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Maria Letizia Raffa
To cite this version:
Maria Letizia Raffa. Micromechanical modeling of imperfect interfaces and applications. Solid me-chanics [physics.class-ph]. Università di Roma ”Tor Vergata”; Aix-Marseille Université, 2015. English. �tel-01382758�
in joint supervision with
Course of Philosophy Doctorate in Solids Mechanics Aix-Marseille University
Micromechanical modeling of imperfect
interfaces and applications
a dissertation submitted in partial fulfillment
of the requirements for the degree of Philosophiæ Doctor by
Maria Letizia Raffa
PhD Committee
Prof. Elio Sacco University of Cassino - Supervisor Prof. Eric Jacquelin University Lyon I - Supervisor Prof. Raffaella Rizzoni University of Ferrara - Examiner
Prof. Djim´edo Kondo University Paris VI - Examiner Prof. Serge Dumont University of Nˆımes - Invited Examiner
Prof. H´el`ene Welemane National Engineering School of Tarbes - Invited Examiner Prof. Giuseppe Vairo University of Rome “Tor Vergata” - Thesis Advisor
Prof. Fr´ed´eric Lebon Aix-Marseille University - Thesis Advisor
PhD Course Coordinator Thesis Advisors
Prof. Paolo Bisegna Prof. Fr´ed´eric Lebon
Prof. Giuseppe Vairo
Department of Civil Engineering and Computer Science Engineering (DICII) ED 353 - Engineering Sciences : Mechanics, Physics, Micro and Nanoelectronics
The crucial role of solid interfaces in structural problems in several engineer-ing fields (e.g., Civil Engineerengineer-ing, Mechanical Engineerengineer-ing, Biomechanics, etc.) is well-established and they represent certainly a scientific topic of great interest. Nowadays, analytical and numerical modeling of structural interfaces are challenging tasks, due to the complex physical phenomena to take into account (such as adhesion, non-conforming contact, microcrack-ing, friction, unilateral contact), as well as to the need of numerical methods suitable for treating small thickness of the interface zones and jumps in the physically relevant fields.
Present PhD thesis aims to develop a consistent and general analytical tool able to overcome some modeling shortcomings of available modeling strate-gies accounting for soft and hard interfaces, and characterized by evolving microcracking. A novel approach, referred to as Imperfect Interface Ap-proach (IIA), is proposed. It consistently couples asymptotic arguments and homogenization techniques for microcracked media in the framework of the Non-Interacting Approximation (NIA). In detail, the micromechan-ical homogenization is exploited to find the effective elastic properties of a microcracked interphase between deformable adherents. Additionally, by employing a matched asymptotic expansion method, the interface laws are deduced, in the limit of a vanishing interphase thickness. In the context of linear elasticity, the IIA is successfully employed to derive a set of imperfect interface, addressing both the case of an initially-isotropic three-dimensional interphase and of initially-orthotropic two-dimensional interphase. In par-ticular, in two-dimensional case, stress/strain-based soft and hard interface laws are properly recovered as depending on the adopted homogenization approach. By generalizing the matched asymptotic expansion method to finite strains, a nonlinear soft interface model has been derived. As a new general application, the IIA is applied to formulate a spring-type model for non-conforming contact, which is successfully compared with theoreti-cal predictions and available experimental data. Finally, numeritheoreti-cal
Le rˆole crucial des interfaces solides dans les probl`emes de structures dans de nombreux domaines de l’Ing´enierie est d´esormais bien connue (G´enie Civil, G´enie M´ecanique, Biom´ecanique, etc.) et c’est certainement un su-jet de grand int´erˆet scientifique. Aujourd’hui, la mod´elisation analytique et num´erique des interfaces structurelles repr´esentent un d´efi du fait des ph´enom`enes physiques tr`es complexes qu’il faut prendre en compte (tels que adh´esion, contact non-conforme, microfissuration, frottement, contact unilat´eral) autant que le besoin d’avoir des m´ethodes num´eriques qui soient capables de traiter `a la fois la faible ´epaisseur des zones d’interface et les sauts dans les champs physiques concern´es.
