Finite-displacement elastic solution due to a triple contact line

At the line of triple contact of an elastic body with two immiscible fluids, the body is subjected to a force concentrated on this line, the fluid-fluid surface tension. In the simple case of a semi-infinite body, limited by a plane, a straight contact line on this plane, and a fluid-fluid surface tension normal to the plane, the classical elastic solution leads to an infinite displacement at the contact line and an infinite elastic energy. By taking into account the body-fluid surface tension (i.e., isotropic surface stress), we present a new and more realistic solution concerning the semi-infinite body, which gives a finite displacement and a ridge at the contact line, and a finite elastic energy. This solution also shows that Green's formulae, in the volume and on the surfaces, are valid (these formulae play a central role in the theory).


I. INTRODUCTION
Surface properties of deformable bodies have many applications, e.g., in adhesion, coating, thin films and nanosciences.When a deformable body is in contact with two fluids (e.g., liquid and air), it is subjected to the fluid-fluid surface tension along the body-fluid-fluid contact line.The classical solution for the deformation of an elastic body, occupying a semiinfinite space limited by a plane, and subjected to a fluid-fluid surface tension concentrated on a straight line of this plane and normal to the plane, leads to an infinite displacement at this line and an infinite elastic energy. 1 However, the presence of a ridge at the contact line was experimentally observed 2 and then confirmed by other experiments and Fourier transform calculations. 3In our previous work, 4 by taking into account the body-fluid surface stresses and surface energies, we showed that there are two equilibrium equations at the contact line: (i) the equilibrium of the three surface stresses (with no contribution of the volume stresses); (ii) a scalar equation involving the surface energies, the surface stresses, and the surface strains, which leads to a generalization of the classical Young's equation (only valid for a rigid body) (see Sec. II).The first equation-as an equilibrium of three forces tangent to the three interfaces-implies a finite displacement and the formation of a ridge at the contact line.This equation was then experimentally confirmed. 5In the present paper, by taking into account the body-fluid surface tension (i.e., isotropic surface stress), we obtain a new solution of the above elastic problem concerning the semi-infinite body, with a finite displacement and a ridge at the contact line, and a finite elastic energy.

