Hypersingular formulation for boundary stress evaluation revisited. Part 1: Smooth boundaries.

Transcription

1 A. Salvadori / Electronic Journal of Boundary Elements, Vol. 6, No. 2, pp ) Hypersingular formulation for boundary stress evaluation revisited. Part : Smooth boundaries. Alberto Salvadori DICATA, University of Brescia, via Branze 43, 2523 Brescia, Italy Abstract This work focuses on the HBIE) hypersingular boundary integral equation, also called traction equation, and on its use to evaluate the stress tensor in linear elasticity. When the field point is moved to the boundary, by means of a it process, free terms come into play. As a common belief, they are due to the strongly singular kernel: indeed it is proved that the hypersingular kernel does not cause any free term when tractions are evaluated on smooth boundaries with respect to the boundary surface normal when the concept of normal makes sense). The stress tensor along the boundary involves surfaces with normal differing from the boundary normal, too. In this case, free terms are proved to be generated also by the hypersingular kernel, aside from the regularity of the boundary: their analysis is the main goal of the present work. Keywords Hypersingular boundary integral equations; stress evaluation; free terms; boundary element method Introduction Boundary integral equations BIEs) of a linear elastic problem in the framework of small strains and displacements stem from Somigliana s identity [], which is the boundary integral representation of displacements at a point x inside an open domain Ω R d, d = 2, 3, made of a homogeneous isotropic material. The traction operator can be applied to Somigliana s identity, in view of the regularity of the operators involved, thus obtaining the boundary integral representation of tractions p at an interior point x Ω on a surface of normal nx) [2]. Such a representation formula called hypersingular identity HI) in [3]) involves Green s functions collected in matrices G pu and G pp ) which describe components p i ) of the traction vector p on a surface of normal nx) due to: i) a unit force concentrated in space point y) and acting on the unbounded elastic space Ω in direction j; ii) a unit relative displacement concentrated in space at a point y), 24

2 A. Salvadori / Electronic Journal of Boundary Elements, Vol. 6, No. 2, pp ) crossing a surface with normal ly) and acting on the unbounded elastic space Ω in direction j). Denoting with Γ the boundary of Ω, considering the quasi-static external actions: tractions px) on Γ p Γ, displacements ūx) on Γ u = Γ\Γ p, and assuming zero domain forces fx) = 0 in Ω, the HI reads: χ Ω x)px) + G pp r; nx); ly))uy) dγ y + ) Γ p + G pp r; nx); ly))ūy) dγ y = G pu r; nx))py) dγ y + Γ u Γ u + G pu r; nx)) py) dγ y, x / Γ Γ p having set r def = x y and χ Ω x) def = {I if x Ω, 0 otherwise}. The traction BIE can be derived from ) by performing the it to the boundary Ω x x 0 Γ. In the it process, singularities of Green s functions are triggered off: kernel G pu shows a strong singularity of Or d+ ); kernel G pp is usually called hypersingular, since it shows a singularity of Or d ) greater than the dimension of the integral [4]. Moreover, the strongly singular kernel G pu generates a free term Dx), that holds /2 I for smooth boundaries. The traction equation, formulated on smooth boundaries, making use of the notions of Hadamard s finite part HFP), see [5], and of Cauchy s principal value CPV), see [6], reads as follows [7]: Dx)px)+ = G pp r; nx); ly))uy) dγ y + 2) Γ p + = G pp r; nx); ly))ūy) dγ y = G pu r; nx))py) dγ y + Γ u Γ u + G pu r; nx)) py) dγ y, x Γ Γ p In equation 2), px) indicates the traction at x Γ referring to the outward normal lx), that is px) stands for px, lx)). This statement implicitly requires smoothness of the boundary, because lx) must be definite at x Γ. In this sense the boundary needed to be smooth, so far: investigations on boundary smoothness and the traction equation, proofs of the nature of HFP, CPV, and free terms in the it process approach, and a short literature review are provided in Sections 2 and 3. The traction and displacement equations form a linear integral problem, whose solution displacement uy) and traction py) fields along the boundary) is usually approximated by the boundary element method [7]. The evaluation of the stress tensor is a fundamental post-processing task, often performed by means of the hypersingular BIE HBIE) for displacement derivatives [9]. In many applications On the opposite, the boundary it of the Somigliana s identity can be considered at a non smooth point [8] 25

3 A. Salvadori / Electronic Journal of Boundary Elements, Vol. 6, No. 2, pp ) Figure : Boundary and its complementary part. see e.g. [0], []) the most interesting points where to evaluate the stress tensor belong to the boundary and a natural way to get the stress tensor is by evaluating equation ) with respect to a surface of normal nx) orthogonal to the boundary normal lx), thus obtaining the in-plane stress components by means of a it to the boundary process Ω x x 0 Γ. The present work aims at analyzing the traction equation when direction nx) differs from the boundary normal, what is missing so far, to the best of my knowledge, with the only partial exception of [2]. This note involves basic questions, such as: does the free term coefficient depend on the chosen direction nx)? does it depend on the smoothness of the boundary? does the hypersingular kernel contribute to it? are HFP and CPV well defined on any boundary? Such questions, some of which have also been tackled by other approaches 2, are dealt with in section 2, for the free term coefficient due to the hypersingular kernel G pp, and in section 3, with respect to the strongly singular kernel G pu. In section 2 it is shown that G pp always generates a free term coefficient, say D pp x, nx)), that vanishes when the chosen direction nx) approaches the outer normal lx) on a smooth boundary. Similarly, the strongly singular kernel G pu gives rise to a free term coefficient D pu x, nx)) that yields /2 I when the chosen direction nx) approaches the outer normal lx) on a smooth boundary. Furthermore, the hypersingular kernel generates free terms called additional in [3]) on every boundary less smooth than C 2. Differently from D pp x, nx)) such a free term, say A pp x, nx)) as in [3], is merely due to the lack of) smoothness of the boundary. The usual way see [8], [4], [9], [3], [5], [6]) to face up the free term analysis and the it to the boundary process Ω x x 0 Γ excludes point x 0 Γ by a neighborhood of arbitrary shapes, say v ε, controlled by a parameter ε; v ε, the so-called vanishing exclusion zone, vanishes as ε 0. The hypersingular identity ) is considered on domain Ω\v ε thus having χ Ω\vε x) = 0), integrals and 2 as in [9], [3]: there it was proved in fact that hypersingular kernel do contributes to the free term coefficient for displacement derivatives but it was not specified how) and implicitly that the free term coefficient does not depend on the chosen direction nx) 26