Cette th`ese vise `a d´evelopper un outil analytique coh´erent et g´en´eral qui soit capable de d´epasser les limitations des strat´egies existantes et concer-nant la mod´elisation des interfaces soft et hard caract´eris´ees par une mi-crofissuration ´evolutive. Une nouvelle approche, appel´ee Imperfect Inter-face Approach (IIA), est propos´ee. Elle couple de mani`ere coh´erente ar-guments de th´eorie asymptotique et techniques d’homog´en´eisation pour les milieux microfissur´es dans le cadre de la Non-Interacting Approxima-tion (NIA). Plus en d´etail, l’homog´en´eisation microm´ecanique est exploit´ee afin de trouver les propri´et´es ´elastiques effectives d’une interphase microfis-sur´ee entre des adh´erents d´eformables. En outre, en employant un proc´ed´e de d´eveloppement asymptotique, des lois d’interface sont d´eduites, pour le probl`eme limite, i. e. pour une ´epaisseur de l’interphase qui tend vers z´ero. Dans le cadre de l’´elasticit´e lin´eaire, l’IIA est employ´ee avec succ`es pour ob-tenir un ensemble de mod`eles d’interfaces imparfaites, r´epondant `a la fois au cas d’une interphase en trois dimensions initialement isotrope et au cas d’une interphase en deux dimensions initialement orthotrope. En particu-lier, dans le cas en deux dimensions, des lois d’interface respectivement soft et hard bas´ees, sont obtenues en fonction de l’approche d’homog´en´eisation adopt´ee (en contraintes ou en d´eformations). En g´en´eralisant la m´ethode de d´eveloppement asymptotique `a la th´eorie ´elastique des d´eformations
fi-non-conforme `a raideurs equivalents, qui est compar´e avec succ`es avec des pr´edictions th´eoriques et des donn´ees exp´erimentales disponibles. Enfin, des simulations num´eriques appliqu´ees `a la ma¸connerie des mod`eles d’interface soft obtenus dans les deux cas lin´eaires et non-lin´eaires, ont ´et´e effectu´ees, montrant l’efficacit´e et la robustesse de la formulation propos´ee.
Il ruolo cruciale che le interfacce solide rivestono nei problemi strutturali in numerosi campi dell’Ingegneria `e ormai consolidato (Ingegneria Civile, Ingegneria Meccanica, Biomeccanica, ecc.), sicuramente esse rappresentano un argomento di grande interesse scientifico. Al giorno d’oggi, la model-lazione analitica e numerica delle interfacce strutturali sono un obiettivo sfidante, sia a causa dei complessi fenomeni fisici da tenere in conto (co-me l’adesione, il contatto non-confor(co-me, la microfessurazione, l’attrito ed il contatto unilaterale), che della necessit`a di avere metodi numerici capaci di trattare simultaneamente l’esiguo spessore delle zone di interfaccia ed i salti nei campi fisici, relativi al problema considerato.
