Decelerator fabric constants required by the generalized form of Hooke's law
Decelerator fabric elastic constants for structural analyses using generalized Hookes law
Alley, V. L., Jr.↗
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Perturbation theory of the Hooke's law model for the two-electron atom
Perturbation theory of Hookes law model for He atom, obtaining ground state energy through third order
Brown, W. B.↗
Perturbation energies for the Hooke's law model of the two-electron atom
Rayleigh-Schroedinger perturbation energies for ground state of two electron atomic Hookes law model through tenth order
Benson, J. M.↗
Quantum Hooke's Law to classify pulse laser induced ultrafast melting
Ultrafast crystal-to-liquid phase transition induced by femtosecond pulse laser excitation is an interesting material's behavior manifesting the complexity of light-matter interaction. There exist two types of such phase transitions: one occurs at a time scale shorter than a picosecond via a nonthermal process mediated by electron-hole plasma formation; the other at a longer time scale via a thermal melting process mediated by electron-phonon interaction. However, it remains unclear what material would undergo which process and why? Here, by exploiting the property of quantum electronic stress (QES) governed by quantum Hooke's law, we classify the transitions by two distinct classes of materials: the faster nonthermal process can only occur in materials like ice having an anomalous phase diagram characterized with dT m /dP < 0, where T m is the melting temperature and P is pressure, above a high threshold laser fluence; while the slower thermal process may occur in all materials. Especially, the nonthermal transition is shown to be induced by the QES, acting like a negative internal pressure, which drives the crystal into a “super pressing” state to spontaneously transform into a higher-density liquid phase. Our findings significantly advance fundamental understanding of ultrafast crystal-to-liquid phase transitions, enabling quantitative a priori predictions.
75 CONDENSED MATTER PHYSICS, SUPERCONDUCTIVITY AND↗
Breakdown of Hooke’s law of elasticity at the Mott critical endpoint in an organic conductor
The Mott metal-insulator transition, a paradigm of strong electron-electron correlations, has been considered as a source of intriguing phenomena. Despite its importance for a wide range of materials, fundamental aspects of the transition, such as its universal properties, are still under debate. We report detailed measurements of relative length changes Δ L / L as a function of continuously controlled helium-gas pressure P for the organic conductor κ-(BEDT-TTF) 2 Cu[N(CN) 2 ]Cl across the pressure-induced Mott transition. We observe strongly nonlinear variations of Δ L / L with pressure around the Mott critical endpoint, highlighting a breakdown of Hooke’s law of elasticity. We assign these nonlinear strain-stress relations to an intimate, nonperturbative coupling of the critical electronic system to the lattice degrees of freedom. Our results are fully consistent with mean-field criticality, predicted for electrons in a compressible lattice with finite shear moduli. We argue that the Mott transition for all systems that are amenable to pressure tuning shows the universal properties of an isostructural solid-solid transition.
Science & Technology - Other Topics↗
Galilean covariant harmonic oscillator
A Galilean covariant approach to classical mechanics of a single particle is described. Within the proposed formalism, all non-covariant force laws defining acting forces which become to be defined covariantly by some differential equations are rejected. Such an approach leads out of the standard classical mechanics and gives an example of non-Newtonian mechanics. It is shown that the exactly solvable linear system of differential equations defining forces contains the Galilean covariant description of harmonic oscillator as its particular case. Additionally, it is demonstrated that in Galilean covariant classical mechanics the validity of the second Newton law of dynamics implies the Hooke law and vice versa. It is shown that the kinetic and total energies transform differently with respect to the Galilean transformations.
