Tresca and von Mises criteria

6.7.1 Tresca and von Mises criteria

In dislocation theory it is usual to consider the flow stress or yield stress of ductile metals under simple conditions of stressing. In practice, the engineer deals with metals under more complex conditions of stressing (e.g. during forming operations) and hence needs to correlate yielding under combined stresses with that in uniaxial testing. To achieve such a yield stress criterion it is usually assumed that the metal is mechanically isotropic and deforms plastically at constant volume, i.e. a hydrostatic state of stress does not affect yielding. In assuming plastic isotropy, macroscopic shear is allowed to take place along lines of maximum shear stress and crystallographic slip is ignored, and the yield stress in tension is equal to that in compression, i.e. there is no Bauschinger effect.

A given applied stress state in terms of the principal stresses σ 1 ,σ 2 ,σ 3 which act along three principal axes, X 1 ,X 2 and X 3 , may be separated into the hydrostatic part (which produces changes

346 Physical Metallurgy and Advanced Materials

(a)

(b)

(c) Figure 6.46 Schematic representation of the yield surface with: (a) principal stresses σ 1 ,σ 2 and

σ 3 , (b) von Mises yield criterion and (c) Tresca yield criterion.

in volume) and the deviatoric components (which produce changes in shape). It is assumed that the hydrostatic component has no effect on yielding and hence the more the stress state deviates from pure hydrostatic, the greater the tendency to produce yield. The stresses may be represented on a stress–space plot (see Figure 6.46a), in which a line equidistant from the three stress axes represents a pure hydrostatic stress state. Deviation from this line will cause yielding if the deviation is sufficiently large, and define a yield surface which has sixfold symmetry about the hydrostatic line. This arises because the conditions of isotropy imply equal yield stresses along all three axes, and the absence

of the Bauschinger effect implies equal yield stresses along σ 1 and −σ 1 . Taking a section through stress space, perpendicular to the hydrostatic line, gives the two simplest yield criteria satisfying the symmetry requirements corresponding to a regular hexagon and a circle.

The hexagonal form represents the Tresca criterion (see Figure 6.46c), which assumes that plastic shear takes place when the maximum shear stress attains a critical value k equal to shear yield stress in uniaxial tension. This is expressed by

(6.32) where the principal stresses σ 1 >σ 2 >σ 3 . This criterion is the isotropic equivalent of the law of

resolved shear stress in single crystals. The tensile yield stress Y = 2k is obtained by putting σ 1 =Y, σ 2 =σ 3 = 0. The circular cylinder is described by the equation:

(6.33) and is the basis of the von Mises yield criterion (see Figure 6.46b). This criterion implies that yielding

(σ 1 −σ 2 ) 2 + (σ 2 −σ 3 ) 2 + (σ 3 −σ 1 ) 2 = constant

will occur when the shear energy per unit volume reaches a critical value given by the constant. This constant is equal to 6k 2 or 2Y 2 , where k is the yield stress in simple shear, as shown by putting σ 2 = 0, σ 1 =σ 3 , and Y is the yield stress in uniaxial tension when σ 2 =σ 3 = 0. Clearly, Y = 3k compared to Y = 2k for theTresca criterion and, in general, this is found to agree somewhat closer with experiment. In many practical working processes (e.g. rolling), the deformation occurs under approximately plane strain conditions with displacements confined to the X 1 X 2 plane. It does not follow that the

Mechanical properties I 347

D −s 1 A s 1

−s 2

Figure 6.47 The von Mises yield ellipse and Tresca yield hexagon. stress in this direction is zero and, in fact, the deformation conditions are satisfied if σ

3 = 2 (σ 1 +σ 2 ), so that the tendency for one pair of principal stresses to extend the metal along the X 3 -axis is balanced by that of the other pair to contract it along this axis. Eliminating σ 3 from the von Mises criterion,

the yield criterion becomes (σ 1 −σ 2 ) = 2k

and the plane strain yield stress, i.e. when σ 2 = 0, given when

σ 1 = 2k = 2Y / 3 = 1.15Y .

For plane strain conditions, the Tresca and von Mises criteria are equivalent and two-dimensional flow occurs when the shear stress reaches a critical value. The above condition is thus equally valid

when written in terms of the deviatoric stresses σ ′ 1 ,σ ′ 2 ,σ ′ 3 defined by equations of the type: σ ′

Under plane stress conditions, σ 3 = 0 and the yield surface becomes two-dimensional and the von Mises criterion becomes

σ 2 1 +σ 1 σ 2 2 2 +σ 2 2 = 3k =Y , (6.34) which describes an ellipse in the stress plane. For the Tresca criterion the yield surface reduces to a

hexagon inscribed in the ellipse, as shown in Figure 6.47. Thus, when σ 1 and σ 2 have opposite signs, the Tresca criterion becomes σ 1 −σ 2 = 2k − Y and is represented by the edges of the hexagon CD and FA. When they have the same sign, then σ 1 = 2k = Y or σ 2 = 2k = Y and defines the hexagon edges AB, BC, DE and EF.