La seguente tesi di dottorato mira a sviluppare uno strumento analitico consistente e generale, che sia capace di superare le restrizioni tipiche delle strategie modellistiche esistenti, riguardanti le interfacce soft e hard caratte-rizzate da una microfessurazione evolvente. Si propone un nuovo approccio, definito Imperfect Interface Approach (IIA). Quest’ultimo, accoppia coe-rentemente argomenti della teoria asintotica e tecniche di omogeneizzazione per materiali microfessurati nel contesto della Non-Interacting Approxima-tion (NIA). Nel dettaglio, l’omogeneizzazione micromeccanica `e impiegata per ottenere le propriet`a elastiche effettive di un’interfase microfessurata situata tra due aderenti deformabili. Inoltre, utilizzando una procedura di sviluppo asintotico, vengono dedotte delle leggi di interfaccia nel problema limite, cio`e per uno spessore di interfase che tende a zero. Nel contesto dell’elasticit`a lineare, l’IIA `e impiegato con successo al fine di ottenere un insieme di modelli di interfacce imperfette, corrispondenti sia al caso di interfase tridimensionale inizialmente isotropo, che al caso di interfase bidi-mensionale inizialmente ortotropo. In particolare, nel caso bidibidi-mensionale si `e dimostrato come si ottengano delle leggi di interfaccia rispettivamente soft e hard, in funzione dell’approccio d’omogeneizzazione adottato (in ten-sioni o in deformazioni). Generalizzando il metodo di sviluppo asintotico alla teoria elastica in deformazioni finite, si ottiene un modello d’interfaccia
stato confrontato con successo sia con le predizioni teoriche che con i dati sperimentali disponibili. Infine, sono state eseguite delle simulazioni nu-meriche, applicando alle strutture in muratura i modelli di interfaccia soft ottenuti nei casi lineare e non-lineare, confermando l’efficacia e la robustezza della formulazione proposta.
1 Introduction 1
2 Generalities 7
2.1 Asymptotic expansion theory . . . 7
2.1.1 Matched asymptotic expansion method . . . 8
2.1.1.1 General notations . . . 9
2.1.1.2 Rescaling process . . . 9
2.1.1.3 Concerning kinematics . . . 11
2.1.1.4 Concerning equilibrium . . . 12
2.1.1.5 Matching external and internal expansions . . . 13
2.1.1.6 Concerning constitutive equations . . . 14
2.1.1.7 Internal/interphase analysis . . . 15
2.2 Homogenization for microcracked media . . . 20
2.2.1 Recall on the Eshelby’s problem . . . 21
2.2.2 Non-interacting approximation (NIA) . . . 23
2.2.3 Non-interacting approximation (NIA) for microcracked media . . 28
2.2.3.1 Stress-based approach . . . 29
2.2.3.2 Strain-based approach . . . 31
2.3 Conclusions . . . 34
3 Soft-type imperfect interface models 37 3.1 Interface model I: 3-d isotropic interphase . . . 39
3.1.1 Matched asymptotic expansion method . . . 39
3.1.2 Homogenization of the microcracked interphase . . . 41
3.1.3 Interface stiffnesses . . . 43
3.2 Interface model II: 2-d orthotropic interphase . . . 44
3.2.1 Matched asymptotic expansion method . . . 45
3.2.2 Homogenization of the microcracked interphase . . . 47
3.3 Generalization to a smoothly-rough interphase . . . 51
3.4 Conclusions . . . 54
4 Hard and nonlinear imperfect interface models 57 4.1 A hard imperfect interface model . . . 58
4.1.1 Matched asymptotic expansion method . . . 58
4.1.2 Homogenization of the microcracked interphase . . . 59
4.1.3 Interface stiffnesses . . . 63
4.2 A St. Venant-Kirchhoff imperfect interface model . . . 63
4.2.1 Matched asymptotic expansion method in finite strains . . . 64
4.2.2 Homogenization of the microcracked interphase . . . 71
4.3 Generalization to a smoothly-rough interphase . . . 72
4.4 Conclusions . . . 75
5 Numerical applications 77 5.1 A simple three-dimensional benchmark . . . 79
5.2 The influence of the rough parameter η . . . 84
5.3 Modeling interfaces at the macroscale . . . 93
5.4 Conclusions . . . 98
6 A non-conforming contact model 101 6.1 Contact model . . . 103
6.1.1 Imperfect interface approach . . . 104
6.1.2 Effective contact stiffness . . . 105
6.2 Results and discussions . . . 109
6.2.1 Model validation . . . 109
