40 G.F.BECKER — FINITE STRAIN IN" ROCKS. 



which is ;ui hyperbola in L and e asymptotic to 



■ , and L 



::/.•--. 1/" :\I,-M 



Thus the fundamental assumption really made in the theory of elas- 

 ticity is thai the load-strain curve is an hyperbola instead of the straight 

 line which Hooke supposed to represenl the relation. The difference, 

 however, as already remarked, is without consequence, so Long as de- 

 ductions from it are confined to very minute deformations/- 



Stress System. — Any force acting on one face of a cube may be resolved 

 into a normal component and two tangential components acting in the 

 directions of the edges of the face. Hence the most general system of 

 forces of constant direction acting on a cube is resoluble into six normal 

 components and twelve tangential ones. If the center of inertia of the 

 cube is at rest, the normal forces on opposite faces must be equal, and 



*Naturt of tin' Proof of Hooke's Liw. — Hooke's law holds good, or, in other words, there ia a linear 

 relation involving a finite parameter betwe ra sm ill str - - and strains, provided — train 



curve fulfills two conditions, viz.. that the curve is continuous both in form and value, and that the 

 tangent of the angle which it makes with the axes at the origin is finite. Ii seems to me that 

 some discussion and even some confusion would have been avoided if -elasticians had taken this 

 u lometrieal view of the functions rather than a purely alg bi deal one. Thus Green simply 

 assumed that the stress-strain function was developable, and that the development contained i 

 term in which only the first power of the variable appeared, while < lebsch seems to have lool 



upon this algebraical relation as a mathematical n ssity. This it certainly is not, for there are 



many continuous functions the development of which contains no term in the first power of the 

 variable. These all represent curves which coincide with one of the axes at the origin : e. g . the 

 hyperbola referred to the vertex as origin.— Mr J. W. [bbetson, in his excellent Mathematical 

 Theorj of Elasticity, makes an attempt to demonstrate Hooke's law by pure reason, independently 

 8f experiment, tie expressly assumes, however, that the eurvi is i luous, and he states, 



without any atl imp! al pi th I i i ite of • n i ition of any traction component with any strain 



rdinal -can never change sign or vanish. '1'his last is equivalent to asserting that the curve 



cannot coincide with either axis at the origin. These two assumptions tog ther cover the whole 

 ground of Hools i's law, and really leave nothing to be | I 5aint-Venant, in his edition of 



Clebsch, p. 10, attempted to show thai if the internal stresses of an elastic mass depend in any 



it in i mas manner on the mutual distances of the molecules, Hooke's law follows. He points out 



thai continuity involves a linear relation between the differentials of a function and th 



differentials of any variable. He then sho on the assumption mad - corresponding small 



and strains are corresponding differentials, and deduces the conclusion stated above. 

 This arg«menl does nol sa al all, for though one maj undoubtedly wri =Adx, 



when- A is con i 'iii md tl on is therefore linear, yet A may have and often do !S have the 



valu iro or infinity. Saint-A maul made no attempt in the passage referred to to show that I 

 must !"■ finite in the case of elastic strains, and seems to have overlooked the necessity for such a 

 proof, 

 [n the ime work, ] elastician forcibly remarks : "Generally and philosophically 



no purely mathematical consid I the mer in which thi ictingonthe 



elements of a bodj an I th ■ ■< mges which they produce depend upon one another." 



Experiment alon md o ily somewhat refined experiment, betr 13 - the I i i thai even the h ir lesl 

 substance? yield somew hat to the smallest pressures, and that the stress-strain curve is con t inn 

 in form as well as value from positive to negative strains. One set of experiments is needful 



how th i • I itl ud of a steel bar which is clamped at the i listorts it, and 



other set is re [tiired to shew that the distortion is of the 9am • absolute amo tnl whether the tly 

 settles on the upper or the lower end id' the i, 



