Von Mises yield condition

An altemative scheme is the von Mises yield condition. In this case, one adopts an approach with a mean-field flavor in which plastic flow is presumed to commence once an averaged version of the shear stresses reaches a critical value. To proceed, we first define the deviatoric stress tensor which is given by,... 

Strains and stresses were computed for the joined specimen cooled uniformly to room temperature from an assumed stress-free elevated temperature using numerical models described in detail previously [19, 20]. The coordinate system and an example of the finite element mesh utilized are shown in Figure 3. Elastie-plastic response was permitted in both the Ni and Al203-Ni composite materials a von Mises yield condition and isotropic hardening were assumed. 

This generalized yield condition, which is usually referred to as the von Mises yield condition, has a simple geometrical visualization in principal stress space of o i,o 2,o 3 shown in Fig. 3.2. There the line making equal angles with the three principal stress axes represents the locus of pure mean normal stress a, along which all deviatoric stresses vanish and no plastic flow can occur. Thus, plastic flow requires a critical deviation from this line in the radial direction away from it... 

The von Mises (or simply Mises) yield condition of yield for the most general case when all stress elements are present becomes... 

Fig. 11.20 Pressure-dependent von Mises yield envelopes under plane-stress and plane-strain condition of loading, calculated with Kj = 1.0 MPa m , for void-free 80 20 PA6/rubber blend with Poisson s ratio v = 0.4 and pressure coefficient = 0.36 (From Bucknall and Paul (2009) reproduced with permission of Elsevier)...

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] ... 

A yielding criterion gives critical conditions (at a given temperature and strain rate) where yielding will occur whatever the stress state. Two main criteria, originally derived by Tresca and von Mises for metals, can be applied to polymers (with some modifications due to the influence of hydrostatic pressure) ... 

Eq. (8.24), and the modified Mohr-Coulomb yield (or failure) criterion, Eq. (8.27). It should be noted that other yield criteria, such as the von Mises criterion, are used to model the flow of bulk solids in hoppers, and more conditions may need to be imposed, such as the Levy flow rule, in order to close the system of equations [Cleaver and Nedderman, 1993],... 

Fig. 11 Calculated surface profiles of the octahedral shear stress at yield assuming a modified Von Mises criterion (a), and of the octahedral shear stress for a glass/epoxy contact under gross sliding condition (b). The grey area delimits the region at the leading edge of the contact where the octahedral shear stress is exceeding the limit octahedral shear stress at yield (a is the radius of the contact area) (from [97])...

In spite of the relative simplicity of the Tresca criterion, conditions for shear yielding in isotropic polymers are best summarized by the von Mises criterion (11),... 

Shear yielding in polymers has much in common with ductility in metals. In polymers, the yielding may be localised into shear bands, which are regions of high shear strain less than 1 m in thickness or the yield zones may be much more diffuse " Under a general state of stress, defined by the three principal stresses Gi, 02 and 03, the condition for yielding is given by a modified von Mises crite-rion l ... 

FEA of the stresses in the UHMWPE cup is difficult, as the stresses exceed the elastic limit. Teoh et al. (2002) considered an 8 mm thick cup with a metal backing, a 32 mm diameter ball and a peak load of 2.2 kN (about 2.5 X body weight) for walking. Using the unrealistic condition that the compressive stress on the ball/UHMWPE interface could not exceed the uniaxial compressive yield stress (of 8 MPa), they predicted the compressive stress to be at this level over a surface region of diameter about 8 mm. However, a von Mises type yield criterion should be used. It requires a pressure of nearly three times the uniaxial yield stress to extrude the PE to the side of the joint (Section 8.2.4). 

Compare this to the prediction of ay/2 from the Tresca criterion. The yield criteria for both the Tresca and Von Mises theories are shown graphically in Figure 6. For simplicity, the plots are shown for conditions of plane stress (ie 03 = 0). We can see that the Von Mises criterion describes an ellipse in stress space, with the Tresca criterion consisting of a series of straight lines bounded by the Von Mises limits. 

Prager and Hodge (195) have defined the internal pressure in a cylindrical vessel that is required to place the elastic-plastic interface on the outside surface of the vessel. This is the pressure required to place all of the vessel wall beyond the yield point. In deriving this relationship it is necessary to establish the condition under which plastic flow is Initiated. A widely used yield criterion is that of Von Mises (205). This criterion can be expressed by the following reliitionship ... 

This law, which also is called von Mises "flow law, can be derived from a general variational principle introduced by Ludwig von Mises. von Mises introduced the hypothesis that stresses corresponding to a given strain field assume such values that the work W becomes as large as possible. That is, the material strives against deformation. From this hypothesis and the yield condition it is easy to derive the yield law. 

Although the relative simplicity of the Tresca criterion is rather attractive it is found that the criterion suggested by von Mises gives a somewhat better prediction of the yield behaviour of most materials. The criterion corresponds to the condition that yield occurs when the shear-strain energy in the material reaches a critical value and it can be expressed as a symmetrical relationship between the principal stresses of the form... 

In three-dimensional principal stress space the modified Tresca cylinder becomes a hexagonal pyramid and the pressure-dependent von Mises cylinder becomes a cone. The significance of the apices of the pyramid and cone is that they define the conditions for which there can be yielding under the influence of hydrostatic stress alone. This is something which cannot happen for materials which obey the unmodified criteria (Section 5.4.3). 

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