Representation of Yield Criteria for Metals | Metallurgy

In this article we will discuss about the three dimensional and two dimensional representation of yield criteria for metals.

Three Dimensional Representation:

The graphical representation gives a better understanding of the two yield criteria mentioned above. For convenience we take the co-ordinate axes along the three principal stresses, i.e. σ₁, σ₂ and σ₃.

In this co-ordinate system, any stress state given by a set of three principal stresses σ₁, σ₂ and σ₃ is represented by a point, von Mises’ yield condition given by Eqn. (4.12) is represented by a cylindrical surface while Tresca’s criterion (Eqn. 4.18) is represented by a hexagonal prism (Fig. 4.1).

Tresca’s hexagonal prism is inscribed inside von Mises’ cylinder because we have taken that the yield strength in tension/compression is same for both the hypotheses, von Mises’ cylinder as well as Tresca’s hexagonal prism have same axis which is equally inclined to the three co-ordinate axes.

The direction cosines of axis of von Mises’ cylinder are 1/√3, 1/√3 and 1/√3. The inclination of this axis with co-ordinate axes is cos^-1 (1/√3) = 54.734°. The radius of von Mises’ cylinder on the plane perpendicular to its axis is given by-

Any point on the axis of the cylinder represents a stress state in which the three principal stresses are equal, which is a case of hydrostatic pressure. A line perpendicular to the axis represents the deviator part of stress tensor. Let us take that a stress state is represented by point P (Fig. 4.2). The vector OP presents the total stress vector. It is divided into two components OR and RP. The vector OR represents the hydrostatic pressure and RP is the deviator component.

If the point P lies inside von Mises’ cylinder, the material is elastic or a rigid body. It becomes plastic only when the point P lies on the surface of cylinder, i.e. when the stress state satisfies the Eqn. (4.12). Points outside the cylinder are not defined. It is taken that the material will suffer plastic deformation when P is on the surface of cylinder and stress increment is directed outside the cylinder. A similar discussion applies to Tresca’s hexagonal prism as well.

Both the yield conditions show that hydrostatic pressure, whatever its value, cannot bring a metal body to the yield point. Because in that condition the stress point will lie on the axis of yield surface and not on its surface.

Now, if the stress point P lies on the yield surface and stress increment is directed tangentially to yield surface we shall only be moving from one state of yielding to another state of yielding and hence no plastic deformation occurs. Plastic deformation occurs only when P lies on the yield surface and the incremental stress vector is directed outside von Mises’ cylinder or Tresca’s hexagonal prism.

Two-Dimensional Representation:

Two-dimensional representations of the two yield criteria are illustrated in Fig. 4.3.

Von Mises’ hypothesis is shown by an elongated ellipse while Tresca’s hypothesis is represented by an elongated hexagon. Here σ₃ has been taken as zero. The figure also gives the equations for different sides of the elongated hexagon. Any one of these equations becomes the operating yield condition when the stress point lies on the side of the hexagon represented by the equation.

According to Tresca’s yield diagram (Fig. 4.3), yielding in the first quadrant occurs when either σ₁ or σ₂ reaches the yield strength of material in tension. In the third quadrant, yielding occurs when σ₁ or σ₂ reaches the yield strength in compression. In the second and fourth quadrants yielding occurs when the maximum shear stress between σ₁ and σ₂ reaches the yield strength in shear.

In the second quadrant σ₂ is tensile while σ₁ is compressive. In fourth quadrant σ₁ is tensile and σ₂ is compressive. The corners of hexagon lie on von Mises’ ellipse, because, we have taken that the yield strength in tension/compression is same for both the yield conditions. If we take that the yield strength in shear is same for both the criteria, Tresca’s hexagon would lie outside Mises’ cylinder.

The two diagrams (ellipse and elongated hexagon) show maximum difference at the points of maximum shear.

Experimental Verification of Yield Criteria:

Several methods have been used to verify the two criteria of yielding of metals. Here we study the technique employed by Taylor and Quinney, in this method a thin tube of the metal to be tested, is loaded by a couple to produce shear stress and a direct axial force is applied to produce axial tension. The material taken for the test should be isotropic and homogeneous.

The tube being thin it can be assumed that there is little variation of stresses across the thickness of tube and we can safely take that the stresses are acting uniformly across the thickness. Both, the shear stress and direct stress may be individually varied till the tube yields.

Thus different combinations of direct stress and shear stress values which can produce yielding of the metal are obtained. Let the axial force produce a direct axial stress σ and the couple (twisting moment) produces a shear stress τ. The principal stresses σ₁, σ₂ and σ₃ may be determined as given below-

This is also an equation of ellipse. Equations (4.21) and (4.22) are plotted in Fig. 4.4 for sake of comparison.

The metals like aluminum, copper, mild steel, when tested as described above give points which lie in between the two curves, however, nearer to the Mises’ curve. In general most of the metals obey von Mises’ yield condition.

Yield Criterion for Anisotropic Materials:

The yield criteria, stress-strain relations etc. all apply only to isotropic materials. However, all metals and alloys are basically anisotropic because of their grain and cell structures.

If the grains are randomly oriented and there are large number of them, the macro-behavior approximates to that of an isotropic body. But if the grains are predominantly oriented in particular directions the material behavior is no longer isotropic.

In bulk plastic deformation of metals the effect of anisotropy has generally been ignored. Because the many approximations involved otherwise in the solutions, it is felt that consideration of anisotropy may not affect the results and be fruitful. However, in case of sheet metal, its formability and deep draw-ability is significantly affected by anisotropy induced in the sheet metal during rolling.

The basic experimental invest­igations on deformation behavior of anisotropic materials are very few. Hill has suggested the following yield criterion for anisotropic materials. This criterion is inspired by von Mises’ condition of yielding and reduces to von Mises’ hypothesis when anisotropy reduces to zero.