4 A. Salvadori / Electronic Journal of Boundary Elements, Vol. 6, No. 2, pp ) their singularities are evaluated by suitably choosing v ε and finally the asymptotic analysis for ε 0 is performed. This approach stems from the identity obtained from eq. )) { ε 0 G pp x 0 y; nx 0 ); ly))uy) dγ y + {Ω\v ε } } G pu x 0 y; nx 0 )) py) dγ y = 0 {Ω\v ε} and is unsuitable for the goal of the present note: it does not permit to distinguish the contribution of each kernel to the free term and, ultimately, to analyze the dependence of the free term coefficient on direction nx). Accordingly, a different strategy is here pursued. Consider an open domain Ω R 2, and its boundary Γ, formed by a finite number of curve elements. Γ is oriented by the field of the unit normal vectors ly), uniquely defined for almost every y Γ, with exceptions of corners. Take x 0 Γ, I ε R 2 a neighborhood of it, and define Γ def ε = Γ I ε see figure ). Equation ) can be written by splitting Γ in and its complementary part. Free terms come out from integrals G pp r; nx); ly))uy) dγ y x Ω 3) G pu r; nx))py) dγ y x Ω 4) when Ω x x 0. The singular part of the previous integrals can be evaluated analytically, exploiting the regularity of the kernels and imposing suitable regularity to the solution. With regard to the integral 3), suppose u C,α ), that is u differentiable at y with its derivatives satisfying a Hölder condition. This requirement is not necessary, see [7], to the existence of the it to the boundary of the hypersingular identity ). However, u C,α ) allows an asymptotic analysis of integral 3), by expanding uy) around x 0 as follows: uy) = ux 0 ) + JAC y u) x0 y x 0 ) + O y x 0 +α) 5) for any 0 < α. The Jacobian term is linked to the strain tensor ɛ by the decomposition: JAC y u) x0 = ɛx 0 ) + SKWJAC y u) x0 ) 6) By assuming that the skew-symmetric part which describes only rigid body modes) vanishes, integral 3) becomes: 27

5 A. Salvadori / Electronic Journal of Boundary Elements, Vol. 6, No. 2, pp ) G pp x y; nx); ly)) uy) dγ = G pp ) [uy) ux 0 ) ɛx 0 ) y x 0 )] dγ + + G pp ) dγ ux 0 ) + G pp ) ɛx 0 ) y dγ x Ω In view of eq. 5) and of the hyper singularity of the kernel G pp, the function G pp x y; nx); ly)) [uy) ux 0 ) ɛx 0 ) y x 0 )] is Lebesgue integrable on Γ ɛ also when x Ω moves to x 0. Free terms arise therefore from the remaining terms, which will be dealt with in Section 2 by means of recently proposed formulae [8]. With regard to the integral 4), assume p C 0,α ). This allows to expand py) around x 0 as follows: py) = px 0 ) + O y x 0 α ) 7) for any 0 < α. In view of equation 7), integral 4) becomes: G pu r; nx)) py) dγ = G pu ) [py) px 0 )] dγ + Γ ε + G pu ) dγ px 0 ) x Ω In view of eq. 7) and of the strong singularity of the kernel G pu, the function G pu x y; nx)) [py) px 0 )] is integrable on Γ ɛ also when x. Free terms arise therefore from the remaining term, which will be considered in Section 3. By this approach the contributions of the hypersingular kernel, A pp x, nx)) and D pp x, nx)), and of the strongly singular kernel D pu x, nx)) to the free term coefficient Dx) are separately obtained; they depend on the elastic properties of domain Ω and on the selected direction nx 0 ) at point x 0. The free term coefficient Dx) is a linear combination of D pp x, nx)) and of D pu x, nx)): Dx) = I + D pp x, nx)) D pu x, nx)) 8) In Section 4 it is shown that, for smooth boundaries, Dx) is independent on the direction nx) and on the material properties, and it holds again /2 I for any direction. Accordingly, equation 2) holds as it stands for any direction nx) on 28

6 A. Salvadori / Electronic Journal of Boundary Elements, Vol. 6, No. 2, pp ) smooth boundaries, provided that px) refers to the direction nx). Such a result is extended to non-smooth boundaries in a companion paper [9]. In that case, the free term coefficient is shown to depend on the material properties of the body, on the surface normal nx) and on the angle at the corner point. It is worth noting that the property of independence of Dx) on the direction nx) and on the material properties of the domain holds for the exact solution fields but it is generally not satisfied by their approximation. A natural question comes into play, does the discretization play a role in the expression of the free term coefficient? Section 5 will show that the free term cannot be taken as /2 I in the stress tensor evaluation, even for smooth boundaries. In fact, a convergent sequence of field points Ω x n x 0 Γ is considered, together with the stress tensor σx n ): it is proved that σx n ) / σx 0 ) when σx 0 ) is evaluated by means of Dx) = /2 I on a smooth boundary. The actual value of the free term due to the discretization is provided. 2 Free terms analysis for the hypersingular kernel This section aims at analyzing the it x Ω) { } G pp r; nx), ly)) dγ y ux 0 ) + G pp r; nx), ly)) ɛx 0 ) y dγ y x x 0 9) providing different smoothness to the curve, by changing the support of the integrals 9) from the curve to the interval I ε ; thereafter, they will be evaluated by means of analytical integration formulae that have been presented in [8] 3. In changing the support, the order of the induced approximation on the integral { } x x 0 G pp r; nx), ly)) dγ y x Ω depends on the smoothness of the boundary. In section 2. suitable smoothness conditions are considered to let the approximation be Oε). Starting from section 2.2, smoothness conditions lead to an O) approximation, thus a free term. With regard to the it x x 0 { G pp r; nx), ly)) ɛx 0 ) y dγ y } x Ω the order of the induced approximation always turns out to be Oε) on smooth boundaries. Nevertheless, free terms may arise which have a different nature. 3 For the formal complexity of such formulae, the computer code MATHEMATICA, release 4, has been extensively used. 29

7 A. Salvadori / Electronic Journal of Boundary Elements, Vol. 6, No. 2, pp ) 2. Smooth boundaries Let ỹs) = {ỹ s), ỹ 2 s)}, 0 s l be the parametric equations of curve with s denoting the curvilinear abscissa; assume ỹ ) differentiable in 0 s l. Consider a local coordinate system L centered at x 0, as in figure 2-a, with axis y, y 2 tangent and normal to at x 0, respectively. Point x 0 is selected such that the parametric equations of with respect to y read yy ) = {y, y 2 y )}, y I ε = def [, ε]. Define with z = def {y, 0}, as in figure 2-b, and with B ε a ball of radius ε centered at x = 0; assume x Ω and the following: Hypotheses 2. Let be defined by y 2 y ) = y α, α 3 For being y 20) = 0 by definition, all results based on hypothesis 2. hold also for all y 2 ) that admit a Taylor expansion around y = 0 with y 2 0) = 0. The following propositions hold. Lemma 2. For any bounded function fx, y) in B ε : ε ) fx, y) x y 2 x z 2 dy = Oε) Proof: The function gx, y ) def = x y 2 and only if) α 3. It turns out therefore that [20] ε gx, y ) dy = x z 2 ε As a consequence, for any bounded function fx, y) in B ε : ε ) fx, y) x y dy 2 x z 2 sup fx, y) B ε and the thesis follows by the mean value theorem. Corollary 2.. It holds: ε fx, y) ε x y 2 dy = is continuous in the set B ε I ε if y 2 z dy 2 0) ε fx, y) x z 2 dy + Oε) y 2 z 2 dy Remarks to lemma 2. It is worth noting that in equation 0) it holds: y 2 z 2 = y2α 2) + y 2α ) which is integrable on I ε if and only if) α > 3/2. The interesting point in equation 0) does not lie on the integrability of the function when α < 3, nor on the existence of the it; instead, it lies on the convergence of the it to the r.h.s. integral. 30

