Biaxial states of stress

Note that this question involved a biaxial state of stress in the material and hence, strictly speaking, the creep curves used are not appropriate. However, creep curves for biaxial states of stress are rarely available, and one possible approach is to calculate an equivalent stress, a , using a van Mises type criterion... 

Division 2 stress analysis considers all stresses in a triaxial state combined in acc-ordance with the maximum shear stress theory. Division 1 and the procedures outlined in this bcx)k c-onsider a biaxial state of stress combined in accordance with the maximum stress theory. Just as you would not design a nuclear reactor to the rules of Division 1, you would not design an air receiver by the techniques of Division 2. Each has its place and applications. The following discussion on categories of stress and allowables will utilize information from Dicision 2, which can be applied in general to all vessels. 

This theory asserts that yielding occurs when the largest difference of shear stress equals the shear yield strength. According to this theory, yielding will start at a point when the maximum shear stress at that point reaches one-half of the uniaxial yield strength, Fy. Thus for a biaxial state of stress where ai > 02, the maximum shear stress will be ((Ti - 2)12. 

For thick-walled vessels (Rm/t < 10), the radial stress becomes significant in defining the ultimate failure of the vessel. The maximum principal stress theory is unconservative for designing these vessels. For this reason, this book has limited most of its application to thin-walled vessels where a biaxial state of stress is assumed to exist. 

Fig. 2.1. Film and substrate separated, but with distributed force / acting on the film edge so that its strain is exactly the mismatch strain. This loading gives rise to an equi-biaxial state of stress at each material point in the film such a state of stress with magnitude a is illustrated on the right side.

Since the in-plane dimension of the film-substrate system is much larger than its total thickness, an equi-biaxial state of stress can be assumed in the system (ignoring edge effects). Therefore, ase z) = arr (z)-Prom (2.14), note that Cm = (os 0 ) (T — To). From (2.19)-(2.21), it is seen that Cq = — Cm/S and k = 3etn/(4fis)- At the interface between the film and the substrate at z = fig/2, the stress values for Mi = Ms = M are... 

For the case of thin films being deformed plastically under the action of equi-biaxial states of stress, the stress history follows a straight line path in stress space. In some of the experimental methods for study of thin film plasticity that have been described in this section, the in-plane stress components are not equal, in general, and the trajectory in stress space during deformation is not a straight line. Consider a film material being deformed under plane strain extension in the a —direction the plane strain constraint is enforced in the. j—direction. The elastic modulus is E and the Poisson ratio is Pf = 0.3. The initial yield locus for the material in terms of tensile yield stress stress (Tyf is the Mises condition. The material... 

Consider an elastic material subjected to an equi-biaxial state of stress, characteristic of a uniform epitaxial thin film coherently bonded to a substrate in the presence of lattice mismatch. This is the state of stress that develops in the film when the alloy with spatially uniform compo-... 

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... 

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. 

The procedure described above is straightforward in principle. However, in practice, great care must be taken in the test fixture design to assnre that applied loads cause a uniform state of stress in the test specimen. Two types of tests that have been developed for this purpose include (1) the simple tensile test for uniaxial states of stress, (2) the thin-walled tube subjected to combined torsion and internal pressure, for biaxial states of stress. Some theoretical aspects of the simple tensile test are developed in the sample problem which follows. For a more detailed discussion on experimental procedures for characterizing the material properties of composite materials, see Caisson and Pipes and Whitney et al. ... 

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