Plastic Response of Crystals and Polycrystals

From the standpoint of a stress-strain curve, the constitutive behavior discussed in the previous section corresponds to a limited range of loads and strains. In particular, as seen in fig. 2.10, for stresses beyond a certain level, the solid suffers permanent deformation and if the load is increased too far, the solid eventually fails via fracture. From a constitutive viewpoint, more challenges are posed by the existence of permanent deformation in the form of plasticity and fracture. Phenomenologically, the onset of plastic deformation is often treated in terms of a yield surface. The yield surface is the locus of all points in stress space at which the material begins to undergo plastic deformation. The fundamental idea is that until a certain critical load is reached, the deformation is entirely elastic. Upon reaching a critical state of stress (the yield stress), the material then undergoes plastic deformation. Because the state of stress at a point can be parameterized in terms of six numbers, the tensorial character of the stress state must be reflected in the determination of the yield surface. 

As will be revealed in more detail in chap. 8, the initiation of plastic deformation can, in most instances, be traced to the onset of dislocation motion. From the 

The calculation goes as follows. If we define the principal axes as ei, C2 and es and the corresponding principal stresses ai, a2 and as, the state of stress may be 

With respect to this coordinate system, we aim to find the plane (characterized by normal n = niei) within which the shear stress is maximum. The traction vector on the plane with normal n is given by = a n. To find the associated shear stress on this plane, we note that the traction vector may be decomposed into a piece that is normal to the plane of interest and an in-plane part. It is this in-plane contribution that we wish to maximize. Denote the in-plane part of the traction vector by r, which implies t = n)n - - r. It is most convenient to maximize 

If we eliminate the unknown ns via the constraint n n = 1, then minimizing of eqn (2.58) with respect to i and 2 results in two algebraic equations in these unknowns. What emerges upon solving these equations is the insight that the planes of maximum shear stress correspond to the set of 110 -type planes as defined with respect to the principal axes, and that the associated shear stresses on these planes take the values r = (ai — o ), which are the differences between the and principal stresses. 

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