Dennis M. Kochmann

Resilient 3D hierarchical architected metamaterials

Significance Fractal-like architectures exist in natural materials, like shells and bone, and have drawn considerable interest because of their mechanical robustness and damage tolerance. Developing hierarchically designed metamaterials remains a highly sought after task impaired mainly by limitations in fabrication techniques. We created 3D hierarchical nanolattices with individual beams comprised of multiple self-similar unit cells spanning length scales over four orders of magnitude in fractal-like geometries. We show, through a combination of experiments and computations, that introducing hierarchy into the architecture of 3D structural metamaterials enables the attainment of a unique combination of properties: ultralightweight, recoverability, and a near-linear scaling of stiffness and strength with density. Hierarchically designed structures with architectural features that span across multiple length scales are found in numerous hard biomaterials, like bone, wood, and glass sponge skeletons, as well as manmade structures, like the Eiffel Tower. It has been hypothesized that their mechanical robustness and damage tolerance stem from sophisticated ordering within the constituents, but the specific role of hierarchy remains to be fully described and understood. We apply the principles of hierarchical design to create structural metamaterials from three material systems: (i) polymer, (ii) hollow ceramic, and (iii) ceramic–polymer composites that are patterned into self-similar unit cells in a fractal-like geometry. In situ nanomechanical experiments revealed (i) a nearly theoretical scaling of structural strength and stiffness with relative density, which outperforms existing nonhierarchical nanolattices; (ii) recoverability, with hollow alumina samples recovering up to 98% of their original height after compression to ≥50% strain; (iii) suppression of brittle failure and structural instabilities in hollow ceramic hierarchical nanolattices; and (iv) a range of deformation mechanisms that can be tuned by changing the slenderness ratios of the beams. Additional levels of hierarchy beyond a second order did not increase the strength or stiffness, which suggests the existence of an optimal degree of hierarchy to amplify resilience. We developed a computational model that captures local stress distributions within the nanolattices under compression and explains some of the underlying deformation mechanisms as well as validates the measured effective stiffness to be interpreted as a metamaterial property.

Stable propagation of mechanical signals in soft media using stored elastic energy

Significance Advances in nonlinear mechanics have enabled the realization of a variety of nontraditional functions in mechanical systems. Intrinsic dissipation typically limits the utility of these effects, with soft polymeric materials in particular being incompatible with meaningful wave propagation. Here we demonstrate a nonlinear soft system that is able to propagate large-amplitude signals over arbitrary distances without any signal degradation. We make use of bistable beams to store and then release elastic energy along the path of the wave, balancing both dissipative and dispersive effects. The soft and 3D printable system is highly customizable and tunable, enabling the design of mechanical logic that is relevant to soft autonomous systems (e.g., soft robotics). Soft structures with rationally designed architectures capable of large, nonlinear deformation present opportunities for unprecedented, highly tunable devices and machines. However, the highly dissipative nature of soft materials intrinsically limits or prevents certain functions, such as the propagation of mechanical signals. Here we present an architected soft system composed of elastomeric bistable beam elements connected by elastomeric linear springs. The dissipative nature of the polymer readily damps linear waves, preventing propagation of any mechanical signal beyond a short distance, as expected. However, the unique architecture of the system enables propagation of stable, nonlinear solitary transition waves with constant, controllable velocity and pulse geometry over arbitrary distances. Because the high damping of the material removes all other linear, small-amplitude excitations, the desired pulse propagates with high fidelity and controllability. This phenomenon can be used to control signals, as demonstrated by the design of soft mechanical diodes and logic gates.

Unidirectional Transition Waves in Bistable Lattices.

We present a model system for strongly nonlinear transition waves generated in a periodic lattice of bistable members connected by magnetic links. The asymmetry of the on-site energy wells created by the bistable members produces a mechanical diode that supports only unidirectional transition wave propagation with constant wave velocity. We theoretically justify the cause of the unidirectionality of the transition wave and confirm these predictions by experiments and simulations. We further identify how the wave velocity and profile are uniquely linked to the double-well energy landscape, which serves as a blueprint for transition wave control.

Analytical stability conditions for elastic composite materials with a non-positive-definite phase

Elastic multi-phase materials with a phase having appropriately tuned non-positive-definite elastic moduli have been shown theoretically to permit extreme increases in multiple desirable material properties. Stability analyses of such composites were only recently initiated. Here, we provide a thorough stability analysis for general composites when one phase violates positive-definiteness. We first investigate the dynamic deformation modes leading to instability in the fundamental two-phase solids of a coated cylinder (two dimensions) and a coated sphere (three dimensions), from which we derive closed-form analytical sufficient stability conditions for the full range of coating thicknesses. Next, we apply the energy method to derive a general correlation between composite stability limit and composite bulk modulus that enables determination of closed-form analytical sufficient stability conditions for arbitrary multi-phase materials by employing effective modulus formulas coupled with a numerical finite-element stability analysis. We demonstrate and confirm this new approach by applying it to (i) the two basic two-phase solids already analysed dynamically; and (ii) a more geometrically complex matrix/distributed-inclusions composite. The specific new analytical stability results, and new methods presented, provide a basis for creation of novel, stable composite materials.

