A direct formulation for three-dimensional elastoplastic boundary elements A. P. Cisilino*,**, M. H. Aliabadi* and J. L. Otegui** *Wessex Institute of Technology, As hurst Lodge, As hurst, Southampton SO40 7AA, UK **INTEMA, UniversidadNational de Mar del Plata -CONICET J.B. Justo 4302, (7600) Mar del Plata, Argentina Abstract In this work an efficient BEM formulation for three-dimensional elastoplasticity is presented. Particular attention is paid to accurate evaluation of the Cauchy Principal Value volume integrals. Nine-noded quadrilateral elements and 27-noded cubic (brick) cells are used for the surface and volume discretization. An explicit initial strain formulation is used to satisfy the non-linearity. The accuracy of the proposed method is demonstrated by solving a number of benchmark problems. Introduction Boundary Element Method (BEM) is now well established for the solution of linear problems and some nonlinear problems in engineering}!]. The application of the boundary element method to two-dimensional plasticity is well documented for example see [2], [3], [4], [5], [6], [7] and [8], but there are not many papers published on its application to three-dimensional problems[9]. One of the main difficulties in the application of BEM to nonlinear problems such as plasticity is accurate integration techniques for higher order discretization. The principal difficulty in this case is the accurate evaluation of the Cauchy principal value volume integrals when computing the interior stresses. The evaluation of these integrals mostly relies on analytical (see Matsumoto &c Yuuki [6]) or semi-analytical integration (see Telles & Brebbia [4]). Unfortunately these techniques are applicable only in plane problems using piecewise constant or, at most, linear cells, since higher order formulations produce very complex integrals. A more general indirect method based on the application of a constant inelastic strainfieldwas proposed by Telles & Brebbia [3], however this procedure has never been used in actual computations. The most general procedure is the direct evaluation of the Cauchy principal value integrals. This technique was first implemented for two-dimensional elastoplasticity problems by Leitao et al. [7]. In this paper an efficient implementation of the BEM for three dimensional elastoplasticity is presented. The Cauchy principal value integrals for the volume cell integration are evaluated using a Taylor series expansion method. Boundary stresses are evaluated by differentiating the nodal displacements. The explicit version of the initial strain formulation is used to deal with the non-linear characteristics of the problem, which is solved by an iterative load-incremental algorithm.
170 Boundary Elements Boundary Integral Formulation Plasticity, as far as this work is concerned, is a time-independent phenomenon. However it may be associated to a time-like parameter, the loading factor, and that is the reason why the rate notation is used. This rate denotes the current value of the variable, that is the former value plus the component resulting from the current load increment. The adoption of an initial strain formulation for three-dimensional inelastic problems leads to (see for example ref. [1]) ajin = [ u*jpj(x)dt - I pljiijdt + / cr/we^dn (1) Jr Jr Jci where tij, PJ and e?^ are the displacement rate, traction rate and initial plastic strain rate, respectively; u*^ and p\- are the fundamental displacement and traction, and the fundamental stressfieldo*-^ accounts for the influence of the plastic strains. Equation (1) is assumed valid for any location of the load point (interior or boundary points) provided Cij and the second boundary integral on the right-hand side are properly interpreted [1]. Within the context of small strain theory, the total strain rate for elastoplastic problems is assumed to be represented by e«= 5(*U+6j,0 = *&+*& (2) where i\^ and e^- are respectively the elastic and plastic parts of the total strain rate tensor. If inelastic strains are considered as initial strains, the application of Hooke's law to the elastic part of the total strain rate tensor results in the following expression for the stress rate components where the plastic volumetric deformation is considered incompressible. Symbols G and v stand for the shear modulus and the Poisson's ratio respectively. Under these assumptions, the stress rates at interior points can be computed using expressions (2) and (3), together with derivatives of expression (1) (3) o\, = / u^kpkdt - I pl^uk(x)dt + / atjm&udsl Jr Jr Jn 9/nr - _ K7 - &/)<% + (1 + 5i/)eSft,-l (4) where u^ and t*^ and cr^/ contain several derivatives of the fundamental solutions ujj and tlj and a*-^ repectevely, together with elastic constants. The limit of the integral equation (4) when the source point goes to the boundary can be taken to compute the stress rates at boundary points. However, this procedure is computationally expensive, because of the occurrence of hyper-singular integrands in the boundary integrals which must be treated in a Cauchy principal value sense. To overcome this problem, boundary stress rates are computed in this work from the boundary traction and displacementfields,following a procedure devised by Telles and Brebbia[3]. Numerical Aspects In order to solve the system of integral equations presented in the previous sections, the boundary F and the part of the domain Q that is likely to yield must be discretized.
