The von Mises yield criterion

The von Mises yield criterion [11] assumes that the yield behaviour is independent of hydrostatic pressure and that the yield stresses in simple tension and compression are equal. It is expressed most simply in terms of the principal components of stress so that 

The constant term in Equation (11.7) can be expressed easily in terms of the yield stress Oy in uniaxial testing. Then we can assign the values Oi = Oy and (72 = (73 = 0, and the constant on the right is found to be 2(7y.  

In rather more sophisticated terms the von Mises yield criterion assumes that the 3(ield criterion depends only on the components of the deviatoric stress tensor obtained by subtracting the hydrostatic components of stress from the total stress tensor. In terms of principal components of stress the deviatoric stress tensor is 

The von Mises yield criterion is often written in terms of the so-called octahedral shear stress Toct, where 

We have seen that the Coulomb 3deld criterion defines both the stresses required 

In an arbitrary 1-2-3 axis set, the criterion is in the form of the invariant expression 

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If the extruder is to be used to process polymer melts with a maximum melt viscosity of 500 Ns/m, calculate a suitable wall thickness for the extruder barrel based on the von Mises yield criterion. The tensile yield stress for the barrel metal is 925 MN/m and a factor of safety of 2.5 should be used. 

The tensile yield stress variation as a function of W for a material which has a von Mises-type yield locus is illustrated schematically in Figure 5. This variation is caused by the fact that as the width of the specimen increases, the biaxiality also increases toward the asymptotic value at plane strain. If the material obeys the von Mises yield criterion exactly, the plane strain yield stress should be 15% higher than it would be for simple tension. On the other hand, if the material obeys the Tresca yield criterion, the plane strain yield stress should be identical... 

The shear component of the applied stress appears to be the major factor in causing yielding. The uniaxial tensile stress in a conventional stress-strain experiment can be resolved into a shear stress and a dilational (negative compressive) stress normal to the parallel sides of test specimens ofthe type shown in Fig. 11-20. Yielding occurs when the shear strain energy reaches a critical value that depends on the material, according to the von Mises yield criterion, which applies fairly well to polymers. 

Using the von Mises yield criterion, shear yielding occurs at a critical value of the octahedral stress ... 

Since the stresses are singular at the crack tip, then clearly the yield oiterion is exceeded in some zone in the crack tip region. If this zone is assumed to be small, then it will not greatly disturb the elastic stress field so that the extent of the plastic zone will be defined by the elastic stresses. If it is assumed that the Von Mises yield criterion is applicable (a reasonable first approximation for polymers), then the shape and e of the plastic zone may be derived from the stresses given in Eq. (15). As-sumii a state of plane strain so that the transverse stress is given by 1/(0 + oee)> then for a yield stress of Oy, the dastic zone radius becomes ... 

Equivalent stresses (a) and strains (e) were derived from surface shear stresses and strains by means of the Von Mises yield criterion ... 

Derive the von Mises yield criterion in the form tod = VC/3 from equation (8.6) and the definition of the octahedral shear stress Tod-... 

Gradually increasing stresses and aj = —ctj are applied to a material with the third principal stress <73 constant and equal to zero and the magnitudes of (Tj and ct2 are found to be a when the material yields. Assuming that the material obeys the von Mises yield criterion, calculate the magnitudes of and [Pg.246]

Von Mises stress is originally formulated to describe plastic response of ductile materials. It is also applicable for the analysis of plastic failure for coal undergoing high strain rate. The von Mises yield criterion suggests that the yielding of materials begins when the second deviatoric stress invariant J2 reaches a critical value. In materials science and engineering the von Mises yield criterion can be also formulated in terms of the von Mises stress or equivalent tensile stress, a scalar stress value that can be computed from the stress tensor ... 

An alternative approach is the von Mises yield criterion, in which the role of all three principal stresses is introduced, i.e.. 

From the expression for the elastic energy of an isotropic solid in terms of the principal stresses and strains, show that the von Mises yield criterion is obtained when the dilatational part is removed, leaving the elastic shear strain energy. 

This can be compared with the von Mises yield criterion, written in a rearranged form as... 