8 A. Salvadori / Electronic Journal of Boundary Elements, Vol. 6, No. 2, pp ) The function G pp x y; nx); ly)) r 2 is bounded around x = 0 for all α, whence the interest in the lemma 2.. Figure 2: Global and local references on for smooth boundaries. Lemma 2.2 Denote by fx, y) = G pp x y; nx); ly)) x y 2 + y 22. It holds: ε fx, y) fx, z) x x 0 x z 2 dy = Oε) ) Proof: By direct substitution, it comes out that fx,y) fx,z) x z 2 respectively: continuous) if α 3 respectively: α > 3). is bounded on B ε Corollary 2.2. It holds: ε x x 0 fx, y) ε x z 2 dy = x x 0 fx, z) x z 2 dy + Oε) By taking x /, lemmas 2. and 2.2 permit to move the integral from curve to interval I ε, controlling the order of the approximation: ε G pp r; nx), ly)) dγ y = G pp r; nx), ly)) + y 2 y ) 2 dy Lemma. 2.) = Lemma. 2.2) = ε ε G pp r; nx), ly)) x y 2 + y 2 y ) 2 x z 2 dy + Oε) G pp x z; nx), lz)) dy + Oε) 3

9 A. Salvadori / Electronic Journal of Boundary Elements, Vol. 6, No. 2, pp ) It is quite natural that the approximation depends on exponent α, because curve approaches interval I ε when α becomes larger and larger. All introduced results lead to the following: Proposition 2. For solving integral 9) on C 3 boundaries, provided that y 2 0) = 0, it is sufficient to solve the following integrals: x +ε x G pp r; nx); ly)) r i dr i = 0, x Ω 2) Proof: Denoting with e the unit vector associated to the y -axis, in L one has x 0) y = y x 0) e + Oy). 2 To evaluate equation 9), one therefore deals with the following integrals: ε G ppx z; nx); lz)) y i dy i = 0, x Ω that can be performed by means of the variable change r = x z, thus providing integrals 2). In [8] it has been proved that for i N 0 it holds: x+ε x G pp r; nx); ly)) r i dr = log r L i) pp + arctan r x 2 ) A i) pp + r 2 Si) pp + r 4 Hi) pp r =x +ε r =x where L i) pp, A i) pp,s i) pp, H i) pp are suitable matrices. By moving point x to the boundary, Ω x x 0, it is straightforward to get: and: L 0) pp = A 0) pp x x 0 = H 0) pp x x 0 r =+ε r 2 S0) pp r = S ) pp x x 0 r 2 = = 2π H ) pp x x 0 arctan x x 0 r r 4 = 0, G ν r 4 = 0, x 2 ) ) n2 n n n 2 r =+ε A ) pp r = r r =ε r = L ) pp = G x x 0 2π ν = G ν ) n ) n2 n n n 2 All singular terms cancel out in the it process, without recourse to any a- priori interpretation in the Hadamard s finite part HFP) sense [5], as mentioned 3) 32

10 A. Salvadori / Electronic Journal of Boundary Elements, Vol. 6, No. 2, pp ) by various authors [9], [2]. However, there exists an intimate relationship [22] between HBIEs and HFP 4, provided by the following proposition Proposition 2.2 For any ε R: x +ε x x +ε x ε = ε G pp r; nx); ly)) dr = = G pp y; n0); ly)) dr 4) G pp r; nx); ly)) r dr = 5) G pp y; n0); ly)) r dr + G ν ) n Proof: Straightforward passages permit to obtain in the local reference L see also [8]): ε ) = G pp y; n0); ly)) dr = G n2 n π ν n n 2 ε ε ) = G pp y; n0); ly)) r dr = r G n2 n =ε log r 2π ν n n 2 = 0 The thesis follows by comparison with eq. 3) r = Corollary 2.2. It holds in L: G pp x y; nx), ly)) dγ y = = G pp y; n0), ly)) dγ y Γ ε G pp x y; nx), ly)) ɛ0) y dγ y = = G pp y; n0), ly)) ɛ0) y dγ y + G ) n 0 ɛ0) e ν 0 0 6) The following term, due to the it 3): D pp x 0, nx 0 )) def = G n 0 ν 0 0 ) ɛx 0 ) e px 0, lx 0 ))) px 0, lx 0 )) 2 7) 4 It has already been proved see [4] and [23] among others) that the hypersingular integral in ) can be interpreted as a HFP in the it as an internal field point x approaches the boundary. In [24], the same conclusion has been obtained by an alternate definition of HFP, without the need for a iting process. Making recourse to the distribution theory [6], the dual BIEs are obtained by the application of a trace operator to the representation formulae ). In such an approach, the strongly singular and hypersingular integrals can be expressed by means of discontinuity jumps also called free terms ) of these integrals on the boundary summed with the values of the integrals on the boundary existing only in the sense of Cauchy Principal Value CPV) or in the sense of the HFP. By exploiting Green s functions properties, the commutativity of the two operators of traction and trace has also been proved, showing the consistency of all different approaches of derivations of the traction BIE. 33

