Mechanical deformation modeling discussion

It will be shown in Chapter 11 that the correlations developed in this monograph can be combined with other correlations that are found in the literature (preferably with the equations developed by Seitz in the case of thermoplastics, and with the equations of rubber elasticity theory with finite chain extensibility for elastomers), to predict many of the key mechanical properties of polymers. These properties include the elastic (bulk, shear and tensile) moduli as well as the shear yield stress and the brittle fracture stress. In addition, new correlations in terms of connectivity indices will be developed for the molar Rao function and the molar Hartmann function whose importance in our opinion is more of a historical nature. A large amount of the most reliable literature data on the mechanical properties of polymers will also be listed. The observed trends for the mechanical properties of thermosets will also be discussed. Finally, the important and challenging topic of the durability of polymers under mechanical deformation will be addressed, to review the state-of-the-art in this area where the existing modeling tools are of a correlative (rather than truly predictive) nature at this time. 

Time-dependent hysteresis effects can also occur in crystalline materials and these lead to mechanical damping. Models, such as the SLS and the generalized Voigt model, have been used extensively to describe anelastic behavior of ceramics. It is, thus, useful to describe the sources of internal friction in these materials that lead to anelasticity. The models discussed in the last section are also capable of describing permanent deformation processes produced by creep or densification in crystalline materials. For polycrystalline ceramics, creep is usually considered from a different perspective and this will be discussed further in Chapter 7. 

The effect of mechanical deformation on a UV-cured urethane-acrylate polymer and on a silicon carbide/ urethane-acrylate model composite was studied by micro-FTIR spectroscopy. This technique was used for the first time to measure the width of the interfacial zone beyond which the fibre has no influence on the matrix, the results were discussed. 14 refs. 

The Gibbs-DiMarzio theory offers predictions including of the change in heat capacity at Tg, and of the dependence of Tg on various variables including molecular weight, cross-link density, mechanical deformation, plasticizer content, and blending with any polymer. These predictions explain well the data which are discussed next, in conjunction with the alternative explanations from the free-volume models. 

An understanding of deformation meehanisms of polymers is important to be able to manage the mechanical characteristics of these materials. In this regard, deformation models for two different types of polymers—semicrystalline and elastomeric—deserve our attention. The stiffness and strength of semicrystalline materials are often important considerations elastic and plastic deformation mechanisms are treated in the succeeding section, whereas methods used to stiffen and strengthen these materials are discussed in Section 15.8. However, elastomers are used on the basis of their unusual elastic properties the deformation mechanism of elastomers is also treated. 

The fundamental concept of the material clock or reduced time is similar to the principle described above in the discussion of time-temperature superposition. In the mechanical constitutive models, however, the change in the stress or deformation induces a shift in the material relaxation time. The fact that the time depends on the state of stress (or strain) or on its history leads to additional non-linearities in behavior from what is expected with, eg, the K-BKZ model. Physical explanations for the shifting material time are often based on free-volume ideas that are often invoked to explain time-temperature superposition. In addition, entropy changes have been invoked as have stress-activated processes. 

The various densification mechanisms at different temperatures can be modelled and displayed in HIP diagrams, in which relative temperature is plotted against temperature normalised with respect to the melting-point (Arzt el al. 1983). This procedure relates closely to the deformation-mechanism maps discussed in Section 5.1.2.2. 

The first section involves a general description of the mechanics and geometry of indentation with regard to prevailing mechanisms. The experimental details of the hardness measurement are outlined. The tendency of polymers to creep under the indenter during hardness measurement is commented. Hardness predicitions of model polymer lattices are discussed. The deformation mechanism of lamellar structures are discussed in the light of current models of plastic deformation. Calculations... 

The following examples explore how paleoaltimetry data may provide critical information about the evolution of mean elevation, averaged relief, and erosion from different models of continental deformation. However, I consider only changes in elevation, relief and erosion that may be predicted by tectonic models and neglect the influence that climate forcing or erosion related feedbacks could exert on such predictions. As discussed previously, the influence of climate, erosion and related feedbacks on tectonic deformation is important and should not be ignored. However, to consider all the complexities of potential interactions on the elevation record is beyond the scope and focus of this paper. In order to best illustrate the relationships between deformation mechanics and elevation, I review a few example elevation histories predicted by several commonly-cited tectonic models. 

In this chapter, two simple cases of stereomechanical collision of spheres are analyzed. The fundamentals of contact mechanics of solids are introduced to illustrate the interrelationship between the collisional forces and deformations of solids. Specifically, the general theories of stresses and strains inside a solid medium under the application of an external force are described. The intrinsic relations between the contact force and the corresponding elastic deformations of contacting bodies are discussed. In this connection, it is assumed that the deformations are processed at an infinitely small impact velocity and for an infinitely long period of contact. The normal impact of elastic bodies is modeled by the Hertzian theory [Hertz, 1881], and the oblique impact is delineated by Mindlin s theory [Mindlin, 1949]. In order to link the contact theories to collisional mechanics, it is assumed that the process of a dynamic impact of two solids can be regarded as quasi-static. This quasi-static approach is valid when the impact velocity is small compared to the speed of the elastic... 

The flow on the suspension visualized and simplified in the model just discussed generates VED and heats the pellets, but does not deform them. Thus, they do not include the dissipative mix-melting (DMM) melting mechanism, only VED. However, with the proper parameter adjustments, they are able to make fair predictions of the overall melting... 

The temperature dependence of the Payne effect has been studied by Payne and other authors [28, 32, 47]. With increasing temperature an Arrhe-nius-like drop of the moduli is found if the deformation amplitude is kept constant. Beside this effect, the impact of filler surface characteristics in the non-linear dynamic properties of filler reinforced rubbers has been discussed in a review of Wang [47], where basic theoretical interpretations and modeling is presented. The Payne effect has also been investigated in composites containing polymeric model fillers, like microgels of different particle size and surface chemistry, which could provide some more insight into the fundamental mechanisms of rubber reinforcement by colloidal fillers [48, 49]. 

The relationship between the structure of the disordered heterogeneous material (e.g., composite and porous media) and the effective physical properties (e.g., elastic moduli, thermal expansion coefficient, and failure characteristics) can also be addressed by the concept of the reconstructed porous/multiphase media (Torquato, 2000). For example, it is of great practical interest to understand how spatial variability in the microstructure of composites affects the failure characteristics of heterogeneous materials. The determination of the deformation under the stress of the porous material is important in porous packing of beds, mechanical properties of membranes (where the pressure applied in membrane separations is often large), mechanical properties of foams and gels, etc. Let us restrict our discussion to equilibrium mechanical properties in static deformations, e.g., effective Young s modulus and Poisson s ratio. The calculation of the impact resistance and other dynamic mechanical properties can be addressed by discrete element models (Thornton et al., 1999, 2004). 

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