Transactions on Modelling and Simulation vol 2, 1993 WIT Press, ISSN X

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1 A general integral-equation formulation for anisotropic elastostatics M.M. Perez"'*, L.C. Wrobel* * Wessex Institute of Technology, University of Portsmouth, Ashurst Lodge, Ashurst, Southampton S04 2AA, UK On leave from Departamento de Engenharia Mecanica, Universidade Federal de Uberlandia, MG, Abstract In this paper a conceptually simple integral-equation formulation for homogeneous anisotropic linear elastostatics is presented. The basic idea of this work is to rewrite the system of differential equations of the anisotropic problem in a slightly different form, to enable the use of the isotropic fundamental solution. This formulation leads to an extended form of Somigliana's identity where a domain term due to the anisotropy of the material appears. To cope with the resulting domain unknowns a supplementary integral equation is then established. Although the solution of these integral equations requires discretization of both contour and domain of the structural component to be analysed, the numerical scheme presented here depends only on the boundary variables of the problem. Encouraging results are presented for two examples, where a rectangular plate is submitted to tension and shear effects, respectively. Introduction The increasing number of structural applications of anisotopic materials has attracted the attention of many researchers concerned with computational modelling. To solve current technological problems that occur, for instance, in the aerospace industry, the use of directionally solidified alloys,

2 90 Boundary Elements metals that have undergone heavy cold pressing and fibre-reinforced laminates is sometimes essential. In fact, many applications of fibre-reinforced laminates, so-called composite materials, are currently seen as conventional practice in engineering design. This paper is concerned with the development of an alternative integralequation formulation for the numerical analysis of homogeneous anisotropic linear elastic problems. The approach presented here consists of rewriting the generalized form of Hooke's law in a slightly different way to enable the use of Kelvin's fundamental solutions for elastostatics as weighting functions. This procedure leads to an extended form of Somigliana's identity which includes a domain term that accounts for the anisotropy of the material. The first primary integral equation of the method is obtained by taking the limiting form of this equation as the interior point approaches the boundary. In order to cope with the domain unknowns arising from this formulation, a supplementary integral equation is derived from the extended form of Somigliana's identity. This supplementary integral equation is then regarded as the second primary integral equation of the proposed method. To solve the system formed by the two primary integral equations, it is necessary to discretize the contour of the mechanical or structural element to be analysed into boundary elements and its interior into domain cells. Two coupled systems of linear algebraic equations are then obtained. The solution of these simultaneous systems of linear equations is done using a technique equivalent to the FEM condensation of internal degrees of freedom (Desai and Abel, 1972), leading to a final solution that is dependent exclusively on the boundary variables of the problem. To assess the accuracy of the proposed formulation two examples are considered at the end of this work. The numerical solutions obtained using quadratic boundary elements along with either constant or quadratic internal cells are presented and compared with the analytical solution to each problem (Lekhnitskii, 1968). The purpose of this work is to investigate whether the use of the isotropic fundamental solutions for linear elastostatics constitutes or not a reliable alternative for the analysis of anisotropic problems. Once the formulation proposed here is verified for two-dimensional problems there are no conceptual difficulties in including dynamic and non-linear (e.g. elastoplasticity) effects in the formulation. Most importantly, it can be directly extended to obtain a general integral-equation formulation for three-dimensional homogeneous anisotropic problems. At this point it is important to mention that although the two-dimensional fundamental solution for anisotropic elasticity presents no particular difficulty in its implementation (Rizzo and Shippy, 1970), the evaluation of the contour integrals for the three-dimensional case is quite

