Buckling and postbuckling of architectured materials: A review of methods for lattice structures and metal foams

Abstract

Recent advances in manufacturing and material science have given rise to numerous architectured materials (archimats), which are tailored for multifunctionality and improved performance. Specifically, lattice structures and metal foams are usually lightweight optimized structural morphologies, which are prone to non-linear instability phenomena, leading to collapse or to a different stable state. This article offers an extensive review of analytical, numerical and experimental methods for investigating buckling and postbuckling in such materials. In terms of analytical modelling, linear elastic and geometrically non-linear models are presented. In numerical analysis, discrete and continuum models are presented, highlighting how numerical modelling can inform design of such materials and finally, experimental methods across different scales are reported, highlighting their merits, depending on the aim of the investigation.

Introduction

Over the past decades, architectured materials (archimats) (please note that other terms are proposed in the literature, such as hybrid materials, metamaterials, multimaterials and tailored materials) have been making their way into material science and engineering. Archimats comprise both material combination and structural configuration and are thus dependent on the observation scale. Hence, an archimat combines several non-miscible materials (or one material and air) in a predefined arrangement such that a representative volume element (RVE) comprises at least one dimension that is very small in comparison with the overall dimensions of the part it composes. This includes, for example, composites, sandwiches, foams, lattices, etc. ¹ Lattice materials, for example, are cellular materials with an open and periodic internal structure and have remarkable potential owing to their multifunctionality. ^2,3 The periodicity of lattice materials ensures that their global material properties mainly depend on the internal architecture. Hence, the overall properties of lattice materials can be tailored towards the requirements of specific applications by designing the lattice structure accordingly. Thus, during the development and tailoring of an archimat, the underlying question on the extent of the combination of the strategies of microstructural and architectural design arises.

This aspect becomes more relevant since advanced manufacturing technologies such as additive manufacturing (AM) allow to produce highly complex and even hybrid lattice structures in customized mass production. ^{4 –6} With the advent of AM technologies, the development of tailored materials has experienced a significant boost. A desired structure can now easily be built in three dimensions (3D) owing to the precise deposition of printable material(s) with microscale accuracy. ^1,7

In lightweight constructions, archimats are used due to their extremely high strength-to-density ratio. The rising demand for energy-saving and, consequently, lighter construction has led to the manufacture of very slender lattice structures. Slender structures are highly prone to loss of structural stability when experiencing compressive stress states. Hence, their compressive strength is not governed by the strut material but by the buckling load of the lattice itself. ^8,9 Experimental work shows that, for example, square lattice materials possess far higher compressive strengths than other lattice structures with comparable relative densities. ^{9 –12}

The design of lattice materials previously focussed on the perfect state, that is, imperfection-free geometry and homogeneous base materials to understand the underlying deformation mechanisms (see refs. ^{13 –16} ). Yet, even small perturbations, for example, due to local manufacturing discrepancies, may result in dramatic changes in the overall structural response of an archimat and may potentially jeopardize its functionality. Thus researchers are currently accounting for imperfections via robust design architectures that minimize the impact of imperfections on the effective overall properties and, ultimately, component performance. ^17,18 So despite the fact that buckling is a phenomenon that has been known for centuries, it is still commonly avoided in engineering design.

However, in 2010, Crosby already emphasized the potential of buckling and the deeper understanding of the (post)buckling mechanisms. ¹⁹ Most recently, structural stability analysis has indeed experienced a research revival. Buckling phenomena are no longer viewed as sources of catastrophic failure but rather as novel opportunities for functionality and design as is manifested in many smart applications surrounding us, for example, sensors, switches and deployable structures,. Thus renewed interest is sparked. ^20,21 For a comprehensive review on how microstructural instabilities can be exploited in archimats, the reader is referred to the article of Kochmann and Bertoldi. ²²

Lattice materials, as an example for an archimat, can be considered as special cases of open-cell foams with a distinct periodic structure. Hence, modelling strategies developed for open-cell foams can be directly applied. These modelling strategies can be categorized into analytical and computational methods. In addition, experimental investigations are mainly undertaken to gain further knowledge on the underlying mechanisms.

