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Concept: Hooke's law


Owing to their atomic-scale thickness, the resonances of two-dimensional (2D) material membranes show signatures of nonlinearities at forces of only a few picoNewtons. Although the linear dynamics of membranes is well understood, the exact relation between the nonlinear response and the resonator’s material properties has remained elusive. Here we show a method for determining the Young’s modulus of suspended 2D material membranes from their nonlinear dynamic response. To demonstrate the method, we perform measurements on graphene and MoS2 nanodrums electrostatically driven into the nonlinear regime at multiple driving forces. We show that a set of frequency response curves can be fitted using only the cubic spring constant as a fit parameter, which we then relate to the Young’s modulus of the material using membrane theory. The presented method is fast, contactless, and provides a platform for high-frequency characterization of the mechanical properties of 2D materials.

Concepts: Materials science, Dynamics, Young's modulus, Elasticity, Hooke's law, String theory, Shear modulus, Poisson's ratio


The elastoplastic deformation behaviors of hollow glass microspheres/iron syntactic foam under tension were modeled using a representative volume element (RVE) approach. The three-dimensional microstructures of the iron syntactic foam with 5 wt % glass microspheres were reconstructed using the random sequential adsorption algorithm. The constitutive behavior of the elastoplasticity in the iron matrix and the elastic-brittle failure for the glass microsphere were simulated in the models. An appropriate RVE size was statistically determined by evaluating elastic modulus, Poisson’s ratio, and yield strength in terms of model sizes and boundary conditions. The model was validated by the agreement with experimental findings. The tensile deformation mechanism of the syntactic foam considering the fracture of the microspheres was then investigated. In addition, the feasibility of introducing the interfacial deboning behavior to the proposed model was briefly investigated to improve the accuracy in depicting fracture behaviors of the syntactic foam. It is thought that the modeling techniques and the model itself have major potential for applications not only in the study of hollow glass microspheres/iron syntactic foams, but also for the design of composites with a high modulus matrix and high strength reinforcement.

Concepts: Tensile strength, Young's modulus, Elasticity, Solid mechanics, Hooke's law, Deformation, Glass microsphere, Syntactic foam


Soft dielectric materials typically exhibit poor heat transfer properties due to the dynamics of phonon transport, which constrain thermal conductivity (k) to decrease monotonically with decreasing elastic modulus (E). This thermal-mechanical trade-off is limiting for wearable computing, soft robotics, and other emerging applications that require materials with both high thermal conductivity and low mechanical stiffness. Here, we overcome this constraint with an electrically insulating composite that exhibits an unprecedented combination of metal-like thermal conductivity, an elastic compliance similar to soft biological tissue (Young’s modulus < 100 kPa), and the capability to undergo extreme deformations (>600% strain). By incorporating liquid metal (LM) microdroplets into a soft elastomer, we achieve a ∼25× increase in thermal conductivity (4.7 ± 0.2 W⋅m(-1)⋅K(-1)) over the base polymer (0.20 ± 0.01 W⋅m(-1)·K(-1)) under stress-free conditions and a ∼50× increase (9.8 ± 0.8 W⋅m(-1)·K(-1)) when strained. This exceptional combination of thermal and mechanical properties is enabled by a unique thermal-mechanical coupling that exploits the deformability of the LM inclusions to create thermally conductive pathways in situ. Moreover, these materials offer possibilities for passive heat exchange in stretchable electronics and bioinspired robotics, which we demonstrate through the rapid heat dissipation of an elastomer-mounted extreme high-power LED lamp and a swimming soft robot.

Concepts: Temperature, Heat, Thermal conductivity, Young's modulus, Heat transfer, Elasticity, Stiffness, Hooke's law


In the present work, we carried out density functional calculations of struvite - the main component of the so-called infectious urinary stones - to study its structural and elastic properties. Using a local density approximation and a generalised gradient approximation, we calculated the equilibrium structural parameters and elastic constants C ( ijkl ). At present, there is no experimental data for these elastic constants C ( ijkl ) for comparison. Besides the elastic constants, we also present the calculated macroscopic mechanical parameters, namely the bulk modulus (K), the shear modulus (G) and Young’s modulus (E). The values of these moduli are found to be in good agreement with available experimental data. Our results imply that the mechanical stability of struvite is limited by the shear modulus, G. The study also explores the energy-band structure to understand the obtained values of the elastic constants.

