Elastic Imperfection Research Articles

Effective Thermoelasticity of Polymer-Bonded Particle Composites with Imperfect Interfaces and Thermally Expansive Interphases

A micromechanical model for the thermoelasticity of polymer-bonded composites is presented. The model is particularly aimed at describing materials where the polymeric binder phase undergoes non-negligible thermal expansion affecting the overall thermoelastic response. Constitutive choices for modeling a mixed binder-void interphase layer are proposed, and an associated decomposition of total eigenstrains into classical, elastic imperfection (damage), and binder thermal expansivity parts is examined within the context of imperfect inter-particle interfaces. A novel temperature dependent modified Eshelby tensor is identified, making possible the development of a temperature dependent modified self-consistent homogenization scheme—what we call the M $\theta$ -SCH model. A method for distinguishing between dispersed and isolated parts of the binder and void phases in the model is also provided, along with a description of particle coating (or interphase) thickness derived from particle morphology and mesoscale effective properties. Although the theory is general, its development is motivated by the need to model anisotropic and highly nonlinear observed thermal expansion behavior of the polymer bonded explosive PBX 9502, for which model simulations are performed and compared with existing measurements.

Imperfect interface effect for nano-composites accounting for fiber section shape under antiplane shear

This paper investigates the elastic responses of fibrous nano-composites with imperfectly bonded interface under longitudinal shear. The proposed imperfect interface model is the shear lag (or the spring layer) model; the presented nano interfacial stress model is the Gurtin–Murdoch surface/interface model; and the three-phase confocal elliptical cylinder model is the geometry model accounting for the fiber section shape. By virtue of the complex variable method, a generalized self-consistent method is employed to derive the closed from solution of the effective antiplane shear modulus of the fibrous nano-composites with imperfect interface. Five existing solutions can be regarded as the limit form the present analytic expression. The influences of the interface elastic constant, the interfacial imperfection parameter, the size of the elliptic section fiber, the fiber section aspect ratio, the fiber volume fraction and the fiber elastic property on the effective antiplane shear modulus of the nano-composites are discussed. Particularly, numerical results demonstrate that the interfacial elastic imperfection will always cause a significant reduction in the effective antiplane shear modulus; and the fiber interface stress effect on the effective modulus of the fibrous nano-composites will weaken with the interfacial imperfection increases.

Inelastic Processes in the Shell

It has been known for some thousands of years that the Moon keeps a nearly constant face to the Earth. Only one explanation has been offered of how this state can have come about, namely tidal friction in the interior of the Moon. Darwin made an extensive study of the effects of tidal friction in the Earth and other planets, with special reference to the apparent secular acceleration of the Moon. Mercury and known satellites appear to keep constant faces to their primaries (there seems to be some d,oubt now about Mercury, but the rotation still appears to be slow). Turbulence in tidal currents in shallow seas seems to account for most of the friction for the Earth; but the other bodies have probably never had shallow seas, and we must infer some imperfection of elasticity at strains of the order of It is very hard to detect such imperfection in the laboratory. Other types however are very conspicuous. Fracture in igneous rocks and well compacted metamorphic ones such as quartzites occurs at strains of the order of and there is a complicated non-linear behaviour when the elastic strains are somewhat less than this; this may lead to permanent deformation far exceeding the direct elastic one. It was shown by various geodesists that mountains produce far smaller disturbances of gravity than would be expected if they were simply added loads. An explanation was given by Airy, who postulated a lower layer of denser but weaker matter. Extra load on a thin crust would push the interface down, and the resulting lack of heavy matter would largely compensate for the extra attraction of the mountain. Later Suess made the crucial inference on geological grounds that most surface rocks are essentially derived from granite or similar rocks with densities about 2.7, and overlie a layer with a density of 3-0 or so. Then A. MohorovilSiC in 1909 detected a duplication of earthquake pulses, as if one pair, Pg and Sg, travelled in an upper layer but another, Pn and Sn, in a lower one with higher velocities. Curiously, none of the continental seismologists at the time appear to mention Suess or Airy. The first mention in geophysics of the geological inference was, I think, in the paper by Winch and me in 1922 on the Oppau explosion. Later comparison with experiment confirmed these early results except that the lower layer seemed to need a density more like 3.3 and the upper was a great deal more complicated. But this is still too simple. Hayford’s survey of the United States assumed exact uniformity of mass per unit area and greatly reduced the residuals in comparison with the hypothesis that extra mass on the surface is not accompanied by defect below. But Helmert noticed that successive residuals along a survey line tended to keep the same sign, and on this ground multiplied Hayford’s uncertainties by about 3. Barrel1 inferred hence that the lower layer cannot be completely weak; he supposed a layer about lOOkm down where the strength is much less than that of surface rocks, but still appreciable. He called this the asthenosphere. Again confirmation came from Turner’s discovery of deep-focus earthquakes in 1921. Earthquakes comparable with the strongest shallow ones occur to depths of about a tenth of the Earth‘s radius. They send out strong S pulses. This is definite evidence that a discontinuous change occurs in the stress-difference and the material must have been able to stand that until the earthquake took place. (Incidentallyl it is hard to

