Arzhang Angoshtari

THERMOELASTIC ANALYSIS OF THICK-WALLED FINITE-LENGTH CYLINDERS OF FUNCTIONALLY GRADED MATERIALS

ABSTRACT A semianalytical thermoelasticity solution for thick-walled finite-length cylinders made of functionally graded (FG) materials is presented. The governing partial differential equations are reduced to ordinary differential equations using Fourier expansion series in the axial coordinate. The radial domain is divided into some virtual subdomains in which the power-law distribution is used for the thermomechanical properties of the constituent components. Imposing the necessary continuity conditions between adjacent subdomains, together with the global boundary conditions, a set of linear algebraic equations are obtained. Solution of the linear algebraic equations yields the thermoelastic responses for each subdomain as exponential functions of the radial coordinate. Some results for the stress, strain, and displacement components through the thickness and along the length are presented due to uniform internal pressure and thermal loading. Based on the results, the gradation of the constitutive components is a significant parameter in the thermomechanical responses of FG cylinders.

A Geometric Theory of Nonlinear Morphoelastic Shells

Many thin three-dimensional elastic bodies can be reduced to elastic shells: two-dimensional elastic bodies whose reference shape is not necessarily flat. More generally, morphoelastic shells are elastic shells that can remodel and grow in time. These idealized objects are suitable models for many physical, engineering, and biological systems. Here, we formulate a general geometric theory of nonlinear morphoelastic shells that describes both the evolution of the body shape, viewed as an orientable surface, as well as its intrinsic material properties such as its reference curvatures. In this geometric theory, bulk growth is modeled using an evolving referential configuration for the shell, the so-called material manifold. Geometric quantities attached to the surface, such as the first and second fundamental forms, are obtained from the metric of the three-dimensional body and its evolution. The governing dynamical equations for the body are obtained from variational consideration by assuming that both fundamental forms on the material manifold are dynamical variables in a Lagrangian field theory. In the case where growth can be modeled by a Rayleigh potential, we also obtain the governing equations for growth in the form of kinetic equations coupling the evolution of the first and the second fundamental forms with the state of stress of the shell. We apply these ideas to obtain stress-free growth fields of a planar sheet, the time evolution of a morphoelastic circular cylindrical shell subject to time-dependent internal pressure, and the residual stress of a morphoelastic planar circular shell.

Differential Complexes in Continuum Mechanics

We study some differential complexes in continuum mechanics that involve both symmetric and non-symmetric second-order tensors. In particular, we show that the tensorial analogue of the standard grad-curl-div complex can simultaneously describe the kinematics and the kinetics of motion of a continuum. The relation between this complex and the de Rham complex allows one to readily derive the necessary and sufficient conditions for the compatibility of displacement gradient and the existence of stress functions on non-contractible bodies.We also derive the local compatibility equations in terms of the Green deformation tensor for motions of 2D and 3D bodies, and shells in curved ambient spaces with constant curvatures.

Convergence analysis of the Wolf method for Coulombic interactions

Abstract A rigorous proof for convergence of the Wolf method (Wolf et al., 1999 [9] ) for calculating electrostatic energy of a periodic lattice is presented. In particular, we show that for an arbitrary lattice of unit cells, the lattice sum obtained via Wolf method converges to the one obtained via Ewald method.

Finite-temperature atomic structure of 180° ferroelectric domain walls in PbTiO3

In this letter we obtain the finite-temperature structure of 180° domain walls in PbTiO3 using a quasi-harmonic–lattice dynamics approach. We obtain the temperature dependence of the atomic structure of domain walls from 0 K up to room temperature. We also show that both Pb-centered and Ti-centered 180° domain walls are thicker at room temperature; domain wall thickness at T=300 K is about three times larger than that of T=0 K. Our calculations show that Ti-centered domain walls have a lower free energy than Pb-centered domain walls and hence are more likely to be seen at finite temperatures.

On the existence of chaotic circumferential waves in spinning disks

We use a third-order perturbation theory and Melnikovs method to prove the existence of chaos in spinning circular disks subject to a lateral point load. We show that the emergence of transverse homoclinic and heteroclinic points lead, respectively, to a random reversal in the traveling direction of circumferential waves and a random phase shift of magnitude pi for both forward and backward wave components. These long-term phenomena occur in imperfect low-speed disks sufficiently far from fundamental resonances.

Effect of external normal and parallel electric fields on 180° ferroelectric domain walls in PbTiO3

We impose uniform electric fields both parallel and normal to 180° ferroelectric domain walls in PbTiO₃ and obtain the equilibrium structures using the method of anharmonic lattice statics. In addition to Ti-centered and Pb-centered perfect domain walls, we also consider Ti-centered domain walls with oxygen vacancies. We observe that an electric field can increase the thickness of the domain wall considerably. We also observe that increasing the magnitude of the electric field we reach a critical electric field E(c); for E > E(c) there is no local equilibrium configuration. Therefore, E(c) can be considered as an estimate of the threshold field E(h) for domain wall motion. Our numerical results show that oxygen vacancies decrease the value of E(c). As the defective domain walls are thicker than perfect walls, this result is in agreement with the recent experimental observations and continuum calculations that show thicker domain walls have lower threshold fields.

Atomic structure of steps on 180° ferroelectric domain walls in PbTiO3

Using the method of anharmonic lattice statics, we calculate the equilibrium structure of steps on 180° ferroelectric domain walls (DWs) in PbTiO3. We consider three different types of steps: (i) Ti–Ti step that joins a Ti-centered DW to a Ti-centered DW, (ii) Pb–Pb step that joins a Pb-centered DW to a Pb-centered DW, and (iii) Pb–Ti step that joins a Pb-centered DW to a Ti-centered DW. We show that atomic distortions due to these steps broaden a DW but are localized, i.e., they are confined to regions with dimensions of a few lattice spacings. We see that a step locally thickens the DW; the defective DW is two to three times thicker than the perfect DW depending on the step type. We also observe that steps distort the polarization distribution in a mixed Bloch–Neel like way; polarization rotates out of the DW plane near the steps. Our calculations show that Pb–Pb steps have the lowest static energy.