Representation of Deformation

There is a large body of literature on the purely elastic interaction of non-pris-matic structural elements within a deformed composite solid (e.g. 63-70) this literature also forms part of the theoretical foundation of the behavior of viscoelastic composite solids. While not generally discussing these foundations three groups of model descriptions of multi-phase materials will be singled out  

The two-shell model of Kerner [65] conforms to the conditions of the second group of models. The dilatation of a spherical inclusion surrounded by a homogeneous medium is derived subject to the condition that displacements and tractions at the surface of the inclusion are continuous. The homogeneous medium is supposed to have the elastic properties of the composite as a whole. The model interrelates shear (Gj) and compressive (Kj) moduli (or Poisson s ratios p ) of an arbitrary number of isotropic elements with the macroscopic moduli Gc and Kc. 

In the case of isotropic inclusions of an elastic shear modulus Gf comparable to that of the matrix, G, Eq. (2.5) describes the variation of the complex shear modulus, G, of the composite as a function of the volume content Vf of the discrete phase  

As will be noted no molecular anisotropies and no effects due to size and size distribution of the particles of the discrete phase are recognized. Through E = 2(1 + p)G Eq. (2.5) can be used to predict also the complex tensile modulus. A good example for the applicability of Eq. (2.5) is furnished by the experimental data obtained by Dickie et al. [75]. For the dynamic tensile modulus of a physical mixture (polymer blend) of 75% by weight polymethylmethacrylate (PMMA, continuous phase) and 25% butylacrylate (PBA, discrete phase) within experimental error correspondence of calculated and measured data was obtained (Fig. 2.13, 

A three-shell model of a two-phase solid was proposed by van der Poel ([67], Fig. 2.14). An interior sphere representing a hard component (Gf, Kf, Vf) is surrounded by a soft shell (Ga, Ka, Va), and by a continuum, for which the elastic 

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Fig. 4.1. Schematic representation of deformation around a short fiber embedded in a matrix subjected to...

One can see that there are several forms for the representation of the constitutive relation of a viscoelastic liquid. Of course, we ought to say that all the types of constitutive relation we discussed in this section are equivalent. We can use any of them to describe the flow of viscoelastic liquids. However, the description of the flow of a liquid in terms of the internal variables allows one to use additional information, if it is available, about microstructure of the material, and, in fact, appears to be the simplest one for derivation and calculation. We believe that the form, which includes the internal variables, reflects a deeper penetration into the mechanisms of the viscoelastic behaviour of materials. From this point of view, all the representations of deformed material can be unified and classified. 

Fig. 5 Schematic representation of deformation information types being obtained using the LICRM setup...

Representations of deformation from Oh symmetry consistent with electron diffraction data for gaseous XeFg. The lone pair of elections is shown to be repelling the fluorine atoms, to give structures with Csv and C2v symmetry. Redrawn with permission from [2]. Copyright 1968, American Institute of Physics. 

FIGURE 4.1 Schematic representation of deformation oftwo sequences of atoms according to the Frenkel model. Positions before (a) and after (b) deformation [3],... 

Adapting Maier s representation of deformability, Steiger [11] established that the plastic strain to fracture couldn t be represented by only one curve and in practice it must be determined by an array of curves. 

Bogatov [12], completing the Maier-Steiger representation of deformability by a third invariant of the stress state (by the Lode-parameter), represented the variation of the plastic strain to fracture as a function of the stress state indexes k,jLi ) by a four-parametrical function (see Equation 10), which describes a surface ... 

Note that the lamina failure criterion was not mentioned explicitly in the discussion of Figure 4-36. The entire procedure for strength analysis is independent of the actual lamina failure criterion, but the results of the procedure, the maximum loads and deformations, do depend on the specific lamina failure criterion. Also, the load-deformation behavior is piecewise linear because of the restriction to linear elastic behavior of each lamina. The laminate behavior would be piecewise nonlinear if the laminae behaved in a nonlinear elastic manner. At any rate, the overall behavior of the laminate is nonlinear if one or more laminae fail prior to gross failure of the laminate. In Section 2.9, the Tsai-Hill lamina failure criterion was determined to be the best practical representation of failure... 

Figure 2 Representation of TLCP deformation process in die exit zone (micro scale). Source. Ref. 33.

The representation of the results from slow strain-rate tests may be through the usual ductility parameters such as reduction in area, the maximum load achieved, the crack velocity or even the time to failure, although as with all tests, metallographic or fractographic examination, whilst not readily quantifiable, should also be involved. Since stress-corrosion failures are usually associated with relatively little plastic deformation, the ductility... 

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). 

Fig. 23. Schematic diagram of the 4-roll mill apparatus. Schematic representation of the flow field within the mill illustrating the deformation of a fluid element [35]...

Fig. 5 Schematic representation of the domain contributions to the tensile deformation of the fibre chain stretching and chain rotation due to shear deformation...

Figure 2.21. Schematic representation of colloid probe-PDMS droplet interaction during the AFM experiment. Solid line depicts the undeformed profile of the PDMS droplet and the rigid colloid probe. Dashed line shows the deformed profile of the PDMS droplet.

Fig. 28. Schematic representation of a splay deformation (a) changes in the components of the director n, defining the orientational change (b) splay deformation of an oriented layer of a...

Fig. 32. Schematic representation of the flexo-electric effect, (a) The structure of an undeformed nematic liquid crystal with pear- and banana-shaped molecules (b) the same liquid crystal subjected to splay and bend deformations, respectively.

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