Tungyang Chen

Derivation of the generalized Young-Laplace equation of curved interfaces in nanoscaled solids

In nanoscaled solids, the mathematical behavior of a curved interface between two different phases with interface stress effects can be described by the generalized Young-Laplace equations [T. Young, Philos. Trans. R. Soc. London 95, 65 (1805); P. S. Laplace, Traite de Mechanique Celeste (Gauthier-Villars, Paris, 1805), Vol. 4, Supplements au Livre X]. Here we present a geometric illustration to prove the equations. By considering a small element of the curved thin interface, we model the interface stresses as in-plane stresses acting along its edges, while on the top and bottom faces of the interface the tractions are contributed from its three-dimensional bulk neighborhood. With this schematic illustration, simple force balance considerations will give the Young-Laplace equations across the interface. Similar procedures can be applied to conduction phenomena. This will allow us to reconstruct one type of imperfect interfaces, referred to as highly conducting interfaces.

Mori-Tanaka Estimates of the Overall Elastic Moduli of Certain Composite Materials

Simple, explicit formulae are derived for estimates of the effective elastic moduli of several multiphase composite materials with the Mori-Tanaka method. Specific results are given for composites reinforced by aligned or randomly oriented, transversely isotropic fibers or platelets, and for fibrous systems reinforced by aligned, cylindrically orthotropic fibers.

Cloak for curvilinearly anisotropic media in conduction

We explore the possibility to cloak a region in curvilinearly anisotropic background materials in the context of conductivity. Materials with curvilinear anisotropy possess constant properties in specific curvilinear coordinate. For cylindrically and spherically anisotropic solids, the cloak center and the origin of material coordinate are generally not collocated. We show that in combination with a rigid-body translation from the cloak center to the material origin, the previous coordinate transformation procedure remains applicable. But now the transformed material specifications depend on the position of cloak center. The validity of the cloak parameters is verified by finite element simulations.

A revisit of a cylindrically anisotropic tube subjected to pressuring, shearing, torsion, extension and a uniform temperature change

Abstract The problem of a cylindrically anisotropic tube or bar was seemed to be first examined by Lekhnitskii (1981) [Lekhnitskii, S.G., 1981. Theory of Elasticity of an Anisotropic Body. (Trans. from the revised 1977 Russian edition.) Mir, Moscow]. Recently, a thorough investigation of the subject was performed by Ting (1996) [Ting, T.C.T., 1996. Pressuring, shearing, torsion and extension of a circular tube or bar of cylindrically anisotropic material. Proc. Roy. Soc. Lond. A452, 2397–2421] in which a formulation akin to that of Stroh’s formalism is employed to resolve the boundary value problem subjected to a uniform pressure, shearing, torsion and uniform extension. In a continuing paper, Ting (1999) [Ting, T.C.T., 1999. New solutions to pressuring, shearing, torsion and extension of a cylindrically anisotropic elastic circular tube or bar. Proc. Roy. Soc. Lond, to appear.] rederived the solutions based on a modified formalism of Lekhnitskii, in which the solutions are in terms of elastic compliances, reduced elastic compliances as well as doubly reduced compliance. The results are much more compact and simpler than those of the earlier one. Independently, in this work, we construct the governing system also under the Lekhnitskii’s framework. Nevertheless, the present work and Ting’s formulation (1999) are not alike. Besides the loads considered in Ting (1996, 1999), we add the effect of a uniform temperature change in the formulation. The assumption that the stresses depend only on r makes it possible to incorporate the various loading cases considered. In addition to the explicit forms of admissible stresses, we derive the admissible displacements which are ensured to be single-valued for a multiply-connected domain. In contrast to the Ting’s works (1996, 1999), which often require superpositions of two or more basic solutions, the present solutions offer complete forms of solutions ready for direct calculations. We also report that, as in rectilinearly anisotropic solids, an entire analogy is observed between the fields of a uniform axial extension and a uniform temperature change in cylindrically anisotropic solids.

Fibrous nanocomposites with interface stress: Hill’s and Levin’s connections for effective moduli

We study the macroscopic behavior of solids containing circular cylindrical nanoinclusions of the same size with surface effects prevailing at interfaces. The overall thermomechanical properties of these solids are shown to comply with two sets of exact connections. The first set, similar to Hill’s universal connections, provides two constraints among the three axisymmetric overall elastic moduli. The second set relates the effective coefficients of thermal expansion to the effective moduli, in analogy with Levin’s formula. In contrast to the classical results, the presence of surface effects makes both sets of connections dependent on the absolute size of the nanoinclusions.