Analytical modelling

Herein, analytical methods comprise all methods where the problem can be stated analytically, no matter whether a closed-form solution is subsequently found or the governing equations are solved with the aid of numerical tools. Initially, linear elastic modelling approaches are presented and their limitations are discussed. Subsequently, geometrically non-linear modelling is presented with its specific application on lattice materials being demonstrated.

Linear elastic modelling

Analytical methods are often based on linear elastic beam theory and have been applied to study various aspects of open-cell foams/lattice materials in a global or homogenized manner. ^23,24 In ref., ²³ relations between the elastic modulus and the relative density in the form of a power law are derived. For the overall homogenized elastic modulus $\overline{E}$ , the relation $\overline{E}\approx E{\overline{\rho }}^m$ is found, where E denotes the elastic modulus of the strut material and $\overline{\rho }$ is the relative density. The parameter m accounts for the deformation mechanism active in the lattice material with respect to the loading condition and is $m=1$ for stretch dominated and $m=3$ for bending dominated lattices. Beam models (see Figure 1) are also used in ref. ²⁴ to investigate the mechanical behaviour of honeycomb structures, where relations for the compressive collapse and fracture toughness are derived. Similar concepts are used in ref. ²⁵

Figure 1.

A hypothetical beam model to describe hexagonal lattice struts buckling under compressive load in the vertical direction.

In this review, the focus lies on the mechanical behaviour of archimats under compressive loading. With respect to this loading scenario, analytical approaches are limited to the evaluation of the critical behaviour, that is, the maximum peak compressive stress ${\sigma }_{\text{pk}}$ . ^10,24,25 In ref. ¹⁰ , the buckling stress ${\sigma }_{\text{pk}}$ for global elastic buckling of the lattice material is determined as follows:

${\sigma }_{\text{pk}}={\sigma \text{'}}_{\text{pk}}+G\hspace{1em}\text{with}\hspace{1em}{\sigma \text{'}}_{\text{pk}}=\frac{{\pi }^2{E}_{\text{s}}}{2}{\left(\frac{a}{H}\right)}^2\overline{\rho }$

1

The peak stress ${\sigma \text{'}}_{\text{pk}}$ is the buckling stress of the struts without the influence of the horizontal bars and depends linearly on the relative density $\overline{\rho }$ . The parameter G denotes the overall shear modulus of the lattice material. Similar equations exist for other lattice materials, see, for example. ^9,25 Such formulations are insufficient for two reasons. First, calculating the compressive peak buckling stress offers no insight into the global postbuckling material response. Second, this approach has served, up to now, only to supplement experimental work. For efficient design and tailoring of the material behaviour, it is, however, necessary to estimate their mechanical behaviour without experimental work.

Thus, geometrically non-linear analytical models are required.

Geometrically non-linear modelling

Geometrically non-linear modelling is necessary to explore the behaviour of structures beyond their critical buckling load. The analytical procedure reviewed herein is the method of minimizing the total potential energy in the form described by Thompson and Hunt. ²⁶ The aim of this convenient procedure is to describe the deformation behaviour of the system under the influence of load with only very few generalized coordinates or so-called degrees of freedom, q_i. The total potential energy V consists of the strain energy U of the system and the work of the external load P along the end-shortening $\mathcal{E}$ :

$V(q_i,P)=U(q_i)-P\mathcal{E}(q_i)$

2

Once this formulation is determined for a certain structure, the stability of the system can be readily investigated. The relationship of the load P versus the displacement and hence the generalized coordinates q_i results from the equilibrium path of the system derived by the total derivative of V with respect to the generalized coordinates, thus:

$\frac{\text{d}V}{\text{d}{q}_i}\stackrel{!}{=}0$

3

The methodology allows for a large deflection formulation and the assessment of both semi-continuous and fully discretized geometric models while parameters can be varied systematically and imperfection studies can be conducted.

