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.↗
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↗
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.↗
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↗
A Data-Driven Method for Modeling Creep-Fatigue Stress- Strain Behavior Using Neural ODEs
In this paper, we introduce a data-driven machine learning approach for modeling one-dimensional stress–strain behavior under cyclic loading, utilizing experimental data from the nickel-based Alloy 617. The study employs uniaxial creep–fatigue test data acquired under various loading histories and compares two distinct neural network-based ODE models. The first model, known as the black-box model, comprehensively describes the strain–stress relationship using a Neural ODE equation. To interpret this black-box model, we apply the Sparse Identification of Nonlinear Dynamical Systems (SINDy) technique, transforming the black-box model into an equation-based model using symbolic regression. The second model, the Neural flow rule model, incorporates Hooke’s Law for the linear elastic component, with the nonlinear part characterized by a Neural ODE. Both models are trained with experimental data to accurately reflect the observed stress–strain behavior. We conduct a detailed comparison with the standard Chaboche model, which includes three back stresses. Our results demonstrate that the neural network-based ODE models precisely capture the experimental creep–fatigue mechanical behavior, exceeding the standard Chaboche model’s accuracy. Furthermore, an interpretable model derived from the black-box neural ODE model through symbolic regression achieves accuracy comparable to the Chaboche model, enhancing its interpretability. The results highlight the potential of neural network-based ODE models to depict complex creep–fatigue behavior, eliminating the necessity for experts to define a specific, material-focused model form.
creep-fatigue↗
Uncovering hidden bias in neutron diffraction residual strain measurements
When calculating residual strain via neutron or X-ray diffraction, uncertainties propagated from the peak fit are often inadequate to describe the true scatter of measurements about a singular strain state, such as one that should describe a macroscopic continuum. Because diffraction is inherently a selective process, orientation-dependent scatter arises from the sub-sampling of strong microstructure and strain gradients. This paper investigates the appropriateness of propagated uncertainties with reference to their original intention, i.e. noise about a mean value. Thirty-six unique orientations of strain measurements are taken at multiple locations within an additive friction-stir deposition component with fine-scale gradients (∼200 µm) of plastic strain, texture and residual elastic strain. Multiple strain and stress calculation pathways are compared: direct substitution of three measurements into Hooke's law, direct inversion of any six unique orientations into the strain state tensor and thirty-six measurement least-squares estimation. For the last two cases, the appropriateness of the uncertainty interval is statistically evaluated on the basis of a physical constraint: common agreement under the strain transformation law. For this sample, the direct inversion of six measurements retains a conservative estimate of the uncertainty. However, propagated uncertainties in the least-squares solution greatly underestimate the true experimental scatter. A simple pathway to estimate appropriate uncertainty intervals is suggested. These results demonstrate that the interpretation of uncertainty in residual strain is strongly dependent on intrinsic sample-dependent effects, and that oversampling orientations and statistical analysis can give more accurate results with realistic uncertainties.
36 MATERIALS SCIENCE↗
Angular correlation in the helium atom.
Angular correlation in helium atom with electron- electron interaction
ELECTRON INTERACTION↗
Experimental investigation of strains in fabric under biaxial and shear forces.
The paper defines the experimental phase of an objective to obtain the mechanical characteristics and coefficients required by the generalized form of Hooke's law for nylon-polyurethane-coated fabric. Test specimens were cylindrical fabric sleeves and were loaded in axial tension by an Instron, in hoop tension by pressurizing, and in shear by a torquing fixture. An extensive amount of strain data is included for a wide combination of the three membrane loads. The tests indicate highly nonlinear stress-strain characteristics of the fabric and a strong dependency on all three membrane loads.
Alley, V. L., Jr.↗
Large amplitude flexural vibration of thin elastic flat plates and shells
The general equations governing the large amplitude flexural vibration of any thin elastic shell using curvilinear orthogonal coordinates are derived and consist of two coupled, nonlinear, partial differential equations in the normal displacement w and the stress function F. From these equations, the governing equations for the case of shells of revolution or flat plates can be readily obtained as special cases. The material of the shell or plate is isotropic and homogeneous and Hooke's law for the two-dimensional case is valid. It is suggested that the difference between the hardening type of nonlinearity in the case of flat plates and straight beams and the softening type of nonlinearity in the case of shells and rings can, in general, be traced to the amount of curvature present in the underformed median surface of the structure concerned.
Pandalia, K. A. V.↗
Determination of the technical constants of laminates in oblique directions
An off-axis tensile test theory based on Hooke's Law is applied to glass fiber reinforced laminates. A corrective parameter dependent on the characteristics of the strain gauge used is introduced by testing machines set up for isotropic materials. Theoretical results for a variety of strain gauges are compared with those obtained by a finite element method and with experimental results obtained on laminates reinforced with glass.
Vidouse, F.↗
An investigation of the isotropy of epoxy polymers
Measurements of the engineering constants E, G, and nu are routinely made for polymeric materials. If these materials are isotropic, these measurements should satisfy the relationship G = E/2(1 nu). However, many past measurements have indicated that this relationship is not satisfied. This raises questions about the assumptions of material isotropy and the applicability of Hooke's law. The methods used to measure these engineering constants for a number of different polymers are first described. Then, new results obtained in the current investigation are presented, indicating that the elastic constants do in fact satisfy the isotropic relationship for strains up to 0.5 percent. However, it is shown that at strain levels above this level, the relationship between stress and strain is nonlinear.
Odom, Edwin M.↗
Ratchetting behavior in viscoplasticity - A technical note
Viscoplastic models that use the Armstrong and Frederick (1966) kinematic hardening relationship are known to generally overpredict the observed accumulation of ratchet strains. The reason for this behavior is a consequence of the mathematics used to describe Hooke's law and the evolution equations for plastic strain and back stress. Conditions for ratchetting exist whenever there is a cycle-averaged mean stress present over a cyclic loading path. Two types of controlled loading paths are often studied: cyclic relaxation is the ratchet mechanism associated with displacement (strain) control, while cyclic creep is the ratchet mechanism associated with load (stress) control. Overcoming the possible adverse trends in predicted ratchetting behavior is a difficult task, and how one ought to go about doing this is not yet completely clear. We therefore choose to discuss the mathematical cause of ratchetting and how it has been dealt with to date, but make no attempt to correct this flaw.
Freed, Alan D.↗