NA TURE 



97 



THURSDAY, DECEMBER i, 1887. 



THE MATHEMATICAL THEORY OF PER- 

 FECTLY ELASTIC SOLIDS. 



An Elementary Treatise on the MatJteniatical Theory of 

 Perfectly Elastic Solids ; with a Short Account of 

 Viscous Fluids. By William John Ibbetson, M.A, 

 (London : Macmillan and Co., 1887.) 



IT is strange that students should have had to wait till 

 the present time for a systematic English text-book 

 on the mathematical theory of elastic bodies. The 

 want has been decidedly felt at Cambridge since the 

 introduction of the subject into the schedule for the 

 Mathematical Tripos in 1873; and though parts of 

 Thomson and Tail's treatise on natural philosophy, and 

 the reprint of Green's papers, had already brought a large 

 amount of useful matter into an accessible form for those 

 who had not time or opportunity to read the original 

 memoirs, still it was found that learners, naturally looking 

 for some compendious account of the whole subject, 

 generally fell back upon M. Lame's treatise. 



The book at present under notice will supply this want 

 satisfactorily. The plan on which it has been written is 

 excellent in idea, and has on the whole been followed out 

 well, though perhaps there is here and there some want 

 of proportion, as for instance in the elaborate and purely 

 mathematical details of Chapter V. 



It is, no doubt, a difficult matter to decide what results 

 of mathematical analysis should be introduced without 

 proof in a treatise on mathematical physics, and there is 

 little question that, as a matter of convenience to the 

 reader, it is wiser to err on the side of assuming too 

 little knowledge rather than too much. On the other 

 hand, wherever questions of pure mathematics are intro- 

 duced and discussed at length, they should be such as 

 have a direct bearing on important parts of the physical 

 subject. Now the general forms of the dynamical 

 equations for an elastic body in terms of curvilmear 

 co-ordinates, which are established in Chapter V. after a 

 considerable amount of preliminary analysis, are so com- 

 plicated as to be practically valueless. Indeed, the one 

 case referred to is dismissed in a single paragraph. The 

 special forms for polar and semi-polar co-ordinates, often 

 to be used with advantage, may be very much more 

 simply established independently. 



To return to the general plan of the book. It com- 

 mences with a short preliminary chapter, headed " Pro- 

 perties of Elastic Solids," in which, after showing that 

 the subject cannot be profitably considered from the point 

 of view of molecular structure, the author defines the 

 ideal solid which must for purposes of analysis replace 

 the real body. 



In Chapter II. the general properties of strain are 

 treated very clearly and at considerable length. A little 

 more consideration might perhaps have been given with 

 advantage to finite homogeneous strain. No fewer than 

 three quadrics are introduced for the purpose of putting 

 results into a geometrical form, viz. the strain ellipsoid, 

 the elongation quadric, and the position ellipsoid ; while 

 in the succeeding chapter, on " The Analysis of Stress," 

 Vol. XXXVII. — No. 944. 



four surfaces — the first, second, third, and fourth stress 

 quadrics — are used for a similar purpose. It can hardly 

 be doubted that so great a number of surfaces will tend 

 rather to confusion than to that clearness of conception 

 of the properties of strain and stress for which they are 

 presumably introduced. 



The nature and mode of specification of stress is care- 

 fully expounded in Chapter III., and the dynamical 

 equations to be satisfied throughout the body and over 

 the boundary are obtained in terms of the stress-com- 

 ponents. Attention should be called to a statement in 

 § 153 of this chapter, as likely to mislead the student. It 

 is to the effect that "the component stresses are to be 

 considered as small quantities of the first order." Though 

 in a certain sense this is true, it is not true that the ratio 

 of the stress per unit area to, say, the weight per unit 

 volume of the body is a small proper fraction, and this 

 surely is the strict sense the words should bear. 



The next chapter, on " The Potential Energy of Strain," 

 is excellent. The method is similar to that used by 

 Thomson and Tait ; and the successive simplifications 

 introduced into the expression for the potential energy of 

 the strained body by considering successively greater 

 degrees of symmetry of structure, leading up to perfect 

 isotropy, are well shown. It is also pointed out that from 

 the definition of an isotropic body, its potential energy is 

 necessarily a function of the invariants of the strain, thus 

 reducing the number of independent elastic constants in 

 this case at once to two. Having thus arrived at a definite 

 conception of the isotropic elastic solid, the author ex- 

 pressly limits all the investigations that follow to the 

 case of such a body. From the expression obtained for 

 the potential energy, forms are deduced for the stress 

 components in terms of the strain components and the 

 elastic constants, and thence finally the dynamical equa- 

 tions are obtained in terms of the displacements. 



In Chapter V., as already stated, these equations are 

 thrown into new forms, and the remainder of the book is 

 devoted to their solution under various conditions as to 

 the nature of the applied forces and tractions and the 

 form of the body. 



As an introduction to the consideration of particular 

 questions the following five general theorems are proved, 

 viz. : — 



" (i.) that a state of strain cannot be maintained 

 unchanged without the action of applied forces or surface 

 tractions : 



"(ii.) that the state of strain maintained by a given 

 system of equihbrating applied forces and surface tractions 

 is therefore perfectly determinate : 



" (iii.) that the most general free motion of an elastic 

 solid consists of a number of superposed harmonic 

 oscillations of the particles about their natural positions : 



" (iv.) that the most general motion of such a body 

 under the action of an equilibrating system consists of a 

 number of superposed harmonic oscillations of the 

 particles about the equilibrium positions that would be 

 maintained by the system : 



" (v.) that a system of applied forces varying as a 

 simple harmonic function of the time gives rise to forced 

 harmonic oscillations of the particles of the same period 

 about their natural positions." 



The proofs given of these theorems, and especially of 

 the third, are rather unnecessarily long ; but, with a view 

 to avoiding repetition later on, it is certainly convenient; 



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