Analytical approaches

A number of important mechanical properties of architected materials with relatively simple topologies can be treated by analytical approaches. If a periodic truss-based lattice material (e.g., the octet lattice^{Reference Deshpande, Fleck and Ashby10}) can be approximated as a network of structural elements (trusses or beams) connected by ideal nodes, then its effective elastic stiffness along different loading directions can be calculated by solving mechanical equilibrium and compatibility equations; similarly, its effective strength can be estimated by relating the macroscale external stress applied to the lattice to the loads carried by each strut, and adopting a yield, fracture, or elastic buckling failure condition at the strut level. This technique has been successfully applied to a number of periodic lattice topologies^{Reference Deshpande, Fleck and Ashby10–Reference Zok, Rathbun, Wei and Evans14} as well as to stochastic foams;^{Reference Gibson and Ashby11} in the latter case, analytical solutions provide the scaling of stiffness and strength with relative density, but absolute values require the fitting of nondimensional constants of order one to experiments or numerical simulations.

More recently, analytical techniques have been extended to plate-based closed-cell periodic architected materials to derive necessary conditions that guarantee elastic isotropy.^{Reference Tancogne-Dejean, Diamantopoulou, Gorji, Bonatti and Mohr15} Furthermore, analytical techniques have been used to extract the effective initial fracture toughness of truss-based cellular materials and foams, and to elucidate the dependence of toughness on relative density, fracture strength of individual struts, and unit-cell size. Also in this case, nondimensional constants of order one are fitted to experiments or simulations.^{Reference Fleck, Deshpande and Ashby16} More complex material behavior can be described by methods of homogenization to extract the linear, nonlinear, or dynamic metamaterial response from averaging over a representative unit cell—this includes both the determination of effective material behavior^{Reference Deshpande, Fleck and Ashby10,Reference Triantafyllidis and Bardenhagen17–Reference Kumar and McDowell20} (such as the stress–strain response) as well as multiscale simulations in which the constitutive behavior on the macroscale is computed on the fly by solving a nested lower-scale problem at the unit cell level. The latter approach can also describe localization and size-dependent phenomena,^{Reference Glaesener, Lestringant, Telgen and Kochmann21} but may require semi-analytical or numerical solution schemes.

Multiple studies using the previously mentioned techniques have unveiled key features of lattice materials:^{Reference Fleck, Deshpande and Ashby16} (1) a sufficient coordination number (number of bars meeting at a node) must exist for the lattice to carry loads primarily by tension or compression of its members (stretching-dominated lattice). When this condition is satisfied, the lattice is mechanically efficient, with stiffness and strength scaling linearly with the relative density; a key example is the octet lattice. Conversely, when this condition is not satisfied, the lattice carries load by bending of its members (bending-dominated lattice), and its stiffness and strength scale with higher powers of relative density (2 and 3/2, respectively). (2) Most stochastic foams are bending-dominated, and hence mechanically inefficient compared to stretching-dominated periodic lattices. (3) The fracture toughness of truss-based lattice materials and foams scales with the square root of the unit-cell size and a topology-dependent power of the relative density, ranging between 1/2 and 2 (the scaling power is 1 for the octet lattice).

Analytical predictions for stiffness, strength, and fracture toughness have been verified against experiments and numerical simulations and have proven to be remarkably accurate for lattices with solid cross sections at relatively low relative densities.^{Reference Deshpande, Fleck and Ashby10,Reference Gibson and Ashby11,Reference Zok, Waltner, Wei, Rathbun, McMeeking and Evans13,Reference Zok, Rathbun, Wei and Evans14,Reference Valdevit, Wei, Mercer, Zok and Evans22–Reference O’Masta, Dong, St.-Pierre, Wadley and Deshpande24} As the relative density is increased and the thickness-to-diameter ratio of hollow truss lattices is decreased, the aspect ratio of the members (length-to-diameter ratio) is reduced, ultimately violating the slender-strut assumptions underlying such calculations.^{Reference Valdevit, Godfrey, Schaedler, Jacobsen and Carter25,Reference Meza, Phlipot, Portela, Maggi, Montemayor, Comella, Kochmann and Greer26} To address these challenges, numerical techniques are required. The same applies to plate- and shell-based architectures, which, in principle, follow similar scaling laws; yet, their structural response cannot easily be described by analytical approaches, so that numerical techniques have become the primary method of choice.

