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∗
120o
∗
h/l = 1
θ= 30oα= 1
Anisotropy tailoring in geometrically isotropic multi-material lattices
Abstract and Figures
This article proposes the concept of anisotropy tailoring in multi-material lattices based on a mechanics-based bottom-up framework. It is widely known that isotropy in a mono-material lattice can be obtained when the microstructure has an isotropic geometry. For example, regular hexagonal lattices with a unit cell comprised of six equal members and equal internal angle of 120^o each, show isotropy in the elastic properties. Such limited microstructural configuration space for having isotropy severely restricts the scope of many multi-functional applications such as space filling in 3D printing. We first demonstrate that there are multiple structural geometries in mono-material lattices that could lead to isotropy. It is shown that the configuration space for isotropy can be expanded by multiple folds when more than one intrinsic material is introduced in the unit cell of a lattice. We explicitly demonstrate different degrees of anisotropy in regular geometrically isotropic lattices by introducing the multi-material architecture. The contours of achieving minimum anisotropy, maximum anisotropy and a fixed value of anisotropy are presented in the design space consisting of geometric and multi-material parameters. Proposition of such multi-material microstructures could essentially expand the multi-functional design scope significantly, offering a higher degree of flexibility to the designer in terms of choosing (or identifying) the most suitable microstructural geometry. An explicit theoretical characterization of the contours of anisotropy along with physical insights underpinning the configuration space of multi-material and geometric parameters will accelerate the process of its potential exploitation in various engineered multi-functional materials and structural systems across different length-scales with the demand of any specific degree of anisotropy but limitation in the micro-structural geometry.
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Supplementary resource (1)
... Honeycomb lattices are extensively used in engineering structures as these provide a structure with minimal density with ecient surface lling, high energy absorption capacity under crushing and impact, high strength to weight ratio, and relatively high out-of-plane compression and shear properties [1,20,21]. In the development of metamaterials by micro-scale architecturing of periodic beam-like networks, heaxagonal lattices and its derivatives (such as square, rhombic and auxetic congurations) have widely been investigated for eective property modulation at the macro scale [22,23,24,25,26]. ...
... Further, since we aim to deal with evaluating the eective linear elastic properties of the lattice here with the assumption of innite (or large) number of unit cells,`size eect' would not have a signicant eect in the current analysis [1,3]. The idea of anisotropy tailoring in lattices with multi-material properties is demonstrated in [22], where the considered unit cell has constituent elements of multiple materials. Active elastic property modulation with the help of voltage has been realized in a recent work [23]. ...
... The overall strain in direction 3 is calculated using equation (22). ...
Analytical investigations to characterize the effective mechanical properties of lattice materials allow an in-depth exploration of the parameter space efficiently following an insightful, yet elegant framework. 2D lattice materials, which have been extensively dealt with in the literature following analytical as well as numerical and experimental approaches, have limitations concerning multi-directional stiffness and Poisson's ratio tunability. The primary objective of this paper is to develop mechanics-based formulations for a more complex analysis of 3D lattices, leading to a physically insightful analytical approach capable of accounting the beam-level mechanics of pre-existing intrinsic stresses along with their interaction with 3D unit cell architecture. We have investigated the in-plane and out-of-plane effective elastic properties to portray the physics behind the deformation of 3D lattices under externally applied far-field normal and shear stresses. The considered effect of beam-level intrinsic stresses therein can be regarded as a consequence of inevitable temperature variation, pre-stress during fabrication, inelastic and non-uniform deformation, manufacturing irregularities etc. Such effects can notably impact the effective elastic properties of lattice materials, quantifying which for 3D honeycombs is the central focus of this work. Further, from the material innovation viewpoint, the intrinsic stresses can be deliberately introduced to expand the microstructural design space for effective elastic property modulation of 3D lattices. This will lead to programming of effective properties as a function of intrinsic stresses without altering the microstructural geometry and lattice density. We have proposed a generic spectral framework of analyzing 3D lattices analytically, wherein the beam-level stiffness matrix including the effect of bending, axial, shear and twisting deformations along with intrinsic stresses can be coupled with the unit cell mechanics for obtaining the effective elastic properties.
... Very pioneer work for evaluating the effective equivalent elastic properties of regular mono-material conventional honeycomb cells is presented by Gibson et al. [1]. Later, Mukhopadhyay et al. [2] explored the possibilities of anisotropy tailoring of properties through the multi-material arrangement in a symmetric honeycomb lattice. Recently, Sinha et al. [3] proposed a lattice to obtain the elastic modulus and consider the effect of residual stresses. ...
