Ductile Materials under Combined Stress. 11 



Tresca, which show that a ductile material eventually behaves 

 like a very viscous fluid, and a few experiments performed by 

 Messrs. W. Voigfc and L. Januszkiewiez on wax rods sub- 

 jected to tension only, and to tension and external fluid pressure 

 (compressed air) combined. In the latter case it was found 

 that at rupture the shearing-stress in the material was the 

 same in the two cases, thus agreeing with Tresca's experiments 

 upon metals. 



14. Theories of Elastic Strength under Combined Stress : 

 the Maximum Stress Theory. — As examples of combined 

 stress continually arise in practice, some working theory as 

 to the criterion of strength, preferably elastic strength, must 

 be assumed and adopted. Two theories hold the field to-day. 

 The first is the assumption, it can hardly be called by any 

 more ambitious name, that the material yields when one of 

 the principal stresses reaches a certain amount, which must, by 

 taking a special case, be the stress determined by a simple 

 tension experiment. This was the theory adopted, in the 

 absence of experimental data, by Rankine, and it is the one 

 used by English and American engineers. 



As all experiments upon torsion give results at wide variance 

 with this theory, it can hardly be considered to be correct ; and 

 if it is used, a different working stress should be adopted in 

 cases of torsion or systems of stress approximating thereto. 



15. The Maximum Strain Theory. — The second theory is 

 that the material yields when the greatest strain reaches a 

 certain amount, which must, taking a special case, be the 

 yield-point strain in simple tension. This theory was first advo- 

 cated by St. Venant as fitting in with that molecular theory 

 which leads to the uniconstant theory of elasticity. Upon 

 this theory the value of Poisson's ratio, or the ratio of lateral 

 contraction to axial extension of a specimen under simple 

 tension, is 0'25 ; upholders of the theory maintaining that 

 variations from this value which experiment exhibits are due 

 to imperfections of the material. The greatest-strain theory 

 is that adopted upon the Continent, and is strongly upheld by 

 many elasticians. 



16. Theorem upon the Limiting Values of cr. — In the case 

 of a cube of elastic material, whose elastic constants are E 

 and a (E being Young's modulus, or the ratio of stress to 

 strain in a simple tension test), subjected to a uniform fluid 



p 

 pressure p, the linear contraction is -£(1 — 2<r). If the con- 

 stitution of the material is to be stable, this must be a positive 

 quantity, and hence the maximum value of a is 0*5. This 

 also shows that an isotropic stable material cannot decrease in 