The above methodology has recently been successfully employed to model fibre-stayed, collinear lattice structures ²⁷ while experimental and numerical investigations confirmed the results. ^28,29 Its potential for further investigations on archimats has been accredited in ref. ³⁰ The framework is particularly advantageous when so-called modal nudges, which are small, deliberate geometric alterations to the structure that were previously determined via information derived by the postbuckled configuration of the baseline idealized structure, ³¹ are to be introduced. By employing these nudges, the postbuckling behaviour of a structure can be altered significantly. That means that through these deliberate alterations, favourable, that is, stable, postbuckling paths can be enforced, yielding predictable and potentially exploitable responses in a structural context.

Digital fabrication and rapid prototyping (commonly known as 3D printing) techniques are currently flourishing. They are thus furthering the development of archimats using this framework since they add powerful physical insight and offer a proof-of-concept. Hence, a shift towards buckliphilia is made ³² focussing its efforts on demonstrating that the postbuckling regime allows for dramatic reconfigurations which can be exploited for function, thus harnessing instabilities. For example, in archimats, local bistable mechanisms triggered by small geometric variations are easily incorporated via AM leading to trapping of elastic strain energy as demonstrated experimentally and numerically in ref. ³³ 3D printing is also regarded as a versatile tool for implementing nudges for tailored buckling configurations. ³⁴ As emphasized in ref., ³⁰ the interplay between analytical, numerical and experimental techniques is of utmost importance, such that the way through a plethora of unstable postbuckling equilibrium can be determined. Thus, hereafter the further important building blocks are presented.

Numerical modelling

Computational methods allow gaining further insight into mechanisms observed in experiments as they are capable of providing detailed information on the local non-linear deformation and stress states within archimats. ^{35 –38} Large deformations, ^39,40 for example, due to buckling, ^{41 –43} as well as material non-linearities, due to plastic yielding, or damage within the struts, ^{35,39,44 –47} can be accounted for. Therefore, the complex failure behaviour of archimats involving crushing under compressive loading ^{35,38,42,43,48,49} or the fracture behaviour under various loading conditions ^47,50,51 can be studied using numerical modelling.

Modelling approaches found in the literature for simulating of the mechanical behaviour of archimats by means of the finite element method (FEM) can be grouped into discrete and continuum modelling approaches. Both approaches will be discussed in more detail in the following subsections.

Discrete modelling

Within discrete modelling approaches, archimats are modelled in their entire complexity, that means, each lattice member is explicitly resolved. Discrete models can be divided into unit cell (UC) models and models of the whole finite-sized archimat.

In general, UC models assume archimats of infinite size and, therefore, employ periodic boundary conditions ⁵² allowing for the application of macroscale stress or strain fields. A UC may consist of one or several base cells of the archimat where the base cell is the smallest geometric microstructure with which the infinite (or finite) structure can be reconstructed by repetitive translation along the axes of the base cell. ⁵³ The number of base cells within a UC depends on the mechanisms to be investigated. For determining the effective linear or non-linear elastic properties of the archimat by means of computational homogenization, one base cell is sufficient. ^42,44,54 To some extent, structural irregularities can also be considered in such models. ^42,50,55 For studying the onset of buckling and the postbuckling behaviour of archimats, the size of the UC, that is, the number of base cells within a UC has to be large enough to capture all relevant microscale deformation mechanisms. ⁴⁸ In this case, UCs are often denoted as RVE. ^{40,55 –57} In the literature, various studies can be found, which use UC models for studying microscale buckling within archimats. ^{35,40,48,53,55 –58} In combination with the Bloch wave method, UC models are capable of accounting for buckling modes with wavelengths (far) larger than the UC size. ^{53,59 –61} This can be exploited in the optimization of archimats with respect to their resistance to buckling. ⁶² For further details on the optimization of archimats, the reader is referred to Osanov and Guest ⁶³ and Thomsen et al. ⁶² as well as the references provided therein. UC models can also provide valuable information on the yield strength ^38,44 and the crushing behaviour of archimats ^35,48 if the UC size is appropriately chosen.

