Distortional strain energy criterion

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... 

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. 

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... 

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... 

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