Finite element analysis of the unit-cell response

Numerical techniques such as the finite element (FE) method are popular approaches in situations where the effective properties can be extracted from a periodic unit cell, but are not within analytical reach. The first step in FE analysis is mesh generation. Unit cells of architected materials have been meshed with truss, beam, shell, and solid elements, depending on the topology and geometry of the unit cell and the desired compromise between computational speed and accuracy. Except for trivial topologies, the unit cell is generated by computer-aided design (CAD) and subsequently meshed in an FE package or via a dedicated mesh generator.^{Reference Zok, Waltner, Wei, Rathbun, McMeeking and Evans13} For situations in which commercial CAD packages are unable to accurately resolve important geometrical features (e.g., the shape of a filleted node in a hollow lattice with stout members), custom-built parametric geometry generators have been developed, which offer the additional benefit of seamless integration with geometry optimizers (albeit at the expense of generality).^{Reference Valdevit, Godfrey, Schaedler, Jacobsen and Carter25}

Recently, stochastic shell-based architected materials with spinodal topologies were shown to possess excellent mechanical properties while being, in principle, amenable to self-assembly by a variety of processing routes. For such topologies, periodic unit cells were generated by solving the Cahn–Hilliard equation for spinodal phase separation for given average phase volume fractions. Interfaces between the resulting spinodal phases were extracted by level-set or image-processing techniques. This approach has generated both isotropic^{Reference Hsieh, Endo, Zhang, Bauer and Valdevit27} and anisotropic^{Reference Vidyasagar, Krödel and Kochmann28} topologies, by suitable choice of a potential energy landscape. Spinodal topologies are characterized by a uniform negative Gaussian curvature, which enables a more efficient distribution of local stresses upon external loading than for lattice-based materials (Figure 1a).

Figure 1.

The mechanical performance of spinodal shell-based architected materials. (a) Comparison of the distribution of the local von Mises stress (${{\rm{\sigma }}_{VM}}$) normalized by the limit stress for the materials (${{\rm{\sigma }}_{limit}}$), in a hollow microlattice unit cell and a spinodal shell topology of the same relative density, both loaded under the same macroscopic uniaxial compressive strain. Notice that the uniform negative Gaussian curvature of the spinodal shell results in a more uniform stress distribution and superior mechanical efficiency. (b) Comparison of elastoplastic stress–strain response obtained by uniaxial compression experiments and finite element (FE) modeling. The samples of spinodal shell architectures at three relative densities, displayed in the insets, are produced by two-photon polymerization direct laser writing at the microscale in an acrylate resin and tested with an Alemnis nanoindenter inside a scanning electron microscope.^{Reference Hsieh, Endo, Zhang, Bauer and Valdevit27}

When long-range effects such as buckling and bifurcation are of no concern, properties of periodic architected materials can be extracted from a single unit cell subject to, for example, periodic boundary conditions. These conditions probe the effective response when all unit cells deform in the same manner, hence approximating the response of a periodic infinite architected material undergoing homogeneous deformation. For example, the stiffness tensor can be extracted via numerical homogenization, a procedure relying on an asymptotic expansion of the governing equations, by computing the linearized incremental stress–strain response.^{Reference Andreassen and Andreasen29,Reference Torquato30} Application of this approach to truss lattices has demonstrated how classical beam theory indeed applies only in the limit of slender structures, whereas stubby beams and, in particular, hollow-strut architectures require the described numerical treatment to extract the effective mechanics as a function of topology and relative density.^{Reference Meza, Phlipot, Portela, Maggi, Montemayor, Comella, Kochmann and Greer26,Reference Portela, Greer and Kochmann31} Recently, similar techniques have been applied to the stiffness tensor of closed-cell plate-based architected materials, which demonstrated that plate-based materials with cubic symmetry exist that are both elastically isotropic and optimally stiff (i.e., they approach the well-known theoretical Hashin–Shtrikman upper bound^{Reference Berger, Wadley and McMeeking32}).

