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Additive manufacturing (AM) is enabling the fabrication of materials with engineered lattice structures at the micron scale. These mesoscopic structures fall between the length scale associated with the organization of atoms and the scale at which macroscopic structures are constructed. Dynamic compression experiments were performed to study the emergence of behavior owing to the lattice periodicity in AM materials on length scales that approach a single unit cell. For the lattice structures, both bend and stretch dominated, elastic deflection of the structure was observed ahead of the compaction of the lattice, while no elastic deformation was observed to precede the compaction in a stochastic, random structure. The material showed lattice characteristics in the elastic response of the material, while the compaction was consistent with a model for compression of porous media. The experimental observations made on arrays of 4 × 4 × 6 lattice unit cells show excellent agreement with elastic wave velocity calculations for an infinite periodic lattice, as determined by Bloch wave analysis, and finite element simulations.

In a previous study (Barham et al 2007 Acta Mech. 191 1–19), the finite deformation of a circular magnetoelastic membrane in an axisymmetric dipole field was calculated by specializing the equations of three-dimensional magnetoelastic equilibrium. The predicted response was found to be similar to the classical limit-point instability occurring in analogous purely mechanical problems. A limit-point instability occurs under conditions corresponding to the incipient non-existence of equilibria. Under such conditions the body is necessarily on the verge of a dynamical state. In the present setting, this corresponds to the occurrence of a maximum in the equilibrium deflection of the membrane with respect to applied field strength and proximity of the field source. The earlier conjecture of a limit-point instability, advanced in Barham et al (2007 Acta Mech. 191 1–19), is confirmed in the present work by using a variational method based on an adaptation of the energy criterion of elastic stability to the magnetoelastic setting.

Stiffness scales linearly with density in stretch-dominated lattice meta-materials offering the possibility of very light yet very stiff structures. Current additive manufacturing techniques can assemble structures from lattice materials, but the design of such structures will require accurate, efficient simulation techniques. Equivalent continuum models have several advantages over discrete truss models of stretch dominated lattices, including computational efficiency and ease of model construction. However, the development an equivalent model suitable for representing the dynamic response of a periodic truss in the small deformation regime is complicated by microinertial effects. This paper derives a dynamic equivalent continuum model for periodic truss structures suitable for representing long-wavelength wave propagation and verifies it against the full Bloch wave theory and detailed finite element simulations. The model must incorporate microinertial effects to accurately reproduce long wavelength characteristics of the response such as anisotropic elastic soundspeeds. The formulation presented here also improves upon previous work by preserving equilibrium at truss joints for simple lattices and by improving numerical stability by eliminating vertices in the effective yield surface.

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