An Advanced Boundary Element Method for Solving 2D and 3D Static Problems in Mindlin s Strain Gradient Theory of Elasticity

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1 INTERNATIONAL JOURNAL FOR NUMERICAL METHOD IN ENGINEERING Int. J. Numer. Meth. Engng 2000; 00:1 6 [Version: 2002/09/18 v2.02] An Advanced Boundary Element Method for olving 2D and 3D tatic Problems in Mindlin s train Gradient Theory of Elasticity G. F. Karlis 1, A. Charalambopoulos 2 and D. Polyzos 3, 1 Department of Mechanical and Aeronautical Engineering, University of Patras GR Patras, Greece 2 Department of Materials cience and Engineering University of Ioannina Dourouti, Ioannina 3 Department of Mechanical and Aeronautical Engineering, University of Patras GR Patras, Greece polyzos@mech.upatras.gr UMMARY An advanced Boundary Element Method (BEM) for solving two (2D) and three dimensional (3D) problems in materials with microstructural effects is presented. The analysis is performed in the context of Mindlin s Form II gradient elastic theory. The fundamental solution of the equilibrium partial differential equation is explicitly derived. The integral representation of the problem, consisting of two boundary integral equations, one for displacements and the other for its normal derivative is developed. The global boundary of the analyzed domain is discretized into quadratic line and quadrilateral elements for 2D and 3D problems respectively. Representative 2D and 3D numerical examples are presented to illustrate the method, demonstrate its accuracy and efficiency and assess the gradient effect on the response. The importance of satisfying the correct boundary conditions in gradient elastic problems, is illustrated with the solution of a simple 2D problem. Copyright c 2000 John Wiley & ons, Ltd. key words: Boundary Element Method, Enhanced Elastic Theories, Mindlin s Form II Gradient Elasticity, Microstructure and Microstructural effects. 1. INTRODUCTION Due to the lack of internal parameters classical theory of linear elasticity fails to describe size and microstructural effects or to describe fields characterized by very high gradients of strains. However, this is possible with the use of other enhanced elastic theories where Correspondence to: Prof. Demosthenes Polyzos, Department of Mechanical and Aeronautical Engineering, University of Patras, GR Patras, Greece, polyzos@mech.upatras.gr Received 3 July 1999 Copyright c 2000 John Wiley & ons, Ltd. Revised 18 eptember 2002

2 2 G. F. KARLI, A. CHARALAMBOPOULO AND D. POLYZO internal length scale parameters correlating the microstructure with the macrostructure are involved in the constitutive equations of the considered elastic continuum. uch theories and the most general are those known in the literature as Cosserat elasticity [1], Cosserat theory with constrained rotations or couple stresses theory [2], [3], strain gradient theory [4], multipolar theory of continuum mechanics [5], higher order strain gradient elastic theory [6], [7], micromorphic, microstretch and micropolar elastic theories [8] and non-local elasticity [9]. Most of the aforementioned theories have been developed in the decade of 60 s and historical reviews as well as comments on the subject can be found in [2], [8], [9], [10], [11], [12], [13]. The present work reports a boundary element formulation of Form II strain gradient elastic theory, which is a special case of Mindlin s general strain gradient elasticity. Mindlin [6], [7] in the middle of 60 s proposed an enhanced elastic theory to describe linear elastic behavior with microstructural effects. To this end, he considered the potential energy density as a quadratic form not only of strains but also of the gradient of strains. However, in order to balance the dimensions of the aforementioned quantities and to correlate the micro-strains with the macro-strains, Mindlin utilized eighteen new constants rendering thus his general theory very complicated from physical and mathematical point of view. In the sequel, considering long wave-lengths and the same deformation for macro and micro structure, Mindlin proposed three new simplified versions of his theory, known as Form I, II and III, utilizing in the constitutive equations seven material and internal length scale constants instead of eighteen employed in his initial model. In Form-I, the strain energy density function is assumed to be a quadratic form of the classical strains and the second gradient of displacement; in Form-II the second gradient displacement is replaced by the gradient of strains and in Form-III the strain energy function is written in terms of the strain, the gradient of rotation, and the fully symmetric part of the gradient of strain. Although the three forms are equivalent and conclude to the same equation of motion, the Form-II leads to a total stress tensor, which is symmetric as in the case of classical elasticity avoiding thus problems associated with non-symmetric stress tensors introduced by Cosserat and couple stress theories. As in the case of classical elasticity, the solution of gradient elastic problems with complicated geometry and boundary conditions requires the use of numerical methods such as the finite element method (FEM) and the boundary element method (BEM). The FEM is the most widely used numerical method for solving applied mechanics problems. hu et al. [14] were the first to use FEM for solving elastostatic problems in the framework of the gradient elasticity theories of Mindlin. ince then, many papers dealing with FEM solutions of gradient elastic problems have appeared in the literature [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29]. The main problem with a conventional FEM formulation is the requirement of using elements with C (1) continuity, since the presence of higher order gradients in the expression of potential energy leads to an equilibrium equation represented by a forth order partial differential operator. Although a displacement formulation is conceptually simpler and the most convenient for implementation, in existing finite element codes, only the works [21], [22], [23], [24], [25], [26], [27], [28] implement C (1) elements with the later being the most comprehensive and complete, since it derives both two and three dimensional C (1) finite elements. The other works avoid the problem mainly via mixed formulations as well as with the use of Lagrange multipliers and penalty methods. Finally, it should be mentioned that from all the above cited papers only [27], [28], [29] deal with three dimensional problems. The BEM is a well-known and powerful numerical tool, successfully used in recent years to solve various types of engineering problems [30], [31]. A remarkable advantage it offers as Copyright c 2000 John Wiley & ons, Ltd. Int. J. Numer. Meth. Engng 2000; 00:1 6

