Re-entrant Lattice Research Articles

Micromechanical modeling of 3D printable interpenetrating phase composites with tailorable effective elastic properties including negative Poisson's ratio

3D printable two-phase interpenetrating phase composites (IPCs) are designed by embedding a 3D periodic re-entrant lattice structure (as the reinforcing phase) in a matrix phase. These IPCs display the cubic or tetragonal symmetry. A micromechanical model is developed to evaluate effective elastic properties of the IPCs. Effective Young's moduli, shear moduli and Poisson's ratios (PRs) of each IPC are determined from the effective stiffness and compliance matrices of the composite, which are obtained through a homogenization analysis using a unit cell-based finite element (FE) model incorporating periodic boundary conditions. The FE simulation results are also compared with those based on various analytical bounding techniques in micromechanics, including the Voigt–Reuss, Hashin–Shtrikman, and Tuchinskii bounds. The effective properties of the IPC can be tailored by adjusting five geometrical parameters, including two strut lengths, two re-entrant angles and one strut diameter, and elastic properties of the two constituent materials. The numerical results reveal that IPCs with a negative PR can be generated by using a compliant matrix material and large re-entrant angles. In addition, it is found that the two re-entrant angles can greatly affect other effective elastic properties of the IPC: the effective shear modulus can be enhanced, while the effective Young's modulus can be enhanced or compromised with the increase of the re-entrant angles. Furthermore, it is seen that by adjusting one of the two re-entrant angles or one of the two strut lengths, the material symmetry exhibited by the IPC can be changed from cubic to tetragonal.

Continuum micro-dilatation modeling of auxetic metamaterials

Metamaterials with re-entrant type lattice structure with cubic symmetry are studied. The main new result is that the continuum theory of porous media generalized to the case of media with cubic symmetry can adequately describe the effective properties of a wide class of lattice structures as continuum media. Identification of the micro-dilatational elastic constants for the cellular metamaterials with negative Poisson's ratio is carried out. Methods for identifying the parameters of the micro-dilatation theory of media through the structural and rigidity parameters of lattice structures are developed. It is shown that material constants of the micro-dilatation theory can be directly linked with the geometrical and elastic parameters of the unit cell of the metamaterial. In particular, “void stiffness” modulus and voids diffusion parameter depend linearly on the stiffness of re-entrant structures in the unit cells and coupling modulus depends significantly only on the geometry of the unit cell. This result is obtained using micro-dilatational analytical solution of the homogeneous elasticity problem and FE modeling of the uniaxial tension, hydrostatic compression and simple shear of the beam lattice metamaterial models. The problem of non-homogeneous uniaxial deformation of the rod and the problem of the pure bending of the beam were used to determine the voids diffusion parameter. It is shown, in particular, that the coupled and scale effects are equivalent for the studied continuum and lattice models of the metamaterials in the considered range of parameters.

Computation of the homogenized nonlinear elastic response of 2D and 3D auxetic structures based on micropolar continuum models

We develop mechanical and numerical homogenization models for the large strains response of auxetic structures in order to derive their effective elastic response accounting for large changes of the geometry. We construct a strain driven nonlinear scheme for the computation of the stress-strain relation of these repetitive networks over a reference representative unit cell (abbreviated as RUC). Homogenization schemes are developed to evaluate the effective nonlinear mechanical response of periodic networks prone to an auxetic behavior, in both planar and 3D configurations, which are substituted by an effective micropolar continuum at the intermediate mesoscopic level. The couple stress part of the homogenized constitutive law takes into account the impact of the local rotations at the mesoscopic level and allows computing the bending response. This methodology is applied to four planar auxetic periodic lattices (the re-entrant hexagonal honeycomb and alternative honeycomb topologies, including the arrowhead, Milton and hexachiral structures), and to the 3D re-entrant and pyramid-shaped lattices. We have considering successively the in-plane and out-of-plane responses of these structures. The transition to an auxetic behavior is shown to be triggered by the imposed strain over the unit cell boundary. The computed evolutions of Poisson’s ratio versus the imposed stretch traduce an enhanced auxetic response as the stretch is increased. A satisfactory agreement is obtained between the homogenized stress-strain responses and the responses computed numerically by finite element simulations performed over a repeating unit cell.