Horzela, Andrzej↗
The stress-strain relations for a nonhomogeneous composite with a circular inclusion
Stress-strain diagram for nonhomogeneous composite with circular inclusion using Hookes law
Hyman, B. I.↗
Finite element stress analysis of polymers at high strains
A numerical analysis is presented for the problem of a flat rectangular rubber membrane with a circular rigid inclusion undergoing high strains due to the action of an axial load. The neo-hookean constitutive equations are introduced into the general purpose TITUS program by means of equivalent hookean constants and initial strains. The convergence is achieved after a few iterations. The method is not limited to any specific program. The results are in good agreement with those of a company sponsored photoelastic stress analysis. The theoretical and experimental deformed shapes also agree very closely with one another. For high strains it is demonstrated that using the conventional HOOKE law the stress concentration factor obtained is unreliable in the case of rubberlike material.
Durand, M.↗
Stress-Dependent Voltage Offsets From Polymer Insulators Used in Rock Mechanics and Material Testing
Dielectric insulators are used in a variety of laboratory settings when performing experiments in rock mechanics, petrology, and electromagnetic studies of rocks in the fields of geophysics,material science, and civil engineering. These components may be used to electrically isolate geological samples from the experimental equipment, to perform a mechanical compliance function between brittle samples and the loading equipment, to match ultrasonic transducers, or perform other functions. In manyexperimental configurations the insulators bear the full brunt of force applied to the sample but do not need to withstand high voltages, therefore the insulators are often thin sheets of mechanically tough polymers. From an instrument perspective, transduction from various types of mechanical perturbation has beenqualitatively compared for a number of polymers [1, 2] and these error sources are readily apparent duringhigh-impedance measurements if not mitigated. However even when following best practices, a force dependent voltage signal still remains and its behavior is explored in this presentation. In this experimenttwo thin sheets (0.25 mm) of high-density polyethylene (HDPE) were set up in a stack, held alternatelybetween three aluminum bars; this stack was placed on the platen of a 60T capacity hydraulic testingmachine. The surface area, A, over which the force is applied to the PE sheets in this sandwich is roughly 40 square cm, each sheet forming a parallel-plate capacitor having roughly 320 pF [3], assuming therelative dielectric permittivity of PE is approximately 2.3. The outer two aluminum bars were connected to the LO input ofthe electrometer and the central aluminum bar was connected to the HI input of a Keithley model 617 electrometer. Once the stack is mechanically well-seated with no air gaps, the voltage offset is observed tobe a linear function of the baseline voltage for a given change in applied force. For a periodically appliedforce of 66.7 kN the voltage offsets were measured as a function of initial voltage, and these data were fitwith a linear function that was constrained to pass through the origin. The best fit solution had a correlation coefficient of R=0.85 and a slope of approximately -0.0228 volts/volt. The voltage offset when normalizedis demonstrated to be constant -2.28% for both positive and negative polarities over nearly 3 orders ofbaseline voltage magnitude. From this, the voltage-force coefficient is derived to be -0.34 ppm/N. Thiscorrelates well to a first-order parallel plate capacitor model that assumes constant area, and smalldeformation such that the polymer may be mechanically modeled by a spring that obeys Hookes law. Thissimple model predicts that the coefficient of proportionality is a function of Youngs modulus E= 0.8 GPaand surface area of the insulator, theoretically -1EA= -0.31 ppm/N. The outcome of this work is animproved insulator made from ultra-high molecular weight (UHMW) polyethylene and other approachestoward the minimization of and compensation for these experimental artifacts.
Electromagnetic Studies of Rocks↗
Large Deformation Constitutive Laws for Isotropic Thermoelastic Materials
We examine the approximations made in using Hooke's law as a constitutive relation for an isotropic thermoelastic material subjected to large deformation by calculating the stress evolution equation from the free energy. For a general thermoelastic material, we employ the volume-preserving part of the deformation gradient to facilitate volumetric/shear strain decompositions of the free energy, its first derivatives (the Cauchy stress and entropy), and its second derivatives (the specific heat, Grueneisen tensor, and elasticity tensor). Specializing to isotropic materials, we calculate these constitutive quantities more explicitly. For deformations with limited shear strain, but possibly large changes in volume, we show that the differential equations for the stress components involve new terms in addition to the traditional Hooke's law terms. These new terms are of the same order in the shear strain as the objective derivative terms needed for frame indifference; unless the latter terms are negligible, the former cannot be neglected. We also demonstrate that accounting for the new terms requires that the deformation gradient be included as a field variable.
