Page:Encyclopædia Britannica, Ninth Edition, v. 7.djvu/831

ELASTICITY 807 most rigid of natural solids (as, for instance, iron). But it is scarcely possible to hold a piece of cork in the hand without perceiving the fallaciousness of this last attempt to maintain a theory which never had any good foundation. By careful measurements on columns of cork of various forms (among them, cylindrical pieces cut in the ordinary way for bottles), before and after compressing them longitudinally in a Bramah s press, we have found that the change of lateral dimensions is insensible both with small longitudinal contractions and return dilatations, within the limits of elasticity, and with such enormous longitudinal contractions as to ^ or | of the original length. It is thus proved decisively that cork is much more rigid, while metals, glass, and gelatinous bodies are all less rigid, in proportion to resistance to compression, than the supposed "perfect solid"; and the practical invalidity of the theory is experimentally demonstrated. By obvious mechanism of jointed bars a solid may be designed which shall swell laterally when pulled, and shrink laterally when compressed, in one direction, and which shall be homo geneous in the same sense (article 40) as crystals and liquids are called homogeneous. 49. Modulus of Simple Longitudinal Strain. In sec tions 45, 46, we examined the effect of a simple longitudinal stress in producing elongation in its own direction, and contraction in lines perpendicular to it. With stresses substituted for strains, and strains for stresses, we may apply the same process to investigate the longitudinal and lateral tractions required to produce a simple longi tudinal strain (that is, an elongation in one direction, with no change of dimensions perpendicular to it) in a rod or solid of any shape. Thus a simple longitudinal strain e is equivalent to a cubic dilatation e without change of figure (or linear dilata tion e equal in all directions), and two distortions consist ing each of dilatation e in the given direction and con traction e in each of two directions perpendicular to it and to one another. To produce the cubic dilatation e alone requires (section 44) a normal traction Ice equal in all directions. And, to produce either of the distortions simply, since the measure (section 43) of each is |e, requires a distorting stress equal to n x |-e, which consists of tangential tractions each equal to this amount, positive (or drawing outwards) in the line of the given elongation, and negative (or pressing inwards) in the perpendicular direction. Thus we have in all normal traction = (& + %n)e, in the direction of the given strain, and normal traction = (&-f?i)e, in every direction perpen dicular to the given strain. Hence the modulus of simple longitudinal strain is k + ^n. 50. Weight-Modulus and Length of Modulus. Instead of reckoning moduluses in units of force per unit of area, it is sometimes convenient to express them in terms of the weight of unit bulk of the solid. A modulus thus reckoned, or, as it is called by some writers, the length of the modulus, is of course found by dividing the weight-modu lus by the weight of the unit bulk. It is useful in many applications of the theory of elasticity, as, for instance, in this result, which is proved in the elementary dynamics of waves in an elastic solid or fluid (chap. xvii. of the Mathematical Theory, below) : the velocity of trans mission of longitudinal 1 vibrations (as of sound) along a bar of cord, or of waves of simple distortion, or of simple longitudinal extension and contraction in a homogeneous (4). 1 It is to be understood that the vibrations in question are so much spread out through the length of the body that inertia does not sensibly influence the transverse contractions and dilatations which (unless the substance have in this respect the peculiar character presented by cork, section 48) take place along with them. isotropic solid, or of sound waves in a fluid, is equal to the velocity acquired by a body in falling from a height equal to half the length of the proper modulus 2 for the case; that is, the Young s Modulus _i_ ) f r the case, the modulus of rigidity (n) for the second, the modulus of simple longitudinal strain (k + ^n) for the third, the modulus of compression k for the fourth. Remark that for air the static " length-modulus of compression " at constant temperature is the same as what is often techni cally called the " height of the homogeneous atmosphere." 51. In reckoning moduluses there must be a definite understanding as to the unit in terms of which the force is measured, which may be either the kinetic unit or the gravitation unit for a specified locality, that is, the weight in that locality of the unit of mass. Experimenters hitherto have stated their results in terms in the gravitation unit, each for his own locality, the accuracy hitherto attained being scarcely in any cases sufficient to require corrections for the different intensities of gravity in the different places of observation. The most useful and generally convenient specification of the modulus of elasticity of a substance is in grammes- weight per square centimetre. This has only to be divided by the specific gravity of the substance to give the length of the modulus. British measures, however, being still unhappily sometimes used in practical and even in scientific statements, we too often meet with reckonings of the modulus in pounds per square inch or per square foot, in tons per square inch, or of length of the modulus in feet or in British statute miles. The reckoning most commonly adopted in British treatises on mechanics and practical statements is pounds per square inch. The modulus thus stated must be divided by the weight of 12 cubic inches of the solid, or by the product of its specific gravity into 4335, 3 to find the length of the modulus in feet. To reduce from pounds per square inch to grammes per square centimetre, multiply by 70 - 31, or divide by 014223. French engineers generally state their results in kilogrammes per square millimetre, and so bring them to more convenient numbers, being 1 ^ of the incon- 2 In sections 73-76 we shall see that changes of temperature produced by the varying stresses cause changes of temperature which, in ordinary solids, render the velocity of transmission of longitudinal vibrations sensibly greater than th;it calculated by the rule stated in the text, if we use the static modulus as understood from the definition there given; and it will be shown how to take into account the thermal effect by using a definite static modulus, or kinetic modulus, according to the circumstances of any case that may occur. 3 This decimal being the weight in pounds of 12 cubic inches of water. The one great advantage of the French metrical system is that the mass of the unit volume (1 cubic centimetre) of water at its temper ature of maximum density (3 945 C.) is unity (1 gramme) to a sufficient degree of approximation for almost all practical purposes. Thus, ac cording to this system, the density of a body and its specific gravity mean one and the same thing; whereas on the British no-system the density is expressed by a number found by multiplying the specific gravity by one numl>er or another, according to the choice of a cubic inch, pint, quart, wine gallon, imperial gallon, cubic foot, cubic yarr), or cubic mile that is made for the unit of volume; and the grain, scruple, gunmaker s drachm, apothecary s drachm, ounce Troy, ounce avoirdupois, pound Troy, pound avoirdupois, stone (Imperial, Ayrshire, Lanarkshire, Dumbartonshire), stone for hay, stone for corn, quarter (of a hundredweight), quarter (of corn), hundredweight, or ton that is chosen for unit of mass. It is a remarkable phenomenon, belonging rather to moral and social than to physical science, that a people tending naturally to be regulated by common sense should voluntarily condemn themselves, as the British have so long done, to unnecessary hard labour in every action of common business or scientific work related to measurement, from which all the other nations of Europe have emancipated themselves. Professor W. H. Miller, of Cambridge, concludes, from a very trustworthy comparison of standards by Kupffer, of St Petersburg, that the weight of a cubic decimetre of water at tem

perature of maximum density is 1000 01 3 grammes.