Stiffness Of Lattice Research Articles

Learning Stiffness Tensors in Self-Activated Solids via a Local Rule.

Mechanical metamaterials are often designed with particular properties for specific load-bearing functions. Alternatively, this study aims to create a class of active lattice metamaterials, dubbed self-activated solids, that can learn desired stiffness tensors from the elastic deformations they experienced, a crucial feature to improve the performance, efficiency, and functionality of materials. Artificial adaptive matters that combine sensory, control, and actuation elements can offer appealing solutions. However, challenges still remain: The designs will rely on accurate off-line and global computations, as well as intricate coordination among individual elements. Here, a simple online and local learning strategy is initiated based on contrastive Hebbian learning to gradually guide self-activated solids to possess sought-after stiffness tensors autonomously and reversibly. During learning, the bond stiffness of the active lattice varies depending only on its local strain. The numerical tests show that the self-activated solid can not only achieve the desired bulk, shear, and coupling moduli but also manifest uni-mode and bi-mode extremal materials by itself after experiencing the corresponding elastic deformations. Further, the self-activated solid can also achieve the desired time-varying moduli when exposed to temporally different loads. The design is applicable to any lattice geometries and is resistant to damage and instabilities. The material design approach and the physical learning strategy suggested can benefit the design of autonomous materials, physical learning machines, and adaptiverobots.

Effect of hydrostatic pressure on the supramolecular assembly of surfactant-cyclodextrin inclusion complexes.

The supramolecular assembly of simple colloids into complex, hierarchical structures arises from a delicate interplay of short-range directional and isotropic long-range forces. These assemblies are highly sensitive to environmental changes, such as temperature variations and the presence of specific molecules, making them promising candidates for nanomachine design. In this study, we investigate the effect of hydrostatic pressure, up to 1800 bar, on the supramolecular assemblies of cyclodextrin/surfactant complexes. Using small-angle neutron scattering, we demonstrate that while the overall structure of the supramolecular aggregates remains largely stable under pressure, the stiffness of the planar lattice formed by the inclusion complexes, the basic structural unit of the supramolecular assemblies, shows a fourfold increase between 250 and 1000 bar. These findings suggest that high-pressure studies can be exploited to better understand the mechanisms of supramolecular assembly processes, thereby aiding in the design of more robust and functional systems.

Compressive performance of 3D printed fault‐tolerant polymer lattice structures

AbstractThe sudden macroscopic damage to the structure is one of the main factors affecting its safe service. This paper investigates the compressive performance of a fault‐tolerant polymer lattice structure prepared by stereolithography apparatus. The lattice structure comprises hexagonal and rhombohedral lattices with varying thicknesses. The quasi‐static compression experiment results show that bi‐lattice has better fault tolerance than single rhombohedral lattice or pure triangular lattice. Compared with the low stiffness of the rhombohedral lattice and the abrupt fracture of the triangular lattice, the bi‐lattice possesses excellent stiffness and substantial deformation capability. When a thinner beam breaks, the thicker beam impedes the transmission of damage, resulting in even dispersion of damage throughout the structure. Consequently, the overall structure remains intact, and its carrying capacity is maintained. This advantage of damage tolerance before macro fracture is significant for the design and safe operation of structures.

Lattice structures in ‘pore shape-stiffness’ framework: A ‘porosity-moment of inertia’ based interpretation

• A novel ‘Porosity-Moment of inertia’ based interpretation. • Helpful to analyze the effect of change in pore shape on the effective stiffness of considered lattice structures at constant porosity levels. • To minimize stress-shielding in scaffolds: A desirable development from the perspective of design of lattices. Existing mathematical models (scaling laws) that correlate stiffness of lattices with their porosity and shape of pores, only suggest the monotonic decrease of stiffness with increase in porosity levels for the given shape of pore. Such mathematical models are of limited use to analyze the effect of change in shape of the pore(s) on the stiffness of lattice at constant porosity levels. To address the highlighted research gap, this work utilizes the archaic concept of classical mechanics (moment of inertia) to propose a normalized parametric tensor for the unified-quantification of the shape of pore(s) in lattices. Micromechanics based results successfully demonstrate the application of suggested framework to improve the understanding, interpretation, and predictability of effect of change in shape of the pore(s) on the stiffness of lattice structures.

