Distortion Strain Energy Theory

Although formulated by Maxwell in 1865 and later in 1904 by Huber who wrote a paper in Polish, this criterion is generally attributed to von Mises and commonly 

In this energy there are two components the hydrostatic or mean stress and the deviatoric one given by 

Huber noted that the hydrostatic component did not produce failure while Henky deduced that if the hydrostatic component was not responsible for yielding then it should have been the other component and offered a physical interpretation of the criterion suggesting that yielding begins when the elastic energy of distortion reaches a critical value. Therefore, by subtracting the hydrostatic component 9.14 from the total strain energy 9.12 the distortion component is obtained 

From Eq. (9.16) it can be derived that failure (yielding) will occur when 

Equation (9.17) defines the equivalent stress Teg according to the von Mises theory. Under plain stress conditions (biaxial) Eq. (9.17) becomes 

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Of the many theories developed to predict elastic failure, the three most commonly used are the maximum principal stress theory, the maximum shear stress theory, and the distortion energy theory. The maximum (principal) stress theory considers failure to occur when any one of the three principal stresses has reached a stress equal to the elastic limit as determined from a uniaxial tension or compression test. The maximum shear stress theory (also called the Tresca criterion) considers failure to occur when the maximum shear stress equals the shear stress at the elastic limit as determined from a pure shear test. The maximum shear stress is defined as one-half the algebraic difference between the largest and smallest of the three principal stresses. The distortion energy theory (also called the maximum strain energy theory, the octahedral shear theory, and the von Mises criterion) considers failure to have occurred when the distortion energy accumulated in the part under stress reaches the elastic limit as determined by the distortion energy in a uniaxial tension or compression test. 

This section incorporates the unpublished work of Palmer and Weaver subsequently the fatigue analysis was included as an integral part of the FMP Shaft Design Guide which Palmer and Weaver compiled.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. 

This theory asserts that the total strain energy is composed of two parts the strain energy required for hydrostatic strain and the strain energy required for distortion. In this theory, it is assumed that yielding will begin when the distortion component is equal to the uniaxial yield strength, Fy. Where ai> 02> (73, yielding will occur when... 

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... 

The stress distribution given by Eq. 15.1 is shown in Fig. 15.1 for a vessel with r /fj = 2.2, The maximum stress is in the hoop direction and is at the inner surface where r = r. As the pressure is increased, the stresses increase until they reach a maximum limiting stress where failure is assumed to occur. For thin vessels the ASME Code assumes that failure occurs when the yield point is reached. This failure criterion is convenient and is called the maximum principal stress theory. In thick vessels the criterion usually applied for ductile materials is the energy of distention theory. This theory states that the inelastic action at any point in a body under any combination of stresses begins only when the strain energy of distortion per unit volume absorbed at the point is equal to die strain energy of distortion absorbed per unit volume at any point in a bar stressed to the elastic limit under a state of uniaxial stress as occurs in a simple tension test. The equation that expresses this theory is given by... 

In this chapter an effective Hamiltonian for a static cooperative Jahn-Teller effect is proposed. This Hamiltonian acts in the space of local active distortions only and possesses extrema points of the potential energy equivalent to those of the full microscopic Hamiltonian, defined in the space of all phonon and uniform strain coordinates. First we present the derivation of this effective Hamiltonian for a general case and then apply the theory to the investigation of the structure of Jahn-Teller hexagonal perovskites. 

The linear optical properties (UV-visible and Raman) of PDA crystals have been thoroughly characterized and are reasonably well-understood. The lowest energy optical transition is typically located at about 2.0 eV and is excitonic in origin. Distortion of the backbone due to deliberately induced disorder or strain caused by side group interactions shifts this transition to higher energies. Crystal strain (e.g. polymer chains in the monomer lattice) can shift the transition either way. More work/ theory and experiment, is needed to sort out and understand these effects. 

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