Review of the matched asymptotic approach of the coupled criterion

Comptes Rendus Mécanique Sara Jiménez-Alfaro, Israel García García and Aurélien Doitrand Review of the matched asymptotic approach of the coupled criterion Volume 353 (2025), p.339-357 Online since: 6 February 2025 https://doi.org/10.5802/crmeca.285 This article is licensed under the Creative Commons Attribution 4.0International License. http://creativecommons.org/licenses/by/4.0/ CENTRE MERSENNE The Comptes Rendus. Mécanique are a member of the Mersenne Center for open scientific publishing www.centre-mersenne.org — e-ISSN : 1873-7234 Comptes Rendus. Mécanique 2025, Vol. 353, p.339-357 https://doi.org/10.5802/crmeca.285 Review article/Article de synthèse Review of the matched asymptotic approach of the coupled criterion Revue de l’approche asymptotique du critère couplé Sara Jiménez-Alfaro ,∗,a,b, Israel García García ,cand Aurélien Doitrand ,d aDepartment of Civil and Environmental Engineering, Imperial College London, Exhibition Road, London, SW7 2AZ, UK bDepartment of Engineering Science, University of Oxford, Parks Road, Oxford, OX1 3PJ, UK cDepartamento de Mecánica de Medios Continuos y Teoría de Estructuras, Escuela Técnica Superior de Ingeniería, Escuela Politécnica Superior, Universidad de Sevilla, Camino de los Descubrimienos s/n, 41092 Sevilla, Spain dUniversité Lyon, INSA-Lyon, UCBL, CNRS, MATEIS, UMR5510, F-69621 Villeurbanne, France E-mails: [email protected]x.ac.uk, s.jimene[email protected] (S. Jiménez-Alfaro), israelgar[email protected] (I. G. García), [email protected] (A. Doitrand) Abstract. Matched Asymptotics is a powerful mathematical technique with broad applicability in various engineering fields. One of its key uses is in Fracture Mechanics, where it provides accurate approximations in the vicinity of the crack tip with low computational complexity. This method can be seamlessly integrated with the Coupled Criterion (CC), which enables the prediction of crack nucleation and propagation in brittle materials. Hence, this paper deeply explains how the MA technique can be applied together with the CC in the context of Fracture Mechanics, providing a detailed literature review of the advances made in the last decade. Résumé. Les développements asymptotiques raccordés constituent une technique mathématique puissante, largement applicable dans divers domaines de l’ingénierie. L’une de leurs principales utilisations se situe en mécanique de la rupture, où ils permettent d’obtenir des approximations précises à proximité de la pointe des fissures tout en maintenant une faible complexité de calcul. Cette méthode peut être intégrée de manière fluide au critère couplé (CC), qui permet de prédire l’amorçage et la propagation des fissures dans les matériaux fragiles. Cet article explique comment la technique des développements asymptotiques raccordés peut être utilisée conjointement avec le critère couplé dans le cadre de la mécanique de la rupture, tout en offrant une revue détaillée de la littérature sur les avancées réalisées au cours de la dernière décennie. Keywords. Matched asymptotic expansion, Coupled criterion, Fracture mechanics. Mots-clés. Développement asymptotique raccordé, Critère couplé, Mécanique de la rupture. Funding. European Union’s Horizon 2020 research and innovation programme under Marie SklodowskaCurie grant agreement No. 861061-NEWFRAC, Iberdrola Foundation under the Marie Sklodowska-Curie Grant Agreement No 101034297, Ministerio de Ciencia e Innovación de España (Project PID2020-117001GBI00/AEI/10.13039/501100011033). Manuscript received 29 October 2024, revised and accepted 14 January 2025. ∗Corresponding author ISSN (electronic): 1873-7234 https://comptes-rendus.academie-sciences.fr/mecanique/ 340 Sara Jiménez-Alfaro et al. 1. Introduction The matched asymptotic (MA) expansion method is an approach that enables solving an equation or a system of equations [1–5]. It is well adapted to solve singularly perturbed differential equations, for which different approximate solutions are determined, each of which being accurate for a given part of the domain under investigation. These solutions are then combined to give a single approximate solution that is accurate for the whole domain under investigation. The domain may generally be divided into two subdomains. In the first one, the solution (called the outer solution) is accurately approximated by an asymptotic series representing a regular perturbation (i.e. by setting to zero a small