Maximum Principal Strain Criterion

The maximum principal strain criterion for failure simply states that failure (by yielding or by fracture) would occur when the maximum principal strain reaches a critical value (ie., the material s yield strain or fracture strain, e/). Again taking the maximum principal strain (corresponding to the maximum principal stress) to be 1, the failure criterion is then given by Eqn. (2.4). 

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No experimental data were available for the aluminium or the mixed joint at 180°C, and only two joints were analysed at this temperature for the FM350NA adhesive. It was thought that the errors in this case could be attributed to the fact that a maximum stress rather than maximum strain criterion had been applied. Given that the FM350NA undergoes reasonable plastic strain at the elevated temperature it may have been more appropriate to use a maximum principal strain criterion. 

It is worthwhile to consider whether the classical theories (or criteria) of failure can still be applied if the stress (or strain) concentration effects of geometric discontinuities (eg., notches and cracks) are properly taken into account. In other words, one might define a (theoretical) stress concentration factor, for example, to account for the elevation of local stress by the geometric discontinuity in a material and still make use of the maximum principal stress criterion to predict its strength, or load-carrying capability. 

If we now allow for non-linear adhesive behaviour, the high adhesive stress concentrations predicted by the linear elastic analysis will be relieved to some extent. Figure 54 shows the predicted spread of the yield zone of adhesive at the tension end of a double-lap joint as the load is increased. As would be expected, plastic flow begins near the adherend corner and the load corresponds to a joint efficiency of 21%. Each subsequent load increment represents an increase in joint efficiency of 4 4%. When elastic perfectly-plastic behaviour is assumed for the adhesive, a maximum strain criterion for failure seems appropriate. In Fig. 55 the joint efficiency is plotted against the maximum principal strain in the adhesive at each end of a double-lap joint. Assuming a failure strain for the adhesive of 5%, the analysis predicts a joint efficiency of 31% for a double-lap joint compared with 16% predicted by the linear elastic analysis. Similarly, the non-linear analysis predicts an efficiency of 39% for the double-scarf joint compared with 20% predicted by the linear elastic analysis. Although the predicted efficiencies are almost doubled by allowing for non-linear behaviour in the adhesive, failure in the adhesive is still predicted to be more probable than failure in the adherends (Table 5). 

As With the shear strength, the maximum shear strain is unaffected by the sign of the shear stress. The strains in principal material coordinates, 1- yi2 be found from the strains in body coordinates by transformation before the criterion can be applied. 

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... 

Criteria of principal strain and of strain energy agree best with test data, maximum principle stress, maximum principle shear, and stress bias criterion correspond less well, maximum dilation the least... 

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