Von Mises yield surface

Fig. 11.13 von Mises yield surface displaying the effect of rate, time or strain hardening. 

Figure 12.11 Tresca and von Mises yield surfaces in principal stress space.

Bulk tensile testing has shown that adhesives generally exhibit plasticity and, hence, nonlinear material properties are required to model their behavior over the fiill load range. Nonlinear properties may also be required for adherends. Thus, a combination of elasto-plastic material models may be used to predict the behavior of adhesive joints under load. The definition of the yield surface is important when using elasto-plastic material models. Von Mises yield surface is commonly used for the analysis of metals, which assumes that the yield behavior is independent of hydrostatic stress. As a result, the yield surface is identical in tension and compression. However, the yield behavior of polymers has been shown to exhibit hydrostatic stress dependence (Ward and Sweeney 2004) as the yielding starts earlier in tension than in compression. Thus, a yield criterion which includes hydrostatic stress effects should be used to determine the yield surface. Various yield criteria with hydrostatic stress dependence such as Drucker-Prager, Mohr-Columb, and modified Drucker-Prager/cap plasticity model have been implemented in commercially available finite element software. 

The yield criteria of polymers have been reviewed by Ward (7) and more recently by Raghava et al. (8). Except for the craze yield criteria of Sternstein and Ongchin (9) and Bowden and Oxborough (10), most of the yield data can be described by a pressure-modified, von Mises-yield criterion. The corresponding yield surface is everywhere convex. A typical yield locus on the [Pg.103]

For macroscopically isotropic polymers, the Tresca and von Mises yield criteria take very simple analytical forms when expressed in terms of the principal stresses cji, form surfaces in the principal stress space. The shear yield surface for the pressure-dependent von Mises criterion [Eqs (14.10) and (14.12)] is a tapering cylinder centered on the applied pressure increases. The shear yield surface of the pressure-dependent Tresca criterion [Eqs (14.8) and (14.12)] is a hexagonal pyramid. To determine which of the two criteria is the most appropriate for a particular polymer it is necessary to determine the yield behavior of the polymer under different states of stress. This is done by working in plane stress (ct3 = 0) and obtaining yield stresses for simple uniaxial tension and compression, pure shear (di = —CT2), and biaxial tension (cti, 0-2 > 0). Figure 14.9 shows the experimental results for glassy polystyrene (13), where the... 

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

It should be emphasized that two fundamentally different types of craze tests were performed in this work. The test described initially, in which the craze stress below a notch was calculated from the slip line plasticity theory, without exposure to solvent, is a test in which the strain is changed as a function of time. The craze stress itself is calculated assuming that both slip line plasticity theory and the simple von Mises yield criterion are both applicable. The second test, used to determine the effect of solvent on crazing, is a surface crazing test under simple tension in which the strain... 

The assumption of material isotropy, which implies that ai, 02 and 02 are interchangeable, means that the Tresca and von Mises yield criteria take very simple analytical forms when expressed in terms of the principal stresses. Thus the yield criteria form surfaces in principal stress space, i.e. that space where the three rectangular Cartesian axes are parallel to the principal stress directions. Points lying closer to the origin than the yield surface represent combinations of stress where yield does not occur points on or outside the surface represent combinations of stress where yield does occur. 

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]

For the von Mises yield criterion, the yield surface is given by equation (3.32),... 

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. 

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... 

The distance from the crack tip, along the X-axis, at which the von Mises equivalent stress falls below the yield stress, defines the size of the plastic zone, r. For the plane stress case of unconstrained yielding, which corresponds to the free surface of the specimen in Figure 4, this gives... 

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])...

ABS and HIPS. The yield stress vs. W/t curves of ABS and HIPS are very similar. They are somewhat surprising because the yield stresses reach their respective maximum values near the W/t (or W/b) where plane strain predominates. This behavior is not predicted by either the von Mises-type or the Tresca-type yield criteria. This also appears to be typical of grafted-rubber reinforced polymer systems. A plausible explanation is that the rubber particles have created stress concentrations and constraints in such a way that even in very narrow specimens plane strain (or some stress state approaching it) already exists around these particles. Consequently, when plane strain is imposed on the specimen as a whole, these local stress state are not significantly affected. This may account for the similarity in the appearance of fracture surface electron micrographs (Figures 13a, 13b, 14a, and 14b), but the yield stress variation is still unexplained. 

Figure 14.9 Section of the yield surface in the plane 03 = 0 choosing the Tresca criterion (hexagonal envelope) and von Mises criterion (elliptical envelope) for polystyrene. The points correspond to experiments performed under pure shear (gi = -CJ2), biaxial tension (oj, 03 > 0), and uniaxial tension and compression. (From Ref. 13.)...

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). 

In the second form the von Mises criterion expresses directly the fact that the yield depends equally on the three shear stresses (cTj — oy)/2. A somewhat simpler criterion, the Tresca yield criterion, makes the slightly different assumption that yield takes place when the largest of these three shear stresses reaches a critical value. The surface in cr-space that represents the criterion is therefore defined by the six equations... 

Thus, in the three-dimensional stress space, the von Mises criterion can be represented by a circular cylinder of radius / 3, with its axis equally inclined to the three principal stress axes. Stress points plotted within the cylinder are wholly elastic and when the stress point reaches the surface of the cylinder, yielding occurs. Strain hardening will be represented by a radial expansion of the cylinder. 

The magnitude and distributions of stress in the knee are different from the hip. In the hip, the spherical contacting surfaces are highly conforming, and the effective (von Mises) stress levels are below yield, and, thus, below the onset of irrecoverable plastic deformation. Consequently, for hip components, UHMWPE can reasonably be considered to behave as an elastic material at the... 

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 ... 

Figure 9.11 shows the effect of surface treatment on extensional viscosity for 30% calcium carbonate filled polystyrene [27]. The data are presented in two forms, namely steady state extensional viscosity vs. extensional rate in Figure 9.11(a) and steady state extensional viscosity vs. tensile stress in Figure 9.11(b). Irrespective of the type of data representation, it is seen that surface treated calcium carbonate reduces the level of extensional viscosity and brings it closer to that of the unfilled polymer. The yield stress value is reduced considerably though the values of the ratio of yield stress in extension to that of shear is still maintained nearer to the von Mises value of 1.73 as can be seen from Table 9.1. Surface treatment tends to modify the forces of particle-particle interaction and hence show reduced yield stress values due to lowering of the interaction forces [2,27]. 

Arcan et aL (1978) proposed a biaxial fixture, commonly known as the Arcan fixture, to produce biaxial states of stress. The compact nature of the Arcan fixture enables obtaining the shear properties in all in-plane directions in a relatively simple manner. The Arcan fixture can be used to apply both shear and axial forces to the test specimen. The adhesive characterization in mixed mode loading allows for the generation of the yield surface of the adhesive in the hydrostatic versus the von Mises plane, which enables one to develop more accurate adhesive models for better simulations. Several modifications to the original test fixture have been proposed to include compression, such as that by El-Hajjar and Haj-Ali (2004). A scheme of the test fixture is shown in O Fig. 19.17. 

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