parameter representing, e.g., a singular perturbation). The second one consists of a region in which this first approximation is inaccurate, due to perturbation terms that are not negligible. This constitutes the inner solution. The outer and inner solutions are then combined through a process called “matching” in such a way that an approximate solution for the whole domain is finally obtained. Asymptotic expansions were used to define the elastic constitutive law of the homogeneous equivalent material of a composite when a tangential slip is allowed on the fiber/matrix interface [6]. It was shown that a limit slip coefficient exists beyond which the stiffness of the material rapidly decreased. They were also used in the framework of homogenization as an alternative to the multiple scalings approach [7, 8]. MA expansions were used in combination to the singularity theory to determine the elastic displacements and stress fields corresponding to a class of junctions between rods and bulk bodies modeled as a flexible clamping in the framework of two-dimensional elasticity [9]. Leguillon analysed the problem of crack branching in a homogeneous elastic but non isotropic material. Based on asymptotic expansions, the energy release rate was computed and a revisited Griffith’s criterion including anisotropic fracture properties was suggested [10]. Sicsic and Marigo studied the propagation of a crack band and derived the conditions for which it behaves like a Griffith’s crack [11]. MA expansions were also used to study the behavior of interface cracks, for instance to further analyse the “Cook and Gordon” [12] interface debonding effect ahead of a primary crack [13], edge debonding in laminates [14] or to analyze the role of residual thermal stresses on the crack deflection or penetration at a bimaterial interface. The 2D and 3D singularities at a bimaterial interface were derived [15], also considering contact and friction between two anisotropic materials [16]. The mode III asymptotic expansions for a crack in or along a joint enabled defining an apparent toughness of the interface to be used for crack propagation [17]. It was also used to derive the stress intensity factors near an angular point on the front of an interface crack [18]. Moreover, the character of the stress singularity at the tip of a classical crack in a homogeneous material was approximated by an asymptotic series for cracks in Mode I, Mode II and Mode III. The first two modes were studied in the work of Williams [19], which is well known by the scientific community, since asymptotic solutions for free–free, clamped–clamped and free– clamped boundary conditions are given therein. MA expansions are particularly relevant when studying fracture and especially crack initiation in a structure. Indeed, the latter can be studied in the framework of MA expansions as the unbroken problem corrected by the crack that initiates (provided its smallness with respect to the structure characteristic dimensions). This idea was actually made effective by Leguillon [20, 21] who proposed to study crack initiation by coupling a stress criterion and an energy criterion. This approach has spread and is now a common way to study crack initiation, as evidenced by numerous applications summarized in the two review papers [22, 23]. The CC can be implemented through several ways, for instance by solving an implicit equation if analytical solutions can be provided for the stress fields and the energy release rate variation as a function of the crack surface [22, 24–26]. A second way to implement the CC is through finite element Sara Jiménez-Alfaro et al. 341 Figure 1. V-notched three-point bending specimen subjected to a force Fwith a crack (length ℓ) initiating at the V-notch tip. Notice that erepresents the notch depth. (FE) simulations of the full structure under investigation including the crack that initiates [27– 30]. In cases where analytical solutions are not available and so as to achieve a numerically more efficient approach than full FE implementation, an effective way to implement the CC is to use MA expansions. The objective of this paper is to give an overview of the matched asymptotic approach of the CC. We first recall the general idea (Section 2) and the formulation (Section 3) of the MA approach. Then, we describe its numerical implementation (Section 4) and provide some application examples (Section 5). 