Mechanical metamaterials at the theoretical limit of isotropic elastic stiffness.

A wide variety of high-performance applications require materials for which shape control is maintained under substantial stress, and that have minimal density. Bio-inspired hexagonal and square honeycomb structures and lattice materials based on repeating unit cells composed of webs or trusses, when made from materials of high elastic stiffness and low density, represent some of the lightest, stiffest and strongest materials available today. Recent advances in 3D printing and automated assembly have enabled such complicated material geometries to be fabricated at low (and declining) cost. These mechanical metamaterials have properties that are a function of their mesoscale geometry as well as their constituents, leading to combinations of properties that are unobtainable in solid materials; however, a material geometry that achieves the theoretical upper bounds for isotropic elasticity and strain energy storage (the Hashin-Shtrikman upper bounds) has yet to be identified. Here we evaluate the manner in which strain energy distributes under load in a representative selection of material geometries, to identify the morphological features associated with high elastic performance. Using finite-element models, supported by analytical methods, and a heuristic optimization scheme, we identify a material geometry that achieves the Hashin-Shtrikman upper bounds on isotropic elastic stiffness. Previous work has focused on truss networks and anisotropic honeycombs, neither of which can achieve this theoretical limit. We find that stiff but well distributed networks of plates are required to transfer loads efficiently between neighbouring members. The resulting low-density mechanical metamaterials have many advantageous properties: their mesoscale geometry can facilitate large crushing strains with high energy absorption, optical bandgaps and mechanically tunable acoustic bandgaps, high thermal insulation, buoyancy, and fluid storage and transport. Our relatively simple design can be manufactured using origami-like sheet folding and bonding methods.

Convective Assembly of a Particle Monolayer.

Recently, the steady-state process of convective assembly has emerged as a viable production route for colloidal monolayers. The present study models the regions of particle assembly: Region I comprises convective concentration of a particle suspension in a liquid below a meniscus, Region II comprises permeation of fluid through the dense particle monolayer, and Region III comprises capillary densification. For each region, the dominant physics and nondimensional groups are identified, and quantitative models are derived to describe the evolution of microstructure in terms of the main process parameters. The concentration profile within the assembly zone of Region I is predicted, including the role of a concentration-dependent diffusion constant and the shape of the meniscus. The fluid flow through the assembled monolayer is treated in Region II, along with a stability calculation to reveal that isolated particle clusters do not survive on top of the monolayer. The physics of capillary crystallization is addressed in Region III, with an emphasis on the density of cracks that emerge. The Peclet number and Capillary number both play important roles but in different regions of the assembly process.

Simulation of the cytoskeletal response of cells on grooved or patterned substrates.

We analyse the response of osteoblasts on grooved substrates via a model that accounts for the cooperative feedback between intracellular signalling, focal adhesion development and stress fibre contractility. The grooved substrate is modelled as a pattern of alternating strips on which the cell can adhere and strips on which adhesion is inhibited. The coupled modelling scheme is shown to capture some key experimental observations including (i) the observation that osteoblasts orient themselves randomly on substrates with groove pitches less than about 150 nm but they align themselves with the direction of the grooves on substrates with larger pitches and (ii) actin fibres bridge over the grooves on substrates with groove pitches less than about 150 nm but form a network of fibres aligned with the ridges, with nearly no fibres across the grooves, for substrates with groove pitches greater than about 300 nm. Using the model, we demonstrate that the degree of bridging of the stress fibres across the grooves, and consequently the cell orientation, is governed by the diffusion of signalling proteins activated at the focal adhesion sites on the ridges. For large groove pitches, the signalling proteins are dephosphorylated before they can reach the regions of the cell above the grooves and hence stress fibres cannot form in those parts of the cell. On the other hand, the stress fibre activation signal diffuses to a reasonably spatially homogeneous level on substrates with small groove pitches and hence stable stress fibres develop across the grooves in these cases. The model thus rationalizes the responsiveness of osteoblasts to the topography of substrates based on the complex feedback involving focal adhesion formation on the ridges, the triggering of signalling pathways by these adhesions and the activation of stress fibre networks by these signals.

Simulation of the contractile response of cells on an array of micro-posts.

A bio-chemo-mechanical model has been used to predict the contractile responses of smooth cells on a bed of micro-posts. Predictions obtained for smooth muscle cells reveal that, by converging onto a single set of parameters, the model captures all of the following responses in a self-consistent manner: (i) the scaling of the force exerted by the cells with the number of posts; (ii) actin distributions within the cells, including the rings of actin around the micro-posts; (iii) the curvature of the cell boundaries between the posts; and (iv) the higher post forces towards the cell periphery. Similar correspondences between predictions and measurements have been demonstrated for fibroblasts and mesenchymal stem cells once the maximum stress exerted by the stress fibre bundles has been recalibrated. Consistent with measurements, the model predicts that the forces exerted by the cells will increase with both increasing post stiffness and cell area (or equivalently, post spacing). In conjunction with previous assessments, these findings suggest that this framework represents an important step towards a complete model for the coupled bio-chemo-mechanical responses of cells.