II. GENERAL EQUILIBRIUM EQUATIONS
In the previous work Ref. 4 (with additional comments in Ref. 6 and mathematical aspects in Ref. 7), following the general variational method of Gibbs, 8 we gave the thermodynamic definition and properties of the surface stress and obtained the following equilibrium equations for any deformable body: 1) On each body-fluid surface: div σs + ρ s ḡ + σ • n + p n = 0 (1) where σ s is the body-fluid surface stress, ρ s the surface mass excess per unit area, ḡ the gravity field, σ the body volume stress, n the unit vector normal to the surface (oriented from the fluid to the body), and p the fluid pressure; σs = ι • σ s , where ι is the natural injection of the tangent plane to the surface T x (S) in the three-dimensional space E (i.e., σs βi = σ αβ s ∂ α x i , with components α and β on the surface, and i in E), and div σs is a special divergence based on the tensorial product of the covariant derivative on the surface and the usual derivative in E, defined in Ref. 7 (i.e., with components: (div σs (in which div σ s is the usual surface divergence and the subscript t indicates the vector component tangent to the surface) and a normal component where l is the curvature fundamental form on the surface, l n = l • n (i.e., with components: The eigenvalues of l n are the principal curvatures, 1 R 1 and 1 R 2 , of the surface (a curvature being positive when its center is on the side of n).If σ s is isotropic, i.e., σ s = σs I (eigenvalue σs and I the identity), we have Similar equations were previously written, but under some particular assumptions, e.g., the existence of a "surface traction field" 9 or in the special case of elastic bodies. 10,11 On the body-fluid-fluid contact line, there are two equations (as previously found in the particular case of the elastic thin plate 12,13 ): The first one is vectorial (three-dimensional) and corresponds to a line fixed on the body (but the line can move because the body is deformable): in which the subscripts b, f, and f ′ respectively denote the body and the two fluids, σ bf is the bf surface stress, ν bf the unit vector normal to the contact line and tangent to the bf surface (directed to the inside of bf), idem for σ bf ′ , ν bf ′ , and ν ff ′ , and γ ff ′ is the ff ′ surface tension.
This equation expresses the equilibrium of the three surface stresses acting on the contact line (with no contribution of the volume stresses), and determines the angles of contact φ f , φ f ′ , and φ b , respectively measured in f, f ′ , and b (satisfying ).This point was then experimentally verified in Ref. 5.
The second equation is scalar and corresponds to a line moving with respect to the body, but fixed in space (i.e., the displacement of the material points of the body, due to the deformation, exactly compensates the displacement of the line with respect to the body, so that the line remains fixed in space): in which τ is a unit vector tangent to the line, σ bf,νν and σ bf,τ ν are respectively the components along ν bf and τ of the bf surface stress acting on the line, idem for σ bf ′ ,νν and σ bf ′ ,τ ν , γ bf and γ bf ′ are respectively the bf and bf ′ surface energies, a νν is the surface stretching deformation, normal to the line, in the bf side, a τ ν the surface shear deformation, parallel to the line, in the bf side, and idem for a ′ νν and a ′ τ ν in the bf ′ side.An equivalent form of this equation is (with the help of Eq. ( 6)) where a r,νν = a ′ νν aνν and a r,τ ν = a ′ τ ν −aτν aνν .
Note that, for a perfectly rigid body, the above Eqs.( 6) and ( 7) cannot be written (since they are based on the possible displacement of the material points, due to the deformation) and, in this case, we obtain the unique scalar equation which is the classical Young's equation (in which γ bf and γ bf ′ are surface energies, and not surface tensions).If the body is deformable, Young's equation is not valid and the valid Eq. ( 8) is the generalization of Young's equation.Indeed, in the rigid body limit, a r,νν = 1, a r,τ ν = 0, and sin φ b = 0), and Eq. ( 8) leads to Young's equation.Also note that, in the case of a very little deformable body, i.e., an almost rigid body, the surface (and volume) stresses become almost infinite, so that the two first terms in Eq. ( 6) are almost infinite and this equation implies that φ b is almost equal to π (in order to equilibrate the finite fluid-fluid surface tension), i.e., that there is almost no ridge on the surface.Finally, note that, if the body is a fluid, then 7) is useless, and Eq. ( 6) expresses the equilibrium of the three fluid-fluid surface tensions.
Eq. ( 7) expresses that the variation of surface energy (e.g., increase in bf surface and decrease in bf ′ surface, when the line moves with respect to the body) is equal to the work of the surface stresses acting on the line (due to the displacement of the material points, on each side of the line), when the line remains fixed in space.This equation gives a condition on the components a r,νν and a r,τ ν of the "relative" surface deformation of the bf ′ side with respect to the bf side (we defined the "relative deformation gradient" in Ref. 14).
If σ bf ′ ,τ ν = 0 (which, e.g., occurs if the surface stress is isotropic), it gives the relative surface stretching deformation a r,νν (normal to the line) of the bf ′ side with respect to the bf side.

III. VALIDITY OF GREEN'S FORMULA: NO CONTRIBUTION OF THE VOLUME STRESSES
The preceding theory is based on Green's formula where σ is the volume stress tensor, w a virtual displacement, V a volume of the body in the neighborhood of the contact line, S the boundary surface, n the unit vector normal to the surface and directed towards the interior of the body, dv an element of volume, and da an element of area.With the help of a first example of solution, we showed in Ref. 7 that: 1) Owing to the singularity at the contact line, the components of σ do not belong to the Sobolev space H 1 (V).
2) Nevertheless, Green's formula remains valid, because all the components ∂ j u i and σ ij are either bounded or subjected to the inequality in V (r is the distance to the contact line; c and d being positive constants), and these inequalities imply that lim ε→0 S(ε) where S(ε) is the boundary surface of a small tubular volume V(ε) of the body, of radius ε, around an element of contact line (V(ε) is bounded by (i) the surface of the body and (ii) a half-cylinder of radius ε around the contact line).The validity of Green's formula is based on Eq. ( 12), which directly expresses that the volume stresses have no contribution at the contact line.
3) The elastic energy is finite (in the neighborhood of the contact line).
These results will be confirmed with the solution given in the present paper.

