Mathematical Research in Materials Science Opportunities and Perspectives (1993) / Chapter Skim
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5 DEFECTS, DEFORMATION, AND INTERFACES
5 DEFECTS, DEFORMATION, AND INTERFACES
Pages 52-65
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From page 52... ...
At the mesoscopic length scale, it has been customary to impose a boundary condition specifying that the solution match the continuum solution. Such a procedure, however, has not been fully justified and may require more attention in view of the sensitivity of solutions to boundary conditions.
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From page 53... ...
in the following sections, additional problem areas are presented, including modeling methodology, mechanics of defects and interfaces, plasticity and fracture, large local fields and instability in random systems, dynamic fracture, liquid crystals, and a few topics concerning the structure of interfaces on the atomistic level and the macroscopic properties that result. They provide a glimpse into some of the current activities in the areas of defects, deformations, and interfaces and point out some issues that may interest mathematical scientists.
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From page 54... ...
Direct simulation by ah initio methods and even atomistic calculations using empirical interatomic potentials are in many cases too expensive to be of practical utility for complex systems (see Chapter 8~. Mathematical analysis aimed at combining direct simulation of local behavior with a mean-field theory is needed to develop descriptions of nonhomogeneous solid phases.
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From page 55... ...
As a material's plasticity is increased, the governing elliptic equations become very stiff, and the hyperbolic limits of these elliptic equations cannot easily be solved by numerical procedures. Hyperbolic equations describe perfectly plastic materials, but the uniqueness of their solutions is not guaranteed.
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From page 56... ...
This is usually a good assumption for metals and many polymers under many conditions, although it does break down under large tensile stress as experienced at the crack tip due to void formation. In more brittle materials, the extent of intrinsic plasticity is usually small, and various damage mechanisms, including friction, microfracturing, and cavitation, often contribute significantly to the apparent plastic strains.
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From page 57... ...
Moreover, weak-link arguments do not usually take into account the inhomogeneous strain field inside a material; in this sense, fracture mechanics and fracture statistics are incompatible. Mechanical failure is merely one example of a large class of instability properties of quasi-homogeneous materials that do not yield in a straightforward manner to solution by the homogenization procedures that have been so successful for modeling effective transport properties and elastic moduli.
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From page 58... ...
in time according to a distribution that is a function of the load history. Monte Carlo simulations using such models yield information on asymptotic distributions for strength and lifetime, scaling phenomena, critical points and transitions, fractal behavior, localization, universality, renormalization, fracture toughness, and interface effects among elements.
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From page 59... ...
Liquid crystalline materials are commonly described in terms of a director field. When the director is correlated in a single spatial direction, the meso-phase is called nematic; the simplest conceptual picture of a nematic phase is aligned rod-like molecules.
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From page 60... ...
An equilibrium theory for low-molar-mass nematic liquid crystals developed by Oseen, Frank, and Ericksen describes static configurations in these materials rather well, including configurations that occur under the competing influences of external magnetic and electric fields and solid surfaces to control alignment in these transversely isotropic liquids; see de Gennes (1974) and Ericksen (1976)
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From page 61... ...
it is known that, at a critical shear rate, the director field in shear flow undergoes a transition from "aligned" to "tumbling." Both limited experimental observations and scaling arguments based on molecular size considerations suggest that director singularities must be "soft"; that is, the elastic coefficients must depend on the distance to a singularity's core. Thermotropic liquid crystalline polymers of possible commercial interest are multiphasic in the melt.
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From page 62... ...
These new experimental capabilities now allow direct observations that require recent theoretical formalisms for interpretation and that allow detailed predictions of theories to be tested (Metois and Heyraud, 1987; Eaglesham et al., 1990; Williams and Bartelt, 1991; Alfonso et al., 1992~. Great advances in computer power now allow sophisticated theoretical tools, such as Monte CarIo and molecular dynamics simulations, fractal and scaling analysis, and transfer matrix and free fermion calculations, to be applied routinely to data interpretation.
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From page 63... ...
Although there are now many alternative and more elaborate lattice microemuIsion models, as well as a large succeeding and closely related literature on complex fluids, vesicles, microemuIsions, biomembranes, and so on, the Wheeler-Widom mode] is presented in some detail here to illustrate the basic ideas of microemuIsion models and to convey a sense of the analysis done in this context.
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From page 64... ...
This peak implies an incipient periodicity in an otherwise isotropic solution, which prefigures the liquid crystalline phases that are so prominent a part of the phase diagrams of real water-oil-amphiphile solutions. The scattering functions implied by the mode]
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From page 65... ...
Computer Simulation Computer simulation has made an enormous contribution to the complex study of grain boundaries; see, for example, Alber et al.
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