... Then, new parametric results are presented for the in-plane elastic modulus. The literature of Gibson and Ashby [1] and Mukhopadhyay et al. [2] are considered to show excellent agreement with elastic modulus E 1 and E 2 for the present formulation. In Gibson's study, axial, bending, and shear modes of deformations have been considered for an auxetic mono-material case, while in Mukhopadhyay's work [2], a multi-material auxetic structure under only bending deformation is analyzed. ...
... The literature of Gibson and Ashby [1] and Mukhopadhyay et al. [2] are considered to show excellent agreement with elastic modulus E 1 and E 2 for the present formulation. In Gibson's study, axial, bending, and shear modes of deformations have been considered for an auxetic mono-material case, while in Mukhopadhyay's work [2], a multi-material auxetic structure under only bending deformation is analyzed. In Fig. 2, the variation of elastic properties (E 1 and E 2 ) under the influence of axial, bending, and shear effects is plotted for a mono-material auxetic honeycomb structure for different cell angles (θ ). ...
... Generally, a unit cell-based approach is adopted to design lattice metamaterials with a single type of intrinsic material [37] (refer to Figure 1(a, d)). Recently, multi-material lattices have been proposed where dierent cell walls in the unit cells have dierent intrinsic material properties (refer to Figure 1(b)) [60]. With the progress of multi-material additive manufacturing, such lattices can be physically realized with adequate precision [61,62,63,64,65]. ...
... If we further apply the condition for multi-material symmetric hexagonal structure (θ 2 =θ 3 = θ), then we can obtain the expressions for deection of cell wall 2 and 3 under only bending deformation, similar to the case of Mukhopadhyay et al. [60]. ...
... The concept of multi-material hexagonal honeycombs (symmetric structure) and concerned analyses of eective elastic moduli were rst presented by Mukhopadhyay et al. [60]. In that analysis, beam-level deformation was considered based on bending with the assumption of thin cell walls. ...
Bi-level tailoring of cellular metamaterials involving a dual design space of unit cell and elementary beam level architectures has recently gained traction for the ability to achieve extreme elastic constitutive properties along with modulating multi-functional mechanical behavior in an unprecedented way. This article proposes an efficient analytical approach for the accurate evaluation of all constitutive elastic constants of asymmetric multi-material variably-thickened hexagonal lattices by considering the combined effect of bending, stretching, and shearing deformations of cell walls along with their rigid rotation. A tri-member unit cell is conceptualized, wherein all nine constitutive constants are obtained through the mechanics under one cell wall direction and subsequent repetitive coordinate transformations. We enhance the design space of lattice metamaterials substantially here by introducing multiple exploitable dimensions such as asymmetric geometry, multi-material unit cells and variably-thickened cell walls, wherein the conventional monomaterial auxetic and non-auxetic hexagonal configurations can be analyzed as special cases along with other symmetric and asymmetric lattices such as a range of rectangular and rhombic geometries. The generic analytical approach along with extensive numerical results presented in this paper opens up new avenues for efficient optimized design of the next-generation multi-functional lattices and cellular metamaterials with highly tailored effective elastic properties.
... While a wide range of effective material properties can be achieved in such passive material microstructures, the properties cannot be tuned in an on-demand basis after they are manufactured. The recent developments in this direction include bi-level topology architected optimum metamaterials [5], hierarchical metamaterials [24], disordered metamaterials [25], anti-curvature metamaterils with programmed curvature [26,27,28,29], multi-material and space-filled lattices [30,31], origami and kirigami-inspired metamaterials [32,19], to mention a few. Recently the concept of pneumatic elastostatics and deployability in mechanical metamaterials has been proposed based on inflatable lattices that can exhibit extreme specific stiffness along with on-demand tunability [33]. ...
... Now, the slack variables have to be positive to keep the iterations running in the feasible domain. For each of γ > 0, we have Equation 30. ...
... method considering slack variables (refer toEquation 29 and Equation30) to tackle the subproblems that arise throughout the iteration. First, an ideal subproblem is put up in the sequential quadratic programming model to look for the following feasible point in the present iteration point. ...
