42. The Maximum Stress Theory, proposed by Poncelet and St. Venant, asserts that extension, rather than stress, is the de ciding factor. If is the (numerically) greatest of the three principal strains, it asserts that failure will occur if where e is the limiting strain in a simple tensile test. This criterion, in virtue of (i8), may be written in the equivalent form which shows that all three of the principal stresses are involved.
This theory, too, is discredited by the evidence which has been cited. Wehage, in 1888, found the criterion to be at variance with results for wrought iron subjected to equal tensions in two direc tions at right angles.
43. The Maximum Shear Theory was first proposed by Coul omb. It asserts that the greatest shearing stress, or (what is the same thing) the algebraic difference between the greatest and least of the principal stresses, is the deciding factor. Thus, if P1, P2, P3 are the principal stresses, and if the theory asserts that failure will occur if where f has the same meaning as before.
Tests made on ductile materials (low-carbon steel, copper, brass) lend considerable support to this theory. In the experi ments of J. J. Guest (Phil. Mag., July, 1900), somewhat inconclu sive evidence was obtained from non-ferrous materials, but the results for steel were in close accord with the "stress-difference" criterion, which is generally accepted to-day as a basis for design in ductile materials. It is commonly known as "Guest's law." From a theoretical standpoint, however, Guest's law is open to the objection that it indicates no limit to the strength of a 'Extension of the theory by the assumption of different limits for tension and compression (the latter determined by means of a com pression test) only partially removes this difficulty.
material when the state of stress is that known as "hydrostatic tension,"—i.e., when the principal stresses are all equal and posi tive. Practical difficulties have so far prevented the imposition of this stress in an actual experiment, but it is impossible to believe that elastic failure could never occur. Moreover, it is known that the limits of proportionality for tension and compression are not always equal, as the law would suggest. In relation to materials like cast iron, the criterion certainly requires modification : we shall consider that modification which has been proposed by 0.
Mohr. (Zeitschr. des Vereines Deutscher Ingenieure, Bd. 44, goo.) 44. Mohr's theory rests on an hypothesis regarding the nature, or inner mechanism, of elastic failure. Just as two bodies having rough surfaces in contact will slip over one another if a force is applied which is sufficient to overcome the resistance of friction, so two parts of the same body, separated by an imaginary surface, are imagined to slip when the tangential stress at this surface attains a certain limiting value. Irreversible or "non-elastic" strain is regarded as the resultant effect of such slips occurring on a large number of planes in the material. And just as the resistance of friction, in the first case, depends on the pressure between the two bodies at their surface of contact, so it is con templated that the limiting intensity of the tangential stress may depend upon the normal stress with which it is compounded. Mohr's theory seeks to determine, by means of simple tests, the general nature of this dependence.
According to the view just stated, if two planes in the material are subjected to the same normal stress, slipping will occur first on that plane which is subjected, in addition, to the greater tan gential stress. Therefore, in Mohr's circle diagram (fig. 4), the state of stress on that plane at which elastic failure originates will be given by some point on the outer circle BAX, which is de termined by the greatest and least of the principal stresses.
Let f t denote the limit of proportionality for the material, as determined by a simple tensile test. Then it is known that elastic failure can occur under a state of stress defined by principal stresses ft, 0, 0, and the normal and tangential components of stress for that surface at which failure originates will be given, for this state, by some point on the circle OAB in fig. 12. Similarly, if is the limit of proportionality as determined by a test of the material in simple compression (where the principal stresses are --1C, 0, 0), we know that the component stresses on the surface at which failure originates will be given by some point on the circle OCD of the same diagram. It is not necessary to draw more than one half of each circle, because the sign of the tangen tial stress, on the hypothesis stated, will not affect the question of failure.