Elastic imperfection sensitivity of hexagonally packed circular–cell honeycombs

The sensitivities of the in–plane macroscopic linear stiffnesses of perfectly elliptical–cell honeycombs to geometric imperfections are derived through an analytical method and a finite–element–based numerical solution. The sensitivities of the honeycomb stiffness to three types of initial imperfections are studied. These imperfections are the deviation from circularity of a cell, a uniform change in thickness of honeycomb cell walls and thickness variations along the honeycomb cell wall. The sensitivities of the in–plane macroscopic stiffnesses to the deviation from circularity of the cell and to the thickness change of the honeycomb cells are derived analytically. The sensitivities of the in–plane properties to thickness variations along the cell wall are obtained numerically.

Complete Asymptotic Expansions of Solutions of the System of Elastostatics in the Presence of an Inclusion of Small Diameter and Detection of an Inclusion

We consider the system of elastostatics for an elastic medium consisting of an imperfection of small diameter, embedded in a homogeneous reference medium. The Lame constants of the imperfection are different from those of the background medium. We establish a complete asymptotic formula for the displacement vector in terms of the reference Lame constants, the location of the imperfection and its geometry. Our derivation is rigorous, and based on layer potential techniques. The asymptotic expansions in this paper are valid for an elastic imperfection with Lipschitz boundaries. In the course of derivation of the asymptotic formula, we introduce the concept of (generalized) elastic moment tensors (Polya–Szego tensor) and prove that the first order elastic moment tensor is symmetric and positive (negative)-definite. We also obtain estimation of its eigenvalue. We then apply these asymptotic formulas for the purpose of identifying with high precision the order of magnitude of the diameter of the elastic inclusion, its location, and its elastic moment tensors.

The inverse bending problem for a thin plate with an elastic imperfection

We consider the inverse problem consisting of determining the unknown shape of an elastic imperfection contained in a thin plate from the condition of equal strength in the stressed state along the phase interface surface. It is shown that such a state is attained in the case of an elliptic imperfection whose shape depends on the values of the applied moments and the mechanical properties of the component phases. It is established that for the geometry found for the imperfection the sum of the moments is constant and the second invariant of the deviator of the stress tensor is superharmonic over the entire plate. Numerical computations are carried out. In special cases the results obtained coincide with known data. One figure. Bibliography: 5 titles.

Dynamic analysis of finite linear elastic solids containing small elastic imperfections: Theory with application to asymmetric circular plates

Eigensolutions, Green's functions, and steady state responses of three-dimensional, homogeneous, isotropic, finite linear elastic solids, containing small linear elastic imperfections, are presented in this paper. The inhomogeneities are treated as perturbations to otherwise homogeneous solids, the spectral eigensolutions of which are known. The exact eigensolutions of the imperfect solid, formulated through the Betti reciprocal theorem, satisfy a set of integral equations from which both analytic perturbation formulas and a numerical iteration scheme are developed. Special emphasis is given to analysis of perfect, homogeneous, elastic solids that are spectrally degenerate. Finally, the vibration of circular plates with evenly spaced radial inclusions, which cause splitting of some of the repeated vibration modes, is analyzed both analytically and numerically for the eigensolutions.