36 MATERIALS SCIENCE↗
Experiments with airplane brakes
This report begins by examining the forces on the brake shoes. For the determination of the load distribution over the shoes it was assumed that the brake linings follow Hooke's law, are neatly fitted and bedded in by wear. The assumption of Hooke's law, that is, the proportionality between compression of the lining and the absorption of force, is fulfilled to a certain extent for the loading, as becomes apparent from the load tests described further on. But there is a material discrepancy at unloading. From the load distribution we merely defined the position of the normal force resultant, while for the rest, the effect of the distribution was disregarded in the comparison of the different shoe dispositions.
Michael, Franz↗
An In-Depth Tutorial on Constitutive Equations for Elastic Anisotropic Materials
An in-depth tutorial on the constitutive equations for elastic, anisotropic materials is presented. Basic concepts are introduced that are used to characterize materials, and notions about how anisotropic material deform are presented. Hooke s law and the Duhamel-Neuman law for isotropic materials are presented and discussed. Then, the most general form of Hooke s law for elastic anisotropic materials is presented and symmetry requirements are given. A similar presentation is also given for the generalized Duhamel-Neuman law for elastic, anisotropic materials that includes thermal effects. Transformation equations for stress and strains are presented and the most general form of the transformation equations for the constitutive matrices are given. Then, specialized transformation equations are presented for dextral rotations about the coordinate axes. Next, concepts of material symmetry are introduced and criteria for material symmetries are presented. Additionally, engineering constants of fully anisotropic, elastic materials are derived from first principles and the specialized to several cases of practical importance.
Nemeth, Michael P.↗
A general methodology for inverse estimation of the elastic and anelastic properties of anisotropic open-cell porous materials—with application to a melamine foam
This paper proposes an inverse estimation method for the characterisation of the elastic and anelastic properties of the frame of anisotropic open-cell foams used for sound absorption. A model of viscoelasticity based on a fractional differential constitutive equation is used, leading to an augmented Hooke's law in the frequency domain, where the elastic and anelastic phenomena appear as distinctive terms in the stiffness matrix. The parameters of the model are nine orthotropic elastic moduli, three angles of orientation of the material principal directions and three parameters governing the anelastic frequency dependence. The inverse estimation consists in numerically fitting the model on a set of transfer functions extracted from a sample of material. The setup uses a seismic-mass measurement repeated in the three directions of space and is placed in a vacuum chamber in order to remove the air from the pores of the sample. The method allows to reconstruct the full frequency-dependent complex stiffness matrix of the frame of an anisotropic open-cell foam and in particular it provides the frequency of maximum energy dissipation by viscoelastic effects. The characterisation of a melamine foam sample is performed and the relation between the fractional-derivative model and other types of parameterisations of the augmented Hooke's law is discussed.
36 MATERIALS SCIENCE↗
Mechanical Properties of Cells
Many physiologic and pathologic processes alter the biomechanical properties of the tissue they affect, and these changes may be manifest at the single cell level. The normal and abnormal mechanical properties of a given cell type can be established with the aid of an atomic force microscope (AFM), nonetheless, consistency in the area of the tip has been a mayor limitation of using the AFM for quantitative measurements of mechanical properties. This project attempts to overcome this limitation by using materials with a known elastic modulus, which resembles the one of the cell, to create force-deformation curves to calculate the area of indentation by means of Hooke s Law (sigma = E(epsilon)), which states that stress (sigma) is proportional to the strain (epsilon) where the constant of proportionality, E, is called the Young s modulus, also referred as the elastic modulus. Hook s Law can be rearranged to find the area of indentation (Area= Force/ E(epsilon)), where the indentation force is defined by the means of the added mass spring calibration method.