Plastic Deformation of Bicrystals Within Continuum Dislocation Theory

Within continuum dislocation theory the plastic deformation of bicrystals under plane strain constrained shear is considered. An analytical solution is found in the symmetric case (for twins) which exhibits the energetic and dissipative thresholds for dislocation nucleation, the Bauschinger translational work hardening, and the size effect. Similar features hold true also for the numerical solution in the general case.

Homogenized mechanical properties of auxetic composite materials in finite-strain elasticity

Careful microstructural design can result in materials with counterintuitive effective (macroscale) mechanical properties such as a negative Poisson’s ratio, commonly referred to as auxetic behavior. One specific approach to achieving auxetic behavior is to elastically connect structural elements with rotational degrees of freedom to result in elastic structures that unfold under uniaxial loading in specific directions, thereby giving rise to bi- or triaxial expansion, i.e. auxetic behavior (transverse expansion under uniaxial extension). This concept has been applied successfully to elastically coupled two-dimensional rigid rotational elements (such as rotating rectangles and triangles) which exhibit a negative effective in-plane Poisson’s ratio under uniaxial (ex)tension. Here, we adopt this fundamental design principle but take it to the next level by achieving auxetic behavior in finitely strained composites made of stiff inclusions in a hyperelastic matrix, and we study the resulting elastic properties under in-plane strain by numerical homogenization. Our results highlight the emergence of auxetic behavior based on geometric arrangement and properties of the base material and demonstrate a path towards simple inclusion–matrix composites with auxetic behavior.

Broadband control of the viscoelasticity of ferroelectrics via domain switching

We show that the viscoelastic properties of polycrystalline ferroelectric ceramics can be significantly altered over a wide range of mechanical frequencies when domain switching is controlled by cyclic electric fields. The dynamic stiffness of lead zirconate titanate is shown to vary by more than 30%, while damping increases by an order of magnitude. Experimental results are interpreted by the aid of a continuum-mechanics model that captures the nonlinear electro-mechanically coupled material response for the full electric hysteresis.

A negative-stiffness phase in elastic composites can produce stable extreme effective dynamic but not static stiffness

We investigate the effective elastic properties and overall stability of four specific two-phase elastic composite systems having a non-positive-definite phase (often referred to as a negative-stiffness phase) to determine whether or not the presence of the negative-stiffness phase can lead to stable extreme overall stiffness. We start with an instructive spring-mass model to illustrate the underlying physical mechanisms before proceeding to the two- and three-dimensional two-phase solids of coated cylindrical and coated spherical inclusions, and we finally study a general particle-matrix composite. For all examples, we correlate effective stiffness with overall stability to demonstrate that the static effective stiffness measures can never reach extreme values due to the inclusion of a negative-stiffness phase in a stable manner, while dynamic loading indeed permits resonance-induced extreme effective stiffness.

Infinitely stiff composite via a rotation-stabilized negative-stiffness phase

We show that an elastic composite material having a component with sufficiently negative stiffness to produce positive-infinite composite stiffness can be stabilized by the gyroscopic forces produced by composite rotation.

Relaxed Potentials and Evolution Equations for Inelastic Microstructures

We consider microstructures which are not inherent to the material but occur as a result of deformation or other physical processes. Examples are martensitic twin-structures or dislocation walls in single crystals and microcrack-fields in solids. An interesting feature of all those microstructures is, that they tend to form similar spatial patterns, which hints at a universal underlying mechanism. For purely elastic materials this mechanism has been identified as minimisation of global energy. For non-quasiconvex potentials the minimisers are not anymore continuous deformation fields, but small-scale fluctuations related to probability distributions of deformation gradients, so-called Young measures. These small scale fluctuations correspond exactly to the observed microstructures of the material. The particular features of those, like orientation or volume fractions, can now be calculated via so-called relaxed potentials. We develop a variational framework which allows to extend these concepts to inelastic materials. Central to this framework will be a Lagrange functional consisting of the sum of elastic power and dissipation due to change of the internal state of the material. We will obtain time-evolution equations for the probability-distributions mentioned above. In order to demonstrate the capabilities of the formalism we will show an application to crystal plasticity.