Graphical Methods Of Representing Some Conditions Of Plasticity

[Two of the most useful and important equations available to the metallurgist for the study of plastic deformation of metals are the Huber-von Mises-Hencky1-3 and the St. Venant7-10 equations. HUBER-VON MISES-HENCKY EQUATION The Huber-von Mises-Hencky equation answers the question: When does plastic flow occur in metal that is subjected to a multi-axial stress? It occurs, according to this equation, when the three principal stresses,? al, a2, and a3 satisfy the equation: (s1 - s2)2 + (s 2 - s 3)2 + (s 1 ? s3)2 = 2K2 [I] Here K is the yield strength of the material as determined in an ordinary tensile test. If the quantities a1, a2, and a3 are assigned to the coordinates of a Cartesian three-dimensional system, this expression describes a cylinder whose axis of symmetry is inclined at equal angles to the three coordinate axes4,5 (Fig. I). The radius of this cylinder is v2/3K. For the purpose of simplicity, it is sometimes advantageous to ascribe the values s 1/K = si, s 2/K = s2, s 2/K = s3 to the three coordinate axes instead of the principal stress alone. In this case, Eq. I becomes (s1 - S2)2 + (S2 - S3)2 + (Sl - S3)2 = 2 [2] and the radius of the cylinder becomes 2/3v. In the latter method of representation, a point lying inside the cylinder indicates that the metal will react elastically, whereas a point lying on the cylindrical surface indicates that the metal will behave plastically. Some of the geometric features of this cylinder are worthy of study. Let us inquire first as to what interpretation may be placed upon what we will choose to call the "circles of latitude" of the cylinder (L in Fig. I) and the "meridians" of the cylinder (M in Fig. I). Circles of Latitude Circles of latitude are formed by allowing a plane, passing perpendicular to the cylinder axis, to intersect with the cylinder. Any point, P, on this circle is presumed to have the coordinates s1, s2, and s3. The coordinates of the point resulting from the intersection of the plane and the cylinder axis will be a, a, and a (since the cylinder axis is equally inclined to the three coordi¬nate axes). Let us now pass a line through the point on the circle and the point on the axis.]