Stiffness and energy absorption of additive manufactured hybrid lattice structures

ABSTRACT Lattice structures have been widely used for lightweight structures and energy absorption applications, yet there remain persistent challenges to develop high-performance lattice structures. Existing lattice structures are mainly made by uniform or graded units, some work has been focused on the hybrid lattice structures that combine two or more different units. In this work, two hybrid lattice block structures, named GO_embeded and GO_enfolded, were designed based on two kinds of basic unit cells (Grid unit and Octet unit) and then additively manufactured, and their performances of stiffness and energy absorption were studied. The results show that the stiffness of the hybrid lattice blocks could be comparable to the uniform lattice blocks when they have the same relative density. However, the hybrid lattice blocks have better energy absorption capability relative to uniform lattice blocks. This study provides a basis for the performance of hybrid lightweight structure.

Mechanical interaction between additive-manufactured metal lattice structures and bone in compression: implications for stress shielding of orthopaedic implants

One of the main biomechanical causes for aseptic failure of orthopaedic implants is the stress shielding. This is caused by an uneven load distribution across the bone normally due to a stiff metal prosthesis component, leading to periprosthetic bone resorption and to implant loosening. To reduce the stress shielding and to improve osseointegration, biocompatible porous structures suitable for orthopaedic applications have been developed. Aim of this study was to propose a novel in-vitro model of the mechanical interaction between metal lattice structures and bovine cortical bone in compression. Analysis of the strain distribution between metal structure and bone provides useful information on the potential stress shielding of orthopaedic implants with the same geometry of the porous scaffold. Full density and lattice structures obtained by the repetition of 1.5 mm edge cubic elements via Laser Powder Bed Fusion of CoCrMo powder were characterized for mechanical properties using standard compressive testing. The two porous geometries were characterized by 750 μm and 1000 μm pores resulting in a nominal porosity of 43.5% and 63.2% respectively. Local deformation and strains of metal samples coupled with fresh bovine cortical bone samples were evaluated via Digital Image Correlation analysis up to failure in compression. Visualization and quantification of the local strain gradient across the metal-bone interface was used to assess differences in mechanical behaviour between structures which could be associated to stress-shielding. Overall stiffness and local mechanical properties of lattice and bone were consistent across samples. Full-density metal samples appeared to rigidly transfer the compression force to the bone which was subjected to large deformations (2.2 ± 0.3% at 15 kN). Larger porosity lattice was associated to lower stiffness and compressive modulus, and to a smoother load transfer to the bone. While tested on a limited sample size, the proposed in-vitro model appears robust and repeatable to assess the local mechanical interaction of metal samples suitable for orthopaedic applications with the bone tissue. CoCrMo scaffolds made of 1000 μm pores cubic cells may allow for a smoother load transfer to the bone when used as constitutive material of orthopaedic implants.

Additively Manufactured Semiflexible Titanium Lattices as Hydrogel Reinforcement for Biomedical Implants

Hydrogels are one of the most widespread biomaterials used in tissue engineering. However, they possess weak mechanical properties and are often unstable in load‐bearing applications in vivo. A novel class of flexible Ti–6Al–4V titanium alloy lattices manufactured using laser powder bed fusion (L‐PBF) serves as a tunable reinforcement for hydrogels, providing them with additional mechanical stability and flexibility, while ensuring biocompatibility. A study on the design parameters of the structural elements of the lattices is performed to evaluate their influence on the mechanical properties of the structure. Mechanical testing of Ti–6Al–4V lattices shows a compressive modulus ranging from 38.9 to 895.5 kPa in the flexible direction. In the other two directions, the lattices are designed to have minimal flexibility. Lattices embedded in a 1% agarose hydrogel show a strain‐rate‐dependent, viscoelastic behavior given by the hydrogel component with the additional stiffness of the titanium lattice. Stress distribution upon loading is simulated using finite element analysis (FEA) and compared to experimental data using multiple regression statistical analysis. As a proof of concept, an intervertebral spinal disc implant is designed with mechanical properties matching the compressive moduli of the nucleus pulposus and anulus fibrosus reported in the literature.

