Materials modeling continuum models

In large measure the paradigm within which work is carried out is strongly influenced by the objectives of the work, the background of the investigator, and the particular materials model under study. From a strictly fluid mechanics, hydrodynamic, or continuum framework, defect issues are not overtly at issue. From a strictly mechanical framework, the defective solid... 

With the advent of nanomaterials, different types of polymer-based composites developed as multiple scale analysis down to the nanoscale became a trend for development of new materials with new properties. Multiscale materials modeling continue to play a role in these endeavors as well. For example, Qian et al. [257] developed multiscale, multiphysics numerical tools to address simulations of carbon nanotubes and their associated effects in composites, including the mechanical properties of Young s modulus, bending stiffness, buckling, and strength. Maiti [258] also used multiscale modeling of carbon nanotubes for microelectronics applications. Friesecke and James [259] developed a concurrent numerical scheme to evaluate nanotubes and nanorods in a continuum. 

The difference between continuum and discrete models stays, as we have said, in the description of the secundary, but larger, portion of the material model, the solvent. This is a delicate point, which deserves some comments to supplement what said in Section 1. 

From the standpoint of the continuum simulation of processes in the mechanics of materials, modeling ultimately boils down to the solution of boundary value problems. What this means in particular is the search for solutions of the equations of continuum dynamics in conjunction with some constitutive model and boundary conditions of relevance to the problem at hand. In this section after setting down some of the key theoretical tools used in continuum modeling, we set ourselves the task of striking a balance between the analytic and numerical tools that have been set forth for solving boundary value problems. In particular, we will examine Green function techniques in the setting of linear elasticity as well as the use of the finite element method as the basis for numerical solutions. 

The continuum mechanics modeled by JAS3D are based on two fundamental governing equations. The kinematics is based on the conservation of momentum equation, which can be solved either for quasi-static or dynamic conditions (a quasistatic procedure was used for these analyses). The stress-strain relationships are posed in terms of the conventional Cauchy stress. JAS3D includes at least 30 different material models. 

For the equivalent continuum, the LBNL research team used a linear elastic material model. A rock-mass Young s modulus 14.77 GPa and a rock-mass Poisson s ratio of 0.21 were adopted from CRWMS M O (1999). These elastic parameters, which represent the bulk rock mass (including the effect of fractures) have been estimated using an empirical method based on the Geological Strength Index (GSI). The adopted rock-mass Young s modulus is about 50% lower than the Young s modulus of intact rock determined on core samples from the site. 

The assignment of the 610 nm absorption was based on the apphcation of continuum models of the F and Fj center [18]. In the continuum model of the F center the electron is bound to the vacancy by a potential of the form e jKr, where K is an effective dielectric constant for the material. The continuum model of the F2 center is based on the analogous assumption that the transitions of the Fj center can be interpreted in terms of transitions of a hydrogen molecular ion immersed in a dielectric medium. The energy levels of the center in this model are the eigenvalues of the Hamiltonian... 

In Odegard s study [48], a method has been presented for finking atomistic simulations of nano-structured materials to continuum models of tfie corresponding bulk material. For a polymer composite system reinforced with SWCNTs, the method provides the steps whereby the nanotube, the local polymer near the nanotube, and the nanotube/ polymer interface can be modeled as an effective continuum fiber by using an equivalent-continuum model. The effective fiber retains the local molecular stractuie and bonding information, as defined by MD, and serves as a means for finking tfie eqniv-alent-continuum and micromechanics models. The micromechanics method is then available for the prediction of bulk mechanical properties of SWCNT/polymer com-... 

The major difficulty in using computational models is the development of material models, which require a complete set of material parameters in constitutive and failure models information (Anderson and Bodner, 1988). Furthermore, a ntrmerical method based on continuum mechanics encounters a fundamental difficrrlty when material failttre is involved, as such numerical methods, for example FE and finite differences (FD), are incapable of deahng with a large ntrmber of discontinrrities. 

Consider the case of an RVE subject to oiJy small strains and linear elasticity (z.e., sti-ess varies Imearly with the infinitesimal strain tensor for each individual material). We assume that the continuum approximation is valid at all points within the separate materials, that the polymer and matrices are perfectly bonded, and that there is no voiding or cracking. Note that the use of these assumptions creates a somewhat idealistic material model. 