3 A DEMONTRATION OF THE INT. J. NUMER. METH. ENGNG CLA FILE 3 compared to other numerical methods, such as the finite difference and finite element methods, is the reduction of the dimensionality of the problem by one. Thus, three dimensional problems are accurately solved by discretizing only two-dimensional surfaces surrounding the domain of interest. In the case where the problem is characterized by an axisymmetric geometry, the BEM reduces further the dimensionality of the problem, requiring just a discretization along a meridional line of the body. These advantages in conjunction with the absent of C (1) continuity requirements, render the BEM ideal for analyzing gradient elastic problems. Tsepoura et al. [32] were the first to use BEM for solving elastostatic problems in the framework of the gradient elastic theories of Mindlin. This work was followed by the publications [33], [34], [35], [36], [37], [38], [39], which are the only papers dealing with two and three dimensional BEM solutions of static and dynamic gradient elastic and fracture mechanics problems. All these papers implement two simple gradient elastic models, with the first being the simplest possible special case of Mindlin s Form II strain gradient elastic theory and the second an enrichment of the simple gradient elastic model with surface energy terms which affect only the boundary conditions of the problem [13]. In the present paper the BEM in its direct form is employed for the solution of twodimensional (2D) and three-dimensional (3D) elastostatic problems in the framework of the Form-II strain-gradient theory of Mindlin. Although the proposed boundary element methodology concerns boundary value problems with smooth and non-smooth boundaries, the numerical examples presented here are confined to smooth boundaries. The BEM solution of gradient elastic problems with non-smooth boundaries is the subject of a forthcoming paper. The paper consists of the following five sections: ection 2 presents in brief the Mindlin s theory implemented in the present paper. In ection 3 the 2D and 3D fundamental solutions of the problem are explicitly derived. ection 4 demonstrates the integral representation of the gradient elastic boundary value problem in the context of Mindlin s simplified Form-II strain gradient elastic theory. ection 5 presents the proposed BEM formulation, while in section 6 three numerical examples (2D and 3D) are presented to demonstrate the accuracy of the method and illustrate the importance of considering the correct boundary conditions imposed by Mindlin s theory. Finally, ection 7 consists of the conclusions pertaining to this work. 2. The Form-II strain gradient elastic theory of Mindlin Mindlin in the Form II version of his strain gradient elastic theory [6] considered that the potential energy density W is a quadratic form of the strains ε ij and the gradient of strains, κ ijk i.e., where W 1 2 λε ii ε jj + µε ij ε ij + ˆα 1 κ iik κ kjj + ˆα 2 κ ijj κ ikk + ˆα 3 κ iik κ jjk +ˆα 4 κ ijk κ ijk + ˆα 5 κ ijk κ kji (1) ε ij 1 2 ( iu j + j u i ), κ ijk i ε jk 1 2 ( i j u k + i k u j ) κ ikj (2) with i denoting space differentiation, u i being displacements and λ, µ and â 1 â 5 being constants explicitly defined in [6]. It should be noted here that the constants λ, µ are not Copyright c 2000 John Wiley & ons, Ltd. Int. J. Numer. Meth. Engng 2000; 00:1 6

4 4 G. F. KARLI, A. CHARALAMBOPOULO AND D. POLYZO the same with the corresponding Lame constants λ, µ of classical elasticity. Both λ, µ have units of N/m 2, while ˆα 1 ˆα 5 have units of force. Thus, this particular case of Mindlin s theory has in total seven elastic constants instead of the eighteen constants of his general theory. trains ε ij and gradient of strains κ ijk are dual in energy with the Cauchy-like and double stresses, respectively, defined as τ ij W ε ij τ ji (3) µ ijk W κ ijk µ ikj (4) which implies that τ ij 2 µε ij + λε ll δ ij (5) and µ ijk 1 2 ˆα 1 [κ kll δ ij + 2κ lli δ jk + κ jll δ ki ] + 2ˆα 2 κ ill δ jk +ˆα 3 (κ llk δ ij + κ llj δ ik ) + 2ˆα 4 κ ijk + ˆα 5 (κ kij + κ jki ) (6) or in vector form τ µ( u + u ) + λ( u)ĩ (7) µ 1 [ ( ) ] ˆα 1 2 u Ĩ + Ĩ u + u Ĩ + u Ĩ + 1 [ ( ) 213 ( ) ] ˆα 3 Ĩ 2 u + Ĩ u + 2 u Ĩ + u Ĩ + 2ˆα 2 u Ĩ + ˆα 4 ( u + u ) ˆα 5 (2u + u + u ) (8) where the symbol indicates dyadic product according to a b a i b jˆx i ˆx j, is the gradient operator, Ĩ the unit tensor and (a b c) 213 a c b. The total stress tensor σ ij is then defined as σ ij τ ij i µ ijk or σ τ µ (9) with µ representing the relative stresses. As it is mentioned in [41], τ represents the Cauchy-like (not Cauchy) stress, while the total stress vector σ is the Cauchy stresses for the present enhanced elastic theory. Taking the variation of (1) and equilibrating with the work done by external and body forces f k, one obtains the following equilibrium equation j (τ jk i µ ijk ) + f k 0 (10) accompanied by the classical essential and natural boundary conditions where the displacement vector u and/or the traction vector p have to be defined on the global boundary of the analyzed domain, the non-classical essential and natural boundary conditions where the normal displacement vector q u/n and/or the double traction vector R are prescribed on, the non-classical boundary condition satisfied only when non-smooth boundaries are dealt with, where the jump traction vector E has to be defined at corners and edges. Copyright c 2000 John Wiley & ons, Ltd. Int. J. Numer. Meth. Engng 2000; 00:1 6