Modelling the archimat structure in its entire size is required if the localization of deformation such as the formation of crush bands under compressive loading ^{38,42,43,48,55,57,64 –66} or the formation of elastic shear bands in the vicinity of crack tips ⁵¹ are to be studied. Other aspects that are included in finite-sized models are boundary conditions similar to those in experiments and effects resulting from free edges. Consequently, these models allow for a direct comparison with experiments. ^{37,38,57,64,65,67} Furthermore, finite-sized archimat models allow considering structural irregularities in a statistical manner. Typical irregularities are a misalignment of struts and vertices, ^{42,43,47,48,55,66,68,69} radius variations within individual struts, ^55,67 porosity of the parent material ⁶⁷ and missing struts or missing clusters of struts. ^43,66,68,69

For the finite element discretization either continuum elements ^{36,38,55 –58,62} or structural elements such as beams ^{40,42 –44,46,49,53,55,61,67,70,71} or shells ^38,66 have been used. Continuum element models provide a detailed representation of the geometry of archimats involving material aggregation at the vertices, ⁵⁴ varying diameters of the individual struts ³⁶ and manufacturing imperfections as obtained, for example, from computer tomography scans. ⁵⁵ They can also account for the thickening of the struts under large deformations due to a non-zero Poisson’s ratio of the parent material. ⁵⁸ Furthermore, they provide highly resolved information about the stress and strain field within lattice members close to the vertices. ^35,36,54,72 Continuum element models are computationally demanding and consequentially are mainly used in UC models ^{35,38,54,56 –58,62} but, due to increasing computational power, they are getting more frequently employed for studying the mechanical behaviour of archimats of finite size. ^36,56,57,72

Using structural instead of continuum elements leads to a much higher computational efficiency, where beam elements being commonly employed. For low-density archimats, beam models have been shown to provide reliable predictions of the mechanical behaviour involving the overall (non-)linear elastic properties, ^40,42,54,71 the onset of buckling, ^{48,53,60,61,70} the collapse and crushing behaviour, ^{35,42 –44,64} as well as the fracture behaviour. ^47,50,51 However, within beam models, a special treatment of the vertices is required to properly account for the locally increased stiffness due to material aggregation as well as the influence of the vertices on the relative density of the archimat. Concepts how these aspects can be considered are presented, for example, in Luxner et al. ⁵⁴ and Smith et al. ³⁵

Structural and continuum element models can also be combined allowing for a detailed representation of the stress and strain states within a limited number of continuum element modelled base cells, whereas the rest of the archimat is discretized using structural elements. ³⁷

Continuum modelling

If a finite-sized archimat structure consists of a large number of UCs or base cells, it can become computationally expensive to resolve the archimat in its entire complexity. In this case, it is more feasible to represent the archimat by a homogeneous material showing the same effective mechanical response.

If the length scale of the macroscopic problem is far larger than the size of the UC, that is, the principle of separation of scales is satisfied, the effective behaviour of the lattice can be described using a Cauchy continuum. The corresponding effective properties of the archimat can then be derived using classical first-order homogenization schemes ⁵² such as asymptotic homogenization ^55,73 or the periodic microfield approach. ^42,54 A comparison of different homogenization schemes for predicting the effective elastic moduli of various archimats can be found in Arabnejad and Pasini. ⁷³

In common engineering applications of archimats, the size of the structural problem and the UC are often of the same order of magnitude leading to noticeable size effects. ^{74 –77} Issues also arise in situations where highly heterogeneous deformations occur, for example, near sharp corners or crack tips ⁵¹ as well as during the formation of crush bands ^{38,48,55,64,65} as a consequence of microscale buckling. These phenomena are related to the absolute size of the microstructure and, therefore, cannot be captured using classical Cauchy continuum theory. To overcome this issue, generalized continua have been used as they introduce a material length scale into the constitutive relations. Micropolar, ^{78 –82} strain-gradient ^{82 –84} or micromorphic ⁸⁵ continua have been employed to describe the effective behaviour of archimats. For a general review and classification of the various generalized continuum theories, the reader is referred to Noor, ⁸⁶ Fatemi et al., ⁸⁷ Forest and Sievert, ⁸⁸ and references therein. For simple lattice materials, the constitutive behaviour of the generalized continuum models can be derived using concepts based on beam theories ^{78 –82,89} which may even lead to closed-form solutions for the tangent stiffness tensor. ^82,89 For complex microstructures and especially if material non-linearities of the parent material are involved, computational homogenization schemes can be employed. ^{85,90 –92}

