Cohesive crack propagation analysis using a dipole BEM formulation with tangent operator
Introduction
The crack propagation anaysis is a field of great relevance in the structural engineering, recognized by the number of different approaches that involve characterizing the ultimate resistance of solids. In this regard, numerous theories that allow the representation of mechanical deformable structures with discontinuities (cracks) are contemplated. Therefore, for analysis of cracked bodies, analytical solutions are restricted to a small set of problems, with certain simplifications in geometry and/or boundary conditions. Consequently, numerical models are required to ensure the accurate modelling of complex engineering structures, since the boundary conditions in these problems change at each crack lengh increment [1].
In the finite element method (FEM), diferencial equations are placed in terms of unknown displacements of elements nodes over the entire domain. However, despite classical FEM is widely disseminated in the scientific community, it is not effective in representing fracture problems. Due to its domain mesh, the remeshing procedure is not a simple task. Moreover, for a better representation of singular fields, a very refined mesh is required in these regions, leading to an increase in the computational cost. Some works can be highlighted, as presented by Carpinteri [2], Bittencourt et al. [3]. Another important formulation based on the FEM is the so-called extended finite element method (XFEM) that offer an elegant alternative tool to model cohesive cracks without the remeshing requirement, since the cohesive forces inclusion does not require the addition of interface elements [4], [5], [6], [7].
On the other hand, the boundary element method (BEM) has been recognized as an alternative method to the FEM and provides a tool for solving a wide range of fracture problems [8]. Furthermore, the BEM is an efficient numerical technique for modelling this type of problems, since the non-requeriment of a domain mesh, naturally describes the stress concentration at the vicinity of crack tips.
The BEM has been extensively used by many researchers in the field of fracture mechanics. One of the first technique based on the BEM formulation for modeling of cracked structures refers to the use of the subregion technique for the crack growth analysis between two boundaries [9]. Another alternative is called displacement discontinnuity method (DDM), which is based on hyper-singular integral equations for accurate model crack propagation [10], [11], [12].
The dual boundary element method (DBEM) is a widely used approach for modelling crack growth problems and it is based on the use of singular and hyper-singular integral equations along the crack path, avoiding the division of the solid in subregions [13]. Some remarkables works can be mentioned, Sollero and Aliabadi [14], Saleh and Aliabadi [15], Chen and Hong [16]. Numerical works based on the DBEM formulation for industrial problems can also be highlighted, Citarella et al. [17], Carlone et al. [18] and Citarella et al. [19].
Other classical formulations using boundary elements have been proposed in the literature to fracture problems. Among these formulations, it is worth mentioning the symmetric Galerkin BEM [20], multi-zone BEM [21], Green’s function [22] and the continuum strong discontinuity approach (CSDA), Peixoto et al. [23]. The latter employed the implicit formulation of the BEM to bidimensional problems, with emphasis on the cohesive fracture mechanics and considering strong discontinuities. More robust approaches, addressing micromechanical aspects and multiscale modeling, can be found in Sfantos and Aliabadi [24] and Benedetti and Aliabadi [25].
An alternative formulation to crack propagation analysis is based in the introduction of an initial stress field to represent the cohesive zone, developed on the concept of dipoles. In this way, the domain term of the classic BEM integral equation is degenerated along the crack path, with the appearance of the dipoles, responsible for correcting the elastic behavior of the structure. Moreover, this formulation is particularly interesting because it is able to represent mathematically the presence of the fracture process zone (FPZ) with only three algebraic equations, related to stress correction, per source point located in the crack path. In contrast, dual formulation requires four algebraic equations, related to displacements and forces, per source point. The multi-domain technique requires eight algebraic equations per source point along the crack path. Therefore, Oliveira and Leonel [26] proposed a BEM formulation based on the concept of dipoles of stress to analyse crack propagation in quase-brittle materials and using the classical Newton-Raphson scheme to solve the nonlinear system.
Most of the BEM formulations for fracture mechanics problems are focused on the use of the classic Newton scheme to solve the nonlinear problem, which lead to a larger amount of iterations to converge. An alternative to solve the nonlinear system is the special scheme called Tangent Operator (TO). In this scheme, the mechanical degradation of the material is considered, since this operator is constructed using the derivative expressions of the cohesive laws. Consequently, the TO is more efficient in obtaining the convergence of nonlinear systems. Some important works that use this methodology associated with the dual BEM formulation in fracture problems should be highlighted. Leonel and Venturini [13] presented a nonlinear formulation with the TO, to analyse crack propagation in quase-brittle materials, and more recently Cordeiro and Leonel [27] presented an anisotropic BEM/TO methodology to analysed cohesive crack propagation in wood structures.