A three-dimensional formulation of the Bingham plastic was developed by Hohenemser and Prager [H19] in 1932 using the von Mises yield criterion (see Reiner [R4] and Prager [P15]). This employs a deviatoric stress tensor T and has the form... 

Von MiSOS Yiold Critorion. The Von Mises yield criterion (also known as the maximum distortional energy criterion or the octahedral stress theory) (25) states that yield will occur when the elastic shear-strain energy density reaches a critical value. There are a number of ways of expressing this in terms of the principal stresses, a common one being... 

A very simple explanation of the effect of notching has been given by Orowan [95], For a deep, symmetrical tensile notch, the distribution of stress is identical to that for a flat frictionless punch indenting a plate under conditions of plane strain [102] (Figure 12.31). The compressive stress on the punch required to produce plastic deformation can be shown to be (2 + 7t)K, where K is the shear yield stress. For the Tresca yield criterion the value is l.Sloy and for the von Mises yield criterion the value is 2.82oy, where 0 is the tensile yield stress. Hence for an ideally deep and sharp notch in an infinite solid the plastic constraint raises the yield stress to a value of approximately 2>Oy which leads to the following classification for brittle-ductile behaviour first proposed by Orowan [95] ... 

Fig. 3.18. Sketch of the yield function / for two varying principal stresses <7i and <72. The curve with /(<71,(72) = 0 is the yield surface for the material and has an elliptical shape. It corresponds to the von Mises yield criterion (cf. figure 3.23(b)), introduced later. For three principal stresses, / is a hypersurface in four-dimensional space which cannot be shown graphically...

The yield surface for the von Mises yield criterion (occasionally called distor-tional strain energy criterion) is cylindrical in the space of principal stresses, with its centre coinciding with the hydrostatic axis = [Pg.90]

The von Mises yield criterion uses the principal invariants of the deviatoric stress tensor, J(, J2, and J3. Using equation (3.26), we find... 

The von Mises yield criterion, equation (3.29), results if we assume that the yield criterion depends only on the second invariant from equation (3.31) ... 

The left-hand side in equations (3.33) and (3.34) is called the equivalent stress <7eq,M- Figure 3.23(b) shows the von Mises yield criterion for a state... 

It is not possible to prove the validity of these yield criteria theoretically. This is obvious if we remember that they are continuum-mechanical approximations of a discontinuous reality. Experiments confirm that both - especially the von Mises yield criterion - satisfactorily describe the observed material behaviour. 

In contrast to metals, the yield strength of polymers is different in compression and tension. Frequently, the yield strength in uniaxial compression is 20% to 30% larger than in uniaxial tension (see also section 8.4). To account for this, the von Mises yield criterion is augmented by terms that depend on the hydrostatic stress state. We will discuss two possible approaches. 

Inserting this into equation (3.42), we find the flow rule (3.40) for the von Mises yield criterion. 

Due to the anisotropy, the von Mises yield criterion must be used in its coordinate-dependent formulation ... 

So far, we still lack a criterion to define whether the material yields. This is provided by the von Mises yield criterion... 

In reality, the stress state is biaxial at the notch root (the radial stress at the surface is zero), so that there is no difference to the uniaxial case if the Tresca yield criterion is used. If the von Mises yield criterion is used, there is a slight difference which is neglected here. 

If we use the Tresca yield criterion, yielding occurs exactly at cri = Rp. With the von Mises yield criterion, the result is /i/2 [(tri — ctc) + cr f + a ] = Rp. Depending on the value of the circumferential stress, the axial stress at which yielding starts may be up to 15.5% larger than with the Tresca criterion (see... 

Use the von Mises yield criterion to decide whether the material yields Can you decide which of the two results is correct Justify your answer In experiments on single crystals, the yield strength of the slip systems was determined as Tc- i = 60 MPa. Use the von Mises yield criterion to check whether a significant amount of slip systems in the polycrystal is activated at the stress value given The Taylor factor is M = 3.1. Calculate the stress deviator a for the given stress state ... 

These equations admit the possibility of yield surfaces when the von Mises yield criterion jr r = Xy is satisfied. An important consequence of Equations 13.8bl-2 in either form is that the only admissible kinematics in regions where the yield criterion... 

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