### Submitted for publicationAtomistic Viewpoint of the Applicability of Microcontinuum Theories

**Abstract**

Microcontinuum field theories, including micromorphic theory, microstructure theory, micropolar theory, Cosserat theory, nonlocal theory and couple stress theory, are the extensions of the classical field theories for the applications in microscopic space and time scales. They have been expected to overlap atomic model at micro-scale and encompass classical continuum mechanics at macro-scale. This work provides an atomic viewpoint to examine the physical foundations of those well established microcontinuum theories, and to give a justification of their applicability through lattice dynamics and molecular dynamics.

**Introduction**

Continuum theories describe a system in terms of a few variables such as mass, temperature, voltage and stress, which are suited directly to measurements and senses. Its success, as well as its expediency and practicality, has been demonstrated and tested throughout the history of science in explaining and predicting diverse physical phenomena.

The fundamental departure of microcontinuum theories from the classical continuum theories is that the former is a continuum model embedded with microstructures for the purpose to describe the microscopic motion or a nonlocal model to describe the long-range material interaction, so as to extend the application of continuum model to microscopic space and short time scales. Among them, Micromophic Theory (Eringen and Suhubi [1964], Eringen [1999]) treats a material body as a continuous collection of a large number of deformable particles, each particle possessing finite size and inner structure. Upon some assumptions such as infinitesimal deformation and slow motion, micromorphic theory can be reduced to Mindlin's Microstructure Theory [1964]. When the microstructure of the material is considered as rigid, it becomes the Micropolar Theory (Eringen and Suhubi [1964]). Assuming a constant microinertia, micropolar theory is identical to the Cosserat Theory [1902]. Eliminating the distinction of macromotion of the particle and the micromotion of its inner structure, it results Couple Stress theory (Mindlin and Tiersten [1962], Toupin [1962]). When the particle reduces to the mass point, all the theories reduce to the classical or ordinary continuum mechanics.

The physical world is composed of atoms moving under the influence of their mutual interaction forces. These interactions at microscopic scale are the physical origin of a lot of macroscopic phenomena. Atomistic investigation would therefore help to identify macroscopic quantities and their correlations, and enhance our understanding of various physical theories. This paper aims to analyze the applicability of those well-established microcontinuum theories from atomistic viewpoint of lattice dynamics and molecular dynamics.

**Applicability Analyses from the Viewpoint of Dynamics of Atoms in Crystal**

Dynamics of Atoms in Crystal

Dynamics of Atoms in Crystal

There are a number of material features, such as chemical properties, material hardness, material symmetry, that can be explained by static atomic structure. There are, however, a large number of technically important properties that can only be understood on the basis of lattice dynamics. These include: thermal properties, thermal conductivity, temperature effect, energy dissipation, sound propagation, phase transition, thermal conductivity, piezoelectricity, dielectric and optical properties, thermo-mechanical-electromagnetic coupling properties.

The atomic motions, that are revealed by those features, are not random., Iin fact they are determined by the forces that atoms exert on each other, and most readily described not in terms of the vibrations of individual atoms, but in terms of traveling waves, as illustrated in Fig.1. Those waves are the normal modes of vibration of the system. The quantum of energy in an elastic wave is called a Phonon; a quantum state of a crystal lattice near its ground state can be specified by the phonons present; at very low temperature a solid can be regarded as a volume containing non-interacting phonons. The frequency-wave vector relationship of phonons is called Phonon Dispersion Relation, which is the fundamental ingredient in the theory of lattice dynamics and can be determined through experimental measurements, such as nNeutron scattering, iInfrared spectroscope and Raman scattering, or first principle calculations or phenomenological modeling. Through phonon dispersion relations, the dynamic characteristics of an atomic system can be represented, the validity of a calculation or a phenomenological modeling can be examined, interatomic force constants can be computed, Born effective charge, on which the strain induced polarization depends, can be obtained, various involved material constants can be determined.

**Optical Phonons**

Optical phonon branches exist in all crystals that have more than one atom per primitive unit cell. In such crystals, the elastic distortions give rise to wave propagation of two types. In the acoustic type (as LA and TA), all the atoms in the unit cell move in essentially the same phase, resulting in the deformation of lattice, usually referred as homogeneous deformation. In the optical type (as LO and TO), the atoms move within the unit cell, leave the unit cell unchanged, contribute to the discrete feature of an atomic system, and give rise to the internal deformations. In an optical vibration of non-central ionic crystal, the relative displacement between the positive and negative ions gives rise to the piezoelectricity. Optics is a phenomenon that necessitates the presence of an electromagnetic field. In ferroelectrics the anomalously large Born effective charges produce a giant LO-TO splitting in phonon dispersion relations. This feature is associated withto the existence of an anomalously large destabilizing dipole-dipole interaction, sufficient to compensate the stabilizing short-range force and induce the ferroelectric instability. Optical phonons, therefore, appears as the key concept to relate the electronic and structural properties through Born effective charge (Ghosez [1995,1997]). The elastic theory of continuum is the long wave limit of acoustic vibrations of lattice, while optical vibration is the mechanism of a lot of macroscopic phenomena involving thermal, mechanical, electromagnetic and optical coupling effects.

**Micropolar theory (Eringen and Suhubi [1964])**

When the material particle is considered as rigid, i.e., neglecting the internal motion within the microstructure, micromorphic theory becomes micropolar theory. Therefore, micropolar theory yields only acoustic and external optical modes. They are the translational and rotational modes of rigid units. For molecular crystals or framework crystal, or chopped composite, granular material et al, when the external modes in which the molecules move as rigid units have much lower frequencies and thus dominate the dynamics of atoms, micropolar theory can give a good description to the dynamics of microstructure. It accounts for the dynamic effect of material with rather stiff microstructure.

Assuming a constant microinertia, micropolar theory is identical to Cosserat theory [1902], Compared with micropolar theory, Cosserat theory is limited to problems not involving significant change of the orientation of the microstructure, such as liquid crystal and ferroelctrics.

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