In this work, the Dipole BEM formulation is addressed to modelling the nonlinear effects related to the structural behavior. Both external boundary and crack elements were discretized with linear isoparametric elements. The crack elements are discontinuous. In the present study, the model proposed by Hillerborg et al. [28] is employed, and the linear, bilinear and exponential cohesive laws are used to represent the relationship between cohesive stress and crack opening. Moreover, it is proposed the use of the TO, which is derived for all cohesive laws, in order to accelarate the nonlinear solution. Four applications are presented in this paper, including multiple crack analysis, to illustrate the robustness of the Dipole BEM/TO formulation. In all applications, the numerical efficiency of the TO is presented. Numerical and experimental results available in the literature is used to validate the proposed formulation.
Section snippets
Cohesive crack model
An important concept in representing models with nonlinear effects that cannot be neglected, is the region known as fracture process zone (FPZ). It is located in front of the fictitious fissure and is characterized by not responding elasticly to external forces, that is, an energy dissipation zone capable of transmitting effort. In this region, it is assumed that there is a uniform (cohesive) closing stress of a value equivalent to yield stress, for representation of the material residual
Dipole BEM formulation
The BEM has been successfully applied in several engineering problems, such as fracture mechanics, due to its efficiency in modelling stress concentration. This work uses an alternative BEM formulation with the introduction of a stress field, and the treatment of the variable called dipole, responsible for modeling the discontinuities. The following formulation is presented according to Oliveira and Leonel [26].
The integral representation of the displacement field considering an initial stress
Algebraic representation
With the definition of the integral equation for stresses, Eq. (20), and the contribution of the dipoles to the crack opening, Eq. (22), it is necessary to represent it in the algebraic form. The dipole distribution field is unknown and similar to what is done with displacements and surface forces, these can also be interpolated along the elements. It is important to mention that the use of linear functions brings some additional advantages, such as the possibility of analytical treatment of
Numerical solution technique
In this work, the strategy based on a constant operator for solving the nonlinear system is used, where the matrices of the algebraic system are kept constant, while the unbalanced surface forces are updated at each iteration. This strategy leads to good results, although requires a significant number of iterations.
The geometric interpretation of correction process of the nonlinear system by the constant operator (linear cohesive criterion) is illustrated in Fig. 9a. As the fracturing process
Flowchart of the dipole BEM model
The main structure of the computational Dipole BEM model is summarized in the following flowchart (Fig. 10). An incremental-iterative procedure is proposed for crack propagation simulation. Moreover, the elastic behavior of the structure must be corrected for all collocation points. The incremental procedure calculates, for each incremental step (), displacements and forces in the elastic behavior, until the maximum number of increments, . Each displacement and force are subdivided into
Applications
Three applications were chosen to illustrate the efficiency of the proposed Dipole BEM formulation in crack growth problems. The first problem deal with the mode I crack propagation. In the last two examples, the multiple crack growth case in mixed mode fracture is addressed. The experimental and numerical results are provided in the literature. For all applications, the performance comparison between CO and TO was performed and the convergence of nonlinear solution was verified with a
Concluding remarks
In this study, a numerical approach was proposed for cohesive crack propagation analysis using an alternative BEM formulation and the consistent tangent operator. The classical approach to solve the linear problem was also applied. Moreover, the domain term present in the classic displacement integral representation was degenerated in order to be non-null along the crack path. This procedure gives rise to a new variable called dipole, which is responsible for modelling the nonlinear effects.
CRediT authorship contribution statement
Luís Philipe Ribeiro Almeida: Conceptualization, Methodology, Software, Validation, Writing - original draft, Visualization. Eduardo Toledo de Lima Junior: Conceptualization, Methodology, Formal analysis, Writing - original draft, Writing - review & editing. João Carlos Cordeiro Barbirato: Conceptualization, Methodology, Writing - review & editing.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The authors are grateful to the Coordination for the Improvement of Higher Education Personnel – CAPES, for the scholarship provided to the first author.
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