Surface areas in solids can be changed by elastic deformation, and this leads to the concept of surface stresses, gab, and surface strains, eS^. Here the lower indices designate two orthogonal coordinate direc­tions, xl and x2, which are tangential to the surface. Greek indices are used here for surface quantities, and they assume the values of 1 or 2. In contrast, Latin indices are employed for vector — and tensor-valued quantities within the bulk material, and they assume values of 1, 2, or 3.

The definition of the surface stress in a Lagrang — ian reference frame is according to Cahn38

dg

gab = [111]

@eab

Here, the specific surface free energy g is assumed to depend now on the elastic strain components tangen­tial to the surface. We note that eqn [Ill] differs from

the Shuttleworth39 equation. The latter is inconsis­tent with continuum mechanics as has been argued by Gutman.40 In contrast, the surface elasticity theory of Gurtin and Murdoch41,42 is compatible with the elasticity theory of bulk solids, and the two are connected by appropriate boundary conditions as described below. The theory of Gurtin and Murdoch requires that the surface energy be a function of the surface strains.

As found from recent ab initio calculations and from atomistic calculations with empirical interatomic potentials, the dependence of the specific surface energy g on surface strains can be written as

g(eSb) = g(°)+ 2r «byd4eeS<5 + °(fi3) [112]

for small surface strains. Here, Gapyg is the surface elastic modulus tensor, and summation is implied over repeated indices. The surface elastic constants have a dimension of N m — while bulk elastic con­stants have the dimension of N m-2. Since the surface is attached to a bulk solid, surface strains eS^ and the bulk strains ejj are connected. Bulk strains are defined relative to a stress-free reference configuration of the bulk solid. However, a stress-free reference state for its surfaces may not coincide with the stress-free reference configuration for the bulk solid. Ab initio calculations on slabs43-47 of metals have in fact revealed that surface layers on stress-free bulk mate­rials possess in general a positive residual strain, e^. The surfaces ofmetals are naturally stretched and will contract if they can deform the underlying solid.

When the solid is elastically strained, the total surface strains become

eSb = <8 + eab [113]

where the strains eab are identical to the bulk strains ejj at the surface.

The surface stresses relative to the stress-free ref­erence configuration of the bulk are now defined as

EaP T Gap-/SeyS

The first term gives the residual or intrinsic surface stresses, and they are connected to the residual sur­face strains by

gtf = G«pyse;s [115]

This mechanical theory of surface stresses, as out­lined above, has been developed previously in its full mathematical rigor by Gurtin and Murdoch for gen­eral finite strains.41 The above linearized version for
small strains has also been presented by these authors, and they have shown42 that for isotropic materials, the residual surface strains are

e*;8 = e;dab [116]

and the surface elastic modulus tensor

Ga[]yS mS(dayd8d T dadg8y) T 8gyd [117]

contains two surface elastic constants m-S and lS. Typ­ically, e* is between 0.01 and 0.1, and the surface stretch modulus is on the order of 2mMd/(1—2nM), where mM and nM are the elastic constants of the bulk material and d is the thickness of the surface layer, about one interatomic distance.

With the above equations, the specific surface energy expansion, eqn [112], can be written as

y(eab) y0 T 2 (mS T As )e eaa T AS eafteaft

T 2^S eaae88

for isotropic materials. The energy associated with the residual surface stresses g^ is included in the term y0. It represents the surface energy of a planar surface on a stress-free solid.