Metamaterials with negative Poisson’s ratio and non-positive thermal expansion

Four metallic metamaterials with tailorable mechanical properties are designed using bi-material star-shaped re-entrant planar lattice structures, which do not involve pins, adhesive, welding or pressure-fit joints and can be fabricated through laser-based additive manufacturing. Three length parameters, one angle parameter and three material combinations are used as adjustable design parameters to explore structure-property relations. For each of the four designed metamaterials, the effects of the design parameters on the Poisson’s ratio (PR), coefficient of thermal expansion (CTE), Young’s modulus and relative density are systematically investigated using unit cell-based finite element simulations that incorporate periodic boundary conditions. It is found that the bi-material lattice structures can be tailored to obtain 3-D printable metallic metamaterials with positive, near-zero or negative PR and CTE together with an uncompromised Young’s modulus. In particular, it is shown that metamaterial # 1 can exhibit both a negative PR and a non-positive CTE simultaneously. These metallic metamaterials can find applications in structures or devices such as antennas and precision instruments to reduce thermomechanical stresses and extend service lives.

Design of cut unit geometry in hierarchical kirigami-based auxetic metamaterials for high stretchability and compressibility

We studied the mechanical response of a recently developed new class of mechanical metamaterials based on the paper art of cutting, kirigami. Specially, the geometrical and structural design of representative cut units, via combined line cut, cut-out, and hierarchy of the structure, was explored for achieving both extreme stretchability and/or compressibility in kirigami metamaterials through experiments, alongside geometrical modeling and finite element simulations. The kirigami design was tested on constituent materials including non-stretchable copy papers and highly stretchable silicone rubber to explore the role of constituent material properties. The cut unit in the shape of solid rectangles with the square shape as a special case was demonstrated for achieving the extreme stretchability via rigid rotation of cut units. We found that compared to the square cut units, the theoretically predicted maximum stretchability via unit rotation in rectangle units (aspect ratio 2:1) increased dramatically from about 41% to 124% for the level 1 cut structure without hierarchy, and from about 62% to 156% for the level 2 hierarchical cut structure, which was validated by both experiments and simulations. To demonstrate the achievement of both extreme stretchability and compressibility, we replaced the solid square cut units with porous squares and re-entrant lattice shapes in silicone rubber based metamaterials. We found that a porous structure can enable an extreme compressibility of as high as 81%.

Analysis of dispersive waves in repetitive lattices based on homogenized second-gradient continuum models

We analyze the dispersion of elastic waves in periodic beam networks based on second order gradient models obtained by the homogenization of the initially discrete network. The lattice beams have a viscoelastic behavior described by Kelvin–Voigt model and the homogenized second gradient viscoelasticity model reflects both the lattice topology, anisotropy and microstructural features in terms of its geometrical and micromechanical parameters. The continuum models enriched with the higher-order gradients of the displacement and velocity introduce characteristic lengths parameters which account for microstructural effects at the mesoscopic level. A comparative study of the dispersion relations and damping ratio evolutions for the longitudinal and shear waves has been done for four lattices (the chiral diamond lattice, the classical and reentrant lattices, and the pantograph). The developed model allows analyzing both the effects of damping and internal length scale through the second order displacement gradients on the wave propagation characteristics. An important increase of the natural frequency due to second order effects is observed. For the pantograph lattice, the phase velocity for the longitudinal and shear modes is identical and is not influenced by the direction of wave propagation. The obtained results show overall that the pantograph lattice present the best acoustic characteristics.

Energy absorption in lattice structures in dynamics: Experiments

Lattice structures offer the potential to relatively easily engineer specific (meso-scale properties (cell level)), to produce desirable macro-scale material properties for a wide variety of engineering applications including wave filters, blast and impact protection systems, thermal insulation, structural aircraft and vehicle components, and body implants. The work presented here focuses on characterising the quasi-static and, in particular, the dynamic load-deformation behaviour of lattice samples. First, cubic, diamond and re-entrant cube lattice structures were tested under quasi-static conditions to investigate failure process and stress–strain response of such materials. Following the quasi-static tests, Hopkinson pressure bar (HPB) tests were carried out to evaluate the impact response of these materials under high deformation rates. The HPB tests show that the lattice structures are able to spread impact loading in time and to reduce the peak impact stress. A significant rate dependency of load-deformation characteristics was identified. This is believed to be the first published results of experimental load-deformation studies of additively manufactured lattice structures. The cubic and diamond lattices are, by a small margin, the most effective of those lattices investigated to achieve this.