2. The idea behind matched asymptotics Before presenting the mathematical formalism of the matched asymptotic (MA) approach of the Coupled Criterion (CC), this section is dedicated to provide the philosophy behind it to further understand how it can be set up and define the main required ingredients. In the sequel, the CC is formulated under linear elasticity and small deformation assumptions. Both inertial effects [31, 32] or dissipation mechanisms other than cracking that may occur during initiation, such as, e.g., plasticity [33], diffuse damage [34] or viscous effects, are disregarded. The MA approach of the CC is useful to efficiently study the problem of a small crack initiating in a complex structure subjected to a mechanical or thermal loading. The objective is to determine the loading level at which the crack is likely to initiate as well as the initiation crack length. As a matter of example, we consider the problem of a crack of length ℓthat initiates at the tip of a V-notch (angle β) in a specimen loaded under three-point bending (Figure 1). Notice that this technique is only valid provided ℓis smaller than a characteristic dimensions of specimen (ℓ≪ein Figure 1), an initial assumption that should be checked after the implementation, once the actual value of ℓis obtained using the coupled criterion. 2.1. The coupled criterion The main idea behind the CC arises from the following observations: •Considering an energy criterion only, it enables assessing the propagation of a crack based on the material critical energy release rate Gc[35–37] but generally fails to study its initiation. •Considering a stress criterion only, it enables assessing crack initiation based on the material tensile strength σcexcept in the presence of a singular point. Stress and energy criteria thus appear as complementary and their combination enables assessing crack initiation in many configurations. The stress criterion of the CC is a condition established in the initial domain before crack initiation (thus without crack). It states that the stress normal to the future crack path must be larger than the material strength attained under a similar principal stress state. For instance, it reverts to comparing the opening stress to material tensile strength under uniaxial tensile loading. For the sake of simplicity, we will consider a brittle 342 Sara Jiménez-Alfaro et al. material that exhibits a Rankine strength surface in the sequel, which enables defining the material strength surface based on a single parameter, its tensile strength. The stress criterion thus requires the calculation of the stress field before crack initiation. In the vicinity of a V-notch, the stress tensor actually writes as an expansion in powers of r, William’s expansion in this case since it is a singularity, Taylor’s expansion for a smooth stress field: σ=Krλ−1s(θ)+o(rλ−1), (1) where rand θare polar coordinates, λis the characteristic exponent of the singularity and sis the stress function derived from the characteristic mode of the singularity u. The characteristic exponent and mode of the singularity are obtained by solving an eigen-value problem [2] . The parameter Kis the Generalized Stress Intensity Factor (GSIF) of the singularity, it represents the magnitude of the loading around the V-notch. Notice that in (1) only the dominant term has been represented, assuming that it is real and has multiplicity one, due to the symmetry of the problem represented in Figure 1. However, this is not the case of mixed mode loadings, for example, where two or more singular terms should be considered, with the associated GSIFs. The energy criterion of the CC is obtained from the energy equilibrium between the states prior to and after crack initiation. The crack surface creation energy GcS, where Sis the crack surface, must be balanced by the variation in external force work (Wext) and in elastic strain energy (Wel) so that: ∆Wel +GcS=∆Wext (2) When solving the CC, the objective is to determine the initiation crack surface Scand initiation imposed loading (for instance the initiation force Fcbased on the example provided in Figure 1) by simultaneously fulfilling both stress and energy criteria. We thus need (i) one calculation on the structure without the crack to compute the stress fields and (ii) several calculations with different crack surfaces to establish the energy equilibrium. If we are considering small cracks in a large structure, this may be computationally costly as fine meshes are required in the area close the crack location. The MA approach provides an alternative