11 A. Salvadori / Electronic Journal of Boundary Elements, Vol. 6, No. 2, pp ) plays the role of a free-term, more precisely it is the contribution of the hypersingular kernel to the free-term Dx) in equation 2). It is evident that D pp depends on the elastic properties of the body and on the selected direction nx 0 ) at point x 0 : when it is taken as the outward normal at x 0, that is nx 0 ) = e 2, then D pp vanishes and, as a common belief, the free term Dx) is solely due to the strongly singular kernel. On the contrary, when the traction is sought with respect to a direction nx 0 ) which differs from the normal at x 0, then the hypersingular kernel contributes to the free term even on smooth boundaries. Contributions by the hypersingular kernel have been observed in literature in [9], [3] for the hypersingular boundary integral equation for displacement derivatives and in [8], [6] in the framework of potential theory, when the boundary curvature and the tangent vector to the boundary are not smooth. Free term 7) has a very different nature. Proposition 2.2 states that the HFP is the outcome of a it process for the hypersingular kernel on every smooth boundary. Therefore, on every smooth boundary, HFP in equation 2) is nothing but a consequence of a property of the hypersingular kernel. 2.2 Almost everywhere C 2 boundaries Let ỹs) = {ỹ s), ỹ 2 s)}, 0 s l be the parametric equations of curve with s denoting the curvilinear abscissa; assume ỹs) C 2 [0, l] with exception of a single point s 0, and ỹs) C [0, l]. Define Γ ε the curve parameterized by ỹs) = {ỹ s), ỹ 2 s)}, 0 s s 0 and Γ + def ε = Γ ε its complementary part described by ỹs) = {ỹ s), ỹ 2 s)}, s 0 s l. A local coordinate system L, centered at x 0, is considered with axis y, y 2 tangent and normal to at x 0, respectively see figure 2). Point x 0 is selected such that the parametric equations of read yy + ) = {y+, y+ 2 y+ )}, y+ I+ def ε = [0, ε] and yy ) = {y, y 2 y )}, y I def ε = [, 0] in the local coordinate system L. In view of the given definition of, the it 9) is here considered also when y ) is not defined in a finite number of points on Γ 5. It is worth rewriting the it 9) as follows x Ω): { G pp r; nx), ly)) dγ y ux 0 ) + G pp r; nx), ly)) ɛx 0 ) y dγ y + x x 0 Γ + ε Γ ε G pp r; nx), ly)) dγ y ux 0 ) + Γ + ε Γ ε } G pp r; nx), ly)) ɛx 0 ) y dγ y 8) The free term analysis that follows pertains to Γ + ε, but results may be trivially generalized to the whole. It will be assumed that: Hypotheses 2.2 Let Γ + ε be defined by y 2 y ) = y α, α 2 5 As an example of such a curve, consider a straight line y 2 y + 2 = y+ )2 : it will be called the prototype curve. = 0 joined with an arc of parabola 34

12 A. Salvadori / Electronic Journal of Boundary Elements, Vol. 6, No. 2, pp ) For being y 20) = 0 by definition, all results based on hypothesis 2.2 hold also for all y 2 ) C 2 0, ε] that admit a Taylor expansion around y = 0. It is trivial to see that the regularity requirements are much lower than hypothesis 2.; as a main consequence, all lemmas in the previous section fail and it is not possible to reduce the hypersingular integral from to I ε without introducing a free term. It is easy to prove, for instance, that equation 0) is no longer valid for the prototype curve. Lemma 2.3 If y 2 y ) = y 2 then two functions g 0 x; y ) C 0 B ε ) and g x; y ) exist such that: G pp x y; nx), ly)) + y 2 y ) 2 = G pp x z; nx), lz))+g 0 x; y )+g x; y ) 9) with g 0; y ) = 0 y I ε \0. Proof: Consider g x; y ) as in appendix. By direct substitution g 0; y ) = 0 when y 0. Furthermore, the function g 0 x; y ) = def G pp x y; nx), ly)) + y 2 y ) 2 G pp x z; nx), lz)) g x; y ) shows to be continuous in B ε Γ + ε. Corollary 2.3. If x Ω, it holds: G pp x y; nx), ly)) dγ y = Γ + ε G pp x z; nx), lz)) dy + g x; y ) dy + Oε) I ε + I + ε Lemma 2.3 permits, similarly to lemmas 2. and 2.2, to move the hypersingular integrals from Γ + ε to I ε +. The induced approximation is stated by the following lemma: Lemma 2.4 It holds: g x; y ) dy = I + ε ) G 3 n n 2 = 2π ν) n 2 n Iε Proof: By direct integration of g x; y ) as it is in appendix.. g x; y ) dy 20) 35

13 A. Salvadori / Electronic Journal of Boundary Elements, Vol. 6, No. 2, pp ) In view of lemma 2.4, the approximation of integral 8) on I ε + is no longer Oε), as in lemmas 2. and 2.2, but O). The r.h.s. of equation 20) acts as a free term 6 : it depends on the material parameters G and ν as well as on the selected direction nx). Lemma 2.4 permits to prove the following propositions. Proposition 2.3 On every C 2 boundary it holds: G pp x y; nx), ly)) dγ y = G pp x z; nx), lz)) dy + Oε) Γ ε I ε As a consequence of lemma 2.3, results in section 2. apply also to C 2 boundaries, because free terms arising in boundary approximation cancel themselves out in the sum. The proposition 2. extends as follows: Proposition 2.4 For solving integral 8) on almost everywhere C 2 -boundaries it is sufficient to solve integrals: x x G pp r; nx); lz)) r i dr i = 0, x Ω 2) Proof: Apply the proof of proposition 2. to I + ε and I ε. In spite of their formal similarity, integral 2) is very different from integral 2). In fact point x is an extremity of integral 2) and it to the boundary Ω x x 0 Γ is not expected to exist. Nevertheless, equation 4) extends as follows: Proposition 2.5 For any ε R: { 0 ε } G pp r; nx); lz)) dy + G pp r; nx); lz)) dy = 22) 0 = G pp r; n0); lz)) dy I ε Proof: With reference to figure 2 the integral x Ω) ε 0 G pp r; nx); lz)) dy = x G pp r; nx); lz)) dr = x r 2 S0) pp + r 4 H0) pp r =x r =x has been solved in [8] by means of the variable change r = x y; matrices S 0) pp, H 0) pp follow in the local reference L: S 0) pp = ) G n2 r 3 n r 2 n r + n 2 r 2 2πν ) n r + n 2 r 2 n 2 r + n r 2 ) n2 r n r 2 n r + n 2 r 2 n r + n 2 r 2 n r 2 n 2 r H 0) pp = G r2 2 π ν) 6 it has the same origin of the hypersingular free term in [3], but it refers to the kernel G pp 36

14 A. Salvadori / Electronic Journal of Boundary Elements, Vol. 6, No. 2, pp ) A similar result holds for the integral x Ω) 0 G ppr; nx); lz)) dy = x +ε G ppr; nx); lz)) dr = x r 2 S0) pp + r 4 H0) pp r =x +ε r =x The it to the boundary Ω x x 0 Γ is taken by means of the norm x 0 in L and of a direction, selected through the angle θ as in figure 3. Figure 3: Limit process Ω x x 0 Γ. It is worth noting that the it ε 0 with: Wθ) = def ε G pp r; nx); lz)) dy = = n2 cosθ)+cos3 θ)) n 3 sinθ)+sin3 θ)) 0 G pp r; n0); lz)) dy + G 2 π ν) Wθ) x cos2 θ) n 2 cosθ) + n 2 sinθ)) cos2 θ) n cosθ) + n 2 sinθ)) 3 n 2 cosθ) cos3 θ))+n sinθ)+sin3 θ)) 2 is not well defined. All inconsistent terms however cancel out in the sum 22), so that the thesis is proved. As a consequence, no free terms arise from the constant hypersingular integral on I ε, similarly to 4). Furthermore, even in the presence of an almost C 2 - boundary, the finite part of Hadamard concept is the outcome of a it process for the hypersingular kernel when applied to a constant term. In view of lemma 2.3 and of proposition 2.5, it is straightforward to prove that: Proposition 2.6 If y 2 ± y ) α ± y 2 then G pp x y; nx), ly)) dγ y = = G pp y; n0), ly)) dγ y + Gα+ α ) 2π ν) ) 3 n n 2 n 2 n ) 37