High Pressure Method for Identification of Crystal Elastic Imperfections - Generation of Defects in Silicon Single Crystals Under Pressure

ABSTRACTNew method for identification of the kind of elastic stress-free crystal imperfections which are not visible by standard X-ray and electron microscopy methods was presented. The method is based on the high pressure annealing creating large stresses at defect sites,followed by X-ray topography. This method was applied in structural defects studies of high quality silicon single crystals and allowed to determine the composition and structure of precipitates in silicon.

On imperfection of elasticity in the Earth's interior

Attempts are made to estimate the distribution of damping with depth, based on the observed damping of P waves, of free vibrations, and of surface waves and partly on a discussion by D. L. Anderson and R. S. Hart. Results are consistent in order of magnitude, but could easily be wrong by a factor of 2. Comments are added on the need for further work. Stress is laid on the damping of the free nutation, which indicates that α in the modified Lomnitz law is about 0.2 on an average.

Creep in the earth and planets

Evidence of imperfectly elastic behaviour in the earth and planets is provided by the following phenomena: 1. (1) damping of the earth's free nutation; 2. (2) damping of elastic waves passing through the earth; 3. (3)secular acceleration of the moon; 4. (4) figure of the moon; 5. (5) figure of the earth; 6. (6) rotations of satellites of the planets; 7. (7) free oscillations of the directions of the moon's axes; 8. (8) eigenfrequencies of the earth; and 9. (9) damping of the S wave as a function of frequency. Darwin attempted to explain the secular acceleration of the moon by viscosity in the body of the earth. This was shown to be unsatisfactory in 1915. A logarithmic law of creep, containing one relevant constant, agrees with some laboratory tests and approximately with a large quantity of seismological data. But it was found that if the constant is chosen to fit the damping of the free nutation (Eulerian nutation or Chandler wobble) the beginning of a transverse seismic wave at distance 80° would be spread over about 70 sec and be unreadable. I propose a law of creep by which for a constant shearing stress P the response is a shearing strain ϵ given by: ϵ = P μ [1 + q α {(1 + at) α − 1 ]} where t is time and μ, q, a, α are constants. This is a particular case of imperfection of elasticity. In it, for disturbances lasting a second or more, there are essentially only two creep parameters to be fixed, α and qa α . These may be determined by the two data: 1. (1) that the relaxation time of the free nutation is 30 ± 20 yyears; 2. (2) that S at 80° reaches half its ultimate value in 2 sec. We obtain α as probably between 0.14 and 0.21, with 0.19 as the most probable value. We then proceed to show that this law of creep, with α = 0.19, can account for approximately the right amount of damping (or internal friction) in all the other phenomena listed above. Laws with α = 1 (elastico-viscous law) and α = 0 (logarithmic law) fail to do so. A consequence of this law of creep is that where it holds convection cannot take place.

Creep in the Earth and Planets (Invited Lecture)

