28 C. A. Sciammarella et al. Observing Fig. 3.8(b) upper row and Fig. 3.9, the image corresponds to the onset of plasticity. At the edge of the specimen, εv (x, P) = 0. The linear relation between iso-derivatives fringes and loads ends. Also, lower row, Fig. 3.8 label (b), εu (x, P) = 0 indicates transition to plasticity. Hence, the transition to plasticity indicated by label (b) in Fig. 3.8 is well detected by the two families of iso-derivatives fringes. 3.6 Transition from Onset of Plasticity to Fracture In Sect. 3.5, it is concluded that the iso-derivative lines can be utilized to detect the onset of plasticity recording either εu (x, P) or εv (x, P). The next question to analyze is the detection of the onset of fracture. This can be done analyzing the frames of Fig. 3.8, label (c) iso-derivative fringes before maximum load, label (d) iso-derivative fringes at the maximum load, and label (e) the iso-derivative fringes in the process of post fracture. One clue in this transition can be extracted from the slopes of the iso-derivative fringes (Fig. 3.10). At label (c) near the maximum load, the slopes of the iso-derivative fringes display asymptotic values that are the same for both families of iso-derivative fringes. The iso-derivative fringes of the U(x,P) have increased in number with respect to the iso-derivative fringes of V(x,P). At label (d), the maximum of the tensile load takes place. The sudden change from (c) to (d) indicates that an abrupt change in the specimen geometry has taken place. The next step is to follow the transition from yielding to fracture. In order to achieve this goal, it is necessary to review some concepts. 3.7 Instability of the Plastic Flow The analysis of the transition from the onset of plasticity to actual fracture is an extremely complex subject because it depends on the concept of the behavior of a material under increasing solicitation. In this paper, we are addressing problems related to metals. Within metals, the behavior will depend on the atomic arrangement of the particular metal aside many other factors. We are further narrowing our field to tensile specimens that are utilized in standards of testing to define material properties. We can analyze the concept of hardening, i.e., if in Fig. 3.4 in place of plotting P vs. δ, one plots a version of the stress σ vs. a corresponding expression of strainε, one obtains a definition of hardening or equivalent to a local modulus, Fig. 3.10 Slopes of the isotachic lines in the process of transition from plasticity to fracture
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