Elastic strain term

Further investigations on the kinetics of cordierite nucleation revealed that the elastic strain term (AGg) in Eq. 1-1, could also be of significance in surface nucleation. In a specialized experiment,... 

Consider a deformation consisting of repeated sinusoidal oscillations of shear strain. The relation between stress and strain is an ellipse, provided that the strain amplitude is small, and the slope of the line joining points where tangents to the ellipse are vertical represents an effective elastic modulus, termed the storage modulus /r. The area of the ellipse represents energy dissipated in unit volume per cycle of deformation, expressed by the equation... 

In textbooks, plastic deformation is often described as a two-dimensional process. However, it is intrinsically three-dimensional, and cannot be adequately described in terms of two-dimensions. Hardness indentation is a case in point. For many years this process was described in terms of two-dimensional slip-line fields (Tabor, 1951). This approach, developed by Hill (1950) and others, indicated that the hardness number should be about three times the yield stress. Various shortcomings of this theory were discussed by Shaw (1973). He showed that the experimental flow pattern under a spherical indenter bears little resemblance to the prediction of slip-line theory. He attributes this discrepancy to the neglect of elastic strains in slip-line theory. However, the cause of the discrepancy has a different source as will be discussed here. Slip-lines arise from deformation-softening which is related to the principal mechanism of dislocation multiplication a three-dimensional process. The plastic zone determined by Shaw, and his colleagues is determined by strain-hardening. This is a good example of the confusion that results from inadequate understanding of the physics of a process such as plasticity. 

Pit formation. If we consider a dissolution nucleus at a screw dislocation intersecting the surface which consists of a cylindrical hole of radius r, one atom layer deep (a), then the free energy of formation of this nucleus will be composed of a volume energy, surface energy, and elastic strain energy term, respectively, as follows ... 

Obviously, if we know experimentally the behavior of the macroscopic ordering parameter with T, we may determine the corresponding coefficients of the Landau expansion (eq. 2.52). However, things are not so easy when different transitions are superimposed (such as, for instance, the displacive and order-disorder transitions in feldspars). In these cases the Landau potential is a summation of terms corresponding to the different reactions plus a couphng factor associated with the common elastic strain. 

Replacing the generalized strain e with strain components X4 and Xg and adding the elastic energy term in the Landau expansion results in equation 5.175. 

Solution. According to Section 19.1.3, the elastic strain energy (per unit volume of platelet) is proportional to c/a. The free energy associated with the platelet can then be written in the usual way as the sum of a bulk term, an elastic energy term, and an interfacial energy term,... 

When describing liquid surfaces, the surface tension was of fundamental importance. If we try to extend the definition of surface tension to solids, a major problem arises [324], If the surface of a liquid increases, then the number of surface atoms increases in proportion. For a solid surface this plastic increase of the surface area is not the only possible process. Usually more important is an elastic increase of the surface area. If the solid surface is increased by mechanically stretching, the distance between neighboring surface atoms changes, while the number of surface atoms remains constant. The change in surface area is commonly described in terms of the surface strain. The total surface strain etot is given by the change in surface area divided by the whole surface area detot = dA/A. The surface strain may be divided into a plastic strain dep and an elastic strain dse so that dstot = dep + dee. 

In general, the nuclei are not spherical. There are several reasons for this. One is that the a- 3 surface energy term, y p, depends on the orientations of the a and 3 phases. Another reason is that when nucleation occurs on oc-oc grain boundaries, the angle of contact, 0, between the a and 3 phases depends on the ratio of y p/yagb- A third reason is that the elastic strain energy is minimized if the precipitating phase is lenticular. 

A shortening in relaxation time in the critically strained region makes some materials tough. The shift of relaxation time is attributed to strain-induced dilatation and can reach as much as five decades. Thermal history, on the other hand, dictates the initial state from which this dilatation starts and may be expressed in terms of excess entropy and enthalpy. The excess enthalpy at Tg is measurable by differential scanning calorimetry. Brittle to ductile transition behavior is determined by the strain-induced reduction in relaxation time, the initial amount of excess entropy, and the maximum elastic strain that the material can undergo without fracturing or crazing. 

The change in the surface area of a solid is often described in terms of the surface strain. The total surface strain tot is given by dtl/Q. = dewt, where 2 is the total area. The total strain may be divided into the plastic strain dep and the elastic strain dse so that deioi = dep + det. 

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