Uniform deformation

Protein structures are so diverse that it is sometimes difficult to assign them unambiguously to particular structural classes. Such borderline cases are, in fact, useful in that they mandate precise definition of the structural classes. In the present context, several proteins have been called //-helical although, in a strict sense, they do not fit the definitions of //-helices or //-solenoids. For example, Perutz et al. (2002) proposed a water-filled nanotube model for amyloid fibrils formed as polymers of the Asp2Glni5Lys2 peptide. This model has been called //-helical (Kishimoto et al., 2004 Merlino et al., 2006), but it differs from known //-helices in that (i) it has circular coils formed by uniform deformation of the peptide //-conformation with no turns or linear //-strands, as are usually observed in //-solenoids and (ii) it envisages a tubular structure with a water-filled axial lumen instead of the water-excluding core with tightly packed side chains that is characteristic of //-solenoids. 

Figure 4.2. Illustration of two extreme models for the bending (and/or twisting) of DNA. Uniform deformation of the native structure on the left is contrasted with bending (and/or twisting) exclusively at sites of local denaturation in an otherwise stiff filament on the right. [From Biopolymers 22,2273-2321 (1983), Ref. 23, with permission.]...

The analysis of Eq. (16) has led to the conclusion 7) that the strain-energy function W in the mode of uniform deformation is parabolic with a minimum potential energy in the unstrained state... 

Hence, we arrive at the conclusion that only in the limit a - 0 the Hookean body is the ideal energy-elastic one (r = 0) and the uniform deformation of a real system is accompanied by thermal effects. Equation (19) shows also that the dependence of the parameter q (as well as to) on strain is a hyperbolic one and a, the phenomenological coefficient of thermal expansion in the unstrained state, is determined solely by the heat to work and the internal energy to work ratios. From Eqs. (17) and (18), we derive the internal energy of Hookean body... 

The thermomechanical equation of state of an isotropic Hookean rod subjected to force f along the rod axis can be obtained analogously to that used for the uniform deformation 3 7 8)... 

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 symmetric local mechanical disturbances (tensions) can be normal and tangential. The source of such disturbances in foam films could be an air stream directed normally to the film surface. Then, a local increase in pressure (the normal component of the disturbance) is observed as well as a tangential force that sprays the stream at the film surface (tangential component). As a result of the former the liquid in the film moves from the disturbance zone to its periphery. The process of liquid outflow from the film involves two stages uniform deformation of the disturbed part (i.e. film extension) and liquid outflow by the usual mechanism. The analysis of Krotov [28] indicates that, if the condition h/rd 1 (rb is the radius of the zone of disturbance) is fulfilled, the rate of film thinning as a result of liquid outflow is negligible compared to the rate of film extension. 

When a specimen is stretched plastically a few percent and then unloaded, x-ray measurements show a line shift indicating residual compressive macrostress in the direction of prestrain. The effect is symmetrical after plastic compression, x-rays indicate residual tensile stress. It is not a surface effect, because x-ray measurements made after successive removal of surface layers show that the stress persists throughout the specimen. On the other hand, dissection measurements show that a true macrostress does not exist, and, in fact, none would be expected after uniform deformation. The stress indicated by x-rays is called pseudo-macrostress, pseudo because it is not a true macrostress causing strain on dissection and macro because it causes an x-ray line shift. Pseudo-macrostress is actually an unusual kind of microstress, in which the portions of the material that are in tension and in compression are unequal in volume. It has been discussed in various reviews [16.26-16.28]. 

An estimate of the energy cost for such uniform deformations can be built on the basis of the crystal periodicity. In particular, a simple model of this energy can be written in the form... 

A typical tme stress-true strain curve of Type (A) specimens is shown in Fig. 2. The tme strain is calculated from the displacement of cross-head. As can be seen, the fracture of the specimens occurs at a strain of about 0.5. Relation between true stress, a, and true strain, , in uniformly deformed Type (A) specimen is given by the following Ludwik law,... 

An example is as follows suppose two phases meet at a planar interface and have some chemical component in common then in equilibrium that component will be partitioned between the two phases in such a way that it has the same chemical potential in both. Now suppose a uniform deformation is imposed on the entire sample, including the interface in what way will the equilibrium be disturbed, and what chemical changes will begin to run ... 

The flow characteristics listed in Table 7.1 indicate that the steady-state flows strongly affect the morphology, whereas the dynamic flows have less influence. The extensional flows are characterized by uniform deformation and lack of vorticity, thus they are the most effective in changing the morphology and orientation of the system. 

At first, all hexagonal cells are deformed as in the case of shear, so that the fracture band behaves as a shear band. The reason of such behavior consists in the value of modulus of elasticity for the valence angle between two chemical bonds. It appears usually one-two orders less than the elastic constant of valence forces [7]. This deformation should be compatible with that of remaining part of the nanotube outside the shear band. Simultaneously with the deformation of valence angles, bond stretching in the band initially parallel to the nanotube axis takes place. In addition to this uniform deformation in the shear band, there begins stretching of the bonds which initially were not parallel to the nanotube axis. 

For uniform deformation, there is a constancy of the volume of the material being deformed, i.e.,... 

For the remainder of this chapter, we will return to the subject of linear elasticity and concentrate on the way it is formally described for macroscopic bodies. At first, uniform deformations in a body will be considered but then we will move to using continuum mechanics for the description of non-uniform deformations. 

If a part or several parts in a body have been strained permanently beyond the limit of plasticity and the external forces and moments are removed, the material in the overstrained region and around it will, in general, be subjected to inherent stresses which are then called residual stresses (Nadai 1963). This description is exact from the macroscopic viewpoint however, from the microscopic viewpoint, the residual stress in each grain may be different, even in a tensile test that induces perfectly uniform deformation from a macroscopic viewpoint as shown in Fig. 1. In such a case, residual stress can be understood as a stress mainly due to the different state of stress existing in the variously oriented crystals before unloading (Johnson and MeUor 1962). 

Sh.M. Kogan, Piezoelectric effect on non-uniform deformation and acoustical scattering of carriers in crystals, Fiz. Tverd. Tela, 5(10), 2829-2831 (1963). [Sou Phys. Solid State 5, 2069 2071, (1964).]... 

A study of band formation in thermotropes by Zachariades et al. [96] commented that deformation appeared to be non-uniform and dependent on the thickness of the sample, and that non-uniform deformation occurs because the domains close to the polymer-wall interface deform more effectively by shearing versus the domains in the bulk of the sample which may slip or rotate... They found band structures after solidification of sheared thin samples, but not sheared thick samples. A personal communication from J.F. Fellers which reported a gap dependence was referenced in [56]. 

For a uniformly distributed load, there are two types of deformation of the subgrade under the loading area the uniform and non-uniform deformation. 

Non-uniform deformation results when the load is applied though a non-rigid medium such as the tyre of a vehicle. In this case, the elastic deflection, w, developed is greater at the centre of the loading area than that developed at the circumference of the loading area (Figure 11.5a). 

Figure II.S Deflection distribution forms, (a) Non-uniform deformation, (b) Uniform deformation.

In case of non-uniform deformation, the maximum elastic deflection (i.e. at the centre of the loading area), w, at any depth, z, for Poisson s ratio t = 0.5 can be calculated from the following equation ... 

---