Biaxial stress envelope

Figure 11.10 Biaxial stress envelope for polyfmethyl methacrylate) at room temperature 03 = 0. The curves indicate onset of crazing and shear yielding, as indicated. Note the intersection of crazing and shear yielding envelopes, and concomitant heavy continuous line (11).

A combination of an energy criterion and the failure envelope has been proposed by Darwell, Parker, and Leeming (22) for various doublebase propellants. Total work to failure was taken from the area beneath the stress-strain curve, but the biaxial failure envelope deviated from uniaxial behavior depending on the particular propellant formulation. Jones and Knauss (46) have similarly shown the dependence of failure properties on the stress state of composite rubber-based propellants. 

Sharma (90) has examined the fracture behavior of aluminum-filled elastomers using the biaxial hollow cylinder test mentioned earlier (Figure 26). Biaxial tension and tension-compression tests showed considerable stress-induced anisotropy, and comparison of fracture data with various failure theories showed no generally applicable criterion at the strain rates and stress ratios studied. Sharma and Lim (91) conducted fracture studies of an unfilled binder material for five uniaxial and biaxial stress fields at four values of stress rate. Fracture behavior was characterized by a failure envelope obtained by plotting the octahedral shear stress against octahedral shear strain at fracture. This material exhibited neo-Hookean behavior in uniaxial tension, but it is highly unlikely that such behavior would carry over into filled systems. 

By careful specimen preparation and testing, it is possible to obtain shear yielding in PS under tension. A PS gave 72.5 MPa in tension and 90.4 MPa in compression. Assuming that PS obeys a pressure-dependent von Mises criterion, wherel C increases linearly with p in eqn 5.14, plot a yield envelope for PS under biaxial stress. 

Fig. 5.7. Failure envelope (heavy continuous line) for PMMA under biaxial stress (<733 = 0) at room temperature, showing intersection of crazing and shear yielding envelopes (after S. Sternstein and L. Ongchin).

The constitutive behavior of masonry under biaxial states of stress cannot be completely described from the ccmstitutive behavior under uiuaxial loading conditions. The influence of the biaxial stress state has been investigated up to peak stress to provide a biaxial strength envelope, which cannot be described solely in terms of principal stresses because masonry is an anisotropic material. Therefore, the biaxial strength envelope of masonry must be described either in terms of the full stress vector in a fixed set of... 

One major diflerence between the two mechanisms of deformation is illustrated in Fig. 5.7, which shows a failure envelope for PMMA under biaxial loading. The pure-shear line, defined by er,j = —<722> marks the boundary between hydrostatic compression and hydrostatic tension. Below this line, crazing and other hole-forming processes do not take place because the pressure component of the stress matrix tends to reduce rather than to increase volume above the line, crazing is the principal mechanism of failure. 

In order to determine which is the most appropriate yield criterion for a particular polymer it is necessary to follow the yield behaviour under a variety of states of stress. This is most conveniently done by working in plane stress = 0) and making measurements in pure shear (o- = -0-2) and biaxial tension (o-i, 02 > 0) as well as in the simple uniaxial cases. The results of such experiments on glassy polystyrene are shown in Fig. 5.28. The modified von Mises and Tresca envelopes are also plotted. In both cases they have been fitted to the measured uniaxial tensile and compressive yield stresses, oy, and oy. It can be seen that the von Mises... 

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