The effect of shape on the adhesion of fibrillar surfaces.

Experimental data have demonstrated that mushroom-shaped fibrils adhere much better to smooth substrates than punch-shaped fibrils. We present a model that suggests that detachment processes for such fibrils are controlled by defects in the contact area that are confined to its outer edge. Stress analysis of the adhered fibril, carried out for both punch and mushroom shapes with and without friction, suggests that defects near the edge of the adhesion area are much more damaging to the pull-off strength in the case of the punch than for the mushroom. The simulations show that the punch has a higher driving force for extension of small edge defects compared with the mushroom adhesion. The ratio of the pull-off force for the mushroom to that of the punch can be predicted from these simulations to be much greater than 20 in the friction-free case, similar to the experimental value. In the case of sticking friction, a ratio of 14 can be deduced. Our analysis also offers a possible explanation for the evolution of asymmetric mushroom shapes (spatulae) in the adhesion organ of geckos.

A finite element simulation scheme for biological muscular hydrostats.

An explicit finite element scheme is developed for biological muscular hydrostats such as squid tentacles, octopus arms and elephant trunks. The scheme is implemented by embedding muscle fibers in finite elements. In any given element, the fiber orientation can be assigned arbitrarily and multiple muscle directions can be simulated. The mechanical stress in each muscle fiber is the sum of active and passive parts. The active stress is taken to be a function of activation state, muscle fiber shortening velocity and fiber strain; while the passive stress depends only on the strain. This scheme is tested by simulating extension of a squid tentacle during prey capture; our numerical predictions are in close correspondence with existing experimental results. It is shown that the present finite element scheme can successfully simulate more complex behaviors such as torsion of a squid tentacle and the bending behavior of octopus arms or elephant trunks.

Geometric optimization of a tissue pattern for semilunar valve reconstruction.

BACKGROUND AND AIM OF THE STUDY: A novel geometric trefoil pattern has been suggested for semilunar valve reconstruction. Optimization of the geometry must rely on an appreciation of normal anatomy and knowledge of the mechanical properties of the tissue used for the reconstruction. METHODS: Computer-assisted design (CAD) was used to create an optimized leaflet geometry based on published dimensions for normal human aortic valves. The optimized leaflet geometry was subjected to finite element analysis (FEA) to study stress distribution with pressure loading of the leaflet. In vitro function of the optimized trefoil tissue pattern is being studied by static testing initially, with physiological saline. RESULTS: An optimized leaflet geometry has been developed by CAD, and further refined by FEA. Static testing of the optimized trefoil tissue pattern shows near-normal anatomy, with no prolapse or pin wheeling, and full valve competence to 90 mmHg pressure. CONCLUSIONS: An initial optimized geometry has been developed for a two-dimensional tissue pattern that can be used to reconstruct diseased semilunar heart valves with human pericardium. Optimization studies are based on the mechanical properties of the tissue, CAD to mimic normal anatomy, FEA to study stress distribution, and static load testing to confirm function.

Structural analysis of the Björk-Shiley Delrin heart valve occluder.

BACKGROUND AND AIMS OF THE STUDY: Impact wear grooves were evident in some Delrin occluder discs of explanted Björk-Shiley Delrin (BSD) heart valves. This study focuses on the finite element analysis (FEA) method to understand the maximum principal stresses experienced during the peak in vivo loading of valves in the closed position. MATERIALS AND METHODS: The maximum pressure difference across the valve was measured to be 130 mmHg in a pulse duplicator simulating normal sinus rhythm obtained clinically by cardiac catheterization. The corresponding measured strain was 1.81 x 10(-3). The FEA model incorporated four points of contact between the disc and the orifice ring to estimate the maximum principal stresses in the disc of the BSD heart valve. A linear pressure distribution averaging 130 mmHg was applied so that the finite element results gave a strain of 1.81 x 10(-3) at the gauge location as experimentally observed. RESULTS: The largest stress in the Delrin disc of the BSD valve occurred when the occluder made four-point contact with the orifice ring struts. The resulting localized compressive stress on the inflow side could be as high as 42 ksi, assuming the Hertzian contact theory. The magnitude of tensile stresses were less, but were highest on the outlet surface opposite the point of contact. The highest tensile stress for an ungrooved disc was found to be 8.35 ksi, which was below the ultimate tensile strength and yield stress in flexure for Delrin. Therefore, it is unlikely that yielding or tensile failure will occur at this level of stress. Maximum tensile stresses were found to be 1.442 and 1.448 ksi for discs with single and multiple grooves respectively. CONCLUSION: The model predicts that as a wear groove is created, the area of contact between the disc and the inlet strut of the BSD valve will increase, thereby reducing average compressive contact stress and hence, the wear rate.