Two-dimensional deformation

Fig. 2.4. Warped Mexican-hat type of potential surface for Jahn-Teller distortion of octahedral MXs molecule. The symmetry of the two-dimensional deformation space is 3 m, and displacements along the mirror iines correspond to distortions of the octahedron that preserve tetragonal symmetry (elongated or compressed octahedron)...

We next briefly describe two-dimensional deformation. Consider a biaxial expansion in the x and y directions caused by applying the stresses tx and ty. We will find the relation between the stresses and the elongation Ux and ay [5]. 

This condition holds for two-dimensional deformations (e.g., in the xz surface). However, as shown in Chapter 5 three-dimensional flexoelectric deformations can occur even with strong anchoring of the molecules to the surfaces of the cell (Section 5.1.1). 

The action of the field on a planar texture of a cholesteric liquid crystal for cell thickness, comparable to the pitch and with rigid anchoring of the liquid crystal molecules to the surface, does not cause the two-dimensional deformations discussed. In this case, one-dimensional periodic patterns are observed [19, 24], with the orientation of the domains depending on the number of half-turns of the helix contained within the cell thickness. The phenomenon of one-dimensional, instead of two-dimensional, deformations... 

Fig. 4 Electro-opto-mechanical effect of a monodomain nematic gel observed (a) without polarizer and (b) with crossed polarizers. A 26-pm thick gel with = 4 mol% is placed in a 40-pm thick EO cell filled with a nematic solvent (5CB). The 5CB content in the gel is 82 vol%. An AC field (E) with an amplitude of 750 V and a fi equency of 1 kHz is imposed in the z-direcUon. The field induces a two-dimensional deformation, i.e., a shortening of ca. 20% in the r-direction, no dimensional change in the y-direction, and a lengthening of ca. 20% in the z-direction (due to volume conservation). The appearance of the gel (and surrounding 5CB) under cross-polarized conditions changes from bright to dark as a result of the almost full rotatirai of the director toward the field direction. A and P stand for the optical axes of analyzCT and polarizer, respectively. An mpeg movie is available in the supporting information of [31]...

A linear correlation is observed, and the slope (—0.13) is almost identical with that (—0.12) of the plot using the birefringence data for the similar specimen (SNE-7). This good agreement indicates that the results obtained by the two different methods are consistent. The linear relation jx sin 0 and the two-dimensional deformation are the two key features of the deformation induced by director rotation for unconstrained nematic elastomers. 

If we consider a straight rod of length I and when a load is applied the length is increased by AZ. Then the ratio of AZ/Z is called the strain A. Looking at a two-dimensional deformation pattern, for a rectangle ABCD, is shown in Figure A.4. The two dimensional strain components are defined... 

The present discussion has a twofold objective First, to review the literature in the stress analysis of adhesive joints using the finite-element method. Second, to present a finite-element computational procedure for adhesive joints experiencing two-dimensional deformation and stress fields. The adherends are linear elastic and can undergo large deformations, and the adhesive experiences linear strains but nonlinear viscoelastic behavior. Following these general comments, a review of the literature is presented. The technical discussion given in the subsequent sections comes essentially from the authors research(i 2> conducted for the Oifice of Naval Research. 

General two-dimensional deformations consisting of a strain, followed by a rotation. Reproduced from J.F. Nye, Physical Properties of Crystals, Clarendon Press. 1957 by permission of Oxford University Press. 

Rondelez, Gerritsma, and Arnould have reported the presence of two-dimensional deformations at the threshold voltage for scattering. Electrohydrodynamic instabilities were first predicted for negative... 

The drapeability describes the deformation of a textile fabric under its own weight without external loads. The textile can be deformed three-dimensionally. The term drapeability also stands for the deformability of a textile until the first wrinkles appear. The wrinkle-free deposition of a textile on a spherical body requires a two-dimensional deformability of the textile. A wrinkle would impair the mechanical properties. When producing hats, upholstery, or textiles for the reinforcement of composites, this property is crucial. 

We limit ourselves here to two-dimensional deformations. A detailed three-dimensional treatment of rheology is beyond the scope of this book. Several excellent treatises are available. " ... 

The practical and computational complications encountered in obtaining solutions for the described differential or integral viscoelastic equations sometimes justifies using a heuristic approach based on an equation proposed by Criminale, Ericksen and Filbey (1958) to model polymer flows. Similar to the generalized Newtonian approach, under steady-state viscometric flow conditions components of the extra stress in the (CEF) model are given a.s explicit relationships in terms of the components of the rate of deformation tensor. However, in the (CEF) model stress components are corrected to take into account the influence of normal stresses in non-Newtonian flow behaviour. For example, in a two-dimensional planar coordinate system the components of extra stress in the (CEF) model are written as... 

The model describing interaction between two bodies, one of which is a deformed solid and the other is a rigid one, we call a contact problem. After the deformation, the rigid body (called also punch or obstacle) remains invariable, and the solid must not penetrate into the punch. Meanwhile, it is assumed that the contact area (i.e. the set where the boundary of the deformed solid coincides with the obstacle surface) is unknown a priori. This condition is physically acceptable and is called a nonpenetration condition. We intend to give a mathematical description of nonpenetration conditions to diversified models of solids for contact and crack problems. Indeed, as one will see, the nonpenetration of crack surfaces is similar to contact problems. In this subsection, the contact problems for two-dimensional problems characterizing constraints imposed inside a domain are considered. 

Texturing. The final step in olefin fiber production is texturing the method depends primarily on the appHcation. For carpet and upholstery, the fiber is usually bulked, a procedure in which fiber is deformed by hot air or steam jet turbulence in a no22le and deposited on a moving screen to cool. The fiber takes on a three-dimensional crimp that aids in developing bulk and coverage in the final fabric. Stuffer box crimping, a process in which heated tow is overfed into a restricted oudet box, imparts a two-dimensional sawtooth crimp commonly found in olefin staple used in carded nonwovens and upholstery yams. 

F1G. 1. Two-dimensional representation of a nearly spherical close-packed arrangement of spherons, with one in the inner core (left), and of an arrangement with prolate deformation, consequent to having two spherons in the inner core (right). 

In numerical analysis, both functions of normal surface deformation and pressure distribution have to be discretized in a space domain over U grid points for a line load, or grid points for two-dimensional distributed load. As an example, the deformation for line loading can be rewritten in discrete form as follows ... 

In this section, we show the morphological changes of stretched NR without filler by AFM. Two-dimensional mappings of topography and elasticity for elongated NR will be given to confirm the breakdown of the long-beheved assumption of affine deformation. 

In textbooks, plastic deformation is often described as a two-dimensional process. However, it is intrinsically three-dimensional, and cannot be adequately described in terms of two-dimensions. Hardness indentation is a case in point. For many years this process was described in terms of two-dimensional slip-line fields (Tabor, 1951). This approach, developed by Hill (1950) and others, indicated that the hardness number should be about three times the yield stress. Various shortcomings of this theory were discussed by Shaw (1973). He showed that the experimental flow pattern under a spherical indenter bears little resemblance to the prediction of slip-line theory. He attributes this discrepancy to the neglect of elastic strains in slip-line theory. However, the cause of the discrepancy has a different source as will be discussed here. Slip-lines arise from deformation-softening which is related to the principal mechanism of dislocation multiplication a three-dimensional process. The plastic zone determined by Shaw, and his colleagues is determined by strain-hardening. This is a good example of the confusion that results from inadequate understanding of the physics of a process such as plasticity. 

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