The Relation of Stress to Strain 21

THE RELATION OF STRESS TO STRAIN 21. We have seen (§ 9) that six independent quantities (the "stress-components") are required, in general, to specify a state of stress; and further (§ 7), that three relations between them (the "stress equations" of motion or equilibrium) can be obtained by an application of dynamical principles. These relations are not sufficient to determine the stress-distribution produced by specified loads; to take a simple example, we cannot, by statical considerations alone, determine the load on each leg of a table, when all four are in contact with the ground. We need addi tional relations, and an obvious solution of the difficulty is to relate, by any arbitrary assumption, the six components of stress with the six components of strain. For we have seen (§ 18) that the latter can all be expressed in terms of three independent quantities—the components of displacement; so, by this pro cedure, we shall be left with three equations relating only three unknown quantities—that is, with information sufficient (i.e., in theory) for a solution.

Hooke's Law.-22. The simplest relation that we can assume is direct proportionality—in other words, a "linear" law. The most general state of stress is defined by six independent com ponents of stress, and the most general state of strain is defined by six independent components of strain : we assume that each one of the components of stress may be expressed in terms of the six components of strain by a formula of the type where p stands for the stress-component, . . . for the com ponents of strain, and al, . . . are constants. On this as sumption, the stress equations remain linear when transformed, first into relations between the coefficients of strain, and thence into relations between u, v and w, the three components of dis placement. Hence, if we obtain a solution—that is to say, ex pressions for u, v, w, at any point in a specified body, in terms of the applied forces—these relations will still be satisfied when we multiply u, v, w, together with the applied forces, by any constant factor. So we deduce from our assumption, that the displacements at every point, and hence the strains, will be pro portional to the "load." Robert Hooke, in 166o, discovered by experiment that this is in fact a property of real materials. He published his discovery

(1676) under the form of an anagram, ceiiinosssttuu and did not until two years later disclose the solution—"ut tensio sic uis (vis)"; that is, "the Power of any spring is in the same propor tion with the Tension thereof." (De Potentia restitutiva, London, 1678.) A more accurate statement of the experimental evidence is that, within certain definite limits of strain (see § 38): (I) when the load is increased, the measured strain increases in the same ratio, (2) when the load is diminished, the measured strain diminishes in the same ratio.

(3) when the load is reduced to zero, no measurable strain persists.

It

will be realized that the assumption represented by (Io), whilst it is consistent with these results, is more precise and of wider scope than any experiments that can be made. No method has been devised for measuring either the strain or the stress in the interior of an elastic body : all that can be done is to relate particular displacements with the resultant applied load. Thus the six expressions of type ( ro) are to be regarded as postulates of the mathematical theory, and the justification for applying this theory to real materials must be found in an increasing ac cumulation of observations in which its predictions are verified.

Aeolotropic and Isotropic Materials.-23.

Since six co efficients (of type a2, . . . etc.) are involved in each of the six expressions of type (1o), our generalized statement of Hooke's will be noticed that six quantities are still involved in the description.

law involves altogether 36 coefficients—the "elastic constants" of the material. An argument based on thermodynamical considera tions indicates that only 21 of the elastic constants are to be re garded as independent ; and on a certain hypothesis concerning the structure of real materials it may be shown that their elastic behaviour will be reproduced in an ideal material for which the number is further reduced to 15'. But a much more drastic re duction can be effected if we assume that our ideal material has the same elastic properties in all directions: this property is termed isotropy.