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Review Articles

# Thermomechanics of Solids With Lower-Dimensional Energetics: On the Importance of Surface, Interface, and Curve Structures at the Nanoscale. A Unifying Review

[+] Author and Article Information
A. Javili

A. McBride

P. Steinmann

e-mail: paul.steinmann@ltm.uni-erlangen.de
Chair of Applied Mechanics,
University of Erlangen–Nuremberg,
Egerlandstr. 5,
Erlangen 91058, Germany

Pyotr Leonidovich Kapitza (1894–1984) was a physicist and Nobel laureate who first proposed the presence of thermal interface resistance corresponding to a discontinuous temperature field across an interface for liquid helium; see Kapitza [82].

Here and henceforth, the subscripts t and 0 shall designate spatial and material quantities, respectively, unless specified otherwise.

Note that inertial forces are omitted. Furthermore, in the absence of mass flux, mass is conserved according to the standard relations given in Table 1.

The reduced dissipation inequality on the interface is alternatively expressed in the literature (see, e.g., Ref. [47]) in one of the following forms:

$D¯⊥=Θ¯[[Q·[1Θ-1Θ¯]]]·N¯=Θ¯[[QΘ·[1-ΘΘ¯]]]·N¯=[[QΘ·[Θ¯-Θ]]]·N¯≥0$

The term volumetric is used in analogy to the bulk. Nevertheless, it has a different meaning on the surface. A volumetric deformation in the bulk is a deformation mode that changes the volume uniformly. A volumetric surface deformation, however, is a deformation mode that changes the area uniformly. In this sense the term spherical seems more appropriate. Nonetheless, for the sake of consistency, the term volumetric is used henceforth.

For the sake of simplicity, it is assumed that the surface is an interface between the fluid and an outside vacuum.

Often in the literature $c∧$ is written as sum of the principal curvatures $c∧$ = −[1/r1+ 1/r2] where r1 and r2 denote the principal radii of curvature. Based on this definition, the curvature is negative if the surface curves away from the normal.

For isotropic thermal conduction in the material configuration $G≡I$ and $G≡I⊗F-t$.

For isotropic thermal conduction in the material configuration $G∧≡I∧and G∧≡I∧⊗F∧-t.$

Numerical investigations of a more general case where the surface, interface, and curve are all considered involve additional complications while providing little additional insight into the problem and, therefore, are not studied here.

1Corresponding author.

Manuscript received May 4, 2012; final manuscript received September 25, 2012; published online March 20, 2013. Editor: Harry Dankowicz.

Appl. Mech. Rev 65(1), 010802 (Mar 21, 2013) (31 pages) Paper No: AMR-12-1027; doi: 10.1115/1.4023012 History: Received May 04, 2012; Revised September 25, 2012

## Abstract

Surfaces and interfaces can significantly influence the overall response of a solid body. Their behavior is well described by continuum theories that endow the surface and interface with their own energetic structures. Such theories are becoming increasingly important when modeling the response of structures at the nanoscale. The objectives of this review are as follows. The first is to summarize the key contributions in the literature. The second is to unify a select subset of these contributions using a systematic and thermodynamically consistent procedure to derive the governing equations. Contributions from the bulk and the lower-dimensional surface, interface, and curve are accounted for. The governing equations describe the fully nonlinear response (geometric and material). Expressions for the energy and entropy flux vectors, and the admissible constraints on the temperature field, all subject to the restriction of non-negative dissipation, are explored. A particular emphasis is placed on the structure of these relations at the interface. A weak formulation of the governing equations is then presented that serves as the basis for their approximation using the finite element method. Various forms for a Helmholtz energy that describes the fully coupled thermomechanical response of the system are given. They include the contribution from surface tension. The vast majority of the literature on surface elasticity is framed in the infinitesimal deformation setting. The finite deformation stress measures are, thus, linearized and the structure of the resulting stresses discussed. The final objective is to elucidate the theory using a series of numerical example problems.

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## Figures

Fig. 1

The domains B0, S0, I0, and C0 and the various unit normals

Fig. 2

The material and spatial configurations of a continuum body and the associated motions, deformation gradients, and velocity measures in the various parts of the body

Fig. 3

The bulk B0 and the canonical control region B0 and its representation in the bulk, and on the surface, interface, and curve

Fig. 4

Transformation of a cube into a sphere due to an incrementally prescribed surface tension of γ∧/μ = 1.0 mm

Fig. 5

Transformation of a unit-cube into an ellipsoid due to a prescribed anisotropic surface tension γ∧/μ = [1 + β∧J∧[n·e]2] mm

Fig. 6

Strip with surface: geometry (left) and applied boundary conditions (right). Dimensions are in mm. The thickness is 0.1 mm.

Fig. 7

The stress distribution for a thermomechanical bulk. The results (a)–(f) correspond to the reference configuration and 20%, 40%, 60%, 80%, and 100% of the final deformation, respectively. The stress depicted is the xx-component of the Cauchy stress tensor. The reference configuration is indicated using a dashed (white) line.

Fig. 8

The temperature distribution for a thermomechanical bulk. The results (a)–(f) correspond to the reference configuration and 20%, 40%, 60%, 80%, and 100% of the final deformation, respectively. The reference configuration is indicated using a dashed (white) line.

Fig. 9

The influence a purely mechanical surface on the stress distribution for μ∧/μ=λ∧/λ = 2 mm. The results (a)–(f) correspond to the reference configuration and 20%, 40%, 60%, 80%, and 100% of the final deformation, respectively. The stress depicted is the xx-component of the Cauchy stress tensor. The reference configuration is indicated using a dashed (white) line.

Fig. 10

The influence of a purely mechanical surface on the temperature distribution for μ∧/μ=λ∧/λ = 2 mm. The results (a)–(f) correspond to the reference configuration and 20%, 40%, 60%, 80%, and 100% of the final deformation, respectively. The reference configuration is indicated using a dashed (white) line.

Fig. 11

The influence of interface conduction on the temperature distribution for μ∧/μ=λ∧/λ = 2 mm and k∧/k = 0–100 mm

Fig. 15

The temperature distribution for varying values of the Kapitza resistance r¯Q on the interface

Fig. 12

The influence of the interface heat capacity on the temperature distribution for μ∧/μ = λ∧/λ = 2 mm and c∧F∧/cF = 0–1 mm

Fig. 13

The influence of the interface heat expansion coefficient on temperature distribution for μ∧/μ=λ∧/λ = 2 mm and α∧/α = 0–0.75

Fig. 14

A cube containing an internal energetic interface. The geometry is show on the left and the two different load cases on the right. The first load case corresponds to a thermal load while the second is a mechanical one.

Fig. 16

The stress distribution without and with an HC interface are shown in (a) and (b), respectively, and compared in (c). The resulting temperature distribution is shown in (d).

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