and efficient method to apply the CC, which is described in the sequel. Notice that in a bidimensional problem (2) can be expressed as ∆Wel +Gcℓ=∆Wext (3) where ℓis the newly created crack length (a priori unknown). At the initiation imposed loading ℓ=ℓc, the initiation crack length. Moreover, it is important to highlight that in problems where there are notches or pre-existing cracks, the crack nucleation is frequently determined by the initiation GSIFs of the singularity, denoted as Ki. These parameters depend on the initiation imposed loading and the geometry of the problem. In the problem represented in Figure 1, there is only one leading term, see (1), and therefore only one initiation GSIF. 2.2. The matched asymptotic approach The MA approach of the CC is based on the fact that the crack can be considered as a small perturbation to the elasticity problem where the structure is subjected to a given loading. It consists in successively considering two problems to be solved at two scales. The first problem, solved in the so-called outer domain, is obtained by considering the full structure and neglecting the crack that initiates. In complement, the second problem focuses only on the inner domain around the crack initiation point, independently of the whole structure under investigation. The final solution is then obtained by matching both problems to obtain the stress and energy balance required for the CC application. Sara Jiménez-Alfaro et al. 343 Figure 2. (a) Outer domain where the crack is disregarded, the contour Γcan be used to calculate the Generalized Stress Intensity Factor acting at the V-notch for a given force F. (b) Inner domain where the whole structure is disregarded, the normalized crack length is 1 and the arrows represent the imposed asymptotic displacement fields prescribed at a fictitious boundary sufficiently far from the crack. Outer domain. In the outer problem, the perturbation (i.e. the crack) is neglected and a solution of the problem without perturbation is provided in the outer domain (i.e. the structure without crack). This solution is valid everywhere except in a zone near the crack initiation location, for which a correction to this solution must be brought. The outer domain corresponding to the example given in Figure 1, is shown in Figure 2a. In the outer domain, the loading is described in terms of prescribed displacement or force. Then, the GSIF of the singular point (here, the Vnotch tip) can be calculated for a given applied force or displacement. Under the assumptions of small deformation and linear elasticity, the GSIF is proportional to the imposed force. For a given imposed force, the GSIF can be computed using a contour integral [2] on a closed path surrounding the singular point (e.g., Γin Figure 2a). The GSIF calculation based on the contour integral can be implemented in 3D [38] or in 2D for isotropic [21, 39] or anisotropic [40, 41] materials, for multi-material configurations [40, 42–44], or even based on displacement fields measured experimentally by digital image correlation [45]. Other approaches also exist to compute the GSIF, such as the quasidual function method [46, 47], least square fitting [48] or an extraction from the strain energy density [49]. The solution obtained in the outer domain is valid except near the crack initiation location, which requires a correction representative of the initial problem (Figure 1). Inner domain. The correction to the solution obtained in the outer domain without the perturbation is obtained through the second problem which is solved in the inner domain. It consists in focusing only in a zone near the crack initiation location, providing a detailed description of the crack around the singular point, regardless of the entire structure itself. The inner domain thus corresponds to the singular point that would lie in an infinite medium and would be subjected to remote asymptotic displacement or stress fields. The prescribed loading is thus described in terms of GSIF. An example of inner domain corresponding to the problem depicted in Figure 1 is shown in Figure 2b. In the inner domain, the space variables are normalized with respect to the crack length so that the normalized initiation crack length is 1. Since the whole structure geometry and boundary conditions are disregarded in the inner domain, the asymptotic displacement fields are prescribed as boundary conditions in order to obtain the stress and energy balance required to solve the CC. The