IV. APPLICATION TO THE SEMI-INFINITE ELASTIC BODY
Let us consider a semi-infinite isotropic elastic body b, occupying the half space x ≥ 0 in the orthonormal frame (Ox, Oy, Oz ′ ), in contact with a fluid f occupying the region x < 0 and y > 0, and another fluid f ′ in the region x < 0 and y < 0. It is subjected to the fluidfluid surface tension γ ff ′ , here denoted σ l , which is a force parallel to Ox and concentrated on the line x = y = 0 (Fig. 1a).Sign convention: σ l > 0 if the direction of the force is opposite to Ox (which is the case of a fluid-fluid surface tension), and σ l < 0 if the direction of the force is Ox (in this case, it is a compression).We suppose that the bf and bf ′ surface stress tensors are isotropic, with the same constant eigenvalue, noted σs > 0 (surface tension), and that the bf and bf ′ surface energies are equal.In this case, Eq. ( 7) only implies that a r,νν = 1 (same surface stretching in the bf side and in the bf ′ side), which will be obviously satisfied owing to the symmetry of the problem with respect to the plane y = 0. 15 The other Eq. ( 6) may be written as where φ = φ b /2 (see Fig. 1b).Clearly, the elastic displacement components u x and u y are only functions of (x, y), u z ′ = 0, and, by symmetry, u x (x, −y) = u x (x, y) and u y (x, −y) = −u y (x, y).In the approximation of small deformations, i.e., when the components of the displacement and their first derivatives are small, we have cos φ ≈ ∂ y u x (0, 0+) (derivative at x = 0, y = 0+), and the preceding equation gives The Eqs. ( 3)-( 4) on the surface x = 0 are (in the absence of gravity and using Eq. ( 5)) (div σ s = 0, because σ s = σs I and σs is constant on the surface).In the case of small deformations, n is approximately directed along Ox, 1 R 1 + 1 R 2 ≈ ∂ yy u x , and these equations lead to σs Although we consider the case of no fluid pressure on the surface (i.e., p = 0), we will represent the tension σ l concentrated on the contact line as a Dirac distribution of "pressure" p = −σ l δ(y), which allows to include Eq. ( 13) in Eq. ( 15), with the unique equation on the surface σs Indeed, integrating this equation on the surface, for −ε ≤ y ≤ ε, and taking ε → 0, gives which leads to Eq. ( 13), because the term involving σ xx is equal to 0, as explained in the preceding section: it is a consequence of Eq. (11) or Eq. ( 12) (which will be confirmed for the solution of the present paper) and expresses that the volume stress σ xx gives no contribution at the contact line.

V. THE ANALYTIC FUNCTIONS F AND H
In the following, z will denote the complex variable x+iy and u the complex displacement u x + iu y (function of the complex variable z).We use the general Kolosov's solution of plane strain elasticity where 16, vol.II, ann.XVI), based on the two analytic functions F and G, which we will write using the new analytic function As consequences, Similarly, Kolosov's expressions of the volume stresses may be written as Owing to the symmetry u(z) = u(z) of our problem, we look for analytic functions F and H with the same property, F (z) = F (z) and H(z) = H(z).
As we expect that σ xx , σ xy , and ∂ yy u x vanish at the infinity of the body, i.e., for |z| → +∞ with x > 0 (this will be confirmed on the final solution), Eqs. ( 14) and ( 16) on the surface x = 0 may be extended at the infinity of the body: σ xy = 0 on x = 0 and at the infinity, (20)   σs i.e., using the above expressions ℑ(F ′ − H ′ − (z + z) F ′′ ) = 0 on x = 0 and at the infinity, σ l δ(y) on x = 0, 0 at the infinity.
Assuming that (z + z) F ′′ and ℜ((z + z) F ′′′ ) vanish at the infinity (which will be confirmed on the final solution) and since z + z = 0 on x = 0, the preceding equations may be written as ℑ(F ′ + H ′ ) = 0 on x = 0 and at the infinity, ( 22) σ l δ(y) on x = 0, 0 at the infinity. (23) B the disk |ζ| < 1, and S the circle |ζ| = 1, Eq. ( 22) becomes and, since the first member is an harmonic function in B, it leads to The analytic function − (1+ζ) 2

2
(Φ ′ + Ψ ′ ) is therefore equal to a real constant a in B, hence Eq. ( 23) then becomes σ l δ(y) on x = 0, 0 at the infinity, i.e., using the variable ζ,  We may take this constant equal to 0, since an additive constant in F (or in H) only produces an additive constant in u but does'nt change the derivatives of u and the stress tensor (according to Eqs. ( 17)-( 19)):

