806 ELASTICITY and the same in amount for all the three ways in which a stress may be thus applied to any one cube, and for different cubes taken from any different positions in the solid. Hence the elastic quality of a perfectly elastic, homogene ous, isotropic solid is fully denned by two elements, its resistance to distortion and its resistance to compression. The first has been already considered (section 43). The second is measured by the amount of uniform pressure in all directions per unit area of its surface required to pro duce a stated very small compression. The numerical reckoning of the first is the compressing pressure divided by the diminution of the bulk of a portion of the sub stance which, when uncompressed, occupies the unit volume. It is sometimes called the " elasticity of bulk," or sometimes the " modulus of bulk-elasticity," sometimes the resistance to compression. Its reciprocal, or the amount of compression on unit of volume divided by the compressing pressure, or, as we may conveniently say, the compression per unit of volume per unit of compressing pressure, is commonly called the compressibility. 45. Strain produced by a single Longitudinal Stress (subject of Young s Modulus). Any stress whatever may 1 be made up of simple longitudinal stresses. Hence, to find the relation between any stress and the strain produced by it, we have only to find the strain produced by a single longitudinal stress, which, for an isotropic solid, we may do at once thus : A. simple longi- tudinal stress P is equivalent to a uniform dilating tension JP in all directions, compounded with two distorting stresses, each equal to JP, and having a common axis in the line of the given longitudinal stress, and their other two axes Fi &- 4 - any two lines at right angles to one another and tc it. The diagram (fig. 4), drawn in a plane through one of these latter lines and the former, sufficiently indicates the syn thesis, the only forces not shown being those perpendicu lar to its plane. Hence if n denote the rigidity, and k the modulus of compression, or the modulus of bulk-elasticity (being the same as the reciprocal of the compressibility), the effect will be an equal dilatation in all directions, amounting, per unit of volume, to 3P fp ...... <, compounded with two equal distortions, each amounting to ...... < s , n /J and having (section 43, footnote) their axes in the directions just stated for the axes of the distorting stresses. 46. The dilatation and two shears thus determined may be conveniently reduced to simple longitudinal strains by following the indications of section 43, thus : The two shears together constitute an elongation amount ing to -- in the direction of the given force P, and equal ip contraction amounting to -- in all directions perpendicu- fit i IP lar to it. And the cubic dilatation y- implies a lineal dilatation, equal in all directions, amounting to applicable ; so that if any distorting stress produced a condensation, an opposite distorting stress would produce a dilatation, which is a violation of the isotropic condition. 1 Mathematical Theoiy, chap. viii. On the whole, therefore, we have linear elongation- -^(3^ + 97;.), in the direction of the applied stress, and linear con traction pendicular to 1 1 - f 6/4~9Zv ni a directions per- 1
- o the applied stress. )
(3) 47. Hence " Young s Modulus" = 9/zifc and when the ends of a column, bar, or wire of isotropic material are acted on by equal and opposite forces, it experiences a lateral lineal contraction equal to ~ of the longitudinal dilatation, each reckoned as usual per unit of lineal measure. One specimen of the fallacious mathematics re ferred to in chap. xvi. of the mathematical theory below is a celebrated conclusion of Navter s and Poisson s that the ratio of lateral contraction to elongation by pull without transverse force is . This would require the rigidity to be f of the resistance to compression, for all solids; which was first shown to be false by Stokes 2 from many obvious observations, proving enormous discrepancies from it in many well-known bodies, and rendering it most improbable that there is any approach to a constancy of ratio between rigidity and resistance to compression in any class of solids. Thus clear elastic jellies and india-rubber present familiar specimens of isotropic homogeneous solids which, while differing very much from one another in rigidity (" stiffness "), are probably all of very nearly the same com pressibility as water, which is about ;> 10 oo per atmosphere. Their resistance to compression, measured by the reciprocal of this, is obviously many hundred times the absolute amount of the rigidity of the stiffest of those substances. A column of any of them, therefore, when pressed together or pulled out, within its limits of elasticity, by balancing forces applied to its ends (or an india-rubber band when pulled out), experiences no sensible change of volume, though very sensible change of length. Hence the propor tionate extension or contraction of any transverse diameter rmist be sensibly equal to half the longitudinal contraction or extension ; and such substances may be practically regarded as incompressible elastic solids in interpreting all the phenomena for which they are most remarkable. Stokes gave reasons for believing that metals also have in general greater resistance to compression, in proportion to their rigidities, than according to the fallacious theory, although for them the discrepancy is very much less than for the gelatinous bodies. This probable conclusion was soon experimentally demonstrated by Wertbeim, who found the ratio of lateral to longitudinal change of lineal dimensions, in columns acted on solely by longitudinal force, to be about | for glass and brass ; and by Kirchhoff, who, by a well-devised experimental method, found 387 as the value of that ratio for brass, and 294 for iron. For copper it is shown to lie between 226 and 441, by experiments 3 quoted below, measuring the torsional and longitudinal rigidities of copper wires. 48. All these results indicate rigidity less in proportion to the compressibility than according to Navier s and Poisson s theory. And it has been supposed by many naturalists who have seen the necessity of abandoning that theory as inapplicable to ordinary solids that it may bo regarded as the proper theory for an ideal perfect solid, and as indicating an amount of rigidity not quite reached in any real substance, but approached to in some of the " On the Friction of Fluids in Motion, and the Equilibrium and Motion of Elastic Solids," Trans. Camb. Phil. .Soc, April 1845. Se-> also Camb. and Dub. Math. Jour., March 1848. 3 " On the Elasticity and Viscosity of Metals " (W. Thomson), Pros.
R. S., May 1865.