3 Boundary Elements 91 complex, especially for the fundamental tractions (Vogel and Rizzo, 1973), and was regarded as too time consuming for routine numerical use by Wilson and Cruse (1975). To avoid this difficulty another approach to the problem has been presented by Kinoshita and Mura (Ref.). However, the need to compute series expansions for the fundamental displacements and tractions at each integration point makes this formulation also unsuitable for extensive computation. These difficulties make the aforementioned extension to three-dimensional problems attractive, in particular to improve computational efficiency. The inspiration for the formulation proposed here was drawn from the tentative approach proposed by Brebbia and Dominguez (1989) for anisotropic elasticity; from the work of Shi (1990), where a supplementary integral equation for the solution of bending and eigenvalue problems of anisotropic plates was derived from the integral equation for the displacements within the domain; and from the work of Telles and Brebbia (1979, 1980), where the complete integral-equation formulation for plasticity was first presented. Moreover, the method proposed here is a direct extension of the formulation proposed by Perez and Wrobel (1992, 1993) for the analysis of homogeneous anisotropic problems in potential theory. Integral Equation Formulation The governing differential equation for linear anisotropic elasticity is expressed by (Balas et a/., 1989): + bi = 0 (1) where Dijki represents the fourth-order tensor of elastic properties; %& denotes the components of the displacement vector and 6, is the body force vector. This equation is sufficient to completely describe the elastic behaviour of the material once boundary conditions are defined. As only homogeneous materials are considered in this paper, the components of the elasticity tensor are regarded as constants throughout the domain. The body force effects, though not causing any difficulty for implementation, are not considered in the present work. To enable the use of isotropic fundamental solutions as weighting functions in the formulation proposed here, the elasticity tensor was divided into two components, i.e.: ^uw = 2%w + ^,w (2) In this equation, D^^ stands for an isotropic reference tensor while Dijki denotes the difference between the actual tensor of elastic constants of the anisotropic material and the isotropic reference one. This reference isotropic tensor can be defined by either averaging the elastic constants

4 92 Boundary Elements of the anisotropic material (Brebbia and Dominguez, 1989), a procedure adopted in the present work for its simplicity, or determining its effective isotropic moduli (Cowin, 1989). Taking Eq.(2) into consideration, the weighted residual statement to the anisotropic problem analysed here can then be written as: / «y Ja where and \ ^re the load andfieldpoints, respectively; %%(,%) denotes the isotropic fundamental displacements; p^((,%) denotes the fundamental tractions; %,(%) and PJ(X) stand for displacements and tractions, respectively, while the superimposed bar denotes prescribed values (uj(x) on I\ and pj(x) on 1^); FI + F2 = F constitute the boundary of the structural component being analysed and fi denotes the region enclosed by F. The appropriate expressions for the reference isotropic fundamental displacements and tractions for two-dimensional plane strain problems are given by (Brebbia et a/., 1984): (3) where Sij is the Kronecker delta; G is the shear modulus; i/ is the Poisson's ratio; r = r(<f, %) denotes the distance between and %; r,,- represents the derivatives of r with respect to the coordinates of the field point, i.e., r,i = dr/dxi(x) = n/r; n is the outward unit-normal to the boundary at x while n,- denotes its direction cosines. The case of plane stress can be analysed through the use of an effective Poisson's ratio (Brebbia and Dominguez, 1989). Integrating by parts the weighted residual statent, Eq.(3), yields: (5) where 6j^((, %) represents the isotropic fundamental strain tensor and /m(x) the actual strain field. The fundamental strains e*^ at any point x» due to (6)

5 Boundary Elements 93 a unit point load applied at in the direction i, can be written as (Brebbia et a/., 1984): * ^+^j W - r,.-fyb + 2r,, r,, r,* ] (7) The constitutive equation relating stresses and strains for a linearly elastic material (generalized Hooke's law) is used along with Eq.(2) to write: (DjMm + Djklm) \m = &jk + &jk (8) where cr^ is the isotropic component of the actual stress tensor while &,& is the residual one. Applying Eq.(8) to the left-hand side term in Eq.(6) yields: Recalling that the isotropic fundamental tensors comply with <7*^- = Dwmtfmi and, from Eq.(S), that a^ = >^e/m, it is possible to rewrite the first term in Eq.(9) in the form: (9), X) 4(%) ^n(x) = *]ki(t, X) em <Kl(x) (10) Integrating by parts the right-hand side term in Eq.(lO) and substituting the result into Eq.(9) leads to: (11) r Jr where the first term on the left-hand side can be shown to reduce to the displacement vector at point (Brebbia et a/., 1984) and the dash through the integration symbols denotes integrals that are to be interpreted in the Cauchy principal value sense. Equation (11) can then be written as: (12)

