Maximum-strain-energy theory

Maximum strain energy theory, which postulates that failure will occur in a complex stress system when the total strain energy per unit volume reaches the value at which failure occurs in simple tension. 

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. 

Manning (1947) has shown that the maximum shear strain energy theory of failure (due to Mises (1913)) gives a closer fit to experimentally determined failure pressures for monobloc cylinders than the maximum shear stress theory. This criterion of failure gives ... 

These moments can be combined in a variety of ways depending on the criterion of operation wide variations exist between answers derived from the different formulae. The authors of the FMP Shaft Design Guide recommend that the maximum elastic shear strain energy theory is used. This results in an equivalent bending moment Af, c given by ... 

The material will have a given strength expressed as stress or strain, beyond which it fails. In order to postulate the failure, it is necessary to have a failure criterion with an associate theory to be able to effect a satisfactory design. Such theories include maximum stress, maximum strain, Tsai-Hill (based on deviatoric strain energy theory) and Tsai-Wu (based on interactive polynomial theory). The Tsai-Wu theory is the most commonly used. 

A tbick-walled prepare vessel having an inside diameter of 8 in. and an ( utside diameter of 16 in. is subjeel to an interna pressure of 15,000 i. Determine the maximum inducted stress sM n ding to the maxunum-principal< t]ress theory, the maxim urn-shear-stress theory, the Imal um-strain thec y, and the mandmum-strain-energy theory. 

Classical theories of failure are based on concepts of maximum stress, strain, or strain energy and assume that the material is homogeneous and free from defects. Stresses, strains, and strain energies are typically obtained through elastic analyses. 

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... 

There are conditions of loading a product that is subjected to a combination of tensile, compressive, and/or shear stresses. For example, a shaft that is simultaneously bent and twisted is subjected to combined stresses, namely, longitudinal tension and compression, and torsional shear. For the purposes of analysis it is convenient to reduce such systems of combined stresses to a basic system of stress coordinates known as principal stresses. These stresses act on axes that differ in general from the axes along which the applied stresses are acting and represent the maximum and minimum values of the normal stresses for the particular point considered. There are different theories that relate to these stresses. They include Mohr s Circle, Rankine s, Saint Venant, Guest, Hencky-Von Mises, and Strain-Energy. 

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... 

It has been theoretically predicted that uniaxial strain in the c-plane should be effective in reducing the DOS at the valence band maximum (VBM) of WZ GaN [5,23,24], According to the theory, the symmetry lowering to C from C6v by uniaxial strain leads to an increased energy splitting between... 

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