Validity conditions of the direct boundary integral equation for exterior problems of plane elasticity

Engineering applications justify considering the exterior problem of a half-space or a halfplane in elasticity. The extension of civil works is indeed small compared to the one of the soil mass, which can be considered as infinite and 2D analyses are of frequent use in geotechnical engineering. One faces however the difficulty of the asymptotic behaviour of the fundamental solution for plane elasticity which increases as a logarithm at infinity. The solution of plane elasticity problems on non-bounded domains requires specific conditions at infinity and appears paradoxical (e.g., [1]). One also has to tackle the difficulty of writing valid boundary integral equations and integral representations. The purpose of the paper is to justify the use of the boundary integral equation method for any case of loading at the boundary, including the case of loading having a non zero resultant.

Let us consider an open part  of a linear elastic, isotropic, homogeneous half-plane, a bounded part  of which having been removed (Fig. 1a). There are no volume forces. To apply the direct method, the boundary integral equation is written on the part of the bounded boundary   where displacements or tractions are prescribed. Such an integral equation is obtained by -writing the integral equation on the domain comprised between   and a half-circle S r -looking for the limit of the integrals on S r when its radius r tends to infinity.

The solutions in displacement and traction are denoted by u and t. Functions U and T are elementary solutions in displacement and traction for the half plane. The functions k i U are defined up to an arbitrary translation. The usual choice [e.g. 2], which is adopted here, corresponds to behaviour at infinity such that:

Where j i A and j i B are constants.

In the 2D case, different authors proposed sufficient conditions on the behaviour of the solution at infinity so that it satisfies a boundary integral equation on   . Watson [3] . Constanda [5,6] and Schiavone and Ru [7] used the hypothesis that u i decrease at infinity as

. Bonnet [2] gave a less restrictive

In conclusion, it seems that the least restrictive sufficient condition that is presently known is the one given by [2]. All the sufficient conditions described above are not at all satisfying because they cannot justify studying the boundary problem related to a point loading or (principle of Saint-Venant) a loading with a non zero resultant force. The purpose of the following is to show that the classical boundary integral equations (1,2) are also valid if the resultant of applied forces is non-zero.

Poulos and Davis [8] stated that: “displacements due to line loading on or in a semi-infinite mass are only meaningful if evaluated as the displacement of one point relatively to another point, both points being located neither at the origin of loading nor at infinity”. Accordingly, to mitigate the difficulties related to the behaviour at infinity, it seems natural to introduce the relative displacements in the formulation of the problem. To this aim, a first step is to build a boundary integral equation whose solution corresponds to displacements with respect to a reference point x 0 taken within  (outside the boundary   ). This is equivalent to setting a supplementary condition of no displacement for this reference point x 0 .

The boundary conditions correspond to prescribed displacements on U   and prescribed tractions on

being complementary parts of   . The displacement is zero at point x 0 . Hence the following conditions are met:

The purpose of this section is to show the following lemma:  is constituted by   , S r and P r (Fig. 1a). Solution u satisfies a boundary integral equation (6) for any bounded open set r  [2]:

This equation ( 6) is valid for Let us introduce the “modified Green functions” defined below by (7,8):

Replacing x by x 0 in (6), making the difference with the original equation ( 6) and using (5c), leads to :

The part of the integral above on S r can be written as: , see Fig. 1b), which leads to :

One can check that k i * U (x,y) is O(1/r) when r(y) tends to infinity, and that

Due to Saint-Venant principle, u and t behave at infinity as the response to the resultant of the forces applied on the boundary. It means that u is O(ln(r)) and t is O(1/r). Using polar coordinates, it can be concluded that the integral given by (11) tends to 0 as r tends to infinity.

By using (10), an integral equation for the relative displacement which is valid for

The boundary integral equation on   is the special case of ( 12 12) yields, for any point of  which is not on its boundary, to:

In conclusion, it is proved that any elastic solution in “relative displacement” satisfies particular forms of boundary integral equation and of integral representation (12, 13). It is worthwhile to mention that this integral representation ensures that the condition u(x 0 )=0 is satisfied. However, this method does not

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