6 94 Boundary Elements This equation can be seen as Somigliana's identity with an additional domain term which takes into account the anisotropy of the material. This additional domain term introduces a set of domain variables represented by the residual stress field o-jk(x)- An integral equation for load points on the boundary is obtained by a limiting process, taking the load point from within the domain to the contour F. The resulting integral equation is expressed by: where the elements of the tensor c,-j( ) are function of the internal angle of the boundary at point. Whereas Eq.(13) provides thefirstprimary integral equation of the proposed formulation, another integral equation is still required to provide, after discretization, the necessary number of linear equations to solve the problem numerically. This supplementary integral equation is obtained by differentiating Eq.(12) with respect to the coordinates of the load point, as expressed by: (13) and then combining these derivatives to obtain the strain tensor ij(x)- This strain tensor is then related to the residual stress tensor <T,-J( ) using the generalized Hooke's law, leading to the second primary integral equation of the method, that is: d a + / - 1 Jr ""* ' * P* * % j i r o /* -A>m.T - U-TTT \.fxn(^) 4 Jn C«(.X) ;«(, X) *w(x) dfi(x) (15)

7 Boundary Elements 95 where the first term on the left-hand side is equal to <T,-J( ). As the load point is considered within the domain in Eq.(15), it is possible to apply the differentiation directly to the kernel of the boundary integrals. Recalling the identity: the following tensors are obtained: and: 2 where C»&((,%) «g^en by Eq.(7) while?;, *( > x) is given by: (18) ~ V (9r 1 m $nk + r,n 5mAr) + 2r,fc 6mn - 8r,m r,» r,* ] > (19) an J However, differentiation of the domain integrals needs further attention due to their singular kernels. Differentiation of the Singular Domain Integrals In order to obtain the final form of the integral equation for the residual stresses &,,(() expressed by Eq.(15) it is necessary to differentiate the extended form of Somigliana's identity, Eq.(12), with respect to the coordinates of the load point. This differentiation can be directly applied to the fundamental solution tensors for the boundary integrals. However, the same procedure cannot be applied to differentiate the domain integrals as the concept of differentiation of singular integrals does not follow the classical rule (Mikhlin, 1962, 1970). The correct differentiation of these integrals yields additional terms, which can be determined analytically through the use of Leibnitz' rule (Brebbia et a/., 1984; Perez and Wrobel, 1992, 1993). According to Telles and Brebbia (1979), where the problem of differentiation of similar domain terms in a BEM formulation for plasticity is presented in detail, the derivative of the domain integrals can be written as:

8 96 Boundary Elements where Y\ defines a circle of unit radius centred at the load point and corresponds to the first term on the left-hand side of Eq.(15). The expression for the derivatives of the domain integral obtained in Eq.(20) can be substituted into the second term on the right-hand side of Eq.(15). This term then assumes the form: (20) n c/xmv?; Vr; <--.-- (21) Combining the domain integrals in Eq.(21) yields: 2 [A) &*n(0 " ' Jtl dxm(t) (22) where: ^ r,n 8r,^ r^ r,jt r,/ ] (23) Further, combining the integrals over T( together, it is possible to express the additional term resulting from the differentiation process in the form: [(6-8v)*«(0 - <7,j((N (24) 16(1 -v)g Finally, Eq.(15) can be rewritten as -*y(0 + ym» f p'^(t> Vr

9 Boundary Elements 97 Equations (13) and (25), together with boundary conditions, provide the necessary integral relations for the numerical solution of the problem. (25) Matrix Formulation Following the approach used by Perez and Wrobel (1992, 1993) for anisotropic problems of potential theory, the numerical solution of the system formed by Eqs.(13) and (25) is obtained by discretizing the contour F into boundary elements and the domain fi into internal cells. Then, by applying the discrete version of Eq.(13) at each boundary node, the first set of linear equations is obtained in the form: HU -GP = EB (26) where H and G are the conventional BEM influence matrices; U and P denote nodal boundary displacement and nodal boundary traction vectors, respectively; E is the matrix resulting from the domain integration; and B represents the vector of domain unknowns <7,j at the internal collocation points. The supplementary set of linear equations can similarly be obtained by applying the discretized version of Eq.(25) at the internal collocation points. This procedure leads to: UU-GP = EB (27) where H and G are matrices concerned with boundary integrals whilst E is a square matrix resulting from the domain terms. To avoid the computation of the residual stresses &,_, at interior points, the coupling of the two equations is done by eliminating the domain unknown vector B. With this purpose Eq.(27) is written in the form: B = E->(HU-GP) (28) The expression for vector B in Eq.(28) is then substituted into Eq.(26), resulting in: (H - EE->7I)U = (G - EE~*G)P (29) This procedure is equivalent to the FEM of freedom (Desai and Abel, 1972). condensation of internal degrees