Electron-phonon interaction and superconductivity in representative AuCu3-type intermetallic compounds

The AuCu3-type intermetallic compounds are fertile grounds for exploring new superconductors. In order to uncover the generality of the origin of superconductivity and refine an empirical rule to direct finding new superconductors with higher Tc in this kind of compounds, detailed calculations of the electronic structure, phonons and electron-phonon coupling of 8 representative AuCu3-type intermetallic compounds, i.e., ASn3 (A = Ca, La, Y, Na, Li) and YB3 (B = Sn, Pb, Tl, In) categories, were systematically performed from first-principles methods. Among these compounds, the ASn3 (A = Na, Li) were predicted to bear superconductivity with higher Tc. Moreover, we identified different roles of atoms at Au sites and that at Cu sites in determining the physics. The related results indicate that superconductivity mainly originates from the conventional electron-phonon interaction which is mostly dominated by the coupling between electrons from Cu-p states and phonons from low frequency of atoms at Cu sites. More importantly, we empirically found that the electron-phonon coupling is mainly governed by the stiffness of lattice, from which we roughly deduced that the Debye temperature ΘD may be regarded as the tuning factor for finding new superconductors with higher Tc in structures of AuCu3.

Tailorable Stiffness Chiral Metastructure

This letter presents a cellular solid characterized by extreme tailorability of its mechanical response over a large range of macroscopic strains. The proposed periodic structure is based on a hexagonal chiral lattice topology, modified by introducing transverse curvature in the ligaments, key to the resulting unconventional behavior. The resulting topology allows to exploit different levels of structural hierarchy for tailoring the mechanical behavior of the continuous medium. The capability of the initially curved ligaments to change shape under global deformations is exploited to alter the microstructural motions and – as a result – the macroscopic response. We explore the effect of the additional geometrical design parameter, namely the transverse ligament curvature, on the strain‐dependent stiffness of the chiral lattice, evaluating its interplay with the conventional attributes defining the chiral topology and assessing the attainable envelope of mechanical responses. Purposely choosing the parameters permits to tailor the behavior of the chiral lattice, enabling extreme variations in stiffness. The possibility to attain effective negative and zero‐stiffness regimes over large compressive strains further exemplifies the potential of this design. These unique characteristics can be used to attain specific wave propagation properties, augment structural damping, control vibrations and noise, and tune the deformation of compliant structures.

Stiffness and strength of a semi-regular lattice

Honeycombs and other lattice materials have the advantage that their topology can be designed to achieve unique combinations of properties, such as high strength at low density. The work presented here is exploratory in nature: we investigated the mechanical properties of a two-dimensional lattice and compared its performances to other topologies. Analytical expressions for the uniaxial stiffness and compressive strength were developed and validated against Finite Element simulations. The results showed that the lattice considered is stiffer and stronger than the diamond lattice, and has a higher resistance to elastic buckling than the triangular lattice.

Radial stiffness characteristics of the overlap regions of sarcomeres in isolated skeletal myofibrils in pre-force generating state

We have studied the stiffness of myofilament lattice in sarcomeres in the pre-force generating state, which was realized by a relaxing reagent, BDM (butane dione monoxime). First, the radial stiffness for the overlap regions of sarcomeres of isolated single myofibrils was estimated from the resulting decreases in diameter by osmotic pressure applied with the addition of Dextran. Then, the radial stiffness was also estimated from force-distance curve measurements with AFM technology. The radial stiffness for the overlap regions thus obtained was composed of a soft and a rigid component. The soft component visco-elastically changed in a characteristic fashion depending on the physiological conditions of myofibrils, suggesting that it comes from cross-bridge structures. BDM treatments significantly affected the soft radial component of contracting myofibrils depending on the approach velocity of cantilever: It was nearly equal to that in the contracting state at high approach velocity, whereas as low as that in the relaxing state at low approach velocity. However, comparable BDM treatments greatly suppressed the force production and the axial stiffness in contracting glycerinated muscle fibers and also the sliding velocity of actin filaments in the in vitro motility assay. Considering that BDM shifts the cross-bridge population from force generating to pre-force generating states in contracting muscle, the obtained results strongly suggest that cross-bridges in the pre-force generating state are visco-elastically attached to the thin filaments in such a binding manner that the axial stiffness is low but the radial stiffness significantly high similar to that in force generating state.

On the relation between the energy of a distorted crystal lattice and the bending modulus of strain gradient elasticity

It is a well known fact that linear elastic fracture mechanics (LEFM) predicts stress singularities at the tips of sharp cracks, at sharp edges, at corners and at the surface of material transitions. However, from the viewpoint of the strengths of atomic bonds it is clear that only finite forces may be present at the tip of a stable crack. Therefore, theories of strain gradient elasticity were developed which reduce the values of stress concentrations. Within these theories a bending modulus is postulated which introduces an increased stiffness of the crystal lattice against bending. In the present study, the value of this bending modulus is evaluated on the basis of the electrostatic energy of a bent crystal lattice. This is done for the face centred cubic structure of NaCl. In fact, results for the bending modulus could be obtained although they depend on the crystal size.