An outstanding problem is to reconcile the Moon's rotation with the persistence of its non-hydrostatic dynamical ellipticities. The first requires imperfection of elasticity under strains of order 10−7; the second apparently little under larger ones.Lomnitz gave experimental evidence that creep is linear at elastic shears from 10−5to 10−4, indicating that a linear rule could be right at still smaller values. Positive evidence for the Earth comes from the damping of the 14-monthly nutation, which has a relaxation time of the order of 30 yr. Most work on imperfect elasticity has assumed that under constant shear stress the strain increases with time either liket(elasticoviscosity) or like logt.If the result from the 14-monthly nutation, with elasticoviscosity, is applied to the Moon, the dynamical ellipticities would have subsided considerably in the last 200 yr. With the logarithmic rule anSpulse at 80° would have its beginning spread out over about 70 s and be unreadable. These contradictions are avoided if the increase under constant stress is about liket0.2. The resulting law involves two constants. Without change of these, applications are made to other phenomena. The rotations of the Moon and of other satellites whose rotations are known are explained; so is the persistence of the Moon's dynamical ellipticities; also the failure to detect three free oscillations that might theoretically exist. Elasticoviscosity would imply rapid disappearance of the non-hydrostatic second and third harmonics in the Earth's gravitational field; this is avoided with the new law. Study of damping of free vibrations of the Earth (including surface waves) has usually assumed the logarithmic law, but it appears that the new law fits the data at least as well, and that it may also explain those that have been interpreted in favour of layers of low velocity. It appears that the damping at depths up to 400 km or so is much more severe than the average for the Earth's shell, and more evidence for shorter periods is much needed.Any law with an index less than 1 would forbid thermal instability (convection) and continental drift.

Vibration-frequency transducer errors resulting from elastic imperfections in their materials

Vibration-frequency transducer errors resulting from elastic imperfections in their materials

Double aging of stainless steels

1. The elastic limit of austenitic-martensitic steels Kh17N7YuT and Kh15N7M2Yu can be increased substantially by double or triple aging. 2. Double aging reduced the development of elastic imperfections in the steels investigated (hysteresis, direct elastic aftereffect).

High-accuracy spring-type dial indicator

1. The creation of an industrial model of an instrument in accordance with the described scheme practically solves the problem of increasing the accuracy of spring-type dial indicators up to values satisfying class 0.1 of GOST 11606-65 with a reserve accuracy of 0.03–0.04%, which is sufficient for compensation of the errors occurfing due to imperfection of the lever system, and the attainment of the scales in the set of accuracy corresponding to class 0.1 per GOST 13712-68. This significantly extends the field of application of spring-type dial indicators. 2. The solution of this problem is provided by simple design means, which is an advantage of the described instrument in comparison with existing models. This instrument incorporates a spring-type force meter stiffness regulator, with the aid of which the indicator is adjusted using a simple method widely used in instruments of various accuracies. All this ensures minimum expenditure of labor in manufacturing the instrument and ensures its being competitive. 3. In this instrument it is not possible to attain the accuracy of quadrant dial indicators, i. e., 0.05% of the greatest measurement limit. Such accuracy, in our opinion, is practicallyunattainable in spring-type dial indicators due to the impossibility of significant reduction of elastic imperfections of the spring-type force meter when manufacturing springs using economically acceptable methods. 4. The problem of creating an instrument of class 0.1, as required by GOST 11606-65, is reduced to providing a basic allowable error corresponding to this class, i. e., obtained under normal test conditions [11]. So far as the reduction of additional errors arising due to changes of temperature and other external factors in the instrument's operating environment are concerned, the problem is no less (if not more) complex and is the subject of independent investigations [3]. 5. An economically desirable means for further increase of the accuracy of spring-type dial indicators is improvement of the supports of axles, in particular the use of miniature ball bearings without an internal ring, or supports kinematically connected to the deformed force meter and transmitting the rotary motion bearing on their axles.

Imperfections of Elasticity and Continental Drift

Imperfection of elasticity in the Earth follows a law that forbids convection and continental drift.

Damping of S Waves

THE modified Lomnitz law of imperfection of elasticity makes the strain under constant stress P: a is probably such that at is large for t = 1 second. For longer times the relevant constants are the rigidity μ, α, and qaa. The data used to estimate α and qaa are the damping of the 14-monthly variation of latitude and the fact that S at a distance of about 80° is clear and attains half the value for pure elasticity in about 2 sec. An alternative, possibly better, is to take the ratio as ¼. These lead to estimates as follows1: These lead to no contradictions with other data, such as have been found in attempts to apply the elastico-viscous law. Since estimates of damping have now been made from surface waves, it appears interesting to compare the results with those found from the foregoing law. In this communication values are found for S waves.