10 98 Boundary Elements The final system of linear equations can be obtained by substituting the boundary conditions and rearranging Eq.(29) in order to obtain an expression of the form: AX = F (30) from where the boundary unknowns X of the problem are computed. Once this solution is obtained, the domain unknowns can be computed, if required, by refering to Eq.(5). The algorithm recently proposed by Guiggiani and Gigante (1990), for evaluation of multidimensional Cauchy principal value integrals, was used to determine the components of matrix matrix E when the integration is performed over the cell that contains the load point. Standard Gaussian numerical integration was used otherwise. In this work quadratic boundary elements were used for the discretization of the contour F while rectangular discontinuous internal cells were used for th discretization of the domain Q. For the latter, either constant or Lagrangian quadratic interpolation functions (Burnett, 1987) were used to approximate the unknowns &y within each cell (Fig.l) , Geometric nodes o Functional nodes (a) (b) Figure 1: Constant (a), and subparametric (b) discontinuous parent internal cells Numerical Examples Two examples are presented in this paper. In these examples a thin plate is submitted to uniform traction (plate stretching) and shear, respectively. The loading is applied on middle plane of the cross-section of the plate, thus causing a state of generalized plane stress. In all examples the length of the plate was / = 10m, its height 6 = 4m, and its thickness h = 0.25m (Fig.2). The load intensity per unit length p was taken as 0.25A/A7?n. The D^ terms are coefficients of deformation

11 Boundary Elements 99 (components of the inverse of the elasticity matrix) (Lekhnitskii, 1963). The xy axes are the reference axes of analysis, x\y\ are the principal directions of elasticity of the material (Lekhnitskii, 1968), and a denotes the angle between them. The elastic properties of the material on the principal axes of elasticity were taken as E*,=, = 144.7S9GPa,,, = 11.72lGPa, G*,y, = 9.653GPa and j^iyi = 0.21, representative of afibre-reinforcedgraphite epoxy (Snyder and Cruse, 1975). Six quadratic boundary elements were used for discretizing the contour of the plate whilst both constant and quadratic discontinuous internal cells were used for domain discretization. In all examples a was made equal to 30. Numerical results obtained were compared with the analytical solution for each case (Lekhnitskii, 1968). Example 1. Plate stretching Figure 2 depicts a rectangular plate subjected to tension by normal forces distributed over its two side edges. D,l pl/h Figure 2: Rectangular plate subjected to uniform tension. The results obtained using one constant internal cell and one subparametric quadratic internal cell are presented in Fig.3, along with the analytical solution of the problem (Lekhnitskii, 1968). In this Figure it is important to notice that the magnitude of the results presented for the top side of the plate, 10"*, is one order smaller than the order for the other sides, 10-*. Example 2. Shear loading In this example, the intensity load per unit area p is tangentially distributed over all edges of the rectangular plate (Fig.4). This loading causes the anisotropic plate to be subjected to both shear on plane xy, which is determined by DQQ, and either elongation or shortening of its sides, depending on the signs of D^Q and D^. The results obtained using constant and

12 O cr Displacement vx10" a. 5* 9 C a- <D m nt rt Coordinate y

13 Boundary Elements 101 subparametric quadratic internal cells are presented in Fig.5, along with the analytical solution to the problem (Lekhnitskii, 1968). y.y, %! & 6 7r Figure 4: Rectangular plate subjected to shear loading , Z HD~bo Coordinate x r o o o o Displacement uxlo «- D O.JO 4 Displacement ux10~ 0.20 \ a ) *0 5~007~50 To~00 Coordinate x OQOOO Analytical results AAAAA Constant internal cell ooooo Quadratic internal cell Figure 5: Results obtained for plate under shear loading. Units in m.

