Principal extensions

It is also comparatively straightforward to-calculate P200, P220, P420 and P o for a biaxially oriented aggregate of transversely isotropic units in terms of the principal extension ratios Xx, X2 and (with X,X2 3 = 1). 

Theoretical (computational) calculations can also offer quantitative descriptors of physicochemical properties of the molecular structures, molecular interactions, and thermodynamics of interactions. Principally, extensive studies on the catalytic site of GP have been exploited in theoretical QSAR studies [4]. The techniques engaged correlate biochemical behaviors with the known crystallographic structures, and map regions around the inhibitor molecule and added water molecules to improve the in silico prediction [106-110]. 

Figure 6-3a). In the general case of a pure homogeneous strain, the cube is transformed into a rectangular parallelepiped (Figure 6-3b). The dimensions of the parallelepiped are A, /I2 and /L3 in the three principal axes, where the are called the principal extension ratios. Choosing the coordinate axes for the chain to coincide with the principal axes of strain for the sample, then... 

If an elastomer sample in the form of a unit cube is deformed by pure shear, then the three principal extension ratios are A, = X, = 1, X = MX. [Compare with the case of simple extension where X = X = 1 NX.] Following the arguments of Section B, derive an expression relating aE and A, where [Pg.208]

For uniaxial deformations, two distinct principal extension ratios Ax and Ap, along and perpendicular to the director respectively, satisfy the relation Xz Xp = 1 because of the incompressibility of rubbers. 

Pure shear is represented in Fig. 5 and is defined as a homogeneous strain in which one of the principal extensions is zero and the volume is unchanged. If the extension ratio A] = a while At = 1. then is /a. 

The description of the mechanical deformation of the membrane is cast in terms of principal force restiltants and principal extension ratios of the surface. The force resultants, like conventional three-dimensional strains, are generally expressed in terms of a tensorial quantity, the components of which depend on coordinate rotation. For the purposes of describing the constitutive behavior of the surface, it is convenient to express the surface resultants in terms of rotationally invariant quantities. These can be either the principal force resultants Ni and Nj, or the isotropic resultant N and the maximum shear resultant Ns- The surface strain is also a tensorial quantity, but maybe expressed in terms of the principal extension ratios of the surface. >.1 and Xj- The rate of surface shear deformation is given by (Evans and Skalak, 1979] ... 

Principal extension ratios The ratios of the deformed length and width of a rectangular material element (in principal coordinates) to the undeformed length and width. 

In case a), the mean values of the chain end-to-end vectors are displaced affinely with the principal extension ratios (p = x, y, z) specifying the macroscopic strain. The fluctuations about these mean values are independent of the sample deformation. Consequently, in the free-fluctuation limit, the transformation of the actual chain vectors is not affine in the K s. The elastic free energy change for deformation results in the expression... 

An explanation was proposed for the PET results which has since been shown to apply equally to oriented polypropylene, high density polyethylene and nylon. The polymer sheet is considered as an oriented continuum characterised by three principal extension ratios Xi, X2, A3). If the isotropic sheet is considered as the state of zero strain, then the oriented polymer has extension ratios A3 in the draw or orientation direction (IDD) and it is convenient to take Ai in the direction of the sheet normal. Thus A3 and A2 define the projection of the strain ellipsoid in the plane of the sheet. When a deformation band forms in the oriented polymer the deformation can be described in terms of two shear... 

Figure 8.19 shows the deformation resistance at 353 K, 6.5 K above Tg, albeit at a 50-fold-increased strain rate that brings the initial response close to Tg. The behavior is nearly completely rubbery in form, as is shown clearly when the strain is considered in terms of its Gaussian dependence on the principal extension ratio, X, as is shown in the plot of Fig. 8.20 when the stress is plotted against g X) =1 — 1 /I, without any dilatancy consideration in the deformation resistance that becomes inoperative above Tg. The dependence of a on g X) is linear with the exception of the region near where g X) 0, for which there is a vestigial, very minor, plastic-like behavior because of the 50-fold increase in strain rate. 

The shear y is related to the two principal extension ratios A and 1/A in the plane of shear as follows ... 

Use the relation between shear strain and principal extension ratios given in Problem 3.6. 

FIGURE 32.10. Stress-strain isotherms represented in terms of the shear modulus G and principal extension ratio a, for both unfilled and filled PDMS networks in pure shear. The filled points represent the data used to test for reversibility. From [69] 1991 American Chemical Society. 

If the interface is aligned with the maximum principal extension ratio, cosai = 0 ... 