15 A. Salvadori / Electronic Journal of Boundary Elements, Vol. 6, No. 2, pp ) Proof: From lemma 2.3 it comes out: G pp x y; nx), ly)) dγ y = { I ε { + { + by proposition 2.5: = Gα+ α ) 2π ν) g x; y ) dy + I ε I ε g 0 x; y ) dy + I + ε I + ε G pp x z; nx), lz)) dy + ) { 3 n n 2 + n 2 n Iε + = G pp y; n0), ly)) dγ y I ε and the thesis follows by noting that: = G pp y; n0), ly)) dγ y = { I ε g 0 x; y ) dy + I + ε g x; y ) dy } + g 0 x; y ) dy } + I + ε g 0x; y ) dy + G pp x z; nx), lz)) dy } I + ε g 0x; y ) dy } + } g 0 x; y ) dy + = G pp y; n0), ly)) dγ y I ε Identity 5) extends as follows: Proposition 2.7 For any ε R: { x G pp r; nx); lz)) r dr + x ε = G pp r; n0); lz)) r dr + } G pp r; nx); lz)) r dr = x ) n x+ε G ν Proof: The thesis is proved by direct substitution, observing again that inconsistent terms cancel out in the sum, with the same arguments of proposition 2.5. See also proposition... with ϕ = 0 in the companion paper [9]. Similarly, identity 6) holds as it stands on almost every C 2 boundaries. The term of equation 7) is again the contribution of the hypersingular kernel to the free-term Dx) in equation 2) for every almost everywhere C 2 boundary. It is worth noting that such a free term does not depend on the regularity of the 38

16 A. Salvadori / Electronic Journal of Boundary Elements, Vol. 6, No. 2, pp ) boundary itself, but is merely due to the choice of a direction nx) different from the boundary normal lx). Free term 20) has a very different nature from 7), because it is only due to the lack of) smoothness of the boundary Γ. Furthermore, propositions 2.5 and 2.7 state that the HFP is the outcome of a it process for the hypersingular kernel on every almost everywhere C 2 boundary. Therefore, HFP in equation 2) is nothing but a consequence of a property of the hypersingular kernel. 2.3 C boundaries Let ỹs) = {ỹ s), ỹ 2 s)}, 0 s l be the parametric equations of curve with s denoting the curvilinear abscissa; assume ỹ ) differentiable in 0 s l. Consider a local coordinate system L centered at x 0, as in figure 2-a, with axis y, y 2 tangent and normal to at x 0, respectively. Point x 0 is selected such that the parametric equations of with respect to y read yy ) = {y, y 2 y )}, y I def ε = [, ε]. Assume further that y 2 y ) C I ε ). By having reduced the regularity conditions on with respect to hypothesis 2.2, all lemmas in the previous section fail and it is not possible to reduce the hypersingular integral from to I ε by means of an integrable function g x; y ) on I ε. Unfortunately, I am not able to prove propositions in what follows for the whole class of C boundaries but only for boundaries y 2 y ) that have an asymptotic behavior y 2 y ) y 3/2. At any rate, it can be conjectured that at least all boundaries having an asymptotic behavior y 2 y ) y α with 3/2 α < 2 share the following basic properties; moreover, even this simple case leads to remarkable conclusions. Lemma 2.5 If y 2 y ) = y 3/2 then functions g 0 x; y ) C 0 B ε ), g 3/2 x; y ), g x; y ), and g /2 x; y ) exist such that: G pp x y; nx), ly)) + y 2 y ) 2 = G pp x z; nx), lz)) + g 3/2 x; y ) + g x; y ) + g /2 x; y ) + g 0 x; y ) with g i x; y ) Oy i ) for i = 3/2,, /2. Proof: Consider g ix; y ) Oy) i for i = 3/2,, /2 as in appendix 2. Furthermore, the function g 0 x; y ) = def G pp x y; nx), ly)) + y 2 y ) 2 G pp x z; nx), lz)) g 3/2 x; y ) + g x; y ) + g /2 x; y ) ) shows to be continuous in B ε Γ + ε. 39

17 A. Salvadori / Electronic Journal of Boundary Elements, Vol. 6, No. 2, pp ) Corollary 2.5. If x Ω, it holds: G pp x y; nx), ly)) dγ y = G pp x z; nx), lz)) dy + Γ + ε I ε + g 3/2 x; y ) dy + g x; y ) dy + g /2 x; y ) dy + Oε) I ε + I ε + I ε + Lemma 2.5 permits to move the hypersingular integrals from Γ + ε to I ε +. However, the function g 3/2 x; y ) is no longer integrable on I ε and the approximation is therefore questionable. The following lemma holds: Lemma 2.6 It holds: ε 0 with: Proof: ε g 3/2 x; y ) + g /2 x; y ) dy = = ε 0 ε = 0 g x; y ) dy = 0 G 2π ν) g 3/2 x; y ) + g /2 x; y ) dy 23) ) 3 n2 3 n 3 n n 2 g 3/2 x; y ) + g /2 x; y ) dy = ) G 3 5 ε) 2 n + 3 ε) n 2 ε π + ν) + 3 ε) n ε) n By direct integration. In view of lemma 2.6, the approximation of integral 8) on I ε + is no longer O) as for almost everywhere C 2 boundaries. In this case in fact a finite part of Hadamard concept comes into play, which turns out to be Oε /2 ); its origin is totally different from the usual finite part integral in hypersingular integrals: it arises because the integral on the boundary is approximated by the integral on the tangent at the boundary. The term 24) is the counterpart of 20) for the curve y 2 = y 3/2. To the best of my knowledge, this should be the first attempt to evaluate a free term on a C only boundary. A general expression of such a free term, for at least a curve of type y 2 = y +β with 0 < β < is not available, so far; it will be the subject of further developments to the present note. Propositions 2.4 and 2.5 apply as they stand also on C boundaries. Denoting with Hy ) the usual Heaviside function, proposition 2.6 may be extended as follows: Proposition 2.8 If y 2 y ) α y 3/2 Hy ) then G pp x y; nx), ly)) dγ y = Gα 2 ) 3 n2 3 n = G pp y; n0), ly)) dγ y + 2π ν) 3 n n 2 24) 40