Concepts: Density functional theory, Young's modulus, Elasticity, Elastic modulus, Hooke's law, Local-density approximation, Shear modulus, Bulk modulus


The modulus of elasticity of soft materials on the nanoscale is of interest when studying thin films, nanocomposites, and biomaterials. Two novel modes of atomic force microscopy (AFM) have been introduced recently: HarmoniX and PeakForce QNM. Both modes produce distribution maps of the elastic modulus over the sample surface. Here we investigate the question of how quantitative these maps are when studying soft materials. Three different polymers with a macroscopic Young’s modulus of 0.6-0.7 GPa (polyurethanes) and 2.7 GPa (polystyrene) are analyzed using these new modes. The moduli obtained are compared to the data measured with the other commonly used techniques, dynamic mechanical analyzer (DMA), regular AFM, and nanoindenter. We show that the elastic modulus is overestimated in both the HarmoniX and PeakForce QNM modes when using regular sharp probes because of excessively overstressed material in the samples. We further demonstrate that both AFM modes can work in the linear stress-strain regime when using a relatively dull indentation probe (starting from ∼210 nm). The analysis of the elasticity models to be used shows that the JKR model should be used for the samples considered here instead of the DMT model, which is currently implemented in HarmoniX and PeakForce QNM modes. Using the JKR model and∼240 nm AFM probe in the PeakForce QNM mode, we demonstrate that a quantitative mapping of the elastic modulus of polymeric materials is possible. A spatial resolution of ∼50 nm and a minimum 2 to 3 nm indentation depth are achieved.

Concepts: Polymer, Polystyrene, Young's modulus, Elasticity, Elastic modulus, Stiffness, Hooke's law, Poisson's ratio


In this work, we report experimental evidence of surface stress effects on the mechanical properties of silicon nanostructures. As-fabricated, top-down silicon nanowires (SiNWs) are bent up without any applied force. This self-buckling is related to the surface relaxation that reaches an equilibrium with bulk deformation due to the material elasticity. We measure the SiNW self-deformation by atomic force microscopy (AFM), and we apply a simple physical model in order to give an estimation of the surface stress. If the equilibrium is altered by a nanoforce, applied by an AFM tip, nanowires find a new equilibrium condition bending down (mechanical bistability). In this work, for the first time, we report a clear and quantitative relationship between the SiNWs' apparent Young’s modulus, measured by force-deflection spectroscopy, and the estimated value of surface stress, obtained by self-buckling measurements taking into account the Young’s modulus of bulk silicon. This is an experimental confirmation that the surface stress is fundamental in determining mechanical properties of SiNWs, and that the elastic behavior of nanostructures strongly depends on their surfaces.

Concepts: Measurement, Force, Young's modulus, Elasticity, Solid mechanics, Linear elasticity, Elastic modulus, Hooke's law


A striking feature of stress relaxation in biological soft tissue is that it frequently follows a power law in time with an exponent that is independent of strain even when the elastic properties of the tissue are highly nonlinear. This kind of behavior is an example of quasi-linear viscoelasticity, and is usually modeled in a purely empirical fashion. The goal of the present study was to account for quasi-linear viscoelasticity in mechanistic terms based on our previously developed hypothesis that it arises as a result of isolated micro-yield events occurring in sequence throughout the tissue, each event passing the stress it was sustaining on to other regions of the tissue until they themselves yield. We modeled stress relaxation computationally in a collection of stress-bearing elements. Each element experiences a stochastic sequence of either increases in elastic equilibrium length or decreases in stiffness according to the stress imposed upon it. This successfully predicts quasi-linear viscoelastic behavior, and in addition predicts power-law stress relaxation that proceeds at the same slow rate as observed in real biological soft tissue.