Boundary Elements 171 In this work the boundary discretization is done using 9-noded continuous isoparametric quadrilateral elements, while for the domain discretization 27-noded isoparametric volume cells (bricks) are used. The matrix representation of the discretized boundary displacements integral equation (1), can be written as follows: Hu = Gp + D P (5) where the matrices H, G and D contain integrals involving the kernel functions p\^ u*j and <jjfa repectevely; vectors u, consist of all nodal displacements and traction components on the boundary and and vector P accounts for the plastic strains at the domain nodes. Rearranging equation (5) according to the boundary conditions, results Ay = t + D^ (6) where vector y contains the boundary unknowns and vector i the contributions of the prescribed values on the boundary. The same procedure can be applied to the internal stress equation (4), yielding 6 = - A'y + t+e& (7) where & is vector containing the stress rates at internal points and matrix E contain integrals involving the kernel function o\-^ Once all the matrices appearing in equations (6) and (7) have been identified an initial strain explicit iterative procedure is employed to solve the problem. The solution strategy can be described briefly as follows: stresses associated with nodes in the discretized domain 17, are obtained assuming an elastic response, i.e. ^ = 0, and load tofirstyield is determined for the most stressed point; a new load level A is found by adding to the previous one a percentage of the load atfirstyield. The stresses associated to this load level are obtained using equations (6) and (7) in its incremental form: Ay =A»f + D( P+A P) (8) 6 = -A'y + \f 4- E( P+A<P) (9) where i? stores the plastic strains up to but not including the current load increment. the procedure continues with the iterative determination of the plastic strain increment, A *\ using expressions (8) and (9), together with the well-known Prandtl- Reuss flow rule. In each iteration A ** is compared with its previous value and if it falls within a prescribed tolerance the process has converged for the particular node. all nodes must be monitored and only when convergence is achieved for all of them another load increment is added. The incremental process is repeated until the prescribed load value is achieved.
172 Boundary Elements 0(8,(p,G) (b) Figure 1: Evaluation of strongly-singular integrals. Coordinate transformation and integration limits Treatment of the Integrals One of the typical and most significant issues in the implementation of BEM elastoplastic codes is the evaluation of the singular integrals appearing in both the boundary and domain integral equations. Integrals in this work present singularities of the form 1/r**, where r is the distance from the collocation point to the element or cell being integrated and n depends on the nature of the kernel. According to the relative position of the collocation point with respect to the element or cell being integrated, four types of integrals can be identified: Regular or non-singular integrals This is the case for all kernels when the collocation point does not lie on the integration element or cell. Integrals are in this case evaluated using the standard Gauss quadrature with an element subdivision technique. The number of subdivisions depends upon the relative proximity of the collocation point and the integration domain. Weakly singular (removable singularity) This kind of integral appears when evaluating the integrals involving the kernels u* (boundary integration) and a* (domain integration). These singularities of order 1/r for u* and 1/7-2 fo,. ^* &re removed using a transformation of variable developed by Lachat and Watson [11]. In the case of boundary integration the procedure consists in splitting the element in a number of triangles which are mapped into squares using appropriate transformation Jacobians. These Jacobians exactly cancel out the 1/r singularity (see [12]), and standard Gaussian quadrature is then applied to the squares. An analogous
Boundary Elements 173 procedure is employed for the domain integrals, where the integration cell is divided into pyramid-shaped subcells and then mapped into cuboids. The resulting transformation Jacob ian, proportional to r\ exactly cancels out the 1/r* singularity. Strongly Singular The kernel p* which is integrated on the model boundary presents a strong singularity proportional to l/r%. The transformation of variable previously described for the case of the weakly singular integrals is used to weaken the 1/r* singularity in order to improve the accuracy of the so called off-diagonal terms, i.e. when the collocation point f is on the integration element but the shape function $>* = 0 at f. In this case the numerical evaluation of the strongly singular kernels presents no difficulty as the shape function effectively weakens the singularity further. The highly singular traction integral when $>* = 1 at is found by rigid body considerations[13). Cauchy principal value volume integrals Volume integrals involving the kernels E* present a singularity proportional to l/r^. These integrals are evaluated applying a method proposed by Guiggiani and Gigante [14] which transforms them in a sum of regular integrals to be evaluated using standard quadrature formulae. This method when applied to volume integration transforms the singular integrals into a sum of a regular three-dimensional integral and a regular twodimensional one. The original form of the integral is after using the limit notation = / 7n = lim Z'ePdtl (10) where 17«is a neighbourhood of radius s of the collocation point, seefigure