Energy dissipation in flexural vibrations is considered by introducing into the Timoshenko beam equation viscous-damping terms proportional to the time rate of extensional strain. Two modes of wave transmission arise from the roots of a quartic equation. On the lower mode, a solution is indicated in terms of characteristic functions which represent the damped natural vibrations. For a specified range of values of damping, the amplitudes of natural vibrations are governed by characteristic damping factors which exhibit a maximum in the region of frequencies where wave velocities are dispersed. There can be no frequency cut-offs. In the higher mode of wave motion, the characteristic damping factors exhibit a monotone increase toward short wave-length limits analogous to those obtained in the Bernoulli-Euler-Sezawa equation.

The objective of the program discussed in this paper was to develop a method, using photoelasticity and low-modulus materials, for studying dynamic stress distributions. Early in the program a number of low-modulus materials were studied and Hysol 8705 (a urethane-rubber compound) was selected as the most promising. A complete study of its mechanical and optical properties was made under static and dynamic loadings. It was established, that Poisson’s ratio ν is independent of rate of loading, the stress fringe value f σ is independent of rate of loading for strain rates greater than 8 in/in/ sec, and both the modulus of elasticity E and the strain fringe value f ϵ were dependent oil the rate of loading. The specific energy loss for the material was about 10 per cent for the stress ranges associated with photoelastic determinations. Experimental observations of photoelastic fringe patterns in a rectangular strut subjected to axial impact were made to illustrate the potential of the method. Three different end conditions were imposed on the unloaded end of the strut: (a) A free end normal to the axis; (b) a fixed end normal to the axis; and (c) a free end inclined 45 deg to the axis. For the cases where the ends were normal to the axis it was found that the fringes followed the same law of reflection as the law for stresses given by elementary wave theory.

Stress-strain relations are given for an initially isotropic material, which is macroscopically homogeneous, but inhomogeneous on a microscopic scale. An element of volume is considered to be composed of various portions, which can be represented by subelements showing secondary creep and isotropic work-hardening in plastic deformation. If the condition is imposed that all subelements of an element of volume are subjected to the same total strain, it is demonstrated that the inelastic stress-strain relations of the material show anisotropic strain-hardening, creep recovery, and primary and secondary creep due to the nonuniform energy dissipation in deformation of the sub-elements. Only quasi-static deformations under isothermal conditions are considered. The theory is restricted to small total strains.

The internal-damping effects of varying composition, heat-treatment, cold work, magnetic-field strength, stress history, and static tension stress have been studied at room temperature using fixed-fixed beam specimens supported in a new bending vibration-decay apparatus. Experimental logarithmic decrements are converted to specific damping energies by a straightforward numerical method; such information is needed to extend a fundamental understanding of the physical nature of damping and to provide useful engineering-design data. Changing hardness from 225 to 388 Bhn was found to cause a wider variation in AISI 403 specimen damping than the application of a static tension stress or d-c magnetic field. Even at relatively low vibration stresses, considerable energy dissipation originates from microscopic plastic flow in soft materials.

A differential stress-strain relation is derived for a medium composed of a face-centered cubic array of elastic spheres in contact. The stress-strain relation is based on the theory of elastic bodies in contact, and includes the effects of both normal and tangential components of contact forces. A description is given of an experiment performed as a test of the contact theories and the differential stress-strain relation derived from them. The experiment consists of a determination of wave velocities and the accompanying rates of energy dissipation in granular bars composed of face-centered cubic arrays of spheres. Experimental results indicate a close agreement between the theoretical and experimental values of wave velocity. However, as in previous experiments with single contacts, the rate of energy dissipation is found to be proportional to the square of the maximum tangential contact force rather than to the cube, as predicted by the theory for small amplitudes.

A theoretical study is made of the structural damping in the bending of a simple built-up beam with thin reinforcing spar caps in which the rivets or screws prevent the sliding motion between the cap and the beam. An analytical expression of the energy loss per cycle of static loading is derived in terms of amplitude of load, stiffness of rivets, and tightness of joint. Experimental measurements on a test beam provide a qualitative verification of the theory.

The temperature distribution in the wake of a heated sphere 0.5 in. diam was determined at a gross stream velocity of 30 fps. Measurements were carried out in a channel 0.70 in. in height and 12 in. in width. The surface temperature of the sphere and the energy dissipation from it were determined as a function of gross stream velocities between 10 and 90 fps. The temperature distribution in the wake was correlated on the basis of methods developed for cylinders and satisfactory agreement was obtained.

The traction, moment-twist relation, torsional compliances, and frictional energy loss per cycle are computed for two like, elastic spheres in Hertz contact acted upon by a small oscillating torsional couple.