5 A DEMONTRATION OF THE INT. J. NUMER. METH. ENGNG CLA FILE 5 Traction vectors p, R, E are defined as p k n j τ jk n i n j Dµ ijk (n j D i + n i D j )µ ijk + (n i n j D l n l D j n i )µ ijk (11) R k n i n j µ ijk (12) E k n i m j µ ijk (13) or in vector form as p ˆn τ (ˆn ˆn) : n µ ˆn ( µ) ˆn ( µ 213 ) ( ˆn)(ˆn ˆn) : µ ( ˆn) : µ (14) R ˆn µ ˆn (ˆn ˆn) : µ (15) E (ˆm ˆn) : µ (16) where ˆn is the unit vector normal to the global boundary, D n l l and D j (δ jl ) n j n l ) l is the surface gradient operator written in vector form as (Ĩ ˆn ˆn. The non-classical boundary condition (13) or (16) exists only when non-smooth boundaries are considered. Double brackets indicate that the enclosed quantity is the difference between its values taken on the two sides of a corner while ˆm is a vector being tangential to the corner line. Also it should be mentioned that Giannakopoulos et al. [22], utilize for the double traction vector p the expression p i n j τ ji n j µ kji,k [D j (D p n p )n j ]n k µ kji, which however is equivalent to (11). Finally, taking into account the form of τ and µ, the equilibrium equation (10) in terms of displacements is written as where ( λ + 2 µ)(1 l ) u + µ(1 l ) u + f 0 (17) l 2 1 2(â 1 + â 2 + â 3 + â 4 + â 5 )/( λ + 2 µ) (18) l 2 2 (â 3 + 2â 4 + â 5 )/2 µ (19) l 2 1 and l2 2 have units of m2 and as it is discussed in [42] and [43] both can be considered as internal length scale parameters, which correlate the microstructure with the macrostructure in irrotational and solenoidal deformations, respectively. 3. 2D and 3D fundamental solutions Adopting the methodology presented in [34], the 2D and 3D fundamental solutions of (17) are explicitly derived in the present section. For an infinitely extended gradient elastic space, the fundamental solutions are represented by a second order tensor ũ (x,y) satisfying the partial differential equation ( λ + 2 µ ) (1 l 2 1 ) ũ (x,y) µ ( 1 l 2 2 ) ũ (x,y) δ (x,y)ĩ (20) where δ is the Dirac δ-function, x is the point where the displacement field ũ (x,y) is obtained due to a unit force at point y and r x y. Copyright c 2000 John Wiley & ons, Ltd. Int. J. Numer. Meth. Engng 2000; 00:1 6

6 6 G. F. KARLI, A. CHARALAMBOPOULO AND D. POLYZO The field u can be decomposed into irrotational and solenoidal parts [34] acoording to ũ φ + A + G (21) where φ(r) is a scalar function, A (r) a vector function and G (r) a dyadic function. Due to the radial nature of the fundamental solution, it is apparent that the vector A (r) should be equal to zero. On the other hand the Dirac δ-function can be written as [ ] δ (r) 2 g (r) g (r) g (r)ĩ (22) with g (r) being the fundamental solution of the Laplace operator, having the form { g (r) 1 2π ln 1 r for 2D 1 4πr for 3D (23) Inserting (21) and (22) into (20) one obtains ) [ {( λ + 2 µ 2 φ(r) l φ(r) ]} { ]} + µ [ 2 G (r) l G (r) [ ] (24) g (r) g (r)ĩ The irrotational and solenoidal nature of φ(r) and G (r) impose that (24) is satisfied if ( λ + 2 µ ) [ 2 φ(r) l φ(r) ] g (r) (25) [ ] µ 2 G (r) l G g (r)ĩ (26) It is not difficult to find one that the solutions of the above two partial differential equations have the following form ( ) 1 r φ(r) ) 4π ( λ + 2 µ 2 + l2 1 r l2 1 e r/l1 (27) r G (r) 1 ( ) r 4π µ 2 + l2 2 r l2 2e r/l2 Ĩ (28) r for three dimensions and [ ] 1 r 2 φ(r) ) 2π ( λ + 2 µ 2 (lnr 1) + l2 1 lnr + l2 1 K 0 (r/l 1 ) (29) [ ] 1 r 2 G (r) 2π µ 2 (lnr 1) + l2 2 lnr + l2 2 K 0 (r/l 2 ) Ĩ (30) for two dimensions. Inserting (27) (30) into (21), the fundamental solution ũ (x,y) obtains the final form ũ 1 ] (x,y) [Ψ (r)ĩ 16πµ (1 ν) (31) or ũ 1 ij 16πµ (1 ν) [Ψ (r) δ ij X (r) ˆr iˆr j ] (32) Copyright c 2000 John Wiley & ons, Ltd. Int. J. Numer. Meth. Engng 2000; 00:1 6

7 A DEMONTRATION OF THE INT. J. NUMER. METH. ENGNG CLA FILE 7 where ν λ 2( λ+ µ) is the Poisson ratio,ˆr x y x y and X (r) 1 [( 3l 2 r + 2 (1 2ν) 1 r 3 + 3l 1 r r [( 3l 2 4 (1 ν) 2 r 3 + 3l 2 r r ) e r/l2 3l2 2 Ψ (r) (3 4ν) 1 [( l 2 r + 2 (1 2ν) 1 r 3 + l 1 r 2 [( l 2 4 (1 ν) 2 r 3 + l 2 r r ) e r/l2 l2 2 r 3 for three dimensions and ( ) ] X (r) + 2 (1 2ν) [K 2 rl1 2l2 1 r 2 ( ) ] 4 (1 ν) [K 2 rl2 2l2 2 r 2 [ ( l1 r Ψ (r) (3 4ν)lnr + 2 (1 2ν) ( ) 4 (1 ν) [K 0 + rl2 l 2 r K 1 r K 1 ( r l 2 ) e r/l1 3l2 1 r 3 ] r 3 ) e r/l1 l2 1 ] l 1 ) l2 2 r 2 r 3 ) l2 1 for two dimensions, with K n (r/l i ) being the modified Bessel functions of the second kind and nth order. Utilizing the expansions of e r/li and K n (r/l i ), it is easy to prove that both functions X, Ψ given by relations (33) (36) are regular as r 0 according to the asymptotic relations X (r) O ( r 2 lnr ), Ψ (r) O (1) for the 2-D case X (r) O (r), Ψ (r) O (1) for the 3-D case ] r 2 ] ] ] (33) (34) (35) (36) (37) 4. Integral representation of the problem Consider a finite elastic body of volume V with microstructural effects according to the gradient elastic theory illustrated in section (2), surrounded by a surface. ymbolizing by u, P, R, E and u, P, R, E two deformation states of the same body, it has been proved [34], [22] that the following reciprocal identity is valid [f u f u ] dv + [P u P u ] d [R u n R u ] d (38) n V for a smooth boundary, and [f u f u ] dv + [P u P u ] d V + C a [R u n R u ] d n C a [E u E u] dc a Copyright c 2000 John Wiley & ons, Ltd. Int. J. Numer. Meth. Engng 2000; 00:1 6 (39)