Other concepts falling within the scope of continuum modelling of archimats are the quasi-continuum theory proposed in Phlipot and Kochmann ⁹³ or the non-linear constitutive models proposed by Vigliotti et al. ^94,95 Alternatively, multilevel schemes can be used to study localization phenomena originating from sharp edges/cracks or local buckling in heterogeneous materials. ^93,96,97 Within these schemes, the archimat is resolved in its full complexity around the localization zones, whereas far away from these ‘hot spots’, a macroscale continuum model is used. Proper coupling between the microscopic model and the continuum model has to be ensured for an accurate representation of the involved variables at different scales. Adaptive mesh refinement is employed to simulate the propagation of the localization zones through the material. ^93,96,97

The different continuum modelling concepts have been successfully applied for simulating localization phenomena observed, for example, during the indentation of archimats, ^41,45,89,93 around circular holes in archimat plates under tension ⁹⁵ and for other (macroscopically) inhomogeneous deformation states. ⁸² Other examples involve the bending of archimat beams, ^77,82 the propagation of cracks within archimats ⁹³ and studies on the complex deformation behaviour of tetra-chiral archimats. ⁸⁵

For the sake of completeness, substructuring techniques shall be mentioned. These techniques have been used to study buckling in periodic frame-like structures with low memory usage ^98,99 and can also be adopted for example to treat topology optimization problems in archimats. ¹⁰⁰

Design of archimats

Computational methods not only provide further insight into mechanisms observed in experiments but can also be utilized for designing archimats towards target applications. Examples are the design of archimats with desired stiffness properties, ^{101 –103} with a desired auxetic behaviour, ^{56,57,104,105} with tailored buckling mechanisms at the microscale ¹⁰⁶ or with an artificially designed anisotropy. ^107,108 For obtaining the desired effective properties of archimats, either the internal architecture ³ in terms of the orientation ¹⁰² and aspect ratios ^106,109 of the individual structural members can be modified or a combination of different base materials can be utilized. ^29,104,107 Additionally, manufacturing defects can have a strong impact on the mechanical behaviour of archimats ^17,18,55 and consequently have to be considered in the design process.

In many cases, the design of archimats is based on the experience of the designers, and new archimats are, for example, developed by modifying the architecture of already existing microstructures ^{56,104,105,109} in terms of a trial-and-error approach. In this context, the FEM is mainly employed to provide a proof-of-concept for the new designs ^56,57,109 as well as to perform parametric studies ^104,108,109 to further investigate or fine-tune the effective mechanical response of the developed archimat. Although this approach often results in the desired mechanical behaviour, it does not guarantee the design to be the optimal one and it does not allow for systematic development of new archimats. Furthermore, parametric studies can become computationally expensive as the number of parameters strongly increases with the number of lattice members in the UC.

Common topology optimization allows for a more strategic design of archimats towards desired properties, ⁶³ such as extreme bulk or shear moduli, ^103,110 or a maximum buckling strength ^62,100 under design constraints, such as the volume fraction, ^103,110 or symmetries in the material distribution. ⁶² Within optimization schemes, typically the FEM is employed for discretizing the design space and for predicting the effective properties where either continuum or structural elements can be used. ⁶³ A detailed review on the application of topology optimization schemes for the design of archimats is provided in Osanov and Guest. ⁶³ In Pasini and Guest, ¹⁸ the consideration of manufacturing imperfections during topology optimization is discussed.

Employing topology optimization for the design of archimats comes along with some issues. First, designing archimats towards specific effective properties is an inverse homogenization problem ¹⁶ and the design space in terms of possible microstructures to be considered is infinite-dimensional. ^111,112 Second, the obtained optimum design is sensitive to the initial choice of the microstructure. ^103,110 and third, the optimization procedure can become computationally expensive ¹¹² when many iterations are required until the optimum design is found.