Concepts: Time, Continuum mechanics, Viscoelasticity, Elasticity, Linear elasticity, Hooke's law, Stress relaxation, Creep


Drawbacks with the commonly used PMMA-based bone cements, such as an excessive elastic modulus and potentially toxic residual monomer content, motivate the development of alternative cements. In this work an attempt to prepare an injectable biomaterial based on isosorbide-alicyclic diol derived from renewable resources was presented. Two novel dimethacrylic monomers ISDGMA - 2,5-bis(2-hydroxy-3-methacryloyloxypropoxy)-1,4:3,6-dianhydro-sorbitol and ISETDMA - dimethacrylate of ethoxylated isosorbide were synthesized and used to prepare a series of low-viscosity compositions comprising bioactive nano-sized hydroxyapatite in the form of a two-paste system. Formulations exhibited a non-Newtonian shear-thinning behavior, setting times between 2.6min and 5.3min at 37°C and maximum curing temperatures of 65°C. Due to the hydrophilic nature of ISDGMA, cured compositions could absorb up to 13.6% water and as a result the Young’s modulus decreased from 1429MPa down to 470MPa. Both, poly(ISDGMA) and poly(ISETDMA) were subjected to a MTT study on mice fibroblasts (BALB/3T3) and gave relative cell viabilities above 70% of control. A selected model bone cement was additionally investigated using human osteosarcoma cells (SaOS-2) in an MTS test, which exhibited concentration-dependent cell viability. The preliminary results, presented in this work reveal the potential of two novel dimethacrylic monomers in the preparation of an injectable biomaterial for bone augmentation, which could overcome some of the drawbacks typical for conventional acrylic bone cement.

Concepts: Young's modulus, Tooth enamel, Elastic modulus, Stiffness, Hooke's law, Osteosarcoma, Monomers, Impulse excitation technique


In-vivo investigation of tendon mechanical properties in healthy subjects using Shear Wave Elastography (SWE) techniques is a relatively new field of study. This work aims to evaluate the elastic properties of the patellar tendon in various knee range of flexion. Twenty healthy adult subjects were enrolled in the study. Shear wave speed (SWS) in the patellar tendon was measured in three different positions: Knee extended, knee semi-flexed (30°), and knee flexed (90°). Mean shear modulus was 50.9 +- 33.1 kPa in knee extension position, 137.5 +- 50.7 kPa in 30° flexion position, and 226.5 +- 60.3 kPa in 90° flexion position. The lowest shear modulus was obtained at rest with the knee in a fully extended position. These results are in agreement with those previously reported on Achilles tendon and triceps muscles. Shear modulus values obtained in our study could be considered as baseline values for further investigations in adults.

Concepts: In vivo, Knee, Young's modulus, Flexion, Extension, Tibia, Tendon, Hooke's law


Auxetics comprise a rare family of materials that manifest negative Poisson’s ratio, which causes an expansion instead of contraction under tension. Most known homogeneously auxetic materials are porous foams or artificial macrostructures and there are few examples of inorganic materials that exhibit this behavior as polycrystalline solids. It is now possible to accelerate the discovery of materials with target properties, such as auxetics, using high-throughput computations, open databases, and efficient search algorithms. Candidates exhibiting features correlating with auxetic behavior were chosen from the set of more than 67 000 materials in the Materials Project database. Poisson’s ratios were derived from the calculated elastic tensor of each material in this reduced set of compounds. We report that this strategy results in the prediction of three previously unidentified homogeneously auxetic materials as well as a number of compounds with a near-zero homogeneous Poisson’s ratio, which are here denoted “anepirretic materials”.There are very few inorganic materials with auxetic homogenous Poisson’s ratio in polycrystalline form. Here authors develop an approach to screening materials databases for target properties such as negative Poisson’s ratio by using stability and structural motifs to predict new instances of homogenous auxetic behavior as well as a number of materials with near-zero Poisson’s ratio.

Concepts: Algorithm, Materials science, Computation, Materials, Ratios, Hooke's law, Poisson's ratio, Auxetics