8 8 G. F. KARLI, A. CHARALAMBOPOULO AND D. POLYZO for a non-smooth boundary, where the vectors f represent body forces and P, R, E are traction, double traction and jump traction vectors, respectively, defined in (11) (16). C a represents the edge lines formed by the intersection of two surface portions when the boundary is non-smooth. For a two dimensional non-smooth boundary, where parts of the global boundary form C a corners, it is easy to prove one that the reciprocal identity (39) obtains the form V [f u f u ] dv + [P u P u ] d [R u n R u ] d n + C a [E u E u] (40) where E k n i t j µ ijk or E (ˆt ˆµ ) : µ (41) with ˆt being the tantential vector to the curves forming the corner. Assume that the displacement field u, appearing in the reciprocal identity (39), is the result of a body force having the form f (y) δ (x y)ê (42) with δ being the Dirac δ-function and ê the direction of a unit force acting at point y. Recalling the definition of the fundamental solution derived in section 3, it is easy to see that the displacement field u due to f can be represented by means of the fundamental displacement tensor ũ (x,y) given by the Eqs (32) and (33) (36), according to the relation u (y) ũ (x,y) ê (43) Inserting the above expression of u in (39) and assuming zero body forces f 0, one obtains {[ P } [δ (x y)ê u (y)] dv y + (x,y) ê] u (y) P(y) [ũ (x,y) ê] d y V + C a { R (y) [ ũ (x,y) n y ] [ ] ê R(x,y) ê } u (y) d y n y { E (y) [ũ [Ẽ ] } (x,y) ê] (x,y) ê u (y) dc y (44) or C a δ (x y)u(y) dv y ê + P T (x,y) u (y) P(y) ũ (x,y) d y ê V ũ T (x,y) n y R (y) R T (x,y) u (y) n y d y ê + E (y) ũ (x,y) Ẽ T (x,y) u (y) dc y ê Ca C a (45) Copyright c 2000 John Wiley & ons, Ltd. Int. J. Numer. Meth. Engng 2000; 00:1 6

9 A DEMONTRATION OF THE INT. J. NUMER. METH. ENGNG CLA FILE 9 with ÃT indicating the transpose of Ã. Considering that relation (45) is valid for any direction ê and taking into account the symmetry of the fundamental displacement ũ, one obtains the boundary integral equation c(x) u (x) + P T (x,y) u (y) ũ (x,y) P(y) d y + C a ũ T (x,y) n y C a R (y) R T (x,y) u (y) n y d y ũ (x,y) E (y) Ẽ T (x,y) u (y) dc y where c(x) is the well known jump-tensor of classical boundary integral representations [34]. Utilizing the symbols Ū, P, Q, R and Ē instead of ũ, P T ũ, T n, R T and, Ẽ T, respectively, as well as q instead of u n eq (46) recieves the form c(x) u (x) + P (x,y) u (y) Ū (x,y) P(y) d y + C a C a Q (x,y) R (y) R (x,y) q(y) d y Ū (x,y) E (y) Ē (x,y) u (y) dc y (46) (47) Recalling (40) and (41), the above integral equation in two dimensions obtains the form c(x) u (x) + P (x,y) u (y) Ū (x,y) P(y) d y Q (x,y) R (y) R (x,y) q(y) d y + C a [Ū (x,y) E (y) Ē (x,y) u (y) ] (48) In case the boundary is smooth and the point x belongs to, then the integral equations (47) and (48) reduce to 1 2 u (x) + P (x,y) u (y) Ū (x,y) P(y) d y Q (x,y) R (y) R (x,y) q(y) d y (49) Observing eq (47), one easily realizes that it contains four unknown vector fields, u (x), P(x), R (x) and q(x) while the boundary conditions are two (classical and non-classical). Thus, the Copyright c 2000 John Wiley & ons, Ltd. Int. J. Numer. Meth. Engng 2000; 00:1 6

10 10 G. F. KARLI, A. CHARALAMBOPOULO AND D. POLYZO evaluation of the unknown fields requires the existence of one more integral equation. This integral equation is obtained by applying the operator / on (49) and has the form c(x) q(x) + P (x,y) u (y) Ū (x,y) P(y) d y Q (x,y) + C a C a R (y) R (x,y) Ū (x,y) n y q(y) d y E (y) Ē u (y) dc y (50) while for two dimensional and non-smooth boundary is written as c(x) q(x) + P (x,y) u (y) Ū (x,y) P(y) d y Q (x,y) R (y) R (x,y) + C a Ū (x,y) n y q(y) d y E (y) Ē u (y) (51) Finally, for smooth boundaries the above equations obtain the form 1 2 q(x) + P (x,y) u (y) Ū (x,y) P(y) d y Q (x,y) R (y) R (x,y) q(y) d y The pairs of integral equations (47) and (50), (48) and (51), (49) and (52) accompanied by the classical and non-classical boundary conditions form the integral representation of any Mindlin s Form II strain gradient elastic boundary value problem with 3D non-smooth boundary, 2D non-smooth boundary and smooth boundary, respectively. (52) 5. BEM Formulation In the present section the boundary element formulation of the gradient elastic problem is demonstrated. As it has been already mentioned in the introduction, the formulation is confined to gradient elastic bodies with smooth boundaries. The goal of the boundary element method is to solve numerically the boundary integral representation of the problem presented in the previous section. To this end, the smooth boundary is discretized into e quadratic continuous isoparametric elements each of which has A(e) nodes, with A(e) 3, 8, 6 when line, quadrilateral and triangular elements are considered. For a nodal point k, the discretized integral equations (49) and (52) for the three dimensional Copyright c 2000 John Wiley & ons, Ltd. Int. J. Numer. Meth. Engng 2000; 00:1 6

