Topological search of the crack pattern from a continuum mechanical computation

Highlights

• The purpose of the presented algorithm is to retrieve the crack path out of a continuum numerical model.

• The algorithm is the first step of a larger procedure aiming at retrieving the crack opening and computing leakage.

• The algorithm finds the ridges of the geometrical space generated by damage or an equivalent scalar variable.

• The algorithm is applied successfully to two examples.

• Retrieving crack paths helps to assess convergence of the damage model upon mesh refinement by comparing crack paths.

Abstract

In this contribution, a method to track the crack path from a continuum finite element simulation using failure material models is presented. The work belongs to a larger project, aiming at extracting the crack opening a posteriori of a continuum finite element computation to deal with durability assessment of concrete structures. Cracking is modelled in a diffuse manner by a distribution of an appropriate scalar state variable associated to material degradation. With the proposed method, the crack path is found in the post-processing phase by means of a step-by-step procedure. At each step, a new location point is computed where the state variable is maximum in a given direction at a given distance. Furthermore a specific procedure is proposed to detect multiple cracks in an automatic way. The proposed method is successfully applied to two studies implying either a single crack or a multi-crack pattern, obtained with different continuum models to highlight the versatility of the proposed approach.

Introduction

A correct evaluation of crack path and opening is important in many concrete engineering applications.

For some type of structures, crack opening estimation is required in relation with permeability issues, in order to limit fluid releases. Examples are confinement vessels and cooling towers of nuclear power plants, dams and liquid natural gas reservoirs [41], [16], [38]. In any case, since concrete is exposed to environmental agents, the knowledge of crack characteristics may be exploited in order to assess durability, in the sense of penetration of external chemical aggressive agents [5] or simply for aesthetic reasons. Finally, even design codes prescribe as a service limit state a maximum crack opening which in turn must be addressed carefully.

The assessment of the mechanical state of structures is more and more often obtained by numerical simulations, the finite element method being by far the most employed technique. The mechanical simulation of a degraded or fractured structure can be performed at present days following different approaches.

A first approach, based on fracture mechanics, assumes the existence of a crack or a set of potential cracks. The crack is then modelled as a discontinuity, sometimes by describing it explicitly in the geometry [19], sometimes by enriching with Heaviside functions the interpolation of displacements in the element, as in the X-FEM method [28], [30], [29]. Due to the direct incorporation of the discontinuity, this approach is here referred to as discrete. Using such a method has the clear advantage, that crack properties (crack path and opening) are retrieved directly, because they are main variables of the model. It is also true that X-FEM has allowed to overcome some traditional shortcomings of discrete approaches, such as necessity of incremental remeshing [3] or constraints on crack path direction for cohesive elements, which otherwise must be known a priori [19]. However, this approach has still a limitation when it comes to describe appropriately the dissipative bulk behaviour (i.e. microcracking and kinematic field across the fracture process zone) on one hand, and crack initiation on the other hand.

A completely different approach, called here continuum approach, consists in adopting a nonlinear constitutive law, where the degradation state of the material is described by at least one state variable [2], [21], [22]. Through continuum models the description of both crack initiation and propagation in the same frame is possible. A further benefit of the continuum with respect to discrete approach lies in the correct description of the degradation process for quasi-brittle materials, from diffuse damage to formation of the macro-crack due to coalescence of micro-cracks. In other terms, in quasi-brittle materials energy dissipation takes place in a volume of non negligible size with respect to the characteristic size of the structure; this volume is usually called Fracture Process Zone (FPZ). Unfortunately, continuum approaches do not directly provide the crack path and opening.

A third alternative, more and more explored by researchers, consists in adopting both approaches at the same time. An embedded or explicitly meshed discontinuity models the macro-crack, while a non-linear constitutive law is used to model the diffuse damage [1], [32]. However, intrinsic difficulties may arise by establishing the transition point between the two kind of approaches whatever the mechanical quantity used for the equivalence [8], [42]. Some recent works provide arguments to overcome this problem by proposing a transition between a damage model and a cohesive zone approach [7], [24], but still there are limitations on the crack direction. Finally, the method of thick level sets [31] seems very promising in the next future.

The choice made in this contribution is to perform a continuum nonlinear computation based on classical finite element method and then retrieve the crack characteristics a posteriori in a post-processing phase. The complete procedure is two fold; in a first step the crack path is estimated and in the second step the crack opening along the found crack path is computed. For the latter phase, it is possible to establish a strain equivalence with a strong discontinuity [11]; another method consists in computing a displacement jump by isolating the inelastic part if any of the strain tensor concurring to open the crack, so ignoring the elastic part released upon unloading [26]. In order to obtain the crack path, a method has recently been developed in [10], inspired by the Global Tracking Method used in [33], [32]; the crack path is an isoline of a scalar field obtained from a secondary gradient problem. However, this method is, so far, limited to radial loading and mode I failure, since it is based on the assumption that, at the time the crack path is computed, the principal direction of the largest principal strain is perpendicular to the crack. Furthermore, this is not verified at the intersection between a FPZ and a rebar in concrete. The aim of this new contribution is thus to obtain the crack path in an automatic way for a wider range of mechanical problems.

In the first part of this paper, the method for tracking the crack path is presented. In a second stage, choices of numerical parameters are discussed. Thirdly, two continuum models used for illustrative purposes are briefly described. Finally, the powerfulness of the method is analysed on two testcases.