Owing to the architected void-filled low-density configurations, metamaterials are prone to defects during the complex manufacturing process, or damages under operational conditions. Recently mechanical cloaking has been proposed to shield the effect of such disorders in terms of homogenized mechanical responses. The major drawback in these studies are that the damage location should be known a priori, and the cloak is designed around that damaged zone before manufacturing. Such postulation does not allow unsupervised damage resilience during the manufacturing and service life of metamaterials by active reconfiguration of the stress field depending on the random and unpredictable evolution of damage. Here, we propose a radically different approach by introducing piezoelectric lattices where the effect of random appearance of any single or multiple disorders and damages with complex shapes, sizes and distributions can be shielded through active multi-physically controlled cloaks by voltage-dependent modulation of the stress fields within the cloaking region. Notably, this can be achieved without breaking periodicity and any additional material in the cloaking region unlike earlier studies concerning mechanical cloaks. The proposed active class of elastic metamaterials will bring a step-change in the on-demand mechanical performance of critically important structural components and unsupervised damage resilience for enhanced durability and sustainability.
This article is part of the theme issue ‘Current developments in elastic and acoustic metamaterials science (Part 1)’.
... Xu et al. [13] modied the traditional regular honeycomb into AuxHex structure (unit cells with both auxetic and hexagonal honeycomb patterns) in their study and derived the eective elastic moduli and plastic collapse stresses of the lattice. Mukhopadhyay et al. [14] developed the theory for heterogeneous multi-material lattices, while Mukherjee and Adhikari [15] further extended the theory to incorporate beam-level axial and shear deformations. Mukhopadhyay et al. [16] derived the closed-form elastic-moduli expressions of multi-material honeycomb lattices considering the eect of ller material. ...
... For auxetic lattice, the geometrical constraint i.e. L R + 2 sin θ > 0 is conformed so that the vertices of the interior cell member remain untouched during the manufacturing and deformation [14,46]. In the gures, unfeasible geometry indicates the combination of those L R and θ where the aforementioned geometric constraints are being violated. ...
After tremendous progress in computationally conceptualizing and manufacturing lattice metamaterials with complex cell geometries over the last decade, a strong rationale has evolved lately to achieve active and on-demand property modulation in real-time with greater sensitivity. Traditionally materials show an uncoupled response between normal and shear modes of deformation. Here we propose to achieve heterogeneous mode coupling among the normal and shear modes, but in conventional symmetric lattice geometries through an intuitive mounting of electro-active elements. The proposed bi-level multi-physically architected metamaterials lead to an unprecedented programmable voltage-dependent normal-shear constitutive mode coupling and active multi-modal stiffness modulation capability for critically exploitable periodic or aperiodic, on-demand and temporally tunable mechanical responses. Further, active partial cloaking concerning the effect of far-field complex stresses can be achieved, leading to the prospect of averting a range of failure and serviceability conditions. The manufacturing flexibility in terms of symmetric lattice geometry, along with actively tunable heterogeneous mode coupling in the new class of metamaterials would lead to real-time control of mechanical responses for temporal programming in a wide range of advanced mechanical applications, including morphing and transformable geometries, locomotion in soft robotics, embedded actuators, enhanced multi-modal energy harvesting, vibration and wave propagation control.
... The review of literature presented herein indicates that bioinspired composite structures have been of central interest to the research community for their potential mechanical advantages. Further, it is noticed that a large volume of literature is focused on different lattice structures, including hexagon, octet, gyroid, primitive, diamond, and many other geometries, as they replicate the structures from nature like starfish [74], butterflies [75], wood microstructures [76][77][78], etc. ...
Inspired by several biological structures available in nature, bio-inspired composite structures are evidenced to exhibit a noteworthy enhancement in various mechanical and multi-physical performances as compared to conventional structures. This article proposes to exploit the architecture of annual ring growth of the stems of trees for developing a new class of bio-inspired composites with enhanced static and dynamic performances, including deflections, stresses, strain energy, and vibration. Concentric circular annual-ring geometries are considered where each layer of concentric circular fibers is analogous to the growth per annum of trees. The annual rings are modeled in a finite element-based computational framework by idealizing each layer as a composite of graphite fibers and epoxy matrix under different boundary conditions. The ratio of deflection to weight and frequency to weight of bio-inspired and traditional composites are compared by considering different parameters such as the number of annual rings, layers, and supporting stiffeners. The numerical results reveal that the proposed bio-inspired composites can enhance and modulate the static and dynamic properties to a significant extent, opening new design pathways for developing high-performance fiber network composites.