18 A. Salvadori / Electronic Journal of Boundary Elements, Vol. 6, No. 2, pp ) Proof: From lemma 2.5 it comes out: G pp x y; nx), ly)) dγ y = I + ε + g x; y ) dy + I + ε { + Iε I + ε g /2 x; y ) dy + g 3/2 x; y ) dy + I + ε G ppx z; nx), lz)) dy + g 0 x; y ) dy + by proposition 2.5 and lemma 2.6 see also appendix 2): ) Gα 2 3 n2 3 n = + g 2π ν) 3 n n 0 x; y ) dy + 2 I ε + + = I + ε I + ε G ppx z; nx), lz)) dy } g 3/2 x; y ) + g /2 x; y ) dy + = G pp y; n0), ly)) dγ y I ε and the thesis follows by noting that if y 2y ) α y 3/2 Hy ): = G pp y; n0), ly)) dγ y = + = I + ε I + ε g 0 x; y ) dy + g 3/2 x; y ) + g /2 x; y ) dy + = G pp y; n0), ly)) dγ y I ε Identity 6) holds as it stands also on every C boundary. 3 Free terms arising from the strongly singular kernel. This section aims to analyze the it { x x 0 G pu r; nx)) dγ y px 0 ) on a smooth, at least C, boundary. } x Ω 25) Proposition 3. For solving integral 25) on smooth boundaries it is sufficient to solve integrals: x+ε x G pu r; nx)) dr x Ω 26) 4

19 A. Salvadori / Electronic Journal of Boundary Elements, Vol. 6, No. 2, pp ) Proof: The thesis follows immediately by assuming as in proposition 2. Proposition 3. can be extended to less smooth boundary: Proposition 3.2 If + y 2 y ) 2 = + Oy ) y [, 0 [ ] 0, ε] then for solving integral 25) it is sufficient to solve integrals: Γ + ε x x G pu r; nx)) dr x Ω Proof: Denoting with r = x y and taking x / Γ + ε, one writes in L + : l ε fy) dγ def fỹs)) fyy )) = ds = dy + Oε) 27) r s 0 r r 0 for any fyy )) C 0 I + ε ) in view of the adopted hypothesis. where: Taking the it to the boundary Ω x x 0, it comes out from [8]: x+ε G pu r; nx)) dr = x x 0 x { + log r L pu + arctan x x 0 r x 2 ) A pu + } r =x +ε r 2 S pu r =x L pu = ) 3 2 ν) n 2 ν) n 2 4π ν 2 ν) n 2 2 ν) n r x x 0 r 2 S =x +ε pu = 0 r =x ) r r =x +ε arctan A pu = ) ν) n2 ν n 28) x x 0 x 2 r =x 2 ν ν) n ν) n 2 By introducing the concept of Cauchy principal value, see e.g. [6], the following identity can be proved in the local reference L [8]: Proposition 3.3 For any ε R: x x 0 x+ε x x0 +ε x 0 G pu r; nx)) dr = 29) G pu x 0 y; nx 0 )) dr + 2 ν n2 ν n ) n n 2 42

20 A. Salvadori / Electronic Journal of Boundary Elements, Vol. 6, No. 2, pp ) Proof: [8]): Straightforward passages permit to obtain in the local reference L see also x0 +ε x 0 G pu x 0 y; nx 0 )) dr = ) 3 2 ν) n 2 ν) n 2 log r 4π ν 2 ν) n 2 2 ν) n The thesis follows by comparison with 28). r =ε r = = 0 Accordingly, the term due to the it 28): D pu x 0, nx 0 )) def = ) ν) n2 ν n 2 ν ν) n ν) n 2 30) is the contribution of the strongly singular kernel to the free-term Dx) in equation 2). D pu depends on the elastic properties of the body and on the selected direction nx 0 ) at the point x 0 : when it is taken as the outward normal at x 0, that is nx 0 ) = e 2, then D pu = /2 I which is the well known amount of the free term coefficient for smooth boundaries. 4 Properties of free terms coefficients As mentioned in sections 2. and 3 the free terms coefficients D pu x 0, nx 0 )) and D pp x 0, nx 0 )) depend on the elastic properties of the body and on the selected direction nx 0 ) at the point x 0. An analysis of their behavior with respect to direction nx 0 ) is performed here in the local reference L. Proposition 4. On every smooth boundary, the free term coefficient Dx) in equation 2) is independent on nx) and on the material properties of the body. Moreover, it holds Dx) = 2 I if px) in equation 2) refers to the direction nx). Proof: Free terms coefficients inherit from kernels G pu and G pp the linearity property with respect to nx 0 ), namely: G pux y; nx)) = n x) G pux y; e ) + n 2x) G pux y; e 2) G pp x y; nx); ly)) = n x) G pp x y; e ; ly)) + n 2 x) G pp x y; e 2 ; ly)) Because at the outward normal nx 0 ) = lx 0 ) = e 2 they hold: D pu x 0, e 2 ) px 0, lx 0 )) = 2 px 0, e 2 ) D pp x 0, e 2 ) = 0 3) 43

21 A. Salvadori / Electronic Journal of Boundary Elements, Vol. 6, No. 2, pp ) merely the case nx 0) = e still needs to be investigated. For the strongly singular contribution one writes in such a case: ) D pux 0, e ) px 0, lx 0)) = n 0 ν x 2 ν ν 0 0) e 2 For the hypersingular contribution, after imposing the linear isotropic elastic constitutive law: = ν + ν tr ) I + E E 32) and the following constraint: σ 33 = νσ + σ 22 ) which is due to the plain strain hypothesis, straightforward passages permit to state the following identity: ) D ppx 0, e ) px 0, nx 0)) = G 0 x ν 0 0 0) e One obtains therefore = 2 ν [ ν)σ νσ 22 ] e D pux 0, e ) px 0, lx 0)) D ppx 0, e ) px 0, nx 0)) = 2 e = px0, e) 2 By the linearity property of the kernels, one concludes therefore that: D pux 0, nx 0)) px 0, lx 0)) D ppx 0, nx 0)) px 0, nx 0)) = px0, nx0)) 33) 2 that is, the free term coefficient for the traction equation on smooth boundaries does not depend on the selected direction nx 0 ). By substituting 33) into equation 8), Dx 0 ) amounts to: Remarks to proposition 4. Dx 0 ) = 2 I As a consequence of proposition 4., equation 2) holds as it stands for any direction nx) on smooth boundaries, provided that px) refers to the direction nx). In the proof of proposition 4., the constitutive equation 32) has been used. Accordingly, proposition 4. holds only for displacement and stress fields which fulfill equation 32). Typically eq. 32) does not apply to the discrete approximations of the displacement field uy) and of the traction field py), say ûy), ˆpy)) that pertain to the boundary element method. Accordingly, proposition 4. does not hold only for such discrete fields when nx 0 ) lx 0 ). 44