Refined constitutive models for continuum elements and cohesive crack interface elements typically include many parameters, which require calibration with data from material tests. Data fi om uniaxial compression tests on concrete cylinders can be used for the calibration of the continuum material models for the concrete, as shown in Fig. 10a, while data from masomy prism tests (uniaxial compression and bond wrench) can be used for the calibration of the constitutive models for the masomy. Interface elements typically include parameters pertaining to mixed-mode fracture, and data from mixedmode fracture tests on concrete or masomy mortar joints need to be used for the calibration (e.g.. 

An even coarser description is attempted in Ginzburg-Landau-type models. These continuum models describe the system configuration in temis of one or several, continuous order parameter fields. These fields are thought to describe the spatial variation of the composition. Similar to spin models, the amphiphilic properties are incorporated into the Flamiltonian by construction. The Flamiltonians are motivated by fiindamental synnnetry and stability criteria and offer a unified view on the general features of self-assembly. The universal, generic behaviour—tlie possible morphologies and effects of fluctuations, for instance—rather than the description of a specific material is the subject of these models. 

In this chapter we shall consider four important problems in molecular n iudelling. First, v discuss the problem of calculating free energies. We then consider continuum solve models, which enable the effects of the solvent to be incorporated into a calculation witho requiring the solvent molecules to be represented explicitly. Third, we shall consider the simi lation of chemical reactions, including the important technique of ab initio molecular dynamic Finally, we consider how to study the nature of defects in solid-state materials. 

One of the simplest ways to model polymers is as a continuum with various properties. These types of calculations are usually done by engineers for determining the stress and strain on an object made of that material. This is usually a numerical finite element or finite difference calculation, a subject that will not be discussed further in this book. 

Contact Drying. Contact drying occurs when wet material contacts a warm surface in an indirect-heat dryer (15—18). A sphere resting on a flat heated surface is a simple model. The heat-transfer mechanisms across the gap between the surface and the sphere are conduction and radiation. Conduction heat transfer is calculated, approximately, by recognizing that the effective conductivity of a gas approaches 0, as the gap width approaches 0. The gas is no longer a continuum and the rarified gas effect is accounted for in a formula that also defines the conduction heat-transfer coefficient ... 

Baskes (1999) has discussed the status role of this kind of modelling and simulation, citing many very recent studies. He concludes that modelling and simulation of materials at the atomistic, microstructural and continuum levels continue to show progress, but prediction of mechanical properties of engineering materials is still a vision of the future . Simulation cannot (yet) do everything, in spite of the optimistic claims of some of its proponents. 

There is a view developing concerning the accomplishments of shock-compression science that the initial questions posed by the pioneers in the field have been answered to a significant degree. Indeed, the progress in technology and description of the process is impressive by any standard. Impressive instrumentation has been developed. Continuum models of materials behavior have been elaborated. Techniques for numerical simulation have been developed in depth. 

The third group is the continuum, models, and these are based on simple concepts from classical electromagnetism. It is convenient to divide materials into two classes, electrical conductors and dielectrics. In a conductor such as metallic copper, the conduction electrons are free to move under the influence of an applied electric field. In a dielectric material such as glass, paraffin wax or paper, all the electrons are bound to the molecules as shown schematically in Figure 15.2. The black circles represent nuclei, and the electron clouds are represented as open circles. 

In literature, some researchers regarded that the continuum mechanic ceases to be valid to describe the lubrication behavior when clearance decreases down to such a limit. Reasons cited for the inadequacy of continuum methods applied to the lubrication confined between two solid walls in relative motion are that the problem is so complex that any theoretical approach is doomed to failure, and that the film is so thin, being inherently of molecular scale, that modeling the material as a continuum ceases to be valid. Due to the molecular orientation, the lubricant has an underlying microstructure. They turned to molecular dynamic simulation for help, from which macroscopic flow equations are drawn. This is also validated through molecular dynamic simulation by Hu et al. [6,7] and Mark et al. [8]. To date, experimental research had "got a little too far forward on its skis however, theoretical approaches have not had such rosy prospects as the experimental ones have. Theoretical modeling of the lubrication features associated with TFL is then urgently necessary. 

---