Bradley, Robert↗
Differential Poisson’s ratio of a crystalline two-dimensional membrane
We compute the differential Poisson’s ratio of a suspended two-dimensional crystalline membrane embedded into a space of large dimensionality d≫1. We demonstrate that, in the regime of anomalous Hooke’s law, the differential Poisson’s ratio approaches a universal value determined solely by the spatial dimensionality d c , with a power-law expansion ν=-1/3+0.016/d c +O(1/d c 2 ), where d c =d-2. Thus, the value -1/3 predicted in previous literature holds only in the limit d c →∞.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Determination of lattice parameters, strain state and composition in semipolar III-nitrides using high resolution X-ray diffraction
In group-III-nitride heterostructures with semipolar or nonpolar crystal orientation, anisotropic lattice and thermal mismatch with the buffer or substrate lead to a complex distortion of the unit cells, e.g., by shearing of the lattice. This makes an accurate determination of lattice parameters, composition, and strain state under assumption of the hexagonal symmetry impossible. In this work, we present a procedure to accurately determine the lattice constants, strain state, and composition of semipolar heterostructures using high resolution X-ray diffraction. An analysis of the unit cell distortion shows that four independent lattice parameters are sufficient to describe this distortion. Assuming only small deviations from an ideal hexagonal structure, a linear expression for the interplanar distances d{sub hkl} is derived. It is used to determine the lattice parameters from high resolution X-ray diffraction 2ϑ-ω-scans of multiple on- and off-axis reflections via a weighted least-square fit. The strain and composition of ternary alloys are then evaluated by transforming the elastic parameters (using Hooke's law) from the natural crystal-fixed coordinate system to a layer-based system, given by the in-plane directions and the growth direction. We illustrate our procedure taking an example of (112{sup ¯}2) Al{sub κ}Ga{sub 1−κ}N epilayers with Al-contents over the entire composition range. We separately identify the in-plane and out-of-plane strains and discuss origins for the observed anisotropy.
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
Connection stiffness and dynamical docking process of flux pinned spacecraft modules
This paper describes a novel kind of potential flux pinned docking system that consists of guidance navigation and control system, the traditional extrusion type propulsion system, and a flux pinned docking interface. Because of characteristics of passive stability of flux pinning, the docking control strategy of flux pinned docking system only needs a series of sequential control rather than necessary active feedback control, as well as avoidance of hazardous collision accident. The flux pinned force between YBaCuO (YBCO) high temperature superconductor bulk and permanent magnet is able to be given vent based on the identical current loop model and improved image dipole model, which can be validated experimentally. Thus, the connection stiffness between two flux pinned spacecraft modules can be calculated based on Hooke's law. This connection stiffness matrix at the equilibrium position has the positive definite performance, which can validate the passively stable connection of two flux pinned spacecraft modules theoretically. Furthermore, the relative orbital dynamical equation of two flux pinned spacecraft modules can be established based on Clohessy-Wiltshire's equations and improved image dipole model. The dynamical docking process between two flux pinned spacecraft modules can be obtained by way of numerical simulation, which suggests the feasibility of flux pinned docking system.
42 ENGINEERING↗
Resonant ultrasound spectroscopy for crystalline samples containing initial strain
Resonant ultrasound spectroscopy (RUS) is a mature and robust technique for the nondestructive characterization of the elastic properties of solids capable of providing the elastic constants of anisotropic crystalline solids. The traditional method is based on assuming that the solid is linear elastic and, therefore, obeys a linearized Hookean constitutive relationship (Hooke’s law). In this work, an alternative constitutive law is provided that allows for an initial strain or prestrain to be present stemming from residual stress. Then, the constitutive relationship is integrated into the RUS framework. The model is demonstrated using a realistic prestrain field obtained by simulating shot-peening processing of a polycrystalline Cu sample. The sensitivity of the resonances to the developed prestrain is established and discussed. This work allows researchers to consider the influence of initial strain or residual stress in their samples and the potential influence on accurate elastic constant estimates. The model also supports the potential of RUS for the nondestructive characterization of prestrain in materials.
42 ENGINEERING↗