A. Blázquez

Contact problems with nonconforming discretizations using boundary element method

Abstract A procedure to solve two-dimensional frictionless contact problems between two bodies not necessarily modeled with identical discretizations is presented. The procedure is based on the boundary element method and follows a mixed incremental and iterative technique. The load is applied in a number of identical increments selected by the user for each problem. An iterative procedure to reach a compatible solution is applied for each of these increments of load, the geometry being updated after each increment. The algorithm developed is applicable to any problem belonging to the three classic contact situations: conforming, advancing and receding. Results corresponding to three cases representative of these three types of problems are finally shown.

A simplified numerical analysis of residual stresses in aluminum welded plates

The residual stresses originated in a welded Al-5083-O aluminum alloy plate are numerically studied. The thermal field is simulated by means of one analytical solution of the infinite plate, after which two thermal approaches are introduced, the temperature not changing along the welding direction in either case. The thermomechanical problem is solved by using a code based on the finite element technique (ABAQUS). The influence on the residual stresses of the strain hardening and the temperature material properties dependency has been considered. A plane stress model is used, the concordance of the numerical results thus obtained with the experimental measurements obtained using the blind hole technique being excellent. A discussion of the inability of a plain strain model to reproduce the experimental results is finally presented. The general conclusions of this study can be extended to annealed aluminum alloys (type O), whose elastic properties are not susceptible to change under heat treatment.

Bem solution of two-dimensional contact problems by weak application of contact conditions with non-conforming discretizations

Abstract Non-conforming discretizations of the surfaces involved in the contact problem are sometimes required, due to the geometry and/or the load. A strong application of the contact conditions directly relating variables (displacements and tractions) of the nodes of the discretizations, a general approach called by the authors node-to-point contact scheme, may lead to unsatisfactory results. In this paper, a weak application of the contact conditions, by means of the principle of virtual work, is developed for boundary integral equations. The formulation is presented for two-dimensional problems without or with friction, using the Coulomb model. The modelization is made using the Boundary Element Method and the problem is solved with an incremental procedure based on a displacement scaling approach. The solution scheme proposed is applicable to any contact problem (with small or large displacements) and is validated in this paper by applying it to receding, conforming and advancing contact problems, the jumps in the contact stresses that appeared in node-to-point contact schemes, not having been found in the problems tested.

ON THE REMOVAL OF RIGID BODY MOTIONS IN THE SOLUTION OF ELASTOSTATIC PROBLEMS BY DIRECT BEM

A theoretical and numerical study of the removal of rigid body motions in the solution of the boundary form of Somigliana identity and of the corresponding discretized linear system of the direct BEM is presented. This study is based on the Fredholm theory of linear operators and mechanical aspects of the problem. Various methods suitable for implementation in BEM codes are analyzed and relations between apparently different methods are shown. The relation between global equilibrium conditions and solvability of the discretized linear system of the direct BEM is discussed.

A linear elastic-brittle interface model: application for the onset and propagation of a fibre-matrix interface crack under biaxial transverse loads

A new linear elastic and perfectly brittle interface model for mixed mode is presented and analysed. In this model, the interface is represented by a continuous distribution of springs which simulates the presence of a thin elastic layer. The constitutive law for the continuous distribution of normal and tangential initially-linear-elastic springs takes into account possible frictionless elastic contact between adherents once a portion of the interface is broken. A perfectly brittle failure criterion is employed for the springs, which enables the study of crack onset and propagation. This interface failure criterion takes into account the variation of the interface fracture toughness with the fracture mode mixity. A unified way to represent several phenomenological both energy and stress based failure criteria is introduced. A proof relating the energy release rate and tractions at an interface point (not necessarily a crack tip point) is introduced for this interface model by adapting Irwin’s crack closure technique for the first time. The main advantages of the present interface model are its simplicity, robustness and computational efficiency, even in the presence of snap-back and snap-through instabilities, when the so-called sequentially linear (elastic) analysis is applied. This model is applied here in order to study crack onset and propagation at the fibre-matrix interface in a composite under tensile/compressive remote biaxial transverse loads. Firstly, this model is used to obtain analytical predictions about interface crack onset, while investigating a single fibre embedded in a matrix which is subjected to uniform remote transverse loads. Then, numerical results provided by a 2D boundary element analysis show that a fibre-matrix interface failure is initiated by the onset of a finite debond in the neighbourhood of the interface point where the failure criterion is first reached (under increasing proportional load); this debond further propagates along the interface in mixed mode or even, in some configurations, with the crack tip under compression. The analytical predictions of the debond onset position and associated critical load are used for several parametric studies of the influence of load biaxiality, fracture-mode sensitivity and brittleness number, and for checking the computational procedure implemented.