VI. THE SOLUTION
By extension of the known solution of the differential equation ( 25) when z is a real variable (see Ref. 18, chap.IV, § § 2, n • 3), we obtain the general solution of this equation when z is a complex variable, as where P (z) is a primitive in C − R − of the function e −z/α ( a 2α z + β α log z), satisfying P (z) = P (z), and b a real constant (since F (z) = F (z)).We then find where Ei is the "exponential integral" function, 19 which we here define as the primitive of the function e z z in C − R + , which coincides with the function x → x −∞ e t t dt when x ∈ R * − .This gives (after addition of the constant −β log α) From this expression, we obtain which shows that b must be equal to 0, in order to satisfy the first assumption preceding Eqs. ( 22)-( 23), because z α e z/α Ei(− z α ) + 1 → 0 at the infinity, for ℜz > 0 (proved in the Appendix).Note that (with b = 0) also tends to 0 at the infinity, satisfying the second assumption.
tends to 0, when z → 0, which leads to the finite limit for the displacement From the above expressions of F ′ (z), F ′′ (z), and Ei(z), we obtain where z = re iθ , r > 0, −π < θ < π, and ε 0 (z) denotes any expression of z which tends to 0 when z → 0. Thus, which shows that lim z→0, θ constant (33) Obviously, the infinite limit of ∂ y u y (and, below, ∂ x u x ) is in contradiction with the assumption of small deformations and, strictly speaking, the solution is not valid for z close to 0.
The solution may rather be considered as a mathematical description of the singularity at z = 0. Note that, on the surface x = 0, with y > 0 (i.e., θ = π/2), the above equation shows that ∂ y u x tends to σ l 2 σs when y → 0, y > 0, in agreement with the above Eq.( 13).Since the initial surface x = 0 is, after deformation, represented by the function y → u x (0, y), there is thus a finite displacement and the formation of a ridge at z = 0 on the surface, as shown in Fig. 2a.In this Fig. 2, the displacements u x (0, y) and u y (0, y) on the surface,  In a similar way, we have which shows that lim z→0 and immediately gives the limits of the strain tensor components ε xx = ∂ x u x , ε yy = ∂ y u y , and We also obtain which shows that where ε ∞ (z) denotes any expression of z which tends to 0 when |z| → +∞, ℜz ≥ 0. Thus, Clearly, the infinite limit of u x is due to the assumption of a force σ l applied on the whole infinite line x = y = 0 (the z ′ axis).Note that, on the surface x = 0, with y > 0 (i.e., θ = π/2), the preceding equation shows that u y tends to σ l 4µ (1 − 2ν) when y → +∞ (see Fig. 2b).Also note that, in the limit case ν → 1 2 (i.e., k + 1 → 0), the expression of u (Eq.( 27)) shows that u y = 0 on the surface x = 0.
In Fig. 4a, we observe that u x increases when the distance to (x, y) = (0, 0) increases.In the xy plane of Fig. 4b, the curves u y = constant show asymptotic directions.There are two lines where u y = 0: the straight line y = 0 (obviously) and a curved line, with an intersection of the two lines at x 0 > 0, y = 0.If x remains constant, x > x 0 , and y increases from 0 to positive values, we observe that u y decreases from 0 to a negative value and then increases to positive values (after crossing the value 0).These observations are consistent with Eq. (39) which gives the asymptotic value u y,∞ of u y as a function of the direction θ: (2µ/β)u y,∞ = 2(1 − 2ν)θ − sin 2θ, which decreases from 0 to a negative value and then increases to the positive value π(1 − 2ν) (after crossing the value 0), when θ increases from 0 to π/2.In fact, the asymptotic direction(s) θ of the curve (2µ/β)u y = c (constant) is (are) given by the equation c = 2(1 − 2ν)θ − sin 2θ.Thus, for y ≥ 0, the curves (2µ/β)u y = c with c > 0 As ν increases, the distances between the curves of Fig. 4a increase, which means that (at 2µα/β constant) the gradient of u x decreases, at each point (x/α, y/α).In Fig. 4b, as ν increases to 0.5, x 0 /α (abscissa of the intersection of the straight line y = 0 and the curved line u y = 0) decreases to 0. Moreover, if θ 0 denotes the asymptotic direction of the curved line u y = 0, i.e., 2(1 − 2ν)θ 0 = sin 2θ 0 , 0 ≤ θ 0 ≤ π/2, clearly θ 0 increases from 0 to π/2 when ν increases from 0 to 0.5.When ν reaches the limit value 0.5, this curved line u y = 0 becomes the straight line x = 0.

FIG. 1 .
FIG. 1. Semi-infinite elastic body b in contact with two fluids f and f ′ , and subjected to the fluid-fluid surface tension σ l normal to the surface x = 0 of the body and concentrated on the line x = y = 0 (a).At the surface of the body, there is a constant surface tension σs .(a) Before deformation; (b) after deformation.

FIG. 3 .
FIG. 3. The classical Flamant's solution: displacements u x (a) and u y (b) on the surface x = 0 as functions of y, for ν = 0.3.Notation: β = σ l 2π .The graph u x (a) represents the form of the surface after deformation.At x = y = 0, u x is infinite and u y is discontinuous.