastic stiﬀness matrix. The detailed formulation of K0and fαcan be found in [20, 27].

The eﬀective material compliance matrix is ¯

C=¯

D−1.Under plane stress assumptions, the eﬀective Young’s and

bulk moduli are calculated using the eﬀective elasticity or compliance matrices, stated as

E=1

C11

,¯κ=¯

D11 +¯

D12

2.(2)

Figure 1: Flowchart for material stiﬀness and strength evaluations based a stiﬀness-optimal orthotropic material with a volume

fraction of 0.2. (a) Homogenization of a periodic material using a representative volume element (RVE), i.e., microstructure. (b)

Calculation of yield strength and illustration of the corresponding irreducible Brillouin zone (IBZ) using uniaxial compression. (c)

Bucking band structure, buckling strength and mode.

tion [6, 23] via

Dαβ =1

|Y|

e=1ZYe˜

εα−ˆ

Beχe

αTDe˜

εβ−ˆ

Beχe

βdY,

K0χα=fα, α =1,2,3,(1)

χα|x=1=χα|x=0,χα|y=1=χα|y=0.

Here |Y|denotes the volume of the microstructure, Prepresents a ﬁnite element assembly operation over all Nele-

ments, the superscript ()Tdenotes the transpose, ˆ

Bewith a size of 3 ×8 is the condensed strain-displacement matrix

of element ein the Q6element formulation, Deis the elasticity matrix of the material in element e,˜

εα=hδαβidenotes

the 3 independent unit strain ﬁelds, fαis the condensed equivalent load vector induced by the αth unit strain and K0

is the global condensed elastic stiﬀness matrix. The detailed formulation of K0and fαcan be found in [31] and [25].

The eﬀective material compliance matrix is ¯

C=¯

D−1. Under plane stress assumptions, the eﬀective Young’s and

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摘要：

Architectingmaterialsforextremalstiness,yieldandbucklingstrengthFengwenWang,OleSigmundDepartmentofCivilandMechanicalEngineering,TechnicalUniversityofDenmark,KoppelsAll´e,Building404,2800Kgs.Lyngby,DenmarkAbstractThispaperproposesamethodologyforarchitectingmicrostructureswithextremalstiness,yield,...

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Architecting materials for extremal stiffness yield and buckling strength

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