Active macro-zone approach for incremental elastoplastic-contact analysis

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

Elastoplastic analysis by active macro-zones with linear kinematic hardening and von Mises materials

In this paper a strategy to perform elastoplastic analysis with linear kinematic hardening for von Mises materials under plane strain conditions is shown. The proposed approach works with the Symmetric Galerkin Boundary Element Method applied to multidomain problems using a mixed variables approach, to obtain a more stringent solution. The elastoplastic analysis is carried out as the response to the loads and the plastic strains, the latter evaluated through the self-equilibrium stress matrix. This matrix is used both, in the predictor phase, for trial stress evaluation and, in the corrector phase, for solving a nonlinear global system which provides the elastoplastic solution of the active macro-zones, i.e. those zones collecting bem-elements where the plastic consistency condition has been violated. The simultaneous use of active macro-zones gives rise to a nonlocal approach which is characterized by a large decrease in the plastic iteration number, although the proposed strategy requires the inversion and updating of Jacobian operators generally of big dimensions. A strategy developed in order to reduce the computational efforts due to the use of this matrix, in a recursive process, is shown

Frictionless contact formulation by mathematical programming technique

The object of the paper concerns a consistent formulation of the classical Signorini's theory regarding the frictionless unilateral contact problem between two elastic bodies in the hypothesis of small displacements and strains. A variational approach, employed within the symmetric Boundary Element Method, leads to an algebraic formulation based on nodal quantities. The contact problem is decomposed into two sub-problems: one is purely elastic, and the other pertains to the unilateral contact condition alone. Following this methodology, the contact problem, faced with symmetric BEM, is characterized by symmetry and sign definiteness of the coefficient matrix, thus admitting a unique solution. The solution of the frictionless unilateral contact problem can be obtained - through a step-by-step analysis utilizing generalized quantities as check elements in the zones of potential contact or detachment. Indeed, the detachment or the contact phenomenon may happen when the weighted traction or the weighted displacement is greater than the weighted cohesion or weighted minimum reference gap, respectively; - through a quadratic programming problem based on the minimum of the total potential energy. In the example, given in the paper, the detachment phenomenon is considered and some comparisons of the solution between the step-by-step analysis and the direct approach which utilizes the quadratic programming will be shown

Multidomain SBEM analysis for two dimensionalelastoplastic-contact problems

The Symmetric Boundary Element Method based on the Galerkin hypotheses has found application in the nonlinear analysis of plasticity and contact-detachment problems, but dealt with separately. In this paper we wants to treat these complex phenomena together. This method works in structures by introducing a subdivision into sub-structures, distinguished into macroelements, where elastic behaviour is assumed, and bem-elements, where it is possible for plastic strains to occur. In all the sub-structures, elasticity equations are written and regularity conditions in weighted (weak) form and/or in nodal (strong) form between boundaries have to be introduced, to attain the solving equation system

Displacements approach with external variables only for multi-domain analysis via symmetric BEM

In the present paper a new displacement method, defined as external variables one, is proposed inside the multidomain symmetric Boundary Element formulation. This method is a natural evolution of the displacement approach with interface variables in the multidomain symmetric BEM analysis. Indeed, the strategy employed has the advantage of considering only the kinematical quantities of the free boundary nodes and the algebraic operators involved show symmetry and very small dimensions. The proposed approach is characterized by strong condensation of the mechanical and kinematical boundary nodes variables of the macro-elements. All the domain quantities, such as tractions and stresses, displacements and strains, are computed through the Somigliana Identities in a subsequent phase. Some examples are shown using the calculus code Karnak.sGbem, by which it was possible to make some comparisons with analytical solutions andothe rapproaches to show the effectiveness of the method propose

Active macro-zones algorithm via multidomain SBEM for strain-hardening elastoplastic analysis

In this paper a strategy to perform strain-hardening elastoplastic analysis by using the Symmetric Boundary Element Method (SBEM) for multi-domain type problems is shown. The procedure has been developed inside Karnak.sGbem code by introducing an additional module

Frctionless contact: step by step analysis and mathematical programming technique

The object of the paper concerns a consistent formulation of the classical Signorini's theory regarding the frictionless unilateral contact problem between two elastic bodies in the hypothesis of small displacements and strains. A variational approach employed in conjunction with the Symmetric Boundary Element Method (SBEM) leads to an algebraic formulation based on generalized quantities [1]. The contact problem is decomposed into two sub-problems: one is purely elastic, the other pertains to the unilateral contact conditions alone [2,3]. Following this methodology, the contact problem, by symmetric BEM, is characterized by symmetry and sign definiteness of the coefficient matrix, thus admitting a unique solution. The solution of the frictionless unilateral contact problem has been obtained: • by means of a quadratic programming problem [2], as optimization problem developed in terms of discrete variables, by using Karnak.sGbem code [4] coupled with MatLab. • through a step by step analysis by using nodal quantities as the check elements. Indeed the detachment or contact phenomenon occurs when the traction or the displacement is greater than the cohesion or reference gap, respectively [3]. The innovative approach is given meanly by the only boundary discretization by using the SBEM approach, by the elastic relation written for each bem-e involving the only quantities of the contact zone. In the examples some comparisons of the two strategies will be shown

Internal spring distribution for quasi brittle fracture via Symmetric Boundary Element Method

In this paper the symmetric boundary element formulation is applied to the fracture mechanics problems for quasi brittle materials. The basic aim of the present work is the development and implementation of two discrete cohesive zone models using Symmetric Galerkin multi-zone Boundary Elements Method. The non-linearity at the process zone of the crack will be simulated through a discrete distribution of nodal springs whose generalized (or weighted) stiffnesses are obtainable by the cohesive forces and relative displacements modelling. This goal is reached coherently with the constitutive relation that describes the interaction between mechanical and kinematical quantities along the process zone. The cracked body is considered as a solid having a “particular” geometry whose analysis is obtainable through the displacement approach employed in (Panzeca et al., 2000; 2002-b) by some of the present authors in the ambit of the Symmetric Galerkin Boundary Elements Method (SGBEM). In this approach the crack edge nodes are considered distinct and the analysis is performed by evaluating all the equation system coefficients in closed form (Guiggiani, 1991; Gray, 1998; Panzeca et al., 2001; Terravecchia, 2006)