Additive manufacturing permits the precision design and meticulous fabrication of lattice structures, thereby unlocking their potential performance capabilities. Herein, to enhance the specific strength of the cubic lattice structure, three design strategies are applied: substituting solid struts with hollow ones, increasing the material proportion of vertical struts, and introducing cross plates into the hollow struts. The results demonstrate that the strategy of substituting solid struts with hollow ones effectively contributes to a substantial enhancement of 113% in the compressive yield strength of lattice structures. Furthermore, increasing the vertical material proportion from 70% to 85% while maintaining a constant relative density of 0.30, a significant enhancement is observed in both yield and ultimate strength. Additionally, introducing cross plates into the hollow struts at a vertical material proportion of 90% results in a remarkable enhancement of the specific yield strength to 230 MPa cm³ g⁻¹, significantly surpassing that of the TC4 matrix. The optimized lattice structures exhibit exceptional uniaxial strength, tremendously exceeding the predicted limits of the Gibson–Ashby model. Thus, the Gibson–Ashby model is modified by considering the vertical material proportion Cv. These findings are expected to establish a foundational framework for the optimizing of anisotropic lattice structures by remarkably enhancing their specific strength.
Two‐dimensional (2D) materials are crucially important nanomaterials because of their exciting multi‐functional properties. However, a single layer of 2D materials may not have a certain property adequately, or multiple application‐specific properties simultaneously to the desired and optimal level. For mitigating this lacuna, a new trend has emerged recently to develop nano‐scale engineered heterostructures by stacking multiple layers of different 2D materials, wherein each of the layers could also be twisted. The vast advantage of combining single layers of different 2D materials with different twisting angles has dramatically expanded this research field well beyond the scope of considering a 2D material mono‐layer, leading to a set of multifunctional physical properties corresponding to each possible combination of number of layers, different 2D materials therein, stacking sequence and the twisting angle of each layer. Effective mechanical properties such as Young's moduli are generally of utmost importance for analyzing the viability of such engineered nano‐heterostructures in various nanoelectromechanical applications. Efficient closed‐form generic formulae are proposed for the effective Young's moduli of twisted multi‐layer heterostructures. Based on this physics‐based analytical approach, a wide range of insightful new results are presented for twisted heterostructures, covering mono‐planar and multi‐planar configurations with homogeneous and heterogeneous atomic distributions.
Mechanical metamaterials are usually designed to exhibit novel properties and functionalities that are rare or even unprecedented. What is common among most previous designs is the quasi-static nature of their mechanical behavior. Here, we introduce a previously unidentified class of strain rate-dependent mechanical metamaterials. The principal idea is to laterally attach two beams with very different levels of strain rate-dependencies to make them act as a single bi-beam. We use an analytical model and multiple computational models to explore the instability modes of such a bi-beam construct, demonstrating how different combinations of hyperelastic and viscoelastic properties of both beams, as well as purposefully introduced geometric imperfections, could be used to create robust and highly predictable strain rate-dependent behaviors of bi-beams. We then use the bi-beams to design and experimentally realize lattice structures with unique strain rate-dependent properties including switching between auxetic and conventional behaviors and negative viscoelasticity.
A novel origami metamaterial with programmable multi-directional auxeticity is proposed by hybridizing the concept of re-entrant honeycomb with the Miura pattern. Normal Miura-based origami metamaterials have been widely shown to exhibit negative in-plane Poisson's ratio. However, negative out-of-plane Poisson's ratio cannot be realized in these set of materials, essentially limiting their capability for prospective applications in many multi-functional devices and systems. Here we propose a hybrid Miura-based metamaterial that can show both in-plane and out-of-plane negative Poisson's ratios. More interestingly, we are able to program the Poisson's ratios to have mild to extreme auxeticity and map their mutual interaction as a function of the microstructural configuration. Besides the single-layer meta-sheets, we have shown that a class of multi-layer stacked metamaterial with uniform and graded configurations can be developed to achieve multi-objective functional goals. Theoretical and experimental analyses are combined in this paper to demonstrate the concepts of modulating multi-directional Poisson's ratios. The fundamental mechanics of the proposed origami based metamaterial being scale-independent, this novel class of deployable hybrid materials can be directly transferred for application in a range of milli-, micro-, and nanometer-size systems, essentially opening avenues for the design of energy absorbers, sensors, actuators, medical stents, catalysis, drug delivery systems, adaptive wings for next-generation of aircrafts and other deployable mechanical and electronic systems at multiple length-scales.