Numerical homogenization has been applied to other linear functional properties (including thermal and electrical conductivity,^{Reference Andreassen and Andreasen29} thermal expansion,^{Reference Sigmund and Torquato33} and fluid permeability^{Reference Guest and Prévost34}) and also to nonlinear behavior (e.g., the large-deformation response of auxetic architectures [i.e., architectures with a negative Poisson effect]^{Reference Kochmann and Venturini35}).

Unit-cell strength calculations are more challenging, and their accuracy and reliability depend on the chosen definition of strength (e.g., as the onset of yielding, plastic collapse, fracture, or elastic buckling). Moreover, unit-cell results only apply here if the periodicity assumption is adequate (i.e., if localized, aperiodic behavior can be excluded). The onset of yielding is estimated by applying a macroscopic strain to the unit cell along the desired loading direction and extracting the macroscopic stress at which the maximum local stress within the unit cell exceeds the yield strength of the base material (modeled as linear elastic).^{Reference Valdevit, Godfrey, Schaedler, Jacobsen and Carter25} The onset of elastic buckling is generally obtained by a Bloch-type eigenvalue analysis of a single cell,^{Reference Valdevit, Godfrey, Schaedler, Jacobsen and Carter25,Reference Shan, Kang, Wang, Qu, Shian, Chen and Bertoldi36} with the constituent material again modeled as linear elastic.

When significant geometric nonlinearities are expected, bifurcation can be triggered by precomputing the effective elastic buckling strength and the associated lowest buckling modes (e.g., from preloading the unit cell up to buckling, followed by an eigenvalue analysis). Once extracted, a linear combination of the first modes can be used to introduce a small geometric imperfection, so that a subsequent simulation of the unit cell until collapse via a nonlinear quasistatic or Riks analysis provides a measure of the elastic/plastic buckling strength of the architected material.^{Reference Valdevit, Hutchinson and Evans37} Post-buckling simulations are generally able to capture the entire stress–strain response of the material quite reliably, even for complex topologies (Figure 1b). Eigenvalue extractions on unit cells will only capture buckling modes with wavelengths smaller than the unit-cell size. While often these modes dominate the strength, this is not universally true, and larger numbers of unit cells may be required for accurate extraction of the effective buckling strength.^{Reference Vigliotti and Pasini38,Reference Vigliotti and Pasini39} An alternative technique for elastic buckling strength calculations is based on the Bloch–Floquet theory, which searches for unstable modes of, in principle, arbitrary wavelength.^{Reference Geymonat, Müller and Triantafyllidis40–Reference Hutchinson and Fleck42} This powerful technique allows for an accurate estimation of the lowest buckling modes (and associated strength) for wavelengths longer than the unit-cell size.

The same Bloch wave ansatz is also widely used to predict the response of architected materials to mechanical and acoustic waves,^{Reference Hussein, Leamy and Ruzzene43–Reference Salari-Sharif, Haghpanah, Guell Izard, Tootkaboni and Valdevit45} enabling the modeling and design of metamaterials with tailored bandgaps for controlled wave attenuation, wave guiding and redirection, or signal processing. Applicable to, in principle, any periodic architecture, the entire space of wave vectors is sampled to extract the wave dispersion relations of the metamaterial, uniquely linking excitation frequencies to wavelengths and directions, and revealing frequencies that do not propagate waves in specific directions (partial bandgaps) or in all directions (full bandgaps). Further computing the group velocity from the dispersion surfaces reveals the directions of energy flow in the medium,^{Reference Zelhofer and Kochmann46} thus effectively predicting the directions in which vibrational energy will flow—making the approach enabling tailored wave guiding (or wave beaming).

