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**Active macro-zone approach for incremental elastoplastic-contact analysis.**
*(English)*
Zbl 1352.74200

Summary: The symmetric boundary element method, based on the Galerkin hypotheses, has found an application in the nonlinear analysis of plasticity and in contact-detachment problems, but both dealt with separately. In this paper, we want to treat these complex phenomena together as a linear complementarity problem.{

}A mixed variable multidomain approach is utilized in which the substructures are distinguished into macroelements, where elastic behavior is assumed, and bem-elements, where it is possible that plastic strains may occur. Elasticity equations are written for all the substructures, and regularity conditions in weighted (weak) form on the boundary sides and in the nodes (strong) between contiguous substructures have to be introduced, in order to attain the solving equation system governing the elastoplastic-contact/detachment problem. The elastoplasticity is solved by incremental analysis, called for active macro-zones, and uses the well-known concept of self-equilibrium stress field here shown in a discrete form through the introduction of the influence matrix (self-stress matrix). The solution of the frictionless contact/detachment problem was performed using a strategy based on the consistent formulation of the classical Signorini equations rewritten in discrete form by utilizing boundary nodal quantities as check elements in the zones of potential contact or detachment.

}A mixed variable multidomain approach is utilized in which the substructures are distinguished into macroelements, where elastic behavior is assumed, and bem-elements, where it is possible that plastic strains may occur. Elasticity equations are written for all the substructures, and regularity conditions in weighted (weak) form on the boundary sides and in the nodes (strong) between contiguous substructures have to be introduced, in order to attain the solving equation system governing the elastoplastic-contact/detachment problem. The elastoplasticity is solved by incremental analysis, called for active macro-zones, and uses the well-known concept of self-equilibrium stress field here shown in a discrete form through the introduction of the influence matrix (self-stress matrix). The solution of the frictionless contact/detachment problem was performed using a strategy based on the consistent formulation of the classical Signorini equations rewritten in discrete form by utilizing boundary nodal quantities as check elements in the zones of potential contact or detachment.

### MSC:

74M15 | Contact in solid mechanics |

74B05 | Classical linear elasticity |

74C05 | Small-strain, rate-independent theories of plasticity (including rigid-plastic and elasto-plastic materials) |

74S15 | Boundary element methods applied to problems in solid mechanics |

### Software:

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\textit{F. Cucco} et al., Int. J. Numer. Methods Eng. 96, No. 7, 425--447 (2013; Zbl 1352.74200)

### References:

[1] | Telles, The Boundary Element Method applied to Inelastic Problems (1983) · Zbl 0533.73076 |

[2] | Beer, An efficient numerical method for modelling initiation and propagation of cracks along material interfaces, International Journal for Numererical Methods in Engineering 36 (21) pp 3579– · Zbl 0794.73075 |

[3] | Martin, Boundary element analysis of two-dimensional elastoplastic contact problems, Engineering Analysis with Boundary Elements 21 pp 349– (1998) · Zbl 0957.74069 |

[4] | Aliabadi, Boundary element hyper-singular formulation for elastoplastic contact problem, International Journal for Numererical Methods in Engineering 48 pp 995– (2000) · Zbl 0974.74072 |

[5] | Polizzotto, Shakedown of elastic-plastic solids with frictionless unilateral contact boundary conditions, International Journal Mechanics Sciences 39 pp 919– (1997) · Zbl 0908.73022 |

[6] | Karami, Boundary element analysis of elasto-plastic contact problems, Compututer and Structures 41 pp 927– (1991) · Zbl 0850.73352 |

[7] | Gun, Boundary element analysis of 3-D elasto-plastic contact problems with friction, Computers and Structures 82 pp 555– (2004) |

[8] | Zito, Incremental elastoplastic analysis for active macro-zones, International Journal for Numererical Methods in Engineering 91 pp 1365– (2012) |

[9] | Panzeca, Lower bound limit analysis by BEM: convex optimization problem and incremental approach, Engineering Analysis with Boundary Elements 37 pp 558– (2013) · Zbl 1297.74152 |

[10] | Gao, An effective boundary element algorithm for 2D and 3D elastoplastic problem, International Journal of Solids and Structures 37 pp 4987– (2000) · Zbl 0970.74077 |

