Maximum Distortion Energy Criterion

The total strain energy density may be subdivided into two parts namely, dilatation and distortion, where dilatation is associated with changes in volume and distortion is associated with changes in shape that result from straining. In other words, ut = Uv + Ud, or Ud = UT - Up. From Eqn. (2.5), the total strain energy density is given 

The strain energy density for dilatation (a ) is given in terms of the hydrostatic stress  

The distortion energy density and the maximum distortion energy criterion for failure, in terms of yielding, are given, therefore, by Eqns. (2.7) and (2.8). 

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This failure criterion is given in terms of the octahedral shearing stress, ft is identical to the maximum distortion energy criterion, except that it is expressed in stress versus energy units. The criterion, expressed in terms of the principal stresses, is given in Eqn. (2.9). 

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... 

Within the context of pressure vessel design codes, the comparison of the allowable strength of the material is always done with respect to the stress intensities. This puts the comparison in terms of the appropriate failure theory either the maximum shear stress theory (Tresca criterion) or the maximum distortion energy theory (von Mises criterion). These failure theories have been discussed in some detail in Chapter 3. 

Criteria 2, 5, and 6 are generally used for yielding, or the onset of plastic deformation, whereas criteria 1,3, and 4 are used for fracture. The maximum shearing stress (or Tresca [3]) criterion is generally not true for multiaxial loading, but is widely used because of its simplicity. The distortion energy and octahedral shearing stress criteria (or von Mises criterion [4]) have been found to be more accurate. None of the failure criteria works very well. Their inadequacy is attributed, in part, to the presence of cracks, and of their dominance, in the failure process. 

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. 

It is usually desirable to run a simple bulk tensile test program and subsequently predict (calculate) shear properties from their tensile counterparts. This approach requires a clearly defined relationship between shear and tensile elastic limit and yield variables and material properties. The elastic limit and yield stress values can be related between tensile and shear conditions by using an appropriate failure criterion, such as maximum normal stress, maximum shear stress, and distortion energy criteria. A material parameter that needs to be converted in addition to the usual elastic properties is the viscosity coefficient. This can be done by using Tobolsky s (1960) assumption of equivalent relaxation times in shear and tension. Application of this assumption results in the relation ... 

In predicting limit (threshold) conditions, such as the elastic Kmit, yield, and failure conditions, classical failure criteria, such as the maximum normal stress criterion, maximum shear stress criterion and the distortion energy (von Mises) criterion can be employed. 

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)... 

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... 

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