Over the last decade, instabilities have been exploited to create beneficial effective properties.^{Reference Kochmann and Bertoldi47} For example, elastic snap-through (i.e., the sudden change of configuration of structural elements as a result of an elastic instability) within carefully designed unit cells upon macroscopic loading has been used as a mechanism to dissipate kinetic energy via the damping of high-frequency vibrations of unit-cell features based on the constituent material’s intrinsic damping. When appropriately designed, this mechanism enormously amplifies the intrinsic damping of the base material and can be used to generate single- or multi-stable periodic architected materials with nearly linear elastic constituents, which exhibit exceptional combinations of strength, stiffness and energy dissipation, energy trapping or force isolation.^{Reference Kashdan, Seepersad, Haberman and Wilson48–Reference Izard, Alfonso, McKnight and Valdevit57} Figure 2 provides an example of this mechanism. Figure 2a shows a laser-cut two-dimensional (2D) periodic architected material, where each unit cell contains rigid elements connected by thin flexible hinges. The analytical prediction of the mechanical response of a single hinge under compression is shown in Figure 2b, which shows the snap-through behavior. The behavior of the architected material consisting of multiple cells is analytically predicted in Figure 2c, as a function of the number of cells. Finally, the experimental response of a sample with seven unit cells is compared to the analytical prediction in Figure 2d. The ability to predict the stress–strain curve of the unit cell during the snap-through (and hence through a regime of negative stiffness) is essential for modeling and designing such architected materials and can be obtained by post-buckling FE analysis at the unit-cell level or by beam theory (Figure 2b–d).^{Reference Shan, Kang, Raney, Wang, Fang, Candido, Lewis and Bertoldi52,Reference Bertoldi, Vitelli, Christensen and van Hecke53,Reference Haghpanah, Salari-Sharif, Pourrajab, Hopkins and Valdevit56}

Figure 2.

Multistable architected material exploiting negative-stiffness elements. (a) Implementation of the topology. (b) Calculated load-displacement response of a single hinge under displacement control as a function of the geometric parameters of the hinge, showing the region of negative stiffness. The directions of load, F, and displacement, Δ, and the boundary conditions are shown in the inset. Notice that analytical and numerical models agree nearly perfectly. (c) Calculated stress–strain response of the material shown in (a), for a single unit cell and for 200 unit cells in series (approaching the continuum response). Loading direction and geometrical parameters are defined in the inset. (d) Experimental stress–strain response of the material shown in (a), with seven unit cells in series. Notice the significant amount of energy dissipation and hysteresis enabled by negative-stiffness mechanisms.^{Reference Haghpanah, Salari-Sharif, Pourrajab, Hopkins and Valdevit56} Note: EI, flexural stiffness of the hinge; E, elastic modulus; w, width; b, out-of-plane thickness.

FE analysis of multicell architected materials

The limits of unit-cell analysis become evident in fracture toughness calculations; the localized presence of a discrete crack in the material violates the assumption of a separation of scales (and hence of the assumed periodic deformation). For the definition of fracture toughness to hold, the sample must be much larger than the crack, which in turn, needs to be much larger than the unit-cell size, making fracture toughness computations for lattices remarkably expensive.

Two key approaches have been successfully used: (1) For 2D brittle lattice materials, a relatively small number of unit cells around the crack tip is meshed with beam elements. The sample is subjected to displacement boundary conditions consistent with the K-field solution for a continuum material with the same effective elastic properties as the lattice material, and the largest stress in a strut in the lattice is extracted. The fracture toughness is calculated as the ratio of the constituent material’s tensile strength to the largest stress in the lattice, multiplied by the applied value of the stress intensity factor.^{Reference Fleck, Deshpande and Ashby16,Reference Quintana-Alonso, Fleck, Daniel, Gdoutos and Rajapakse58–Reference Fleck and Qiu60} (2) A single edge notch bend (SENB) specimen with dimensions consistent with ASTM standards is meshed with a lattice material, generally using beam elements. The sample is loaded in three-point bending, and the load-crack extension curve is extracted. Standard formulas are employed to compute the initial fracture toughness and the resistance (R) curve of the lattice material. This approach has been used for 2D and three-dimensional (3D) lattices, including the octet lattice.^{Reference O’Masta, Dong, St.-Pierre, Wadley and Deshpande24} Both techniques have been used to validate analytical estimates and calculate the values of topology-specific nondimensional constants that are included in, but not defined by, the analytical models.

