ABSTRACT
All the relevant interfacial quantities can be expressed as integrals over the threedimensional pressure tensor P of the interface regarded as a three-dimensional body. The
pressure tensor is simply the negative of the three-dimensional mechanical stress tensor σ (to be distinguished from the two-dimensional mechanical stress tensor σs). The pressure tensor must, of course, satisfy the general mechanical equilibrium condition:
(127)
which, for a spherical interface, can be written as: d(r2PN)/d(r2)=PT
(128)
or r2[dPN/d(r2)]=PT−PN
(129)
with PT and PN denoting the components of the pressure tensor in the tangential and radial directions, respectively. The corresponding equation for a cylindrical interface has the following form:
d(rPrr)/dr=Pφφ (130)
where Prr is the radial component and Pφφ is the tangential component perpendicular to the cylinder axis. Let us next introduce the excess tensor: Ps=P−PαβU
(131)
where U is the three-dimensional (or bulk) identity tensor, related to Us in the following way:
U=Us+nn (132)
bearing in mind that Us=a1a1+a2a2. Furthermore, we introduce the conventional notation:
(133)
where λ is a coordinate along the surface normal that equals zero at the dividing surface, Pα is the homogeneous isotropic pressure on the α-side of the interface, and Pβ is the corresponding pressure on the β-side. As elaborated earlier, in particular by Kralchevsky [18], we then have the following micro-mechanical definitions of the two surface tension tensors:
(134)
where: χ (1−λH)2−λ2D2=1−2λH+λ2K
(135)
is recognized as the local dilation factor for the area of a surface that is parallel with the dividing surface. The tensor L is defined as:
L=(1−2λH)Us+λb (136)
and serves the following purpose. In order to calculate the tangential tensions and the bending moments, we need to know the forces acting on a vectorial surface element dSorth of a sectorial strip that is perpendicular to the dividing surface. Then, according to Eliassen [23]:
dSorth=v·Ldλdl (137)
where v is a vector in the tangent plane of the dividing surface, normal to the abovementioned perpendicular sectorial strip at λ values different from zero. Thus, v·L is a vector perpendicular to the sectorial strip. Furthermore, because we have the following definitions:
(138)
(139)
which, due to the orthogonality of Us and q, are consistent with: γs=γUs+ζq
(140) σs=σUs+ηq
(141)
It follows that
(142)
(143)
and
(144)
(145)
Furthermore, the bending moment tensor is defined by:
(146)
which is a tensor in three dimensions. The corresponding two-dimensional surface tensor equals:
(147)
as was already anticipated above. Because the bending moment B equals 1/ 2Us:M and the torsion moment Θ equals 1/2q:M, it follows that:
(148)
(149)
It should be mentioned that Eqs. (148) and (149) as well as Eqs. (142) and (143) were first obtained by Ivanov and Kralchevsky [46]. Below, we treat the special cases of spherical and cylindrical interfaces. Most of the formulas quoted have been published earlier (cf. Ref. 37).