22 A. Salvadori / Electronic Journal of Boundary Elements, Vol. 6, No. 2, pp ) Note, in fact, that at the outward normal nx 0 ) = lx 0 ) = e 2, proposition 4. is proved in view of equation 3) and the constitutive law 32) is no longer required. Accordingly, the traction equation on the boundary 2) holds with Dx 0 ) = 2 I in the BEM solution along the boundary. On the contrary, the traction equation does not hold for the approximated stress tensor evaluation on the boundary as a post processing task see section 5). 5 Approximation of the free term Let ûy), ˆpy) be discrete approximations of the displacement field uy) and of the traction field py), respectively: N u ûy) = ψ u hy) û h ˆpy) = ψ p h y) ˆp h 34) h= The discretization of the unknown fields permits to transform the BIEs into sets of algebraic equations. Two main techniques have been successfully developed to this aim: the collocation [25] and the symmetric Galerkin [7] methods SGBEM). The evaluation of the stress tensor is a post-processing task, after the determination of the unknown sets û h, ˆp h. The constitutive equation 32) does not apply to the approximation fields 34); therefore, the property of independence of Dx) on the direction nx) and on the material properties of the domain, stated in section 4 for the problem solution, does not hold for the approximated evaluation of the stress tensor. To explain this fact by an example, consider the square domain of figure 4, with sides of length 2. The lower horizontal side is constrained by ūy) = 0. The upper horizontal side is subjected to the vertical load py) = x 2 )λ + 2G) e 2, while the two vertical sides are loaded by py) = x 2 )λ ly), denoting with λ and G the Lamè constants and by ly) the outward normal. The analytical solution of the problem reads: uy) = x x 2 ) e 2 σy) = x 2 ) λ λ + 2G λ Having approximated the problem via the SGBEM through the discretization of figure 4-a, the stress tensor has been evaluated at points x n = { 0. n, 0.735}: the stress component σ 22 x n ) is reported in table. Moreover, the stress tensor has been evaluated on the boundary at x = {, 0.735}, by means of the traction equation 2) having set Dx ) = 2 I. Despite the sequence x n x, table and figure 4-b clearly show that σx n ) / σx ). The approximated solution is obtained by using analytical integrations [8], therefore consistency errors cannot be invoked for the lack of convergence. Instead, it is due to the approximation of the free term coefficient Dx ), because equation 33) does not apply anymore. To figure out this last statement, define shape N p h= 45

23 A. Salvadori / Electronic Journal of Boundary Elements, Vol. 6, No. 2, pp ) a) geometry and discretization by means b) relative error at σx ) by of 6 equal linear boundary elements evaluating it directly on the boundary by Eq. 2) and by a it process Figure 4: A numerical benchmark for the stress tensor evaluation on the boundary. x n) σ 22 x n ) x n) σ 22 x n ) x Table : Convergence of the stress tensor σx n ). functions ψh ux) and ψp h y) in equation 34) as follows. Let Γ h be a decomposition of the polygonal boundary Γ of figure 4 with nodes {P h, h =, 2,..., N h }. Let T j be the generic segment of Γ h, and let l j be half the length of T j. Choose over T j a local basis {ϕ 0, ϕ,..., ϕ Nj }. Here ϕ j is a polynomial usually lagrangian) of degree N j defined on a subset of {P h } of N j + nodes in T j. Collect in set T h = {T j } N j= the two at most, i.e. N 2) segments having the common vertex P h see figure 5); then ψ h is defined as: ψ h y) := { ϕn y), y T j, j =,..., N 0, elsewhere 35) where the index n selects the local basis function on T j such that ϕ n P h ) =. By construction, ψ h y) is continuous over Γ h, and its support coincides with T h. Integrals in the HI ), thus taking x Ω, take the form see [8] for a larger 46

24 A. Salvadori / Electronic Journal of Boundary Elements, Vol. 6, No. 2, pp ) Figure 5: Local ϕ n x) and global ψ h x) shape function. presentation) s = u, p: N G ps ) ψ h y) dγ y = Γ s j= F j psx) F j psx) = def T j G ps ) ϕ n y) dγ y 36) Denote by L {y, y 2 } the local coordinate system defined in section 2 see figure 2), with the origin in the midpoint of T j. If y T j then the outward normal ly) = 0, ), y 2 = 0, l j y l j and ϕ n y) = y T a n, having defined: y T = def {, y, y 2,..., y N j } at n = def {a 0) n, a ) n, a 2) n,..., a N j) n } By means of the variable change r = x y and the binomial expansion rule, it is straightforward to get y T = r T X, where i, j =, 2,..., N j + ): ) r T def = {, r, r, 2..., r Nj } X def ij = ) i j x j i) i Equation 36) becomes therefore, ) F j psx) = K j Xan 0 psx) 0 Xa n x+l j ) K j psx) def r T = G ps ) 0 x l j 0 r T dr r2=x2 with x Ω. Integrals K j ps have been analytically solved in [8]. They take the following expression in the local coordinate system L: 47

25 A. Salvadori / Electronic Journal of Boundary Elements, Vol. 6, No. 2, pp ) K j pux) = 4π ν [ logr 2 ) L pu + arctan K j ppx) = G 4π ν [ logr 2 ) L pp + arctan r x 2 r x 2 ) A pu + r 2 S pu + P pu ] r =x +l j r =x l j ) A pp + r 2 S pp + r 4 H pp + P pp ] r =x +l j r =x l j where r 2 = r 2 + x 2 2 and L pu, A pu, S pu, P pu, L pp, A pp, S pp, H pp and P pp are suitable matrices, collected in [26] for ϕ n y) of degree 5. Moving x on the boundary, kernels G pu x y, nx)) and G pp x y, nx), ly)) shows to be singular with respect to y depending on the position of x with respect to T j. With regard to the hypersingular kernel, by taking x T j and by the definition of the finite part of Hadamard, it is straightforward to get: lj ) r T x 2 0 Kj ppx) = = G pp x y; nx); ly)) 0 l j 0 r T dy + + G n [ ] ν) Analytical integrations suggest that the term ) G n 0 [0,, 0,..., 0] Xa ν 0 0 n ) û h = G ν 37) 38) ) n 0 dϕ u n x) û 0 0 h 39) dx must be considered as an approximation of the hypersingular free term 7) on the boundary Γ p : D pp x, nx)) px, nx)) = G ) n 0 ɛx ν ) e 40) It is evident that the isotropic linear constitutive law 32) can neither be applied to the approximated fields 34) nor to the free term approximation 39). For the strongly singular kernel, a similar path of reasoning leads to the following identity: lj ) r T 0 dy x Kj pux) = G pu x y; nx)) 2 0 l j + 2 ν) 0 r T [ n2 ν) 0... n ν 0... n ν) 0... n 2 ν) 0... Analytical integrations suggest that the term [ ] ) n2 ν) 0... n ν 0... Xan 0 ˆp 2 ν) n ν) 0... n 2 ν) Xa h = 4) n ) n2 ν) n ν ϕ p 2 ν) n ν) n 2 ν) nx) ˆp h ] 48