11 A DEMONTRATION OF THE INT. J. NUMER. METH. ENGNG CLA FILE 11 case have the form 1 2 u ( x k) A(e) E 1 1 A(e) 1 1 E A(e) 1 1 E A(e) 1 1 E P ( x k,y e (ξ 1, ξ 2 ) ) N a (ξ 1, ξ 2 )J (ξ 1, ξ 2 ) dξ 1 dξ 2 u e a R ( x k,y e (ξ 1, ξ 2 ) ) N a (ξ 1, ξ 2 )J (ξ 1, ξ 2 ) dξ 1 dξ 2 q e a Ū ( x k,y e (ξ 1, ξ 2 ) ) N a (ξ 1, ξ 2 )J (ξ 1, ξ 2 ) dξ 1 dξ 2 P e a Q ( x k,y e (ξ 1, ξ 2 ) ) N a (ξ 1, ξ 2 )J (ξ 1, ξ 2 ) dξ 1 dξ 2 R e a (53) 1 2 q( x k) A(e) E 1 1 A(e) 1 1 E A(e) 1 1 E A(e) 1 1 E P ( x k,y e (ξ 1, ξ 2 ) ) N a (ξ 1, ξ 2 )J (ξ 1, ξ 2 ) dξ 1 dξ 2 u e a R ( x k,y e (ξ 1, ξ 2 ) ) N a (ξ 1, ξ 2 )J (ξ 1, ξ 2 ) dξ 1 dξ 2 q e a Ū ( x k,y e (ξ 1, ξ 2 ) ) N a (ξ 1, ξ 2 )J (ξ 1, ξ 2 ) dξ 1 dξ 2 P e a Q ( x k,y e (ξ 1, ξ 2 ) ) N a (ξ 1, ξ 2 )J (ξ 1, ξ 2 ) dξ 1 dξ 2 R e a (54) with N a representing shape functions, the first summation is over the elements, the second summation over the element nodes and J is the Jacobian of the transformation from the global coordinate system to the local coordinate system of the element. Finally, u e a, qe a, P e a and Re a are the values of the unknown fields at the nodes of element e. Copyright c 2000 John Wiley & ons, Ltd. Int. J. Numer. Meth. Engng 2000; 00:1 6

12 12 G. F. KARLI, A. CHARALAMBOPOULO AND D. POLYZO For the two dimensional case, the corresponding discretized equations are 1 2 u ( x k) E A(e) 1 A(e) 1 E A(e) 1 E A(e) E 1 P ( x k,y e (ξ) ) N a (ξ)j (ξ) dξ u e a R ( x k,y e (ξ) ) N a (ξ)j (ξ) dξ q e a Ū ( x k,y e (ξ) ) N a (ξ)j (ξ) dξ P e a Q ( x k,y e (ξ) ) N a (ξ)j (ξ) dξ R e a (55) 1 2 q( x k) A(e) E 1 A(e) 1 E A(e) 1 E A(e) 1 E P ( x k,y e (ξ) ) N a (ξ)j (ξ) dξ u e a R ( x k,y e (ξ) ) N a (ξ)j (ξ) dξ q e a Ū ( x k,y e (ξ) ) N a (ξ)j (ξ) dξ P e a Q ( x k,y e (ξ) ) N a (ξ) J (ξ) dξ R e a with the same notation used for eqs (53) and (54). Next, a global numbering scheme is adopted by assigning a number β to each point (e, a). Then the above equations become 1 2 uk qk + L H β k uβ + β1 L β k u β + β1 L K k β qβ β1 L T k β q β β1 L G k β Pβ + β1 L Ṽβ k R β + β1 (56) L L k β Rβ (57) β1 L β1 W k β R β (58) where L is the total number of nodes. Note that eqs (57) and (58) are valid for both the 2D and the 3D case. However, the integrals are different in each case. Namely, for the 3D case H k β 1 1 P ( x k,y e (ξ 1, ξ 2 ) ) N a (ξ 1, ξ 2 )J (ξ 1, ξ 2 ) dξ 1 dξ 2 (59) Copyright c 2000 John Wiley & ons, Ltd. Int. J. Numer. Meth. Engng 2000; 00:1 6