[11] | Gao, Boundary element programming in mechanics (2002) |

[12] | Panzeca, Computational aspects in 2D SBEM analysis with domain inelastic actions, International Journal for Numerical Methods in Engineering 82 pp 184– (2010) · Zbl 1188.74077 |

[13] | Zito, On the computational aspects of a symmetric multidomain BEM approach for elastoplastic analysis, Journal of Strain Analysis for Engineering Design 46 pp 103– (2011) |

[14] | Signorini, Sopra alcune questioni di elastostatica, Atti della SocietĂ Italiana per il Progresso della Scienza 2 pp 231– (1993) |

[15] | Panzeca, The symmetric Boundary Element Method for unilateral contact problems, Computer Methods in Applied Mechanics and Engineering 197 pp 2667– (2008) · Zbl 1194.74500 |

[16] | Zhang, The boundary element-linear complementarity method for the Signorini problem, Engineering Analysis with Boundary Elements 36 pp 112– (2012) · Zbl 1245.74047 |

[17] | Salerno, Frictionless contact-detachment analysis: Iterative linear complementarity and quadratic programming approaches, Computational Mechanics 51 pp 553– (2013) · Zbl 1312.74020 |

[18] | Polizzotto, An energy approach to the boundary element method. Part I: Elastic solids, Computer Methods in Applied Mechanics and Engineering 69 pp 167– (1988) · Zbl 0629.73069 |

[19] | Polizzotto, An energy approach to the boundary element method. Part II: Elastic-plastic solids, Computer Methods in Applied Mechanics and Engineering 69 pp 263– (1988) · Zbl 0629.73070 |

[20] | Panzeca, Macro-elements in the mixed boundary value problems, Computational Mechanics 26 pp 437– (2000) · Zbl 0993.74078 |

[21] | Terravecchia, Revisited mixed-value method via symmetric BEM in the substructuring approach, Engineering Analysis with Boundary Elements 36 pp 1865– (2012) · Zbl 1351.74139 |

[22] | Simo, Computational Inelasticity (1998) |

[23] | Ortiz, Symmetry-preserving return mapping algorthms and incrementally extremal paths: a unification of concepts, International Journal for Numerical Methods in Engineering 28 pp 1839– (1989) · Zbl 0704.73030 |

[24] | Bilotta, A high-performance element for the analysis of 2D elastoplastic continua, Computer Methods in Applied Mechanics and Engineering 196 pp 818– (2007) · Zbl 1120.74804 |

[25] | Panzeca, Symmetric Boundary Element Method versus Finite Element Method, Computer Methods in Applied Mechanics and Engineering 191 pp 3347– (2002) · Zbl 1101.74370 |

[26] | Panzeca, Domain decomposition in the symmetric boundary element method analysis, Computational Mechanics 28 pp 191– (2002) · Zbl 1076.74568 |

[27] | Bonnet, Regularized direct and indirect symmetric varational BIE formulation for three-dimensional elasticity, Engineering Analysis with Boundary Elements 15 pp 93– (2005) |

[28] | Frangi, Symmetric BE method in two-dimensional elasticity: evaluation of double integrals for curved elements, Computational Mechanics 19 pp 58– (1996) · Zbl 0888.73069 |

[29] | Terravecchia, Closed form coefficients in the symmetric boundary element approach, Engineering Analysis with Boundary Elements 30 pp 479– (2006) · Zbl 1195.74263 |

[30] | Holzer, How to deal with hypersingular integrals in the symmetric BEM, Communication Numerical Methods in Engineering 9 pp 219– (1993) · Zbl 0781.65091 |

[31] | Bui, Some remarks about the formulation of three-dimensional thermoelastoplastic problems by integral equation, International Journal of Solids and Structures 14 pp 935– (1978) · Zbl 0384.73008 |

[32] | Cucco, The program Karnak.sGbem Release 2.1 (2002) |

[33] | Perez-Gavilan, A symmetric Galerkin BEM for Multi-connected bodies: A new apprach, Engineering Analysis with Boundary Elements 25 pp 633– (2001) · Zbl 1065.74631 |

[34] | Freddi, Symmetric Galerkin BEM for bodies with unconstrained contours, Computer Methods in Applied Mechanics and Engineering 195 pp 961– (2006) · Zbl 1121.74060 |

[35] | Vodicka, On the Removal of the non-uniqueness in the solution of the elastostatic problems by symmetric Galerkin BEM, International Journal of Solids and Structures 66 pp 1884– (2006) |

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