Homogenization of 1D and 2D magnetoelastic lattices

This paper investigates the equivalent in-plane mechanical properties of one dimensional (1D) and two dimensional (2D), periodic magneto-elastic lattices. A lumped parameter model describes the lattices using magnetic dipole moments in combination with axial and torsional springs. The homogenization procedure is applied to systems linearized about stable configurations, which are identified by minimizing potential energy. Simple algebraic expressions are derived for the properties of 1D structures. Results for 1D lattices show that a variety of stiffness changes are possible through reconfiguration, and that magnetization can either stiffen or soften a structure. Results for 2D hexagonal and re-entrant lattices show that both reconfigurations and magnetization have drastic effects on the mechanical properties of lattice structures. Lattices can be stiffened or softened and the Poisson’s ratio can be tuned. Furthermore for certain hexagonal lattices the sign of Poisson’s ratio can change by varying the lattice magnetization. In some cases presented, analytical and numerically estimated equivalent properties are validated through numerical simulations that also illustrate the unique characteristics of the investigated configurations.

Dynamics of geometrically reconfigurable one dimensional and two dimensional magneto-elastic metamaterials

Periodic structures are presented as metamaterials that exhibit multistability due to the nonlinearities of magneto-elastic interactions and structure geometry. The multistability of these structures affords them the ability to adapt their properties though geometric reconfiguration, bringing about changes in stiffness and Poisson's ratio, and introducing anisotropy. These changes in structural properties cause drastic changes in wave propagation, which is of interest for wave control. The dynamic transformation of one-dimensional (1D) and two-dimensional (2D) lattices between stable states are studied through nonlinear numerical simulations. The analysis is conducted using a lumped mass system of magnetic particles. The structures studied include hexagonal, re-entrant, and kagome lattices. Changes in plane wave propagation properties are predicted by applying Bloch theorem to lattice unit cells with linearized interactions. Results from Bloch analysis are then verified through direct numerical simulations. The propagation of plane waves in these lattices before and after topological changes is compared, and large differences are evident.

Modeling of uniaxial compression in a 3D periodic re-entrant lattice structure

In this study, the behavior of a parametric 3D re-entrant dodecahedron lattice structure with negative Poisson’s ratio was studied. Four geometrical configurations for the re-entrant dodecahedron were designed, and the relationship between the mechanical properties and the design parameters was determined through beam theory. Samples were fabricated successfully via electron beam melting. Compressive tests as well as finite element analysis (FEA) were performed, and the results were compared with theoretical predictions. The modeling yielded explicit analytical equations of various mechanical properties including Poisson’s ratios, modulus and strength, and the compressive strength and the modulus from the prediction match well with the experiments, as well as the FEA results. The methodology used by this study also demonstrated a feasible approach to design 3D auxetic cellular structure for various applications.

Compressive properties of Ti–6Al–4V auxetic mesh structures made by electron beam melting

In this current work, a Ti–6Al–4V 3-D re-entrant lattice auxetic structure is manufactured by the electron beam melting (EBM) process. Four different design configurations (two negative Poisson’s ratio values×two relative densities) were manufactured and tested under compressive stress. Two failure modes were observed whose occurrence appeared to be dependent on the ratio of vertical strut length to re-entrant strut length. A small deflection analytical model is presented that predicts yield strength and modulus for one type of design with good accuracy. Results also show that the re-entrant lattice structure possesses superior mechanical properties compared to regular foam structures. Limitations of the analytical model are also discussed.

Composites with auxetic inclusions showing both an auxetic behavior and enhancement of their mechanical properties

Composite materials made of auxetic inclusions and giving rise overall to negative Poisson’s ratio are considered, adopting a two-steps micromechanical approach for the calculation of their effective mechanical properties. The inclusions consist of periodic beam lattices, whose equivalent mechanical properties are calculated by a discrete homogenization scheme in a first step. The hexachiral and hexagonal reentrant lattices are considered as representative of the two main deformation mechanisms responsible for auxeticity. In a second step, the equivalent properties of the composite are calculated from numerical homogenization using the finite element method. It is shown that both an auxetic behavior and enhanced moduli can be obtained for not too slender micro-beams.