13 A DEMONTRATION OF THE INT. J. NUMER. METH. ENGNG CLA FILE 13 K k β G k β L k β k β T k β Ṽβ k W β k R ( x k,y e (ξ 1, ξ 2 ) ) N a (ξ 1, ξ 2 ) J (ξ 1, ξ 2 ) dξ 1 dξ 2 Ũ ( x k,y e (ξ 1, ξ 2 ) ) N a (ξ 1, ξ 2 )J (ξ 1, ξ 2 ) dξ 1 dξ 2 Q ( x k,y e (ξ 1, ξ 2 ) ) N a (ξ 1, ξ 2 )J (ξ 1, ξ 2 ) dξ 1 dξ 2 ( P x k,y e (ξ 1, ξ 2 ) ) N a (ξ 1, ξ 2 ) J (ξ 1, ξ 2 ) dξ 1 dξ 2 ( R x k,y e (ξ 1, ξ 2 ) ) N a (ξ 1, ξ 2 )J (ξ 1, ξ 2 ) dξ 1 dξ 2 Ũ ( x k,y e (ξ 1, ξ 2 ) ) N a (ξ 1, ξ 2 )J (ξ 1, ξ 2 ) dξ 1 dξ 2 ( Q x k,y e (ξ 1, ξ 2 ) ) N a (ξ 1, ξ 2 )J (ξ 1, ξ 2 ) dξ 1 dξ 2 (60) (61) (62) (63) (64) (65) (66) and for the 2D case H k β K k β G k β L k β k β P ( x k,y e (ξ) ) N a (ξ)j (ξ) dξ R ( x k,y e (ξ) ) N a (ξ)j (ξ) dξ Ũ ( x k,y e (ξ) ) N a (ξ)j (ξ) dξ Q ( x k,y e (ξ) ) N a (ξ)j (ξ) dξ ( P x k,y e (ξ) ) N a (ξ)j (ξ) dξ (67) (68) (69) (70) (71) Copyright c 2000 John Wiley & ons, Ltd. Int. J. Numer. Meth. Engng 2000; 00:1 6

14 14 G. F. KARLI, A. CHARALAMBOPOULO AND D. POLYZO T k β Ṽ k β W k β ( R x k,y e (ξ) ) N a (ξ) J (ξ) dξ Ũ ( x k,y e (ξ) ) N a (ξ)j (ξ) dξ ( Q x k,y e (ξ) ) N a (ξ) J (ξ) dξ When β k the above integrals are non-singular and can be easily computed through Gauss quadrature. In the case of β k the integrals become singular and special treatment is required. This is accomplished by applying the methodology proposed by Guiggiani [40] for singular integrations. Thus, eqs (57) and (58) form the following linear system of algebraic equations ] [ ] G L [ 1 2Ĩ + H K 1 2Ĩ + T ] [ u q which after the application of the classical and non-classical boundary conditions and rearranging, a final linear system is obtained of the form A X B, where the vectors X and B contain all the unknown and known nodal components of the boundary fields respectively. Finally, the above linear system is solved via a typical LU-decomposition algorithm and the vector X, comprising of all the unknown nodal values of u, q, R and P is evaluated. Ṽ W ] [ P R (72) (73) (74) (75) 6. Numerical Examples In this section some 2D and 3D benchmarks that demonstrate the accuracy of the proposed BEM are presented. The first numerical example concerns a 3D/2D problem dealing with the tension of a cylindrical/rectangular gradient elastic bar shown in Figure 1. The analytical solution of the problem is known for the one dimensional case [32] and for material properties where only the Young modulus E and internal length ˆα 4 are non-zero. Thus in the present BEM solution the Poisson ratio has been taken equal to zero, ˆα 1 ˆα 2 ˆα 3 ˆα 5 0 and the height of the cylinder/rectangle much smaller than its diameter/width (d 4.2m, h 1.2m) so that to compare analytical and numerical results across the axis of symmetry. In order to avoid a BEM formulation for non-smooth boundaries, the edges of the cylinder and the corners of the rectangle have been rounded with radius r e 0.05m. The bar is subjected to a tension of T 2.1GPa on its top and bottom sides while its material properties are E 2.1GPa and ˆα MNt. The non classical boundary condition applied to the top and bottom faces of the bar is (q x, q y, q z ) (0, 0, 0). The side surface of the bar is left traction free by imposing (P x, P y, P z ) (0, 0, 0) and (R x, R x, R x ) (0, 0, 0). The 3D problem has been solved with octant symmetry while in the 2D case of the rectangular bar only the one quarter of the domain has been discretized. For the 2D case, approximately 50 elements have been used, whereas for the 3D case up to 150 elements have Copyright c 2000 John Wiley & ons, Ltd. Int. J. Numer. Meth. Engng 2000; 00:1 6

15 A DEMONTRATION OF THE INT. J. NUMER. METH. ENGNG CLA FILE 15 Figure 1. Geometry of the gradient elastic bar in three and two dimensions Axial displacements on internal points D Form II Analytical solution l l l Axial displacements on internal points D Form II Analytical olution l1 0.1 l l z coordinate z coordinate (a) (b) Figure 2. Axial displacements of the internal points for (a) the 3D case and (b) the 2D case been utilized. A set of internal points has been placed along the central vertical axis of the bar. In Figures 2 the axial displacements of the internal points are presented. For all the displayed results, the relative error with respect to the analytical solution [32] is below 0.2%. The second numerical example deals with a hollow cylinder subjected to internal and external pressure as it is shown in Figure 3. The problem is solved under plane strain conditions. The material properties of the cylinder are described in Table I. The internal and external radii and pressures are r i 1.05m, r o 2.1m and T a 100KPa, T b 200KPa, respectively. Double tractions are considered to be equal to zero everywhere. Figure 4 shows the radial displacements on the internal points of the hollow cylinder with respect to the distance r from the center of the cylinder. The numerical results are compared with the corresponding analytical ones [29]. Furthermore, the normal derivative of the displacements, q has been calculated on the boundary and found to be equal to the corresponding analytical result with an error of 0.14%. The relative error for the internal displacement with respect to the mesh used, is presented in Table II. In order to demonstrate the importance of considering the correct integral equations and Copyright c 2000 John Wiley & ons, Ltd. Int. J. Numer. Meth. Engng 2000; 00:1 6

16 16 G. F. KARLI, A. CHARALAMBOPOULO AND D. POLYZO Figure 3. Hollow cylinder under internal and external pressure: plain strain problem Table I. Material constants for the hollow cylinder Young s modulus 4.0 GPa 4.0 GPa 4.0 GPa Poisson s ratio Mindlin s ˆα MNt MNt 2.0 KNt Mindlin s ˆα MNt MNt KNt Mindlin s ˆα MNt 1.36 MNt 0.1 KNt Mindlin s ˆα MNt MNt 1.63 KNt Mindlin s ˆα MNt MNt 0.1 KNt Radial displacement of internal points D Form II Analytical solution l 1 l l 1 l l 1 l Radial distance r Figure 4. Radial displacements on the internal points Copyright c 2000 John Wiley & ons, Ltd. Int. J. Numer. Meth. Engng 2000; 00:1 6