Interpretation of the problems found in applying contact conditions in node-to-point schemes with boundary element non-conforming discretizations

Several procedures for considering contact problems between deformable bodies using boundary elements with non-conforming discretizations are discussed in this paper. All the procedures considered (both those formerly proposed and a new one presented here) are based on an approach of strong imposition of the contact conditions, relating values of displacements and tractions of points and nodes of both bodies to force equilibrium and compatibility equations. All the procedures lead to similar results when the same general rule is followed in all cases: the body whose nodes control the displacements of the contact zone is discretized with more refined meshes. There are, nevertheless, cases, not previously referenced in the literature, where these procedures produce wrong definition of the subzones of adhesion and sliding, as well as some jumps in the stresses. These problems, whose appearance is shown to be due to the intrinsic nature of the node-to-points contact schemes, are discussed and recommendations are given to avoid them. Three classical examples belonging to the conforming, receding and advancing contact problems are studied, considering the presence and absence of friction.

Analytical and Numerical Models of Postbuckling of Orthotropic Symmetric Plates

Two analytical perturbation methods which give approximate solutions of postbuckling behavior of orthotropic simply supported plates are considered in this work: the method of Chandra and Raju and the method of Shen and Zhang. The reproduction of the algebraic developments of these methods by the Mathematica symbolic manipulator program has revealed that there are errors in the formulas included in the original paper by Chandra and Raju. After a revision and correction of these errors, the analytical results of both methods for a set of 23 orthotropic plates are compared, an excellent agreement being found for a wide range of values of geometrical and mechanical parameters in which many actual plates lie. A numerical simulation performed on a reduced sample of six plates using finite-element code ABAQUS has validated analytical results. The present work is intended as a first step in the investigation of the possibility of using reliable analytical formulas in the design of composite plates.

Comparing the conventional displacement BIE and the BIE formulations of the first and second kind in frictionless contact problems

There are several formulations of boundary integral equations (BIEs) used in the general numerical procedure known as boundary element method (BEM). There are also several approaches to deal with contact problems using BEM. In this paper, a comparison between the following procedures: the conventional discretization of the displacement BIE by collocations, the Galerkin discretizations of the symmetric BIE formulation of the first kind and the non-symmetric BIE formulation of the second kind, is performed. Although several aspects of these procedures are discussed, the emphasis is put on the accuracy of the results obtained with identical meshes. The comparison is carried out including problems with analytical solutions or in the presence of singularities, covering conforming, advancing and receding contact problems. Linear elements, conforming discretizations of surfaces in contact and absence of friction define the frame where the study is performed.

Influence of the location of the contact point in node-to-point contact approaches using non conforming boundary element discretizations

The study of contact between deformable bodies with non conforming discretizations inherently involves the possibility of a node having contact with an element at any position along it. This paper presents a study of the influence that the position along the element of this contacting point has on the results The numerical technique used is the Boundary Element Method and the classic problem of the compression of a deformable cylinder on a deformable foundation, which belongs to the class of advancing contact problems, will be used as a reference to evaluate the errors induced by the position of the contacting point.

Surface-based and solid shell formulations of the 7-parameter shell model for layered CFRP and functionally graded power-based composite structures

ABSTRACT In this study, we present the extension of the so-called 7-parameter shell formulation to layered CFRP and functionally graded power-based composite structures using two different parametrizations: (i) the three-dimensional shell formulation, and (ii) the solid shell approach. Both numerical strategies incorporate the use of the Enhanced Assumed Strain (EAS) and the Assumed Natural Strain (ANS) methods to alleviate locking pathologies and are implemented into the FE code ABAQUS. The applicability of the current developments is demonstrated by means of several benchmark examples, whose results are compared with reference solutions using shell elements of ABAQUS, exhibiting an excellent level of accuracy