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A 2D Boundary Element Method (BEM) Formulation to obtain the constitutive response of heterogeneous materials based on the Representative Volume Element (RVE) concept is developed to deal with dissipative phenomena over the domain and the phase debonding on the interface zone. Despite the formulation is written within a multiscale framework, in this work the analyses are performed only at microstructure level, but if a full coupled multiscale analysis is performed the computational cost becomes very expensive. Usually this kind of modelling is performed considering the Finite Element Method (FEM), but it is important to find other numerical solutions which are less expensive, but still accurate. To analyse metal matrix composites the RVEs are assumed to contain a ductile matrix, rigid inclusions and an interface zone where the phase debonding is modelled by a cohesive fracture model using embedded cohesive contact finite elements into the boundary element mesh. The RVE domain is discretized into cells to compute the domain integral written in terms of the inelastic forces, which have a part related to the plasticity phenomenon and another one referred to the cohesive-contact problem. The homogenized results are compared to a FEM model in order to validate the proposed formulation. [ABSTRACT FROM AUTHOR]

Abstract A boundary element formulation to obtain the constitutive response of heterogeneous microstructures is proposed, considering a RVE-based multiscale theory. The sub-region technique is adopted to model the RVE (Representative Volume Element), where voids and inclusions can be defined inside the matrix and different elastic properties and constitutive models can be assumed for each phase. Triangular cells are adopted to approximate the dissipative forces over the RVE domain, where they are assumed constant. After a strain vector is imposed to the microstructure boundary, its elastic forces field is obtained and the respective strain field computed. Then, according to the RVE-based multiscale theory, the RVE equilibrium problem must be solved what requires an iterative procedure to find the displacement fluctuations field that auto-equilibrates the microstructure stress field. After the RVE solution, the boundary tractions must be recalculated to consider the displacement fluctuations field as well as the dissipative forces. Then, from the boundary tractions the homogenized stress vector can be computed. The homogenized constitutive tensor is evaluated considering the constitutive tensors of all cell nodes and also the consistent tangent operator. To validate the proposed model, the homogenized responses of different microstructures are compared to the responses obtained via finite element model. [ABSTRACT FROM AUTHOR]

A BEM formulation to perform elastic analysis of heterogeneous microstructures is proposed in the context of multi-scale analysis. The microstructure, also denoted as RVE (Representative Volume Element), is modeled as a zoned plate where voids or inclusions can be considered inside a matrix. Thus, each sub-region represents either the matrix or an inclusion, where different values of Poisson's ratio and Young's modulus can be defined. The RVE equilibrium equation is solved in terms of displacement fluctuations according to the formulation proposed in de Souza Neto and Feijóo (2006). Only elastic behavior is considered for matrix and inclusions, although the proposed model can be extended to consider dissipative phenomena. To make the micro-to-macro transition necessary in a multi-scale analysis, the homogenized values for stress and constitutive tensor have to be computed adopting homogenization techniques. Some numerical examples of heterogeneous microstructures are presented and compared to a FEM model to show the accuracy of the proposed model. [ABSTRACT FROM AUTHOR]

Abstract: In this work, a plate bending formulation of the boundary element method (BEM) based on the Reissner''s hypothesis to perform linear analysis of plates reinforced by rectangular beams is extended to consider the beams not displayed over their middle surface. Therefore eccentricity effects are taken into account. The building floor structure is modelled as a stiffened plate which is treated as a single body without dividing it into beam and plate elements. Moreover the equilibrium and compatibility conditions are automatically imposed by the integral equations. In the proposed model the final system of equation is obtained by coupling the bending problem to the stretching problem. Besides, in order to reduce the number of degrees of freedom, both the displacements and tractions are approximated along the beam width, leading to a model where the values are defined on the beams axis. In order to validate the proposed formulation, the numerical results are compared to a well know finite element code. [Copyright &y& Elsevier]

Abstract: In this work, a boundary element method (BEM) formulation to perform linear bending analysis of building floor structures where slabs and beams can be defined with different materials is presented. The proposed formulation is based on Kirchhoff''s hypothesis, the building floor being modelled by a zoned plate, where the beams are treated as thin sub-regions with larger rigidities. This composed structure is treated as a single body, the equilibrium and compatibility conditions being automatically taken into account. In the final integral equation, the tractions are eliminated along the interfaces, therefore reducing the number of degrees of freedom. The displacements are approximated along the beam cross-section, leading to a model where the values remain defined on the beam skeleton line instead of their boundaries. The accuracy of the proposed model is shown by comparing the numerical results with a well-known finite element code. [Copyright &y& Elsevier]

Abstract: In this work, a numerical model to perform non-linear analysis of building floor structures is proposed. The presented model is derived from the Kirchhoff''s plate bending formulation of the boundary element method (BEM) for zoned domains, in which the plate stiffness is modified by the presence of membrane effects. In this model, no approximation of the generalized forces along the interface is required and the compatibility and equilibrium conditions along interfaces are imposed at the integral equation level. In order to reduce the number of degrees of freedom, the Navier–Bernoulli hypothesis is assumed to simplify the strain field for the thin sub-regions (rectangular beams). The non-linear formulation is obtained from the linear formulation by incorporating initial internal force fields, which are approximated by using the well-known cell sub-division. Then, the non-linear solution of algebraic equations is obtained by using the concept of the consistent tangent operator. The Von Mises criterion is adopted to govern the elasto-plastic material behaviour checked at points along the plate thickness and along the rectangular beam element axes. The numerical representations are accurately obtained by either computing analytically the element integrals or performing the numerical integration accurately using an appropriate sub-elementation scheme. [Copyright &y& Elsevier]

In this work the BEM non-linear formulation for Kirchhoff''s plates in bending applied to model reinforced concrete slabs is discussed. The integral equations of displacements, bending and twisting moments and shear forces are derived and modified properly to model non-linear behaviours. Two models were implemented to analyse reinforced concrete slabs: a simple elastoplastic criterion defined by adopting the Von Mises surface to govern the concrete behaviour in compression complemented by assuming tension cut off; and the Mazars'' continuum damage model. For both cases, the reinforcement is governed by the uniaxial stress x strain elastoplastic relationship with constant hardening. Internal forces are approximated by numerical integrals along the plate thickness using a Gauss'' scheme, after verifying the non-linear criterion at each point. [ABSTRACT FROM AUTHOR]