17 A DEMONTRATION OF THE INT. J. NUMER. METH. ENGNG CLA FILE 17 Table II. Relative error with respect to the number of elements for the plain strain hollow cylinder Number of elements % Error % % % % satisfying the correct boundary conditions in a gradient elastic problem, the problem of Figure 3 is solved for T a 100KPa, T b 0 and first considering an artificial interface in the circumferential direction (Figure 5(a)) and next an interface separating the cylinder into two equal parts (Figure 5(b)). In both cases, continuity on displacements u, tractions P, normal derivatives of displacements q and double tractions R has been considered at interfaces. The corresponding deformation profiles of the cylinder in both cases are depicted in Figires 5(c) and 5(d), respectively. It is apparent that the obtained solutions are not the same. The displacement profile of Figure 5(c) is the same with the corresponding one taken when no interfaces are considered in the solution of the problem. On the other hand the displacement profile of the cylinder in Figure 5(d) violates the radial symmetry of the problem and the solution appears significant errors near to the interface. The obvious explanation for these two different solutions of the same problem is that in the first case the two considered subregions have smooth boundaries, while in the second case they have not. Thus, for the second case, first the BEM formulation for non-smooth boundaries should be taken into account and second a continuity condition of the jump traction vectors E at the circumferential edges is required. The third numerical example deals with a gradient elastic sphere of radius a 0.5m under an external uniform displacement u r 0.01m. The material characteristics of the sphere are the same used in the previous example (Table I). In order to model the problem octant symmetry has been used and the same classical and non-classical boundary conditions have been applied to all elements. Namely, (u r, u θ, u φ ) (0.01, 0, 0) and (q r, q θ, q φ ) (0, 0, 0), with the subscripts r, θ and φ indicating spherical coordinates. A set of internal points have been placed inside the sphere, along its radius. Figure 6 shows the radial displacement on the internal points with respect to their distance from its center as compared to the analytical solution provided by [35]. In Table III the relative error of the internal radial displacement is presented with respect to the analytical solution, for various mesh sizes. 7. Conclusions A boundary element method for solving two and three-dimensional static, strain gradient elastic problems has been developed. Microstructural effects have been taken into account by means of the Mindlin s Form II strain gradient elastic theory. The equation of equilibrium, all the possible boundary conditions (classical and non-classical), the fundamental solution and the reciprocity identity of the gradient elastic problem are explicitly presented. Both, Copyright c 2000 John Wiley & ons, Ltd. Int. J. Numer. Meth. Engng 2000; 00:1 6

18 18 G. F. KARLI, A. CHARALAMBOPOULO AND D. POLYZO (a) (b) (c) (d) Figure 5. A hollow cylinder divided into two regions (a) by a circular interface and (b) by a straight interface, as well as the boundary displacements for the (c) circular interface case and (d) for the straight interface. Table III. The relative error of the internal radial displacements, with respect to the analytical solution, for the gradient elastic sphere Number of elements % Error % % % % Copyright c 2000 John Wiley & ons, Ltd. Int. J. Numer. Meth. Engng 2000; 00:1 6

19 A DEMONTRATION OF THE INT. J. NUMER. METH. ENGNG CLA FILE Radial displacement u r D Form II Analytical olution BEM Radial distance Figure 6. The radial displacements of the internal points of a sphere subjected to radial displacement fundamental solution and reciprocity identity have been used to establish the boundary integral representation of the problem consisting of one equation for the displacement and another one for its normal derivative. This integral representation concerns problems dealing with both smooth and non-smooth boundaries. The BEM formulation is confined to boundary value problems with smooth boundaries, since the treatment of non-smooth boundaries is the subject of a forthcoming paper. The numerical implementation of the problem is accomplished by discretizing the external boundary into quadratic line or quadrilateral elements and employing advanced integration algorithms for the highly accurate evaluation of the singular integrals. Three representative two and three dimensional numerical examples have been presented to illustrate the method and demonstrate its high accuracy. Finally, with the solution of a simple two-dimensional gradient elastic problem, has been shown that attention should be paid on the handling of the boundary conditions of the problem, especially when non-smooth boundaries are considered. REFERENCE 1. Cosserat E, Cosserat F. Theorie des Corps Deformables Cornell University Library, Mindlin RD, Tiersten HF. Effects of couple stresses in linear elasticity Arch. Rat. Mech. Anal. 1962; 11: Koiter WT. Couple stress in the theory of elasticity I-II Proc. Kon. Nederl. Akad. Wetensch. 1964; B67: Toupin RA. Theories of elasticity with couple-stress Arch. Rat. Mech. Anal. 1964; 17: Green AR, Rivlin R. Multipolar continuum mechanics Arch. Ration. Mech. Anal. 1964; 17: Mindlin RD. Micro-structure in linear elasticity Arch. Rat. Mech. Anal. 1964; 16: Mindlin RD. econd gradient of strain and surface-tension in linear elasticity International Journal of olids and tructures 1965; 1: Eringen AC. Microcontinuum Field Theories I: Foundations and olids pringer-verlang, New York, Eringen AC. Vistas of nonlocal continuum physics Int. J. Engng ci. 1992; 30: Tiersten HF, Bleustein JL. Generalized elastic continua. In R. D. Mindlin and Applied Mechanics, Hermann G (ed). Pergamon Press:New York, 1974; Copyright c 2000 John Wiley & ons, Ltd. Int. J. Numer. Meth. Engng 2000; 00:1 6