26 A. Salvadori / Electronic Journal of Boundary Elements, Vol. 6, No. 2, pp ) must be considered as an approximation of the free term 30) on the boundary Γ u : D pu x, nx)) px, lx)) = ) ν) n2 ν n px, lx)) 42) 2 ν ν) n ν) n 2 In the example studied above, for being x Γ p, the free term coefficient 42) was exactly evaluated. On the contrary, the hypersingular free term coefficient 40) was approximated by the term 39), thus giving rise to the emerged lack of convergence. 6 Concluding remarks. In the present work the hypersingular formulation for boundary stress evaluation has been revised, exploiting recently developed analytical integration formulae [8]. Differently from the vanishing exclusion zone approach see e. g. [5], [6]), in the present work: i) an asymptotic analysis of the singular integrals has been considered; ii) analytical integrations have been performed; iii) the it to the boundary Ω x x 0 Γ has been taken. By this approach the contributions of the hypersingular kernel and of the strongly singular kernel to the free term coefficient Dx) have been separately obtained. As a first conclusion of the present note, the hypersingular kernel contributes to the free term in the stress tensor evaluation on the boundary even when it is smooth. More generally, a hypersingular free term comes into play when the traction vector is evaluated with respect to a normal that differs from the boundary surface aside from the regularity of the boundary: such a phenomenon has been observed even on a straight boundary. A deep investigation has been devoted to free terms coming out from the hypersingular kernel due to an insufficient smoothness of the boundary. It has been proved that free terms arise when the boundary is less smooth than C 2. Furthermore, a closed form free term has been evaluated for a boundary which is only C smooth. In such an analysis on the hypersingular integral operator, it has been proved that the concept of the finite part of Hadamard is consistent in the traction equation, in the sense that it comes out from a it to the boundary process of the hypersingular integral. Such a consistency is questionable in the presence of corners, as will be seen in the companion paper [9], because the finite part formulation of the traction equation 2) in the presence of corners makes sense in view of a property of the global elastic integral operator but not for the hypersingular operator itself. Furthermore, equation 2) has been proved to apply only to fields that comply with the constitutive law 32). For all fields that do not fulfill the constitutive law, equation 2) may not hold. For instance, equation 2) leads to the boundary element method by considering discrete approximation for displacement and trac- 49

27 A. Salvadori / Electronic Journal of Boundary Elements, Vol. 6, No. 2, pp ) tion fields: for the discrete traction equation on the boundary, it has been proved that the free term coefficient on smooth boundaries does not hold 2 I. Such items are not only of theoretical interest: indeed the stress tensor on the boundary, which is the most interesting data in many problems of fracture propagation and bifurcation [0], [], may largely differ from the expected value. As seen in an example, the error due to the free term approximation is of the same order of the error of the approximation scheme. The present work is preinary to the analysis of engineering problems, for which the stress tensor along the boundary is a fundamental item. In particular, problems of fracture initiation, propagation and bifurcation in soil-structure interactions [0], in composite [] and biological materials and tissues are under investigation [27]. In a companion paper, which is the natural prosecution of the present one, an investigation of the integral formulation of the linear elastic problem is considered in the presence of corners, thus extending all results contained in the present work. Acknowledgements Part of the present work was performed in the first semester of 2008 at the Oak Ridge National Laboratory: I am grateful to Professor L.J.Gray for the opportunity of visiting and working with him. The support of the Italian Ministry of University and Research MIUR) under grant ex 60% 2007: Analisi non lineari nella meccanica dei continui e della frattura mediante il metodo degli elementi al contorno is gratefully acknowledged. References [] F.J. Rizzo. An integral equation approach to boundary value problems of classical elastostatics. Quart. Appl. Math., 40:83 95, 967. [2] K.H. Hong and J.T. Chen. Derivations of integral equations of elasticity. ASCE J. Engrg. Mechanics, 4: , [3] O. Huber, A. Lang, and G. Kuhn. Evaluation of the stress tensor in 3D elastostatics by direct solving of hypersingular integrals. Computat. Mech., 2:39 50, [4] M. Diligenti and G. Monegato. Finite-part integrals, their occurence and computation. Rend. Circ. Mat. Palermo, 33II):39 6, 993. [5] J. Hadamard. Lectures on Cauchy s problem in linear partial differential equations. Yale Univ. Press, New Haven, Conn., USA, 923. [6] A.H. Zemanian. Distribution theory and transform analysis. Dover,

28 A. Salvadori / Electronic Journal of Boundary Elements, Vol. 6, No. 2, pp ) [7] M. Bonnet, G. Maier, and C. Polizzotto. Symmetric Galerkin boundary element method. Applied Mechanical Review, 5: , 998. [8] V. Mantič. On computing boundary iting values of boundary integrals with strongly singular and hypersingular kernels in 3d bem for elastostatics. Engng. Anal. Boundary Elem., 3:5 34, 994. [9] M. Guiggiani. Hypersingular formulation for boundary stress evaluation. Engng. Anal. Boundary Elem., 3:69 79, 994. [0] S.T. Slowik, J.M. Chandra Kishen, and V.E. Saouma. Mixed mode fracture of cementitious interfaces. Part : experimental results. Engineering Fracture Mechanics. Mech., 60:83 94, 998. [] S.T. Smith and J.G. Teng. Interfacial stresses in plated beams. Engineering Structures, 23:857 87, 200. [2] C. Fiedler. On the calculation of boundary stresses with the Somigliana stress identity. Int. J. Numer. Methods Eng., 38: , 995. [3] M. Guiggiani. Hypersingular boundary integral equations have an additional free term. Comp. Mech., 6: , 995. [4] A. Young. A single domain bem for 3d elstostatic crack analysis using continuous elements. Int. J. Numer. Methods Eng., 39: , 996. [5] M. Guiggiani. Formulation and numerical treatment of boundary integral equations with hypersingular kernels. In V. Sladek and J. Sladek, editors, Singular integrals in boundary element methods, Advances in Boundary Elements Series, pages Computational Mechanics Publications, Southampton and Boston, 998. [6] V. Mantič and F. Paris. Existence and evaluation of the two free terms in the hypersingular boundary integral equation of potential theory. Engng. Anal. Boundary Elem., 6: , 995. [7] P.A. Martin and F.J. Rizzo. Hypersingular integrals: how smooth must the density be? Int. J. Numer. Methods Eng., 39: , 996. [8] A. Salvadori. Analytical integrations in 2D BEM elasticity. Int. J. Numer. Methods Eng., 537):695 79, [9] A. Salvadori. Hypersingular formulation for boundary stress evaluation revisited. part 2: Corners. Technical Report 5, University of Brescia, [20] L. Amerio. Analisi matematica, volume II. UTET, Milano, 987. in italian. [2] S. Miccoli. Spectral properties of Galerkin boundary element matrices. Comp. Mech., 25: ,