l(a). To achieve the desired regularization, two steps are needed. The first one is the introduction of a spherical coordinates system (p, 0, </?) centred at the collocation node and the second one is the introduction of a Taylor series expansion about the collocation node for the components of the distance r and its derivatives (see Aliabadi et al. [10]). The coordinate transformation leads to the following form for expression (10) where for the sake of simplicity and accuracy when determining integration limits, integral evaluation is carried after dividing cell is into Np pyramid shaped subcells depending on the position of the collocation node, as shown in figure l(b). In expression (11) the integrand function F** stands for the product of the kernel by all the other terms namely, shape functions, Jacobians of both transformations and the initial strain tensor. The terms 0, 0J> <Pi and (p% denote the initial and final angles and a" and p" the initial and final radii, which depend on 9 and (p. Notice that the integrand function F is of O(r)~* since the original singularity is O(r)~^ of the kernel has been weakened by the well known spherical-coordinates transformation Jacobian, which is proportional to r*. The use of the Taylor series expansion about the collocation node for the components of the distance r and its derivatives leads to the definition of a function /"(#,< >), which when divided by p has the same asymptote as the integrand function F (p,0,y?) when p»> 0. In this case the remaining singularity of O(r)~* is completely separated from the rest of the integral. Consequently, the original integral in polar coordinates (11) can be represented by: (11)
174 Boundary Elements Figure 2: Thick pressurized cylinder. Model geometry and discretization. where the asymptotic function /"(#,%>) was subtracted and added back. The behaviour of the auxiliary function / (#,</?) in the vicinity of the singularity makes possible the regularization of the integral After performing the limiting process the final expression for the integral is: (12) where j3 (0) is the asymptotic expression of a"(s, (13) Examples Thick pressurized cylinder Thisfirstexample is a thick pressurized cylinder of inner radio a = 100mm and outer radio b = 200mm as shown infigure2. Due to the symmetry of the problem only one quarter is modelled. Appropriate boundary conditions are also applied in order to make the analysis under a plane strain condition in the z direction. The boundary surface is discretized using 46, 9-noded elements, while the volume is only partially discretized up to 60% of the cylinder thickness, using 10 internal cells (hatched volume infigure2). The constitutive model is considered as elastic-perfectly plastic. The material data is: modulus of elasticity, E = 120GPa, Poisson's ratio, v = 0.3, yield stress, Oyieid = 240MPa, and plastic strain hardening parameter* H' = 0. The internal pressure, p = 181.5MPa, is applied in 15 load increments after load to first yield is determined. This final value of pressure corresponds to the one that makes 60% of the cylinder thickness go under elastoplastic regime. *The parameter H' is the slope of the uniaxial stress-strain diagram of the material when replotted as stress versus plastic strain.
Boundary Elements 175 Numerical results are compared with those obtained analytically by Prager and Hodge [15]. In figure 3, the internal and external radial displacements, together with the pressure, are plotted as function of the radius of elastic-plastic boundary p, while in figure 4, pressure is plotted as function of the external radial displacement. The stress distribution for a position p/a 1.6 of the elastic-plastic boundary is shown in figure 5 for internal and boundary nodes. Although the general agreement between the results is very good, some deviations are found ( note for example the OQQ stress component at boundary nodes in figure 5). This deviations can be attributed to the inaccuracy when determining the exact position of the elastic-plastic boundary, since its position as well as the applied load are given by discrete increments. 2Gu(a)/aOy 2Gu(b)/aa BEM results.00 1.20 1.40 1.60 1.80 2.00 p/a Figure 3: Internal and external radial displacements and pressure vs the radious of theelastic boundary Analytical solution BEM 2Gu(b)/aa yield Figure 4: Internal pressure vs externalradialdisplacement Perforated strip In this example the problem of a perforated strip of strain-hardening material is analysed. The geometry of the problem as well as its discretization are illustrated in figure 6. Due to the symmetry only a quarter of the problem is discretized using 122, 9-noded boundary elements and 25 internal cells (hatched volume infigure6). An uniaxial tension a = 140MPa is applied in 13 equal load increments afterfirstyield. The material properties are: elastic modulus, E = 70<7Pa, Poisson's ratio, v = 0.2, yield stress, oyield = 243MPa and plastic strain hardening modulus, H' = 2240MPa. In figure 7 stress-displacement plots for some key positions on the strip (see figure 6) are shown. The BEM results are compared against FEM ones. FEM results were obtained using MYSTRO-LUSAS package with 20-noded brick elements and the same discretization that in the BEM analysis. All curves shown infigure7 exhibit very good agreement between FEM and BEM results. In figure 8 stress component o^x at the net section of the strip at a load level a = 0.47(7yie/d is plotted. Results are compared to the ones obtained by the FEM analysis and experimentally by Theocaris and Marketos[16], Again close agreement is found between the FEM and BEM results, but both show some deviation from experimental.
