Unloading curves

Finally, Fig. 8.3 shows a third form of elastic behaviour found in certain materials. This is called anelasfic behaviour. All solids are anelastic to a small extent even in the regime where they are nominally elastic, the loading curve does not exactly follow the unloading curve, and energy is dissipated (equal to the shaded area) when the solid is cycled. Sometimes this is useful - if you wish to damp out vibrations or noise, for example you... 

The Oliver-Pharr data analysis procedure [59] begins by fitting the unloading curve to the power-law relation... 

Fig. 39—Loading-unloading curve and displacement of the tip for microscratch test. 1-applied normal force 2-X direction displacement.

Figure 8 Loading and unloading curves for a single 163 pm diameter ion-exchange resin particle (DOWEX 1X8-200, Sigma-Aldrich, UK) obtained at a speed of 22.4 pm s 1 (data provided by T. Liu).

Fig. 5a,b Schematic representation of a the tip-sample contact upon high loading b the according compliance curve. In the case of perfectly plastic response the unloading curve is identical to the vertical line intersecting with the abscissa at hmax. In general, some viscoelastic recovery occurs and the residual impression depth hy is smaller than hmax. The difference hc—hy represents the extent of viscoelastic recovery. Ap and Ae denote the dissipated and the recovered work, respectively. Ap=0 for perfect elastic behaviour, whereas Ae=0 for perfect plastic behaviour. The viscoelastic-plastic properties of the material may be described by the parameter Ap(Ap+Ae) l. The contact strain increases with the attack angle 6. Adapted from [138]... 

Several approach curves, e.g. curve 3, present a periodic curve superimposed on the contact line. Such a modulation has been interpreted through a simple model [272]. The border wall can be seen as a pile of globular PMMA cluster-like structures. At the beginning of the loading phase, the tip pushes on one of these globular structures, and the force increases while the tip deforms and/or removes the cluster. Once the cluster has been removed, the force decreases until the tip meets another particle. This process is repeated several times. Also withdrawal curves on high border walls show a periodic curve superimposed on the unloading curve [272]. 

As could be expected, the mechanical properties of a crazed polymer differ from those of the bulk polymer. A craze containing even 50% microcavities can still withstand loads because fibrils, which are oriented in the direction of the load, can bear stress. Some experiments with crazed polymers such as polycarbonate were carried out to get the stress-strain curves of the craze matter. To achieve this aim, the polymer samples were previously exposed to ethanol. The results are shown in Figure 14.24 where the cyclic stress-strain behavior of bulk polycarbonate is also illustrated (32). It can be seen that the modulus of the crazed polymer is similar to that of the bulk polymer, but yielding of the craze occurs at a relatively low stress and is followed by strain hardening. From the loading and unloading curves, larger hysteresis loops are obtained for the crazed polymer than for the bulk polymer. 

The elastic modulus (E) was calculated directly from the experimental loading-creep-unloading curve (as in Fig. 2.14) according to the procedure of Doemer Nix (1986). 

Fig. 2. Typical load vs. displacement curve for a nanoindentation in tooth enamel. The contact depth (Dp), area and stiffness (S) were all determined at the maximum load (P) by fitting a polynomial expression to the upper 70% of the unloading curve, and they were used to quantify hardness (H) and Young s modulus (E) for each nanoindentation according to standard routines [13]. 

Fig. 4.19 Schematic force — displacement curve obtained in a nanoindentation experiment. The tip is pressed into the sample with a load force F to a maximum displacement hmax. The tangent of the unloading curves at hmax represents the stiffness S. (Reprinted with permission from [44]. Copyright 2006. Elsevier)...

It can be seen from Fig.3 that chromium films differ from molybdenum films in the mechanical behavior essentially. So, unloading curve for chromium at depth about 330 nm shows displacement discontinuity that, as is known, testifies to phase transition in silicon under loading (a metal phase of high pressure Si II [6]). On unloading curve for molybdenum the phase transition in silicon is not fixed. Besides average contact pressure in molybdenum film is lower, than in a chromium film more than in 2 times and this distinction increases with reduction of depth of contact. 

Fig. 5. Progress of the pressure-induced L. /L phase transition in hydrated phosphatidyiethanolamine monitored by time-resolved X-ray diffraction. Included in the figure is the changing scattered X-ray intensity in the (001) lamellar reflection, pressure and in-sample temperature following a 9.64 MPa (96.4 atm, 1400 psi) pressure-jump applied in the load and unloading directions. The data clearly illustrate a recurring limitation in many of these measurements, namely, the control of the transition by heat flux into and out of the sample. This is shown in the load curve. However, heat flow need not always be a limitation as is evident in the unload curve. (Unpublished observations, M. Caf-frey and A. Mencke)...

At low fields, the movement of the domain walls is very much like an elastic band (or a pinned dislocation line) that stretches reversibly. If the field is removed at that point, the unloading curve is coincident with the loading curve and the process is completely reversible. 

A representative load-displacement cmve for unloading is drawn in Figure 10.36b. The maximum load applied is F . and the corresponding maximum displacement is hmax- The initial slope of the unloading curve, AF/dh, is a measure of the stiffness, initial, and is given by ... 

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