Ductile Materials under Combined Stress

Abstract

The Author further considers the results from some earlier tests made on mild steel bars, 3/4 inch diameter, and 30 inches effective length, under combined bending and torsion. It is pointed out that the yield-point is usually selected as the criterion of strength, because it is more easily determined than the elastic limit, it is less affected by special treatment of the material, and it is assumed that the failure of Hooke's Law between the elastic limit and the yield-point is due to local yielding. The elastic limit is the correct point, and is used throughout because the intermediate state mentioned above does not appear in bending. The results of tests on steel and copper tubes under combined bending and torsion are also given. All the results indicate that the maximum stress and maximum strain laws do not apply to ductile materials. The stress difference or shear stress law is approximately true, but there is, in each case, a deviation from the law which is opposed to the other theories mentioned. The deviations from the shear stress law are considered. In an earlier paper the Author suggested a formula for combined bending and torsion which allows for the fact that the bending moment is always greater than the torque. The internal friction hypothesis was also shown to be untenable. The three laws are now expressed in terms of the principal stresses, P1, P2, and P3, of which P1 is the greatest, and P3 is the least. Guest's experiments proved that P2 does not appreciably affect the failure of a material. The maximum stress law states that P1=constant; the maximum strain theory that P1-ηP3 = constant, in which η is Poisson's Ratio; and the stress difference or shear stress hypothesis is expressed in the form P1 - P3 = constant. In the general equation P1-mP3 = constant, the value of m indicates which law is most nearly true for the material. The Author's tests appear to be the only experiments in which bending was adopted, and for these the values of m are 1.57, 1.37, 1.26. The figures apply to different materials, but are all greater than unity. The results obtained by other observers, who employed different methods and combinations of loading, are also examined. For Guest's steel tubes, m varies from 0.9 to 1.87; his copper tubes give 1.04 and 1.09. For brass tubes the values are 1.03 and 1.34. Smith's tests on steel, three series, show m to vary from 0.775 to 1.09. The shear stress law appears to state the average behaviour of ductile materials, but there are considerable deviations from the law, which are usually opposed to the other theories. Other tests by the Author indicate that brittle materials obey the maximum stress law, and it is therefore suggested that the value of m depends chiefly on the degree of ductility of the material considered, and to a lesser extent on the system of loading.