Von Mises Theory

Using von Mises Theory from equation 4.58, the probabilistic requirement, P, to avoid yield in a ductile material, but under a biaxial stress system, is used to determine the reliability, R, as ... 

For most practical purposes, the onset of plastic deformation constitutes failure. In an axially loaded part, the yield point is known from testing (see Tables 2-15 through 2-18), and failure prediction is no problem. However, it is often necessary to use uniaxial tensile data to predict yielding due to a multidimensional state of stress. Many failure theories have been developed for this purpose. For elastoplastic materials (steel, aluminum, brass, etc.), the maximum distortion energy theory or von Mises theory is in general application. With this theory the components of stress are combined into a single effective stress, denoted as uniaxial yielding. Tlie ratio of the measure yield stress to the effective stress is known as the factor of safety. 

The octahedral shear stress criterion, also called the maximum energy of distortion criterion or the Von Mises theory, would predict the following torque at yield. For the shaft the octahedral shear stress would be (from Equation 3)... 

R. Von Mises, Theory of Flight, Dover, New York, 1945, pp. 139 et seq. This book from before the jet era gives a thorough, readable treatment of basic aerodynamics. 

The computational predictions for the metal components were evaluated according to the von Mises theory, subject to ductile materials. Figure 27.7 depicts the results for the interconnector plate located at the bottom of the assembly (Figure 27.2). The results for the metal components comprise aU six path lines defined, hence they are the same locations as used in the sealant analyses and an additional representation is omitted. 

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. 

The von Mises theory for yielding given in Chapter 2 can be written in the form. 

Equation (9.17) defines the equivalent stress plain stress conditions (biaxial) Eq. (9.17) becomes... 

For a component subject to multiaxial cyclic stress, the effective alternating and mean stresses can be calculated following the Von Mises theory. In a special zero-max-zero biaxial cyclic stress profile in both x and y directions, the effective alternating stress can be calculated using Equation 4.7 ... 

In a similar fashion, the distortional energy density (or von Mises) theory of failure can be stated as follows A material subjected to any combination of loads will yield whenever the distortional energy density at any point in the material exceeds the value of the distortional energy density in a simple tensile test at yield. The distortional energy density of a linear elastic isotropic material can be expressed in terms of principal stresses by the equation... 

R von Mises. Mathematical Theory of Probability and Statistics. New York Academic Press, 1964. 

Of all the theories dealing with the prediction of yielding in complex stress systems, the Distortion Energy Theory (also called the von Mises Failure Theory) agrees best with experimental results for ductile materials, for example mild steel and aluminium (Collins, 1993 Edwards and McKee, 1991 Norton, 1996 Shigley and Mischke, 1996). Its formulation is given in equation 4.57. The right-hand side of the equation is the effective stress, L, for the stress system. 

Finite-additive invariant measures on non-compact groups were studied by Birkhoff (1936) (see also the book of Hewitt and Ross, 1963, Chapter 4). The frequency-based Mises approach to probability theory foundations (von Mises, 1964), as well as logical foundations of probability by Carnap (1950) do not need cr-additivity. Non-Kolmogorov probability theories are discussed now in the context of quantum physics (Khrennikov, 2002), nonstandard analysis (Loeb, 1975) and many other problems (and we do not pretend provide here is a full review of related works). 

Burgers, J. M. (1948). A Mathematical Model Illustrating the Theory of Turbulence. In Advances in Applied Mechanics, 1. Ed. von Mises and von Karman. New York Academic Press. Cooper, D. W. and Freeman, M. P. (1982). Separation. In Handbook of Multiphase Systems. Ed. G. 

The solution of this system of equations is a mathematical problem similar to the one encoimtered in the study of the propagation of shock layers in compressible fluids. The shock layer theory developed by von Mises [5] and by Gilbarg [6] can be applied. The treatment is similar to the one previously discussed in Chapter 14, in the case of a single component. The concentration profiles in the shock are given by the system of two nonlinear differential equations (i = 1,2)... 

The Cochran formula, Equation (9), estimates the triplet phase only exploiting the information contained in the three moduli hl, kU h+kl- The representation theory proposed by Giacovazzo " indicates how the information contained in all reciprocal space could be used to improve the Cochran s estimate of The conclusive conditional probability distribution has again a von Mises expression ... 

The Principle of Relativity, Albert Einstein, Henrik A. Lorentz, Hermann Minkowski and Hermann Weyl. 2.00 Experimental Researches in Electricity, Michael Faraday. Cloth-bound. Two-volume set 22.50 Thermodynamics, Enrico Fermi. 2.00 Theory of Elasticity, M. Filonenko-Borodich. 1.75 The Analytical Theory of Heat, Joseph Fourier. 2.50 Die Differential- und Integralgleichungon der Mechanik und Physik, Philipp Frank and Richard von Mises. Clothbound. Two-volume set 15.00... 

Mises, L. von. The Theory of Money and Credit [1912]. Translated by H. E. Bateson. Indianapolis LibertyClassics, 1980. 

This section incorporates the unpublished work of Palmer and Weaver subsequently the fatigue analysis was included as an integral part of the FMP Shaft Design Guide which Palmer and Weaver compiled. Results are quoted, for brevity the reader is referred to references dealing with the Distortion Energy Theory of Failure (also called deviatoric stress, octahedral, von Mises, or shear strain) for a complete analysis. 

Tresca and R. von Mises criteria are for isotropic materials. In 1948, Rodney Hill provided a quadratic yield criterion for anisotropic materials. A special case of this criterion is von Mises criterion. In 1979, Hill proposed a non-quadratic yield criterion. Later on several other criteria were proposed including Hill s 1993 criterion. Rodney HiU (1921-2011) was bom in Yorkshire, England and has tremendous contribution in the theory of plasticity. 

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... 

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... 

We consider /2-flow theory of elastoplasticity, defined by the plastic evolution equations (10)-(13) and the classical von Mises yield snrface... 

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