The energy absorption and structural isolation performance of axially-compressed sandwich structures constructed with stiff face plates separated with an auxetic lattice core metamaterial is studied. Advances in additive manufacturing increasingly allow bespoke, carefully designed, structures to be included within the core lattice to enhance mechanical performance. Currently, the internal structure of the lattice core is deliberately disrupted geometrically to engineer suitable post-buckling behavior under quasi-static loading. The desirable properties of a high fundamental stiffness and a practically zero underlying stiffness in the post-buckling range ensure that energy may be absorbed within a limited displacement and that any transfer of strain to an attached structure is minimized as far as is feasible. It is demonstrated that such disruptions can be arranged to enhance the panel performance. The concept may be extended to promote cellular buckling where the internal lattice buckles with densification occurring at defined locations and in sequence to absorb energy while maintaining a low underlying mechanical stiffness.
Doping in pristine 2D materials brings about the advantage of modulating wide range of mechanical properties simultaneously. However, intrinsic defects (such as Stone-Wales and nanopore) in such hybrid materials are inevitable due to complex manufacturing and synthesis processes. Besides that, defects and irregularities can be intentionally induced in a pristine nanostructure for multi-synchronous modulation of various multi-functional properties. Whatever the case may be, in order to realistically analyse a doped graphene sheet, it is of utmost importance to investigate the compound effect of doping and defects in such 2D monolayers. Here we present a molecular dynamics based investigation for probing mechanical properties (such as Young’s modulus, post-elastic behaviour, failure strength and strain) of doped graphene (C¹⁴ and Si) coupling the effect of inevitable defects. Spatial sensitivity of defect and doping are systematically analyzed considering different rational instances. The study reveals the effects of individual defects and doping along with their possible compounded influences on the failure stress, failure strain, Young’s modulus and constitutive relations beyond the elastic regime. Such detailed mechanical characterization under the practically relevant compound effects would allow us to access the viability of adopting doped graphene in various multifunctional nanoelectromechanical devices and systems in a realistic situation.
Carbon-based nano hetero-structures are receiving increasing attention due their ability in multi-synchronous modulation of a range of mechanical and other critically desirable properties. In this paper, the vibration characteristics of two different graphene based heterostructures, graphene-hexagonal boron nitride (hBN) and graphene-molybdenum disulfide (MoS), are explored based on atomistic finite element approach. Such vibrational characteristics of nanostructures are of utmost importance in order to access their suitability as structural members for adoption in various nano-scale devices and systems. In the current analysis, the developed atomistic finite element model for nano-heterostructures is extensively \textcolor{black}{validated} first with the results available in literature considering elastic responses and natural frequencies. Thereafter a range of insightful new results are presented for the dynamic behaviour of various configurations of graphene-hBN and graphene-MoS heterostructures including their size, chirality and boundary dependence. The investigation of tunable vibrational properties along with simultaneous modulation of other mechanical, electronic, optical, thermal and chemical attributes of such nano-heterostructures would accelerate their application as prospective candidates for manufacturing nanosensors, electromechanical resonators, and a wide range of other devices and systems across the length-scales.
Lattice materials are characterised by their equivalent elastic moduli for analysing mechanical properties of the microstructures. The values of the elastic moduli under static forcing condition are primarily dependent on the geometric properties of the constituent unit cell and the mechanical properties of the intrinsic material. Under a static forcing condition, the equivalent elastic moduli (such as Young’s modulus) are always positive. When dynamic forcing is considered, the equivalent elastic moduli become functions of the applied frequency and they can be negative at certain frequency values. This paper, for the first time, explicitly demonstrates the occurrence of negative equivalent Young’s modulus in lattice materials experimentally. Using additively manufactured titanium-alloy lattice metastructures, it is shown that the real part of experimentally measured in-plane Young’s modulus becomes negative under a dynamic environment. In fact, we show that the onset of such negative Young’s modulus in lattice materials can be precisely determined by capturing the sub-wavelength scale dynamics of the system. Experimental confirmation of the negative Young’s moduli and the onset of the same as a function of frequency provide the necessary physical insights and confidence for its potential exploitation in various multi-functional structural systems and devices across different length scales.