176 Boundary Elements Boundary nodes r/a Figure 5: Stress distribution for boundary and domain nodes (p/a=1.6) r 0100mm - 360mm Figure 6: Perforated strip. Model geometry and discretization
Boundary Elements 177 060 060 050 050 040 030 040 o>.5, 030 0.20 020 010 0.00 0.00 0.25 0.50 0.75 1 00 1.25 150 (u/r)x100 0.00 000 025 050 075 (v/r)x100 Figure 7: Perforated strip. Displacements at positions A, B, C and D However, this deviation is acceptable if it is considered that in both models a rather coarse discretizationcan was used. Conclusions In this paper a direct elastoplastic formulation for three-dimensional elastoplastic boundary elements has been presented. The formulation uses the displacement boundary equation to describe the problem boundary. Internal stresses are computed directly from the integral equation while boundary stresses are calculated from the boundary displacement solution using numerical differentiation. Initial strain formulation was used to satisfy the nonlinearity. Singular kernels arising in the surface and volume integrals are in all cases regularized applying suitable coordinate transformations. The Cauchy-principal-value volume integrals were evaluated using a Taylor series expansion method. The formulation was successfully employed to solve benchmark problems. Acknowledgements Authors wish to express his thanks for funding provided by the Consejo Nacional de Investigaciones Cientificas y Tecnicas de la Republica Argentina (CONICET) and the Foreign and Commonwealth Office of the United Kingdom. Part of this project was founded by the European Commission INCO-DC program, project HIPSIA. The partners in the project are Wessex Institute of Technology, COPPE (Brazil), INTEMA (Argentina) and CESGA (Spain).
178 Boundary Elements 1.4 1.2 1.0 0.8 D 0.6 0.4 0.2 Experimental (Theokaris & Marketos) FEM BEM 0.0 1.0 1.1 1.2 1.3 1.5 y/r 1.6 1.7 1.8 1.9 2.0 References Figure 8: Perforated strip. Stress variation on net section of plate [1] Brebbia C.A., Telles J.C.F and Wrobel,C.A., Boundary Element Techniques, Springer-Verlag, (1992) [2] Mukherjee S., Corrected boundary integral methods in planar thermoelastoplasticity, Int. J. Solids and Structures, 13, 331-335 (1997) [3] Telles J.C.F. and Brebbia, C.A., On the application of the boundary element method to plasticity, Appl Math. Modelling, 3, 466-470, (1979) [4] Telles J.C.F., The Boundary Element Method Applied to Inelastic Problems, Springer-Verlag, Berlin, (1983) [5] Banerjee P.K. and Cathie D.N., A direct formulation and numerical implementation of the boundary element method for two-dimensional problems of elasto-plasticity, Int. J. Mech. Sci, 22, 233-245 (1980) [6] Matsumoto T., and Yuuki R., Accurate boundary element analysis of twodimensional elasto-plastic problems, Boundary Element Methods in Applied Mechanics, M. Tanaka and T.A. Cruse, eds., Pergamon Press, Oxford, pp.205-214, (1988) [7] Leitao V.M., Aliabadi M.H. and Rooke D.P., The dual boundary element formulation for elastoplastic fracture mechanics, Int.J.Numer. Methods for Eng., 38, 315-333 (1995) [8] Corradi S., Aliabadi M.H. and Marchetti M., A variable stiffness dual boundary element method for mixed-mode elastoplastic crack problems, Theoretical and Applied Fracture Mechanics, 25, 43-49 (1996) [9] Gupta A., Delgado H.E. and Sullivan J.M., A three-dimensional BEM solution for plasticity using regression interpolation within the plasticfield,int. J. Numer. Methods for Eng, 33, 1997-2014, (1992) [10] Aliabadi M.H., Hall W.S. and Phemister T.G., Taylor expansions for singular kernels in the boundary element method, Int. J. Numer. Methods for Eng., 21, 2221-2236 (1985)
Boundary Elements 179 [11] Lachat J.C. and Watson J.O., Effective numerical treatment of boundary integral equations: A formulation for three-dimensional elastostatics, Int. J. Numer. Methods for Eng, 10,, 991-1005, (1976) [12] Aliabadi M.H., Hall W.S. and Hibbs T.T., Exact cancelling for the potential kernel in the boundary element method, Comm. Appl. Num. Methods in Engng, 3, 123-128 (1987) [13] Brebbia C.A. and Dominguez J., Boundary Elements: An Introductory Course, Computational Mechanics Publications & Me Graw-Hill Company, (1992) [14] Guiggiani M. and Gigante A., A general algorithm for multidimensional Cauchy principal value integrals in the boundary element method, Journal of Applied Mechanics, 57, 906-915, (1990) [15] Prager W. and Hodge P.G., Theory of Perfectly Plastic Solids, Dover Publications,^!) [16] Theocaris P.S. and Marketos E., Elastic-plastic analysis of perforated thin strips of a strain-hardening material, J. Mech. Phys. Solids, 12, 370-390 (1964).