Effect of Kapitza contact and consideration of tube-end transport on the effective conductivity in nanotube-based composites

Recent studies reported that the theoretical predictions of the effective thermal conductivity of nanotube-based composites by conventional micromechanical models are anomalously higher than those measured experimentally and suggested that the contact resistance on the interface could be the contributing factor to the lower measured value. We explore theoretically whether the large disagreement could be attributed to the effect of Kapitza contact resistance. Our simulations show that the thermal contact resistance on the lateral surfaces of the nanotubes could not be a major factor of this marked disparity. By contrast, the heat transport mechanisms at the ends of the nanotubes could be a significant factor to influence the value. We propose a few simple models to simulate the thermal conductivity at the ends of the nanotubes, similar to two springs in serial and/or in parallel. Under the propositions, we find that the experimental data can be better predicted than the conventional theory and that the tub...

Thermoelastic properties and conductivity of composites reinforced by spherically anisotropic particles

Abstract The present paper is concerned with the overall thermoelastic properties and conductivity of composites reinforced by spherically anisotropic particles. Based on the concept of the replacement particle an equivalent bulk modulus, thermal expansion coefficient and thermal conductivity are derived for the spherically anisotropic particle. Such equivalent properties can be employed in the micromechanical models to predict the overall behavior of the composite. In addition to these, the shear loading is considered. The effective shear modulus is evaluated on the basis of Mori and Tanakas approximations and the dilute phase concentration factors are derived from exact solutions of an auxiliary boundary value problem.

Thermal conduction of a circular inclusion with variable interface parameter

Abstract An imperfect bonding problem associated with a solitary circular inclusion embedded in an infinite matrix under a remotely applied uniform intensity is considered. Specifically, we study the effect of imperfect interfaces which are either of weakly or of highly conducting type and that the interface parameter could vary arbitrarily along the interface. By using the orthogonality properties of the trigonometric series, we show that the solution field is governed by a linear set of algebraic equations with an infinite number of unknowns. The governing matrix for the unknowns is primarily composed of elements which are simple combinations of the Fourier coefficients of the interface parameter. Solutions of the boundary-value problem are employed to estimate the effective conductivity tensor of a composite consisting of dispersions of circular inclusions with equal size. The effective properties solely depend on two particular constants among an infinite number of unknowns. It is demonstrated that, even for a composite with isotropic dispersions of inclusions, the composite may become effectively anisotropic due to the presence of a variable interface parameter. Further, we present two microstructure independent properties regarding the effective conductivity of the considered system. We first show that the effective conductivity tensor for a composite with variably imperfect interfaces is always diagonally symmetric. This is accomplished by means of a reciprocal relation that is established in such systems. Next, we present dual relations for the effective conductivities of two-dimensional composites with variably imperfect interfaces. The latter result is a direct consequence of the existence of a dual relation for the local fields in such composites, as pointed out by Benveniste and Miloh (Benveniste, Y., Miloh, T., 1999. J. Mech. Phys. Solids 47, 1873–1892).

An invariant treatment of interfacial discontinuities in piezoelectric media

Abstract Interfacial discontinuities are examined in piezoelectric media of general anisotropy. The analysis relies on decomposing second-rank tensors at an interface into exterior and interior parts, and on decomposing first-rank tensors at an interface into normal and tangential parts. The solutions depend on three independent interfacial operators, one fourth-rank, one third-rank, and one second-rank tensors. By constructing appropriate augmented matrices, the topic can be treated systematically as that of uncoupled field equations. This leads to a concise organization of the solutions of inclusion and inhomogeneity problems. In applications exact results are given for the matrix interfacial quantities under uniform boundary conditions and/or under uniform transformation fields in the inclusion.

Materials with constant anisotropic conductivity as a thermal cloak or concentrator

An invisibility cloak based on transformation optics often requires material with inhomogeneous, anisotropic, and possibly extreme material parameters. In the present study, on the basis of the concept of neutral inclusion, we find that a spherical cloak can be achieved using a layer with finite constant anisotropic conductivity. We show that thermal localization can be tuned and controlled by anisotropy of the coating layer. A suitable balance of the degree of anisotropy of the cloaking layer and the layer thickness provides a cloaking effect. Additionally, by reversing the conductivities in two different directions, we find that a thermal concentrating effect can be simulated. This finding is of particular value in practical implementation as a material with constant material parameters is more feasible to fabricate. In addition to the theoretical analysis, we also demonstrate our solutions in numerical simulations based on finite element calculations to validate our results.