Equivalent mechanical properties of biological membranes from lattice homogenization

The goal of this manuscript is to set up a novel methodology for the calculation of the effective mechanical properties of biological membranes viewed as repetitive networks of elastic filaments, based on the discrete asymptotic homogenization method. We will show that for some lattice configurations, flexional effects due to internal structure mechanisms at the unit cell scale lead to additional flexional effects at the continuum scale, accounted for by an internal length associated to a micropolar behavior. Thereby, a systematic methodology is established, allowing the prediction of the overall mechanical properties of biological membranes for a given network topology, as closed form expressions of the geometrical and mechanical micro-parameters. The peptidoglycan and the erythrocyte have been analyzed using this methodology, and their effective moduli are calculated and recorded versus the geometrical and mechanical lattice parameters. A classification of lattices with respect to the choice of the equivalent continuum model is proposed: The Cauchy continuum and a micropolar continuum are adopted as two possible effective medium, for a given beam model. The relative ratio of the characteristic length of the micropolar continuum to the unit cell size determines the relevant choice of the equivalent medium. In most cases, the Cauchy continuum is sufficient to model membranes in most of their configurations. The peptidoglycan network may exhibit a re-entrant hexagonal lattice, for which micropolar effects become important. This is attested by the characteristic length becoming larger than the beam length for such configurations. The homogenized moduli give accurate results for both membranes, as revealed by comparison with experimental measurements or simulation results from the literature at the network scale. A first insight into the nonlinear mechanical behavior of the hexagonal and triangular networks is lastly investigated using a perturbative method.

Direct measurements of the correlation between reentrant ferromagnetism and lattice expansion in FeCuZr alloys

Amorphous metastable alloy of nominal composition [Fe_(0.5)Cu_(0.5)]_(87)Zr_(13) has been synthesized by high-energy ball milling. The alloy exhibit a ferromagnetic behavior with a Curie Temperature of T_(C)=255 K, as determined from low-field measurements whereas no transition to a paramagnetic state is observed under high- enough applied magnetic fields. The evolution of hysteresis loops with temperature as well as thermoremanence measurements indicate an anomalous magnetic behavior characterized by a spontaneous increase in the magnetization values as well as by a magnetic hardening when the temperature is increased above T_(C). These effects are strongly correlated with a dilation of the Fe-Fe nearest-neighbor distances, as determined from extended x-ray absorption fine structure (EXAFS) studies. EXAFS results indicate an almost negligible thermal expansion at temperatures below T_(C) while normal thermal expansion takes place at higher temperatures. Such expansion seems to promote a reinforcement of the ferromagnetic interactions among Fe-Fe atoms that would account for the observed spontaneous increase in the magnetization as well as for the evolution of the coercive field.

Instability of Square Vortex Lattice ind-Wave Superconductors is due to Paramagnetic Depairing

The effects of paramagnetic depairing on structural transitions between vortex lattices of a quasi-two-dimensional d-wave superconductor are examined. We find that, when the Maki parameter alphaM is of order unity, a square lattice induced by a d-wave pairing is destabilized with increasing fields, and that a reentrant rhombic lattice occurs in higher fields. Further, a weak Fermi surface anisotropy competitive with the pairing symmetry induces another structural transition near Hc2. These results are consistent with the structure changes of the vortex lattice in CeCoIn5 in H parallel c determined from recent neutron scattering data.

Homogenization and equivalent in-plane properties of two-dimensional periodic lattices

The equivalent in-plane properties for hexagonal and re-entrant (auxetic) lattices are investigated through the analysis of partial differential equations associated with their homogenized continuum models. The adopted homogenization technique interprets the discrete lattice equations according to a finite differences formalism, and it is applied in conjunction with the finite element description of the lattice unit cell. It therefore allows handling of structures with different levels of complexity and internal geometry within a general and compact framework, which can be easily implemented. The estimation of the mechanical properties is carried out through a comparison between the derived differential equations and appropriate elasticity models. Equivalent Young’s moduli, Poisson’s ratios and relative density are estimated and compared with analytical formulae available in the literature. In-plane wave propagation characteristics of honeycombs are also investigated to evaluate phase velocity variation in terms of frequency and direction of propagation. Comparisons are performed with the values obtained through the application of Bloch theorem for two-dimensional periodic structures, to show the accuracy of the technique and highlight limitations introduced by the long wavelength approximation associated with the homogenization technique.

Analysis of in-plane wave propagation in hexagonal and re-entrant lattices

Plane wave propagation in hexagonal and re-entrant lattices is investigated through the application of Bloch's theorem. Two-dimensional dispersion relations are estimated and analyzed to illustrate peculiar properties of re-entrant configurations and investigate the directional behavior of hexagonal lattices for varying construction angles. Special attention is devoted to the determination of phase and group wave velocities. Velocity plots are obtained for varying frequency and direction of wave propagation to analyze the dispersion characteristics of the lattices and further investigate their directional characteristics.