Numerical solutions of the momentum and energy equations are presented for particular types of laminar boundary-layer flow analogous to the Hartree “wedge flows.” Variation of the viscosity and of the thermal conductivity is considered under the circumstances of no dissipation, favorable pressure gradient, and the product of conductivity and density a constant. The solution is based on approximate representations of the velocity and temperature profiles in the boundary layer and these are of such character that the labor of calculation is minimized and the accuracy of the results preserved. The differential equations are reduced to two algebraic equations which rapidly yield the skin friction and the heat transfer in terms of the wall to free-stream temperature ratio for the desired value of Prandtl number. Numerical results are given for a range of wedge flows with gases of Prandtl number 0.70 and 1.0. These results reveal that when the free-stream velocity is variable the temperature difference between the wall and the free stream exerts a substantial effect on the velocity distribution in the boundary layer and on the skin-friction coefficient. Alternatively, the heat-transfer coefficient is not affected radically. A calculation method is presented for the determination of the heat transfer and skin friction for a flow with an arbitrary variation of velocity over an isothermal surface. This method utilizes the results of the present analysis for the variable property wedge flows.

An investigation is made of the phenomena occurring at the contact surfaces of like elastic spheres subjected to a variety of applied forces. Owing to the presence of slip, with its associated energy dissipation and permanent set, the changes in tractions and displacements depend not only upon the initial state of loading, but upon the entire past history of loading and the instantaneous relative rates of change of the normal and tangential forces. On the basis of three of the simplest cases of varying oblique forces, a set of rules of procedure is assembled and then applied to two types of problems. In the first, the initial tangential compliances are calculated for a variety of past histories and instantaneous rates of loading; in the other, a detailed history of a varying oblique force is investigated.

Energy of vibration may be dissipated by microscopic slip-on interfaces where machine elements are joined in a press fit. In this paper slip damping is studied as an agent in reducing turbine-blade resonant stresses and prolonging turbine life. A general theory of slip damping is developed and an expression for the energy loss per cycle of oscillation is found. The predictions of the theory are compared with the results of controlled experiments. It appears that the theory is in satisfactory agreement with experiment and with measurements made on turbine blades elsewhere in this country and abroad. The implications of the general theory in the design of turbine blades are discussed. It appears that slip damping is capable of being an effective agent in reducing resonant stresses, especially in the “stall-flutter” condition where aerodynamic damping is inadequate. The design of a slip-damping joint which would achieve theoretically possible energy decrements much larger than are present in existing commercial construction is shown to depend on the maintenance of an optimum contact pressure.

The ignition threshold of an energetic material (EM) quantifies the macroscopic conditions for the onset of self-sustaining chemical reactions. The threshold is an important theoretical and practical measure of material attributes that relate to safety and reliability. Historically, the thresholds are measured experimentally. Here, we present a new Lagrangian computational framework for establishing the probabilistic ignition thresholds of heterogeneous EM out of the evolutions of coupled mechanical-thermal-chemical processes using mesoscale simulations. The simulations explicitly account for microstructural heterogeneities, constituent properties, and interfacial processes and capture processes responsible for the development of material damage and the formation of hotspots in which chemical reactions initiate. The specific mechanisms tracked include viscoelasticity, viscoplasticity, fracture, post-fracture contact, frictional heating, heat conduction, reactive chemical heating, gaseous product generation, and convective heat transfer. To determine the ignition threshold, the minimum macroscopic loading required to achieve self-sustaining chemical reactions with a rate of reactive heat generation exceeding the rate of heat loss due to conduction and other dissipative mechanisms is determined. Probabilistic quantification of the processes and the thresholds are obtained via the use of statistically equivalent microstructure sample sets (SEMSS). The predictions are in agreement with available experimental data.

The Betz process for minimizing the energy loss of single propellers is extended to the propeller operating in radially varying inflow, with a view toward the development of optimum stern propulsion for bodies of revolution. Assuming the usual lifting-line treatment of the propeller blades, an application of the isoperimetric problem in the calculus of variations is directed toward minimizing the energy loss while maintaining constant total thrust. Resulting expressions describe the optimum radial distribution of induced velocity in terms of an undetermined multiplier. The magnitude of the required total thrust, in turn, fixes the value of this multiplier, making the solution determinate. Inclusion of blade-profile drag at a later point in the development serves to determine an optimum propeller diameter for minimum over-all energy loss. The method gains in accuracy as the ratio of blade number to advance ratio increases, due to an approximation in the relation between the circulation and induced velocity.