solution derived in the inner domain is thus accurate in a zone near the crack initiation location. Matching inner and outer problem solutions. Solving the problems in the outer and inner domains yields two solutions (displacement fields) that accurately represent the initial problem 344 Sara Jiménez-Alfaro et al. Figure 3. Examples of configurations: (a) Inclusion, (b) crack ahead of a V-notch or (c) cavity close to a free edge, that can be studied applying the MA approach of the CC. For display purposes in the representation ℓis purposely not small compared to any dimensions of the structure. of the structure containing a small crack respectively far from and close to the crack initiation location. Matching both solutions also requires a common description of the applied loading. Since it is only described by the GSIF in the inner domain, it justifies the need of calculating the relation between the GSIF and the applied force or displacement in the outer domain. The next step in the MA approach consists in combining both solutions to finally solve the initial problem. This is done by matching both displacement fields in a zone that is (i) sufficiently far from the singular point in the inner domain and (ii) sufficiently close to it in the outer domain. The matching conditions finally enable obtaining the stress and energy balance corresponding to the initial problem under investigation (Figure 1) and further apply the CC for studying crack initiation. Solving the CC. The matching of the inner and outer problem solutions provide a general solution that is accurate over the whole domain under investigation. It yields the displacement fields in the whole structure in presence of a crack. It thus enables calculating the stress fields before crack initiation (Equation (1)) as well as the elastic strain energy variation due to crack initiation (Equation (2)) for a given loading. It finally yields all the ingredients required to solve the CC. The remaining step consists in determining the minimum imposed loading and the corresponding crack length for which both stress and energy conditions are fulfilled. 3. Formulation of the approach The matched asymptotic expansion is used in mechanical engineering to predict the solution, i.e. the displacement field Uℓ(x1,x2) (where (x1,x2) represents the Cartesian coordinates) in the vicinity of an element that can be an inclusion, a crack or a cavity, see Figure 3. This element is frequently called perturbation, since it is assumed that its size ℓis small compared to any dimensions of the structure. As an example to illustrate the formulation of the problem, a small cavity located at the tip of a V-notched is considered, see Figure 4, where the notation of the problem that is approximated is represented. The domain Ωℓhas an outer contour Γ=ΓV∪ΓN∪ΓD∪Γℓ. The contour ΓDis characterized by an imposed displacement ¯ U, whereas the contours ΓN,ΓVand Γℓhave a stressfree boundary conditions. The notation ΓVis referred to the contour of the V-notch and Γℓto the one of the small perturbation. Hence, the set of equations that defines the actual problem is −∇x·σℓ=0 in Ωℓ, (4) σℓ=C:∇xUℓ, (5) σℓ·n=¯ hon ΓN, (6) Sara Jiménez-Alfaro et al. 345 Figure 4. Representation of the notations in (a) the inner domaine and (b) the outer domain for the example of a cavity ahead of a V-notch. σℓ·n=0 on ΓV∪Γℓ, (7) Uℓ=¯ Uon ΓD, (8) where ∇xis referred to the coordinates system of the actual domain x1,x2. In the MA approach, a twofold representation of Uℓ(x1,x2) is proposed in the form of an outer and an inner expansion. Notice that it is assumed that the specimen is in the absence of body forces. Outer expansion. In this approximation, the actual solution is represented as Ul(x1,x2)=U0(x1,x2)+ · · · (9) whereU0(x1,x2) is the solution of the same elasticity problem considering that the perturbation is not observable in the domain, i.e., solved in an unperturbed domain Ω0. As an example, Figure 4b represents Ω0associated with the actual domain of Figure 4a. The second term in (9) denoted with an ellipsis is a “small correction” that decreases to 0 as ℓ→0. The solution U0(x1,x2) is a good approximation of Uℓ(x1,x2) far away from the perturbation. For this reason, it is called the outer field. The set of equations that defines U0(x1,x2) is −∇x·σ0=0 in Ω0, (10) σ0=C:∇xU0, (11) σ0·n=hon ΓN, (12) σ0·n=0 on ΓV, (13) U0=¯ Uon ΓD. (14) Notice that a better approximation of the outer expansion can be achieved by considering higher order