That is, the interfacial area increases exponentially with time. This is attributed to Erwin [22]. Equation (6.14) forms the basis of the view that uniaxial elongational flow is important to achieve good mixing. For a simple shear flow, the principal extension ratios A, An, and Am may be shown to be [28] ... 

A major limitation of the model in the formulation of [71] is the prediction of stress and strain in dependency of temperature for only small unidirectional deformations of about 10%. As principal extension to large finite strains, the same authors published an improved 3-D, thermoviscoelastic approach to a phenomenological temperature dependence of the viscosity [87]. It allowed successful reanalysis of the experimental data of [71]. 

The quantities X, are the principal extension ratios, which specify the strain relative to an isotropic state of Reference Chapt, 5. The cycle rank 4 was first introduced by Flory and is the number of independent circuits in the network or the number of chains which have to be cut to reduce the network to an acyclic structure or tree. Subsidiary quantities called the number of effective chains and junctions noted, respectively, and Pe can be defined by the relationship... 

An unswollen sample is deformed uniaxially. The principal extension ratio along the stretching direction is denoted X.. The sample is then immersed in solvent, and swelling occurs in directions normal to the stretching axis this is accompanied by a lateral change in dimensions. The quantities are the two principal extension ratios in the lateral directions with respect to the unswollen sample of volume Vq. The product of the three principal extension ratios is the inverse of the volume fraction of polymer, v, in the swollen network... 

The A,i are the principal extension ratios, and and B have been defined in equation... 

It would be unreasonable for a physical quantity such as energy to depend on the choice of axes. The use of principal extension ratios, with values independent of the axis set, goes some way to ensuring that this is not the case. However, the choice of subscripts 1, 2 and 3 is arbitrary, so the chosen form must be a S3onmetric function of Ai, A2 and A3. For simplicity it should also become zero when Ai = A2 = A3 = 1, i.e. for zero strain. A further requirement is that for small strains we should obtain Hooke s law for simple tension and the equivalent equation for simple shear. 

As discussed in Section 2.2, we can restrict our discussion to the case of normal strain without loss of generality. We choose principal extension ratios Ai, A2 and A3 parallel to the three rectangular coordinate axes x, y and z. The affine deformation assumption implies that the relative displacement of the chain ends is defined by the macroscopic deformation. Thus, in Figure 3.7 we take a system of coordinates x, y and z in the undeformed body. 

Rivlin s choice of a strain-energy function U that involved the squares of the extension ratios arose because he assumed that negative values of the extension ratios A were a mathematical possibility, whereas it was necessary for U always to be greater than zero. We have seen, however, that by choosing suitable rotations of coordinate axes the most general deformation can be described in terms of pure strain, i.e. three principal extension ratios Ai, A2, A3 which are all positive (although some are necessarily less than unity, because A1A2A3 = 1). 

Such a surface has the property that the reciprocal of the square of its central radius is proportional to the extension of a line in that direction. The lines in the unstrained state for which the extension is a maximum or a minimum or is stationary with being a true maximum or minimum are the principal axes of the strain and the extensions in the direction of these axes are the principal extensions. In our theoretical approach the direction of the largest shortening in the successive phases of the contraction is taken as the dynamic fibre orientation. 

The excursion into x and y directions is much shorter than Rp = bN, the principal extension of the chain in the z direction. It is premature to say that the Gaussian chain resembles a football, however. The cross section of the Gaussian chain is not circular, as shown below. 

Ronca and Allegra, and independently Flory, advanced the hypothesis that real rubber networks show departures from these theoretical equations as a result of a transition between the two extreme cases of behaviour. In subsequent papers Floryl >l and Flory and Ermanl derived a theory based on this concept. At small deformations the fluctuations of the network junctions are constrained by the extensive interpenetration of neighbouring, but topologically remote chains. The severity of these constraints is characterized by the value of the parameter k (k - 0 corresponds to the phantom network, k = to the affine network). With increasing deformation these constraints become less restrictive in the direction of the principal extension. The parameter t describes the departures from affine transformation of the shape of the domains of constraints. The resulting stress-strain relation also takes the form of Eq. (7) with... 

The state of strain in large deformations is commonly described either by the principal extension ratios, Xi, X2, X3, deflned in the notation of Chapter 1 as X,- = 1 + Uilxi, with the coordinate axes suitably oriented, or by three strain invariants whose values are independent of the coordinate system. In simple extension, Xi = 1 + e, where e is the (practical) tensile strain U jx cf. equation 8 above), not to be confused with the e in equations 3,4 and 6. Most of this section is concerned with simple extension. 

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