Engineering measurements of internal friction in metals have been obtained from the decay characteristics of tuning forks of metals of interest. A damping program at Battelle Memorial Institute, conducted for the Office of Scientific Research and Development, called for such measurements at temperatures up to 1500 F, and it was necessary to develop suitable apparatus for obtaining accurate results at those elevated temperatures. A photoelectric method of measuring tine amplitude decay, using shutters on a vibrating fork to modulate light directed at a photocell, was developed. It proved to give very satisfactory results and had none of the mechanical difficulties in operation and uncertainties in results that were encountered at high temperatures when platinum-wire Sauereisen-cemented strain gages were used. It was necessary to reduce energy losses from the fork to its support by using a compliant coupling. This suspension worked so well that the lower limit of decrement measurements was probably determined by acoustic losses. It is believed that such losses added less than 0.00004 to the decrements of steel forks. This report includes details of construction of the equipment and reviews tests performed to establish its accuracy.

When an elastic sphere collides with another perfectly elastic body, part of the initial kinetic energy is lost in starting elastic waves in the two bodies. The energy thus dissipated can be calculated by previous analytical methods only when it is a small fraction of the initial kinetic energy. This paper develops an approximate analytical method for this calculation which is applicable even when the greater part of the energy is dissipated. Thus in a particular case where the energy dissipated is 90 per cent of the original kinetic energy, an error of only 0.7 per cent is made. The new element in this analysis is the introduction of a “normalized” interaction force which is necessarily quite insensitive to one’s ignorance of the exact interaction force. The powerfulness of the method is illustrated by a complete survey of the problem of impact of spheres with beams fixed at each end. Graphs are constructed showing the variation of the coefficient of restitution with the length of the beam and the mass of the sphere. Impacts with multiple blows are excluded. It is found that when the mass of the sphere is kept constant, the coefficient of restitution has a minimum for a certain optimal beam length, and is independent of the beam length for values greater than twice the optimal length. The coefficient is a minimum when the period of the fundamental mode of vibration is approximately equal to 2.4 times the time of contact. The method is also applied to the impact of spheres with large thin plates. The semiempirical formula of Raman is derived. Although this analytical method cannot be applied to impacts with multiple blows, it nevertheless gives the conditions for such multiple blows.

The subject matter considered in this paper deals with the mathematical investigation of heat flow in an annular disk of uniform thickness. Originally, the investigation was carried on in connection with the design of fins for increasing the heat transfer in various kinds of heat exchangers and engines. The results of the study, however, might easily be applied to a number of other problems, since by altering the boundary conditions slightly one may use the same basic equations for calculating the temperature distribution and heat transmission in grinding wheels and disk clutches. The study of the problem in connection with fin design has brought forth other solutions for special cases of the general proposition considered here. The particular results obtained by previous investigators can be readily found from the general equations given in this paper. In order to assist in the numerical solution of the somewhat complicated equations a chart for evaluating the mathematical expressions has been included.

Fatigue resistance is crucial for the engineering application of metals. Polycrystalline metals with highly oriented nanotwins have been shown to exhibit a history-independent, stable, and symmetric cyclic response [Pan et al., 2017, Nature 551, pp. 214-217]. However, a constitutive model that incorporates the cyclic deformation mechanism of highly oriented nanotwinned metals is currently lacking. This study aims to develop a physically based model to describe the plastic deformation of highly oriented nanotwinned metals under cyclic loading parallel to the twin boundaries. The theoretical analysis is conducted based on non-uniform distribution of twin boundary spacing measured by experiments. During cyclic plasticity, each twin lamella is discretely regarded as a perfect elastoplastic element with a yielding strength depending on its thickness. The interaction between adjacent nanotwins is not taken into consideration according to the cyclic plasticity mechanism of highly oriented nanotwins. The modeling results are well consistent with the experiments, including the loading-history independence, Masing behavior, and back stress evolution. Moreover, the dissipation energy during cyclic deformation can be evaluated from a thermodynamics perspective, which offers an approach for the prediction of the fatigue life of highly oriented nanotwins. The cyclic plasticity modeling and fatigue life prediction are unified without additional fatigue damage parameters. Overall, our work lays down a physics-informed framework that is critical for the precise prediction of the unique cyclic behaviors of highly oriented nanotwins.

Two types of non-holonomic constraints (imposing a prescription on velocity) are analyzed, connected to an end of a (visco)elastic rod, straight in its undeformed configuration. The equations governing the nonlinear dynamics are obtained and then linearized near the trivial equilibrium configuration. The two constraints are shown to lead to the same equations governing the linearized dynamics of the Beck (or Pflüger) column in one case and of the Reut column in the other. Although the structural systems are fully conservative (when viscosity is set to zero), they exhibit flutter and divergence instability. In addition, the Ziegler's destabilization paradox is found when dissipation sources are introduced. It follows that these features are proven to be not only a consequence of “unrealistic non-conservative loads” (as often stated in the literature); rather, the models proposed by Beck, Reut, and Ziegler can exactly describe the linearized dynamics of structures subject to non-holonomic constraints, which are made now fully accessible to experiments.