Deformation model

Deformation and rearrangement of the morphological structure have been described by Peterlin [201-203] and Samuels [183]. During the initial step of deformation, a crystalline fiber undergoes an affine transformation, i.e., the strains are uniform throughout the material. However, even at still smaller deformations the morphological inhomogeneity of the 

FIGURE 3.26 Crystallization start temperature versus spin-line stress for polypropylene. (From Spruiell, J.E. White, J.L. Polym. Eng. Sci., 1975,15, 660. With permission.) 

The microscopic deformation described above can proceed macroscopically in two different ways. The fiber may deform uniformly or it may deform by the well-known necking process. The mode of deformation depends on the properties of the spun fiber and the conditions of drawing. Shimizu et al. [199] found that fibers spun at speeds greater than 3000 m/min did not exhibit necking when drawn. Nadella et al. [204] found that the tendency to form necks was reduced in fiber spun under high spin-line stress. 

---

It is instructive to describe elastic-plastic responses in terms of idealized behaviors. Generally, elastic-deformation models describe the solid as either linearly or nonlinearly elastic. The plastic deformation material models describe rate-independent behaviors in terms of either ideal plasticity, strainhardening plasticity, strain-softening plasticity, or as stress-history dependent, e.g. the Bauschinger effect [64J01, 91S01]. Rate-dependent descriptions are more physically realistic and are the basis for viscoplastic models. The degree of flexibility afforded elastic-plastic model development has typically led to descriptions of materials response that contain more adjustable parameters than can be independently verified. 

Nevertheless, as response data have accumulated and the nature of the porous deformation problems has crystallized, it has become apparent that the study of such solids has forced overt attention to issues such as lack of thermodynamic equilibrium, heterogeneous deformation, anisotrophic deformation, and inhomogeneous composition—all processes that are present in micromechanical effects in solid density samples but are submerged due to continuum approaches to mechanical deformation models. 

Note that the term y in Eqs. 2-15 and 2-16 has a different significance than that in Eq. 2-14. In the first equation it is based on a concept of relaxation and in the others on the basis of creep. In the literature, these terms are respectively referred to as a relaxation time and a retardation time, leading for infinite elements in the deformation models to complex quantities known as relaxation and retardation functions. One of the principal accomplishments of viscoelastic theory is the correlation of these quantities analytically so that creep deformation can be predicted from relaxation data and relaxation data from creep deformation data. 

NL Thomas, AH Windle. A deformation model for case II diffusion. Polymer 21 613-619, 1980. 

In a non-atom-centered deformation model, due to Hellner and coworkers (Hellner 1977, Scheringer and Kotuglu 1983), the bonding density is described by charge clouds located between bonded atoms and in lone-pair regions. 

The aspherical density formalism of Hirshfeld is a deformation model with angular functions which are a sum over spherical harmonics. It will be described in more detail in section 3.2.6. All three models have been applied extensively in charge density studies (for a comparison, see Lecomte 1991). 

Hirshfeld (1971) was among the first to introduce atom-centered deformation density functions into the least squares procedure. Hirshfeld s formalism is a deformation model, in which the leading term is the unperturbed IAM density, and the deformation functions are of the form cos" 0jk, where 9jk is the angle between the radius vector r7 and axis k of a set of (n + l)(n + 2)/2 polar axes on each atom /, as defined in Table 3.8 (Hirshfeld 1977). The atomic deformation on atom j is described as... 

Kurki-Suonio K (1977) Charge density deformation models. Isr J Chem 16 115... 

Centerpattern Growth model Deformation model... 

In a subsequent theoretical study, Stamenovic [60] obtained an expression for the shear modulus independent of foam geometry or deformation model. The value of G was reported to depend only on the capillary pressure, which is the difference between the gas pressure in the foam cells and the external pressure, again for the case of <)> ca 1. Budiansky et al. [61] employed a foam model consisting of 3D dodecahedral cells, and found that the ratio of shear modulus to capillary pressure was close to that obtained by Princen, but within the experimental limits given by Stamenovic and Wilson. 

The presence of phase transitions at 19 and 30°C provides an opportunity to test the proposed deformation model. Below 19°C the lattice contracts into a triclinic structure witli strong intermolecular interaction. 5,26 sjamplcs deformed below 19°C should develop off-c-axis orientation while samples deformed above 30°C should not. Figures 1.12 and 1.13 show inverse pole figures for samples deformed at 2 and 70°C. The observed orientation agrees with our proposed model. - With tlris set of experiments, it is possible to activate the oblique slip process or, alternatively, to deactivate it in the high-temperature phase above 30°C. 

B4] Ivins, James P. and John Pomll, A deformable model of the human Ms for measuring small three-dimensional eye movements, Machine Vision and Applications, vol. 11, str. 42-51,1998... 

The stochastic process models can be transformed by the use of specific theorems as well as various stochastic deformed models, more commonly called diffusion models (for more details see Chapter 4). In the case of statistical models, we can introduce other grouping criteria. We have a detailed discussion of this problem in Chapter 5. 

The viscoelasticity of a polishing pad has been shown to play an important role in the CMP process [59]. Fu and Chandra [55] in a viscoelastic pad deformation model showed that MRR decay for an unconditioned pad is strongly influenced by the viscoelasticity of the pad. Guo et al. [60] investigated the effects of pad viscoelasticity on dishing and erosion without considering pad roughness (asperity height distribution). 

To illustrate the usefulness of birefringence measurements in orientation studies, we now briefly discuss two simple models of orientation leading to different expressions of the second moment of the orientation function the affine deformation model for rubbers and the pseudo-affine model more frequently used for semi-crystalline polymers. 

Figure 3.8 shows a diagram of the deformation. The sample initially possesses dimensions Xq, yo, zq, and when deformed it takes on dimensions X, y, z. According to the affine deformation model, the end-to-end distance vector of the chains of the strained network will change its components in... 

It is actually possible within the framework of the statistical theory of elasticity to deduce an expression similar to Eq, (3.33) that considers the experimentally observed decrease in modulus. This is done by using a model different from the affine deformation model, known as the phantom network model. In the phantom network the nodes fluctuate around mean... 

Comparison of Eqs. (3.36a) and (3.33) indicates that the value of modulus G obtained from the affine deformation model is two times the value corresponding to the phantom network. This would mean that the latter model is more applicable in the region of moderate deformations and the affine model is more suitable in the region of low deformations. 

---