Machine learning allows to address most of the issues arising with common topology optimization schemes ^111,113 and has successfully been employed in the design of archimats, ^{101,107,111,112} alloys, ¹¹³ composite materials ^114,115 and frame structures. ^116,117 Machine learning can be employed for property prediction ^107,116,118 as well as for optimizing the microstructure of archimats towards target applications. ^107,111 In the design of archimats, it can be used for topology optimization, ^101,111 for finding the optimum composition of UCs consisting of lattice members with different materials, ¹⁰⁷ for designing archimat families ¹¹² as well as for multiscale system design, ¹¹² for example, to obtain functionally graded structures showing a distinct deformation. Machine learning techniques provide a numerically efficient way in the design of archimats but rely on a sufficiently large data set for training the underlying algorithms. This training data sets consist of different realizations of the arichmat’s microstructure under consideration of the design variables and the information on the respective mechanical response of each realization. The training data set is often constructed by means of finite element simulations ^{101,111,113,114} and consequently requires some numerical effort. However, the number of realizations to be considered within a training data set is by orders of magnitudes smaller than the number of possible realizations ^107,114 of the archimats microstructure. General overviews on the application of machine learning techniques in materials design and discovery can be found, for example, in Agrawal and Choudhary ¹¹⁹ and Liu et al. ¹²⁰

Experimental investigations

Experimental investigations on archimats vary across scales and material type. Usually used for material property characterization, detection of imperfection and investigation of failure propagation, these experimental methods can offer rich data onto which numerical simulations can be informed and calibrated and analytical models verified. Typically archimats, as porous materials, have shown a tendency to reduce in volume progressively under compression while absorbing energy. The aforementioned characteristics have been investigated experimentally for several materials from the UC level (micro) to assembly level (meso) up to studies on the macro level where the performance of archimats in composite configurations as aggregated in Table 1.

Table 1.

Experimental investigations in compression exhibiting buckling and energy absorption across different scales.

	Scale
Type	Micro (UC)	Meso (assembly)	Macro (structural component)
Metal foams closed cell		Uniaxial stress–strain behaviour in constrained and unconstrained conditions 121 –125	Global and local buckling of sandwich plates with metal foam core 126
Metal or metal foam hollow spheres	Mechanical behaviour of single spheres 127,128	Compressive properties of steel foam hollow sphere assemblies 129 –131	Global buckling of sandwich struts with steel foam hollow spheres in the core 132
APM	Mechanical behaviour of single spheres 133	Compression testing of uniform and graded assemblies 134,135
Pyramidal/truss	Mechanical behaviour of single pyramid 136	Mechanical behaviour in compression 11,137 –139	Global buckling of structural elements reinforced with lattices 140
Open-cell lattices		Mechanical behaviour of lattices in compression 9	Buckling and failure in cellular- and lattice-reinforced columns 6,141
Metal foams (open cell)		Buckling and crushing in compression 142,143
Honeycombs and chirals		Mechanical behaviour under compression 49,144,145	Experimental crashworthiness and buckling 146

UC: unit cell; APM: advanced pore morphology.

When an archimat comprises UC patterns in a periodic or random packing setting, such, advanced pore morphology aluminium foam, steel hollow spheres, cuboidal and pyramidal lattices in 3D and chiral, honeycomb and square planar patterns, understanding the mechanical response of the UC is imperative, to create adequate analytical models and numerical simulations. Such UCs can be described by a single sphere or pyramid of a number of struts connected on an apex or even more of these elements in spatial combinations to understand the interactions of the different packing configurations. Considering the load to be compressive, the experimental investigation can be undertaken in a small testing rig under displacement control, measuring load and displacement. Owing to the size of the cells, it might prove impractical to have clip gauge extensometer or strain gauges, but non-contact X-ray tomography or surface digital image correlation (DIC) might be of use. It is expected to see some limited elastic compressive deformation locally followed by elastoplastic buckling of the ligaments or cell walls until the complete loss of strength and crushing of the UC as shown in Figure 2.

Figure 2.

Hypothetical UCs under compressive loading: (a) open-cell buckling as struts buckle in in Euler type buckling and (b) closed UC, indicating cell wall buckling, which includes double curvature and thus more stable. UC: unit cell.