20 20 G. F. KARLI, A. CHARALAMBOPOULO AND D. POLYZO 11. Exadaktylos GE, Vardoulakis I Microstructure in linear elasticity and scale effects: a reconsideration of basic rock mechanics and rock fracture mechanics Tectonophysics 2001; 335: Tekoglou C ize effects in cellular solids, Ph.D. thesis Institure for Metal Research, Vardoulakis I and ulem J Bifurcation Analysis in Geomechanics Blackie/Chapman and Hall, London, hu JY, King WE, Fleck NA Finite elements for materials with strain gradient effects Int. J. Numer. Meth. Engng. 1999; 44: Amanatidou E, Arava N Mixed finite element formulations of strain-gradient elasticity problems Comput. Methods Appl. Mech. Engng. 2002; 191: Matsushima T, Chambon R, Caillerie D Large strain finite element analysis of a local second gradient model: application to localization Int. J. Numer. Meth. Engng 2002; 54: oh AK, Wanji C Finite element formulations of strain gradient theory for microstructures and the C 0 patch test Int. J. Numer. Meth. Engng. 2004; 61: Imatani, Hataday K, Maugin GA Finite element analysis of crack problems for strain gradient material model Philosophical Magazine 2005; 85: Askes H, Gutierrez MA Implicit gradient elasticity Int. J. Numer. Meth. Engng 2006; 67: Dessouky, Masad E, Little D, Zbib H Finite-element analysis of hot mix asphalt microstructure using effective local material properties and strain gradient elasticity Journal of Engineering Mechanics 2006; 158: Akarapu, Zbib HM Numerical analysis of plane cracks in strain-gradient elastic materials Int. J. Fracture 2006; 141: Giannakopoulos AE, Amanatidou E, Aravas N A reciprocity theorem in linear gradient elasticity and the corresponding aint-venant principle Int. J. olids truct. 2006; 43: Markolefas I, Tsouvalas DA, Tsamasphyros GI Theoretical analysis of a class of mixed, C 0 continuity formulations for general dipolar gradient elasticity boundary value problems Int. J. olids truct. 2007; 44: Markolefas I, Tsouvalas DA, Tsamasphyros GI, Mixed finite element formulation for the general antiplane shear problem, including mode III crack computations, in the framework of dipolar linear gradient elasticity Comp. Mech. 2009; 43: Askes H, Bennett T, Aifantis EC A new formulation and C 0 -implementation of dynamically consistent gradient elasticity Int. J. Numer.Meth. Engng 2007; 72: Zervos A, Papanastasiou P, Vardoulakis I A finite element displacement formulation for gradient elastoplasticity Int. J. Numer. Meth. Engng 2001; 50: Zervos A Finite elements for elasticity with microstructure and gradient elasticity Int. J. Numer. Meth. Engng 2008; 73: Papanicolopulos A, Zervos A,, Vardoulakis I A three dimensional C 1 finite element for gradient elasticity Int. J. Numer. Meth. Engng. 2009; 77: Zervos A, Papanicolopulos A, Vardoulakis I Two finite element discretizations for gradient elasticity J. Engrg. Mech. (ACE) 2009; 135: Beskos DE, Boundary element methods in dynamic analysis Appl. Mech. Rev. AME 1987; 40: Beskos DE, Boundary element methods in dynamic analysis. part II ( ) Appl. Mech. Rev. AME 1997; 50: Tsepoura KG, Papargyri-Beskou, Polyzos D, Beskos DE tatic and dynamic analysis of a gradient elastic bar in tension Arch. Appl. Mech. 2002; 72: Tsepoura KG, Polyzos D tatic and harmonic bem solutions of gradient elasticity problems with axisymmetry Comput. Mech. 2003; 32: Polyzos D, Tsepoura KG, Tsinopoulos V, Beskos DE, A boundary element method for solving 2-d and 3-d static gradient elastic problems. part I: Integral formulation Comput. Meth. Appl. Mech. Engng 2003; 192: Tsepoura KG, Tsinopoulos V, Polyzos D, Beskos DE, A boundary element method for solving 2-d and 3-d static gradient elastic problems. part II: Numerical implementation Comput. Meth. Appl. Mech. Engng. 2003; 192: Polyzos D, Tsepoura KG, Beskos DE Transient dynamic analysis of 3-d gradient elastic solids by BEM Comput. truct. 2005; 83: Polyzos D, 3D frequency domain bem for solving dipolar gradient elastic problems Comput. Mech. 2005; 35: Karlis GF, Tsinopoulos V, Polyzos D, Beskos DE, Boundary element analysis of mode i and mixed mode (I and II) crack problems of 2-d gradient elasticity Comput. Methods Appl. Mech. Engrg. 2007; 196: Karlis GF, Tsinopoulos V, Polyzos D, Beskos DE 2D and 3D boundary element analysis of mode-i cracks in gradient elasticity, CME: Computer Modeling in Engineering & ciences 2008; 26: Copyright c 2000 John Wiley & ons, Ltd. Int. J. Numer. Meth. Engng 2000; 00:1 6

21 A DEMONTRATION OF THE INT. J. NUMER. METH. ENGNG CLA FILE Guiggiani M Formulation and numerical treatment of the boundary integral equations with hypersingular kernels Computational Mechanics 1998; 448, Inc., outhampton. 41. Polizzotto C Gradient elasticity and nonstandard boundary conditions International Journal of olids and tructures 2003; 40(26): Ben-Amoz M A dynamic theory for composite materials Z. Angew. Math. Phys. 1976; 27: Vavva MG, Protopappas VC, Gergidis LN, Charalambopoulos A, Fotiadis I, Polyzos D Velocity dispersion of guided waves propagating in a free gradient elastic plate: Application to cortical bone JAA 2009; 125(5): Copyright c 2000 John Wiley & ons, Ltd. Int. J. Numer. Meth. Engng 2000; 00:1 6