Multiscale techniques

Where the effective behavior cannot be extracted from a unit cell, multiscale techniques become the method of choice, which aim toward the accuracy of unit-cell resolution while providing the efficiency of macroscale calculations. One such approach is the quasicontinuum (QC) method, originally introduced for coarse-graining atomistic lattice calculations^{Reference Tadmor, Ortiz and Phillips61} and later extended to truss lattices.^{Reference Beex, Peerlings and Geers62–Reference Phlipot and Kochmann65} Truss-based QC models the full discrete lattice, but introduces kinematic constraints in regions requiring lower resolution (thus retaining the accuracy of individual structural members where needed, while coarsening the description away from those regions into an efficient continuum manner—with both regions being adaptively updated as needed). A prime application is the aforementioned fracture toughness predictions of truss lattices, for which discrete truss-level resolution is required near the crack tip, while large samples are required to apply loading conditions remotely. Here, full resolution is retained at the crack tip (and evolved adaptively based on regions of highest stresses), while an efficient coarsened description is used in the far field, thus enabling simulations of several orders-of-magnitude larger samples than achievable by discrete structural FE calculations. Figure 3c–e shows illustrative examples^{Reference Phlipot and Kochmann65} of coarse-grained fracture specimens using the truss QC method with and without adaptive mesh refinement to produce full resolution in regions of interest.

Figure 3.

Examples of multiscale modeling approaches for truss metamaterials. Replacing a discrete truss network (a) by a continuum modeled efficiently by finite elements^{Reference Glaesener, Lestringant, Telgen and Kochmann21} (b) considerably reduces computational costs. The quasicontinuum (QC) method^{Reference Phlipot and Kochmann65} (c) retains full discrete resolution near a crack tip while coarsening away from the notch; mesh adaptivity within the QC method refines locally where stress concentrations are expected in 2D truss lattices made of kagome (d) and triangular (e) unit cells (two sets of graphics visualize the progression of refinement under static loading).

An alternative is to compute the effective local material behavior from a unit cell on the fly and to feed such information into a macroscale FE simulation.^{Reference Vigliotti and Pasini38,Reference Vigliotti, Deshpande and Pasini66} Owing to the computational expenses of such homogenization-based approaches (incurred by the nested large- and small-scale simulations), their use has been restricted to relatively simple geometries and problems (see Figure 3a–b for an example of a 3D truss replaced by an efficient FE model).

Emergence of size effects

Recent progress in advanced manufacturing has enabled fabrication of architected materials with unit cells at the micron scale and sub-cell geometric features (e.g., truss diameters) at the nanoscale.^{Reference Bauer, Meza, Schaedler, Schwaiger, Zheng and Valdevit67} When made out of metallic and ceramic materials at this scale, interesting metamaterial size effects may emerge. For example, polycrystalline metals with nanoscale grain size can be readily processed, resulting in increased yield strength (Hall–Petch Law) relative to the bulk.^{Reference Torrents, Schaedler, Jacobsen, Carter and Valdevit68–Reference Vyatskikh, Delalande, Kudo, Zhang, Portela and Greer70} Similarly, virtually flaw-free ceramic materials approaching the theoretical strength can be produced at those small scales.^{Reference Meza, Zelhofer, Clarke, Mateos, Kochmann and Greer4,Reference Zhang, Vyatskikh, Gao, Greer and Li71–Reference Bauer, Schroer, Schwaiger, Tesari, Lange, Valdevit and Kraft74} Architecting cellular metamaterials with such exceptionally strong nanoscale constituents allows designers to capitalize on these size effects at a large scale (the scale of the architected material). Moreover, interesting new effects may emerge from architectures in which the feature sizes (e.g., strut diameter or length in trusses) are on the same order as the material features (e.g., grain size).

Our ability to predict the processing-specific microstructural evolution of the constituent material and model these size effects has remained highly limited. Likewise, predicting the effective response when material-level and structural-level phenomena cannot be treated separately anymore is a key open challenge: FE analysis (popular for structure-level simulations) does not capture the small-scale material response and microstructure, while atomistic techniques (capable of accurately describing material microstructure and nanoscale mechanisms) cannot be applied to micron-sized unit cells due to computational costs. The integration of modeling techniques that enable the co-design of material and architecture, by combining tools of Integrated Computational Materials Engineering (ICME) with continuum-level modeling and topology optimization, remains a promising, but widely untapped opportunity.