In the case of metal foams, where a single cell or a small number of them are not representative of the structure, meso (or assembly) scale investigation is needed. The same applies to adhesively bonded hollow spheres when the adhesive bonds can play a role in crushing of subsequent cells. According to the standard method for the compression of porous metals, ¹⁴⁷ a mesoscale specimen should have at least 10 cells on average in each dimension. If the packing is regular (i.e. body-centered cubic [BCC], face-centered cubic [FCC] or hexagonal close packed [HCC]), measuring the external deformations using DIC techniques can suffice in capturing the onset of failure and its early propagation. Having said that, as ligaments or cell walls collapse, large deformations will not be captured by the DIC system, losing some of that data, which can be partially recovered from the cross head displacement monitoring. ¹³⁰ For smaller specimens or when there is interest to map the internal structure of a metal foam, X-ray tomography can capture the internal 3D structure during quasi-static loading. ¹²⁴ This can be done by having at least three sources and receivers in three different positions aiming at the specimen in the loading rig which is placed securely in a lead-reinforced room. The other case is having one source and receiver, spinning around the specimen under load.

The stress–strain response of porous archimats under uniaxial compression usually comprises three parts. A quasi-elastic region that is usually linear (stage I) followed by a plateau close to the first peak stress (stage II). The plateau is actually a succession of crush bands forming as deformation progresses in the assembly. According to the standard, ¹⁴⁷ the plateau mean stress can be calculated as the average stress between 20% strain to 30 or 40% strain, depending when stage III begins. Once all the ligaments or cell walls in each crush band have collapsed, the stiffness of the specimen rises steeply as all the pores close and densification takes place. If the material is brittle and exhibits little or no cohesion, the plateau would be much smaller in extent and densification will not occur as the specimen will crumble at reduced load as depicted in Figure 3.

Figure 3.

Typical stress–strain diagram of archimats in compression. Metal foams and metal foam hybrid assemblies will go through all three stages to densification while brittle archimats will not exhibit a plateau. ¹¹

Buckling and postbuckling at a macroscale have been investigated when archimats are used within components of composite structural elements. ^49,146 Using lattices, honeycombs or metal foams as cores in sandwich configurations or filling square tubes and cylinders with archimats, can provide the integrated structure with bending and buckling resistance, improved energy dissipation and crashworthiness. The presence of archimats can absorb strain energy upon compressive or combined loading and affect local buckling modes, by reducing the amplitude of the stress–strain snaking as the buckle pattern changes, thus increasing energy absorption. If archimats are exposed, DIC techniques can offer valuable insight with surface strain maps ⁶ ; otherwise, the effect of archimats can be measured by comparing the strength, stiffness and deformation patterns to a control configuration, such as empty hollow sections of the same dimensions. ¹⁴⁸

Conclusions

In this article, an overview of analytical, numerical and experimental methods for investigating the buckling and postbuckling behaviour of archimats has been presented. Analytical methods can be split into two strategies: linear and geometrically non-linear methods. If the former is pursued, no insight into the postbuckling behaviour will be obtained. If the archimat is, however, to be investigated or even aimed to be manipulated in the postbuckling range, geometrically non-linear modelling is to be pursued. With the aid of geometrical models with few degrees of freedom, an archimats may even be tuned to follow certain desirable structural responses. AM may furthermore be useful to proof the concepts derived using geometrically non-linear models.

Numerical methods can be grouped into discrete and continuum modelling approaches. Within discrete models, the archimat is represented in its entire complexity. These models allow for a detailed investigation of the deformation and stress states at the microscale of the archimat but come with high computational requirements. Consequently, continuum models are employed for simulations of archimats within structural applications. Mainly, generalized continuum theories are used to describe the effective mechanical behaviour of the archimat at the macorscale. These theories introduce a material length scale into the constitutive relations and therefore allow to account, for example, for size effects or the buckling-induced localization of deformations observed for archimats.

In experimental methods, a scale and material type-dependent strategy is required to quantify behaviour from the UC level all the way to structural component level. When the internal structure needs to be identified, X-ray tomography methods can be used during testing; otherwise if deductions can be made from surface strains, then DICs techniques can be of use.