terms. Particularly, Leguillon et al. considered the second outer term in [50]. Inner expansion. A second expansion can be used to approximate the actual solution by introducing the change of variables yi=xi/ℓand ρ=r/ℓ. In the limit when ℓ→0 we obtain an unbounded domain Ωin in which the dimensionless characteristic length of the perturbation is now equal to 1, see Figure 5 as an example, where the chosen characteristic length is the diameter of the cavity. The inner expansion is therefore expressed as Uℓ(x1,x2)=Uℓ(ℓy1,ℓy2)=F0(ℓ)V0(y1,y2)+F1(ℓ)V1(y1,y2)+ · · · (15) 346 Sara Jiménez-Alfaro et al. Figure 5. Scheme of the inner problem. The set of equations related to the two terms V0(y1,y2) and V1(y1,y2) are −∇y·˜ σ0=0 in Ωin, ˜ σ0=C:∇yV0 ˜ σ0·n=0 on ΓV∪Γℓ −∇y·˜ σ1=0 in Ωin, ˜ σ1=C:∇yV1 ˜ σ1·n=0 on ΓV∪Γℓ The problems indicated above are well-posed when the so-called matching conditions are added to these sets of equations. As a results, it is obtained an inner expansion that is a good approximation of the actual solutionUℓ(x1,x2) in the neighbourhood of the perturbation. Matching conditions. Since the outer expansion is a good approximation of the actual solution far away from the location of the perturbation and the inner expansion is a good approximation in the vicinity of the perturbation, there must exist an intermediate region where both expansions are valid. In that region the matching conditions are defined. The behaviour of the far field near the origin can be described by an expansion in powers of the distance to the singular point r, that can be the Taylor’s expansion in the case of a smooth stress field or the Williams’ expansion in case of a singularity. Assuming the example of the cavity in a V-notch highlighted in Figure 4, the William’s expansion can be applied, normally expressed in polar coordinates as U(r,θ)=U(0,0)+K rλu(θ)+o(rλ), (16) assuming that the dominant term is real and have multiplicity one. The matching conditions can be expressed as F0(ℓ)V0(y1,y2)≈U(0,0), when ρ→ ∞ (17) F1(ℓ)V1(y1,y2)≈Kℓλρλu(θ), when ρ→ ∞ (18) where the term ≈means “behaves like” and ρ=r/ℓ=qy2 1+y2 2. It can thus be set: F0(ℓ)=1 and V0(y1,y2)≈U(0,0), when ρ→ ∞ (19) F1(ℓ)=Kℓλand V1(y1,y2)≈ρλu(θ). when ρ→ ∞ (20) However, it can be shown that the matching condition over V1(y1,y2) does not fulfill the LaxMilgram theorem, since it has an infinite energy in the unbounded domain Ωin, while it should decrease to 0 at infinity to have a finite energy. For this reason, the superposition principle is applied, V1(y1,y2)=ρλu(θ)+ˆ V1(y1,y2) (21) where ˆ V1(y1,y2) is the solution to a well-posed problem. The set of equations that defines the new term ˆ V1(y1,y2) is: −∇y·ˆ σ1=0 in Ωin, ˆ σ1=C:∇yˆ V1, Sara Jiménez-Alfaro et al. 353 The main disadvantages are (i) the solutions are still limited to crack length at onset that has to be very small compared with the characteristic lengths of the problem and (ii) it is necessary to introduce special assumptions when nonlinearities are involved. Other methods have been used to predict crack initiation in the stress singularities or in the problems target of the combination of matched asymptotics and finite fracture mechanics: Cohesive Zone Models (CZM) prescribe a cohesive law between a pair of surfaces, relating force and separation, see e.g. [80]. This method is very versatile and presents good agreement with experiments, but requires setting crack geometry before initiation and typically involve nonlinear computational models. In the last decades gradient-damage-based models such as that named Phase Field have been extensively developed. These models are based on the regularization of the crack through a regularization length. It has been proven that when this length vanishes, the result of LEFM is recovered. Then, for V-notches and related problems, these models present the same problems as LEFM. To overcome this issue, several strategies have been proposed, such as assuming that the regularization length is a material parameter [100], defining a CZM [101], or understanding this regularization length in the context of the CC [90]. Declaration of interests The authors do not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and have declared no affiliations other than their research organizations. Dedication The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Acknowledgments This paper is entirely dedicated to our friend and colleague, Pr. Dominique Leguillon. 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