Twist of rhombohedra, spin helix, and “giant” polarization in a CaMn7O12 multiferroic crystal

A qualitative assessment of axial vector Γ z in invariant P z (McurlM)Γ z proposed earlier for the description of the thermodynamic potential of a CaMn7O12 multiferroic material with electric polarization P z along the z axis of the helix of magnetic moments M has been performed based on the fan-shaped structure of the atomic cell of the crystal. The dependence of Γ z on the permanent dipole moment of MnO6 octahedra (rhombohedra), which are inevitably distorted in the paraelectric phase due to the structural transformation, on the mechanical stiffness of the crystal lattice, and on the projected area of the cross section of rhombohedra with bevels undergoing a shift has been determined from dimensional considerations. It has been demonstrated how these factors contribute to the twist of rhombohedra.

Effect of clathrin light chains on the stiffness of clathrin lattices and membrane budding.

Clathrin-dependent transport processes require the polymerization of clathrin triskelia into polygonal scaffolds. Together with adapter proteins, clathrin collects cargo and induces membrane bud formation. It is not known to what extent clathrin light chains affect the structural and functional properties of clathrin lattices and the ability of clathrin to deform membranes. To address these issues, we have developed a novel procedure for analyzing clathrin lattice formation on rigid surfaces. We found that lattices can form on adaptor-coated convex-, planar- and even shallow concave surfaces, but the rate of formation and resistance to thermal dissociation of the lattice are greatly enhanced on convex surfaces. Atomic force microscopy on planar clathrin lattices demonstrates that the stiffness of the clathrin lattice is strictly dependent on light chains. The reduced stiffness of the lattice also compromised the ability of clathrin to generate coated buds on the surface of rigid liposomal membranes.

The stiffness and strength of the gyroid lattice

Recently, a nanoscale lattice material, based upon the gyroid topology has been self-assembled by phase separation techniques (Scherer et al., 2012) and prototyped in thin film applications. The mechanical properties of the gyroid are reported here. It is a cubic lattice, with a connectivity of three struts per joint, and is bending-dominated in its elasto-plastic response to all loading states except for hydrostatic: under a hydrostatic stress it exhibits stretching-dominated behaviour. The three independent elastic constants of the lattice are determined through a unit cell analysis using the finite element method; it is found that the elastic and shear modulus scale quadratically with the relative density of the lattice, whereas the bulk modulus scales linearly. The plastic collapse response of a rigid, ideally plastic gyroid lattice is explored using the upper bound method, and is validated by finite element calculations for an elastic-ideally plastic lattice. The effect of geometrical imperfections, in the form of random perturbations to the joint positions, is investigated for both stiffness and strength. It is demonstrated that the hydrostatic modulus and strength are imperfection sensitive, in contrast to the deviatoric response. The macroscopic yield surface of the imperfect lattice is adequately described by a modified version of Hill’s anisotropic yield criterion. The article ends with a case study on the stress induced within a gyroid thin film, when the film and its substrate are subjected to a thermal expansion mismatch.

Comprehensive Stiffness of Prestressed Lattice Materials

Several approaches to obtain the comprehensive stiffness of finite frameworks are present in literature; yet, the formulation has not been addressed for lattice materials and infinitely periodic structures. The objective of this paper is to introduce a systematic method to calculate the comprehensive stiffness of prestressed, infinitely periodic, structures and lattice materials with pin- and rigid-jointed connectivity. We first derive the comprehensive stiffness of a finite framework through the superposition of its material and nonlinear geometrical stiffness. By using the Bloch's theorem, we derive the irreducible form of the stiffness system of the finite framework, which represents the stiffness behaviour of the corresponding infinite, periodic assembly. Finally, the comprehensive stiffness of the infinite lattice is homogenized to generate the stiffness characteristics of the lattice material. A detailed example is provided to show the application of the methodology. Closed-form expressions of the elastic properties are presented for 12 planar lattices.