14 102 Boundary Elements Concluding Remarks In this paper a system of singular integral equations for solving elasticity problems in homogeneous anisotropic materials is formulated and solved numerically. The proposed method uses Kelvin's fundamental solutions for isotropic elastostatics in a direct boundary element method approach. It can be seen from Figs.3 and 5 that encouraging results were obtained for both examples (plate stretching and shear) presented here. However, the results obtained for some flexural problems analysed were so far not satisfactory. The research is currently directed towards identifying the causes of such behaviour. Acknowledgements The first author would like to acknowledge the financial support received from the Conselho Nacional de Desenvolvimento Cientifico e Tecnologico, CNPq, of the Ministry for Science and Technology of Brazil. References Balas, J., Sladek, J. and Sladek, V., 1989, Stress Analysis by Boundary Element Methods, Studies in Applied Mechanics 23, Elsevier. Brebbia, C. A., Telles, J. C. F., and Wrobel, L. C., 1984, Boundary Element Techniques. Theory and Applications in Engineering, Springer- Verlag, Berlin. Brebbia, C. A. and Dominguez, J., 1989, Boundary Elements. An Introductory Course, Computational Mechanics Publications, Southampton, and McGraw-Hill Book Company, New York. Burnett, D. S., 1987, Finite Element Analysis, Addison-Wesley Publising Company. Cowin, S. C., 1989, Properties of the Anisotropic Elasticity Tensor, em Quarterly Journal of Mechanics and Applied Mathematics, Vol.42, Part 2, pp Desai, C. S. and Abel, J. F., 1972, Introduction to the Finite Element Method, Van Nostrand Reinhold Company. Guiggiani, M. and Gigante, A., 1990, A General Algorithm for Multidimensional Cauchy Principal Value Integrals in the Boundary Element Method, Transactions of the ASME, Journal of Applied Mechanics, Vol.57, pp

15 Boundary Elements 103 Kinoshita, N. and Mura, T., Green's Functions for Anisotropic Elasticity, Department of Civil Engineering and Materials Research Center, Northwestern University, Evanston, Illinois. Lekhnitskii, S. G., 1963, Theory of Elasticity of an Anisotropic Elastic Body, Holden-Day, Inc., San Francisco. Lekhnitskii, S. G., 1968, Anisotropic Plates, Gordon And Breach Science Publishers, Inc., New York. Mikhlin, S. G., 1962, Singular Integral Equations, American Mathematical Society Transactions, series 1, iv, pp Mikhlin, S. G., 1970, Mathematical Physics, an Advanced Course, North- Holland Series in Applied Mathematics and Mechanics, Vol.11, North-Holland Publishing Company. Perez, M. M. and Wrobel, L. C., 1992, A General Integral Equation Formulation for Homogeneous Orthotropic Potential Problems. Engineering Analysis with Boundary Elements, Vol.10, No.4, pp Perez, M. M. and Wrobel, L. C., 1993, The Use of Isotropic Fundamental Solutions for Heat Conduction in Anisotropic Media, International Journal of Numerical Methods for Heat and Fluid Flow, Vol.3, No.l, pp Rizzo, F.J. and Shippy, D.J., 1970, A Method for Stress Determination in Plane Anisotropic Elastic Bodies, Journal of Composite Materials, Vol.4, pp Shi, G., 1990, Boundary Element Method in Bending and Eigenvalue Problems of Anisotropic Plates, Boundary Elements in Mechanical and Electrical Engineering, Computational Mechanics Publications, Southampton, and Springer Verlag, Berlin. Snyder, M. D. and Cruse, T. A., 1975, Boundary-Integral Equation Analysis of Cracked Anisotropic Plates, International Journal of Fracture, Vol.11, No.2. Telles, J. C. F. and Brebbia, C. A., 1979, On the Application of the Boundary Element Method to Plasticity, Research Note, Applied Mathematical Modelling, Vol.3, pp Telles, J. C. F. and Brebbia, C. A., 1980, The Boundary Element Method in Plasticity, New Developments in Boundary Element Method, C. A. Brebbia (ed.), pp , Computational Mechanics Ltd., Southampton. Vogel, S.M. and Rizzo, F.J., 1973, An Integral Equation Formulation of Three Dimensional Anisotropic Elastostatic Boundary Value Problems, Journal of Elasticity, Vol.3, No.3, pp Wilson, R. B. and Cruse, T. A., 1978, Efficient Implementation of Anisotropic Three Dimensional Boundary-Integral Equation Stress Analysis, International Journal for Numerical Methods in Engineering, Vol.12, pp