Hookean mechanics

Cantilevers in AFM function as force transducers converting unknown force to measurable deflection. The value of the unknown force can then be expressed by Hookean mechanics following spring constant calibrations. In addition to static point loads, cantilevers can also be vibrated, e.g., by an oscillation piezo to which the fixed end of the beam is attached (or by other approaches). Excitation frequency, oscillation amplitude, and phase relationships are variables that govern dynamic tapping (intermittent contact) imaging. This problem will be discussed in the next section. 

We begin the discussion by defining the technique itself. Rheology is a branch of science concerned with the characterization of viscoelastic fluids. A more scientific definition of rheology is the study of materials which are not completely described by either classic Hookean mechanics, which describe solids, or Newton s law of viscosity, which describe liquids, but instead fall somewhere in-between. So, when we talk about the rheology of viscoelastic solids/liquids, what we mean is that we are monitoring the viscosity, elasticity, and other flow properties, and the change in those properties over time in the presence of stimuli. 

In general, the behavior of all classes of polymer behavior is Hookean before the yield point. The reversible recoverable elongation before the yield point, called the elastic range, is primarily the result of bending and stretching of covalent bonds in the polymer backbone. This useful portion of the stress-strain curve may also include some recoverable uncoiling of polymer chains. Irreversible slippage of polymer chains is the predominant mechanism after the yield point. 

Fig. la and b. Mechanical work W (1), elastic heat Q (2), internal energy change AU (3) and heat to work ratio q (4) as a function of strain e (uniform deformation) or s (unidirectional deformation) for quasi-isotropic Hookean solid 8. a — positive a and P b — negative a and p. The arrows indicate inversion points (see text)... 

The most straightforward rheological behaviour is exhibited on the one hand by Newtonian viscous fluids and on the other by Hookean elastic solids. However, most materials, particularly those of a colloidal nature, exhibit mechanical behaviour which is intermediate between these two extremes, with both viscous and elastic characteristics in evidence. Such materials are termed viscoelastic. 

Relevant to this chapter is that the rheological behaviour (property) of any viscoelastic food can be well approximated by an arrangement (structure) of two mechanical elements springs and dashpots. In these models the Hookean elastic contribution is represented by a spring (with modulus E) and the viscous component by a dashpot (operating with a liquid of viscosity /i). 

This equation describes Hookean elasticity, and Po = G (G is the modulus of rigidity). In Fig. 9, the classical mechanical spring model representing Eq. (14) is illustrated. If, however, it is assumed that jSi is the only nonzero constant in Eq. (13), then ... 

In order to derive some simple linear viscoelastic models, it is necessary to introduce the mechanical equivalents for a Newtonian and a Hookean body. 

Fig. 9 Steel spring as the mechanical model for an ideal Hookean body the length of the spring increases proportionally to the force applied, which is here represented by a weight that stretches the spring.

Theories for polymer dynamics of dilute polymer solutions include the elastic (Hookean) spring model (Kuhn, 1934) which considers that the system is mechanically equivalent to a set of beads attached with a spring. The properties are then based on a spring constant between beads and the friction of beads through solvent. The viscosity of a Hookean system is then described by... 

Fig. 12 Result of mechanical measurement performed on a magnetoelast filled with randomly distributed carbonyl iron. The concentrations of the filler particles are indicated on the figure. The cross-linker content was 3 wt % in each case. Neo-Hookean (left figure) and Mooney-Rivlin representations (rightfigure) are plotted...

The simplest model is the statistical theory of rubber-like elasticity, also called the affine model or neo-Hookean in the solids mechanics community. It predicts the nonlinear behavior at high strains of a rubber in uniaxial extension with Fq. (1), where ctn is the nominal stress defined as F/Aq, with F the tensile force and Aq the initial cross-section of the adhesive layer, A is the extension ratio, and G is the shear modulus. 

Our discussion thus far has focused in a rather superficial way on the general evolution of the important area of fracture mechanics. The basic objective of fracture mechanics is to provide a useful parameter that is characteristic of the given material and independent of test specimen geometry. We wUl now consider how such a parameter, such as G (, is derived for polymers. In doing so we confine our discussion to the concepts of linear elastic fracture mechanics (LEFM). As the name suggests, LEFM apphes to materials that exhibit Hookean behavior. 

The ideal elastie response is typified by the stress-strain behavior of a spring. A spring has a constant modulus that is independent of the strain rate or the speed of testing stress is a funetion of strain only. For the pure Hookean spring the inertial effects are neglected. For the ideal elastic material, the mechanical response is deseribed by Hooke s law ... 

The response of rubbery materials to mechanical stress is a slight deviation from ideal elastic behavior. They show non-Hookean elastic behavior. This means that although rubbers are elastic, their elasticity is such that stress and strain are not necessarily proportional (Figure 14.3). 

As discussed earlier for a Hookean solid, stress is a linear function of strain, while for a Newtonian fluid, stress is a linear function of strain rate. The constants of proportionality in these cases are modulus and viscosity, respectively. However, for a viscoelastic material the modulus is not constant it varies with time and strain history at a given temperature. But for a linear viscoelastic material, modulus is a function of time only. This concept is embodied in the Boltzmann principle, which states that the effects of mechanical history of a sample are additive. In other words, the response of a linear viscoelastic material to a given load is independent of the response of the material to ary load previously on the material. Thus the Boltzmann principle has essentially two implications — stress is a linear function of strain, and the effects of different stresses are additive. 

In the study of Smith et al. [1999], a high temporal and spatial resolution microscopy is used to reveal features that previous studies could not capture [Alon et al., 1995]. They found that the measured dissociation constants for neutrophil tethering events at 250 pN/bond are lower than the values predicted by the Bell and Hookean spring models. The plateau observed in the graph of the shear stress vs. the reaction rate suggests that there is a force value above which the BeU and spring models are not valid. Since the model proposed so far considers the ceU as a rigid body, whether the plateau is due to molecular, mechanical or ceU deformation is not clear at this time. 

The fundamental principle on which fracture mechanics is based is that cracklike defects exist in all materials and that when critical conditions are attained at the crack tip, the crack will begin to propagate and the material will fracture. In linear elastic fracture mechanics (LEFM) the assumption is made that the material deforms elastically (i.e. is Hookean) at all times, thereby greatly simplifying definition of the elastic energy stored in the material prior to fracture. The most... 

The mechanical property represented by Eq. (5.50) is often called neo-Hookean. 

The two extreme cases of mechanical behavior can be reproduced very well by mechanical models. A compressed Hookean spring can serve as a model for the energy-elastic body under load (Figure 11-11). On releasing the load, the compressed spring immediately returns to its original position. The relationship between the shear stress (021) = Oe, the shear modulus Ge, and the elastic deformation ye is given by Hooke s law [Equation (11-1)] ... 

The emphasis in this chapter will be on elastic properties at strains small enough for departures from Hookean behaviour to be neglected. Dynamic mechanical data will be quoted to supplement elastic measurements, but the principles of this type of experiment will not be discussed, since a very detailed review of this field (not confined to anisotropic specimens) has been given by McCrum er al. ... 

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