Study on the Valence State of Eu Ions in Sr3Al2O6:Eu Phosphor

Europium-activated strontium aluminate phosphor, Sr3Al2O6:Eu2+, was prepared by solid-state reaction. The main concerns on the formation of a stable Sr3Al2O6 phase and the complete reduction of europium ions in this crystal phase, which is considered controversial in the literature and is available for white light-emitting-diode application, were addressed. It was found that the single phase of Sr3Al2O6 could be well formed and the valence state of europium ions depended on treatment temperature, flux, reduction gas, and the resultant stiffness of the host lattice. The redshift emission of Eu2+ in the Sr3Al2O6 host was not marked in our experimental regime. However, the green emission of Sr3Al2O6:Eu2+ increased gradually with the mole fraction of Al2O3 to SrCO3 in the starting materials owing to the complete formation of the Sr3Al2O6 phase.

Mass and lattice effects in trapping: Ar, Kr, and Xe on Pt(111), Pd(111), and Ni(111)

Abstract The trapping probabilities of argon, krypton, and xenon on Pd(1 1 1) and Ni(1 1 1) have been investigated using supersonic molecular beam techniques. The trapping probability of argon exhibits normal incident energy in a similar fashion on both Pd(1 1 1) and Pt(1 1 1) because the mass of argon is significantly less than the surface mass of either Pd or Pt. In contrast, dynamic corrugation in the gas–surface potential is observed for krypton trapping on Pt(1 1 1) and Pd(1 1 1), resulting in a decreased angular dependence of the trapping probability compared to argon. For xenon trapping on Pd significant lattice deformation during the gas–surface collision appears to give rise to total energy scaling. The trapping probability of xenon on Pd(1 1 1) remains high at unusually high incident kinetic energies due to the overall enhanced energy transfer from the incident atom to the lattice. Trapping probabilities of Ar, Kr, and Xe are significantly lower on Ni(1 1 1) than on either Pt(1 1 1) or Pd(1 1 1) despite the lower surface mass of the Ni atoms. This result is attributed to the lower binding energy of the rare gases on Ni(1 1 1) and the higher Debye temperature of Ni. The energy scaling of Ar trapping on Ni(1 1 1) is determined by static corrugation, but the energy scaling for Kr and Xe on Ni(1 1 1) may involve the effects of dynamic corrugation. In the latter cases, the greater stiffness of the nickel lattice decreases the dynamic corrugation relative to Pt(1 1 1) and Pd(1 1 1).

Prestin and the Dynamic Stiffness of Cochlear Outer Hair Cells

The outer hair cell (OHC) lateral wall is a unique trilaminate structure consisting of the plasma membrane, the cortical lattice, and subsurface cisternae. OHCs are capable of altering their length in response to transmembrane voltage change. This so-called electromotile response is presumed to result from conformational changes of membrane-bound protein molecules, named prestin. OHC motility is accompanied by axial stiffness changes when the membrane potential of the cell is altered. During length changes, intracellular anions (mainly Cl-) act as extrinsic voltage sensors. In this study, we inquired whether the motor proteins are responsible for the voltage-dependent axial stiffness of OHCs, and whether ACh, the neurotransmitter of efferent neurons, modulates the stiffness of the cortical lattice and/or the stiffness of the motor protein. The experiments were done on isolated guinea pig OHCs in the whole-cell voltage-clamp mode. Axial stiffness was determined by loading a fiber of known stiffness onto the apical surface of the cells. Voltage-dependent stiffness and cell motility disappeared, and the axial stiffness of the cells significantly decreased after removal of intracellular Cl-. The result suggests that the stiffness of the motor protein is a major contributor to the global axial stiffness of OHCs. ACh was found to affect both the motor protein and other lateral wall stiffness components.

Effect of centrifugation on the crystal structure of barium nitrate

The structure of barium nitrate crystals grown from supersaturated aqueous solutions under normal conditions and during centrifugation at 11.8×103g is studied for the first time by a high-precision powder neutron diffraction method. It is established for the first time that centrifuging leads to displacement of the nitrate oxygen atoms in the xy plane (resulting in rotation of the NO3 trigonal prism about the third-order axis) and to variation of the bond lengths and angles in the coordination polyhedron of Ba atoms. The latter changes result in the appearance of a torsional mode in the phonon spectrum of barium nitrate and in increasing stiffness of the crystal lattice, as evidenced by a significant increase in microhardness of the samples grown in the centrifuge.