XXV. Discussion

I have been asked to make a written contribution to the discussion, but have nothing new to say. My views will be found in the 1961 version of The Earth and in my lecture to the Royal Astronomical Society last October, which will be published in their Quarterly Journal. My main points are that the only type of imperfection of elasticity considered in convection and drift theories is the elastico-viscous law, which has been found to lead to numerous contradictions when confronted with actual evidence. Different phenomena led to values of the effective viscosity differing by factors of millions. On the other hand, a modified law, chosen to fit two quantitative data and applied far beyond the range of periods related to those data, has steered its way nicely among the other evidence for some sort of imperfection of elasticity, without giving any contradiction. But it does forbid convection and continental drift. I should be disposed to agree that inability to explain an alleged phenomenon is not necessarily a disproof of that phenomenon; but it does require a higher standard of scrutiny of the evidence for that phenomenon. The standard actually applied to evidence for continental drift seems to be considerably lower than is usual for a new phenomenon, and is not associated with any alternative explanations of things that can be explained.

Observations on the X-Ray Structure of Rubber and the Size and Shape of Rubber Crystallites

Abstract In former times, we used to be painfully aware of the shortcomings and elastic imperfections of Hevea rubber. With its disappearance, we have come to think of it as having an ideal balance in physical properties for a rubberlike material which it has been difficult to approach with synthetic polymers. for this reason, it is still important to investigate the molecular structure of Hevea rubber and to try to understand the characteristics of this structure which are responsible for its physical properties. X-Ray diffraction methods can be applied to the problem of the molecular structure of Hevea rubber and a few synthetic rubbers, such as Butyl rubber and Neoprene, because crystallization occurs upon stretching. A detailed description of the x-ray diffraction results with rubber is available in a review article and need not be repeated here. It should be pointed out that the story obtained from the x-ray structure is not complete because there are important aspects of the structure which are not revealed by this means. It is not possible to measure directly the length of the chain molecules. The nature of the amorphous phase, such as the system of cross-linking of the long chain molecules on vulcanization, does not become evident in x-ray patterns. The physical properties of Hevea rubber must depend on a delicate balance of primary and secondary valence forces. The x-ray method does not permit any direct measurement of these forces but merely shows the geometrical arrangement which results from the molecular forces. Even with these limitations, much valuable information can be secured on the nature of the molecular rearrangements which occur upon stretching. Deductions can be drawn from the x-ray diffraction results regarding the form and spatial relationships of the long chain polymeric molecules and the manner in which they interact under stress. Correlations can then be looked for between the crystallization and the physical properties.

On plasticity and creep in solids

1. When a distortional stress is applied to a solid it produces an instantaneous deformation, which would then remain constant so long as the stress was kept constant if the solid was perfectly elastic. All solids, however, show imperfection of elasticity under sufficiently large stresses, and many do so even when the stress is small. Such imperfection takes the form of a continuous increase of the deformation after the stress has been applied. In some cases the deformation appears to increase indefinitely with the time so long as the same stress is kept on; and in these cases a permanent deformation remains after the stress is removed. In other cases it tends to a finite limit, which is not surpassed unless the stress is further increased; and in many of these, when the stress is removed, the body first returns towards its original state by an amount equal to the original elastic distortion, but then proceeds to creep slowly in the same direction until the whole, or nearly the whole, of the deformation has been recovered. The former type of behaviour, involving permanent deformation, may be called plasticity, the latter elastic afterworking or creep. The distinction is important, but in some accounts of experimental work it does not appear that the stress was maintained long enough to ascertain whether the deformation was tending towards a definite limit, or that the body was sufficiently studied after removal of the stress to find out whether the original state was ultimately recovered; and in consequence phenomena have in some cases been attributed to plasticity that are really due to creep. The present paper gives first a new derivation of the equations of plastic flow and discusses its relation to some earlier work; second, a theory of creep and a discussion of its relation to experimental evidence; and third, some geophysical applications of the results.