6.2.2 Comparison with the contact model of Krolikowski and Szczepec 112 6.2.3 Influence of model parameters . . . 114
6.3 Conclusions . . . 117
2.1 Asymptotic procedure . . . 10
2.2 Comparison among NIA and dilute limit behavior . . . 24
2.3 Comparison between homogenization techniques for microcracked media 35 3.1 Problem statement . . . 39
3.2 Sketch of the adopted method . . . 41
3.3 REV with a penny-shaped crack . . . 42
3.4 REA with a linear crack . . . 48
3.5 Asymptotic procedure . . . 52
5.1 Geometry, boundary conditions and mesh detail . . . 79
5.2 Deformed shape in LNL model . . . 82
5.3 Comparison in terms of reaction force . . . 82
5.4 Comparison in terms of displacement jump in x1-direction . . . 83
5.5 Comparison in terms of displacement jump in x3-direction . . . 84
5.6 Von Mises stresses in LNL model . . . 84
5.7 Von Mises stresses in NL2 model . . . 85
5.8 Shear tests . . . 86
5.9 Experimental curves of full-brick triplet [1, 2] . . . 86
5.10 Experimental curves of hollow-brick triplet [1, 2] . . . 87
5.11 Failure modes . . . 87
5.12 Deformed shape with zooms . . . 89
5.13 Deformed shape . . . 89
5.14 Influence of roughness shape . . . 90
5.15 Evolution of l . . . 91
5.16 Identification of A∗ . . . 92
5.17 Influence of A∗ on the global response . . . 92
5.19 Geometry and boundary conditions . . . 94
5.20 Geometry and mesh detail . . . 96
5.21 Deformed shape in mm . . . 96
5.22 Von Mises stresses in MPa . . . 97
5.23 Shear stresses in MPa . . . 97
5.24 Reaction force vs. total strain . . . 98
6.1 Problem statement . . . 104
6.2 Limit behaviors of the contact and non-contact areas . . . 106
6.3 Behaviors of the contact and non-contact radii . . . 107
6.4 Normalized contact area vs. closure pressure . . . 108
6.5 Normal contact stiffness vs. closure pressure . . . 110
6.6 Tangential contact stiffness vs. closure pressure . . . 110
6.7 Contact stiffnesses vs. closure pressure ¯p . . . 111
6.8 Tangential-to-the-normal stiffness ratio as function of the closure pressure113 6.9 Contact stiffnesses vs. closure pressure ¯p . . . 114
6.10 Contact stiffnesses. Model sensitivity to model parameters γN,T0 . . . 115
6.11 Tangential-to-normal stiffness ratio. Model sensitivity to model parame-ters γN,T0 . . . 115
6.12 Contact stiffnesses. Model sensitivity to model parameters γN,T0 . . . 116
6.13 Tangential-to-normal stiffness ratio. Model sensitivity to model parame-ters γN,T0 . . . 116
5.1 Values of degrees of freedom (dof) and solution times (in seconds) for all the analyzed numerical models. . . 81 5.2 Experimentally-determined mechanical properties of the three-fold masonry
constituents [2]. . . 85 6.1 Values of the ratios γ0
N,num/γN,th0 and γ 0
T ,num/γT ,th0 with respect to the history parameter h. . . 112
Introduction
The study of interfaces between deformable solids significantly developed thanks to the rising interest of scientists and industries in mechanics of composite materials. Those first studies, in particular, focused on the presence of matrix-fiber interfaces in compos-ite media and their effect on the determination of the effective thermoelastic properties of this kind of materials. Within the framework of these theories on mechanical be-havior of composites, a commonly adopted assumption was the requirement of the continuity in terms of stresses and displacements at the interfaces among the principal constituents. The stress-based interface condition origins from the local equilibrium and the displacement-based interface condition derives from the hypothesis of perfect bonding. Such an interface condition was defined as perfect interface. Nevertheless, the assumption of perfect interfaces is established to be inappropriate in many mechanical problems. Indeed, the interface between two bodies or two parts of a body, defined as adherents, is a favorable zone to complex physico-chemical reactions and vulnerable to mechanical damage. Goland and Reissner [3], in the forties, were surely the first to modeling a thin adhesive as a weak interface, i. e. they were the first to assume that the adherents were linked by a continuous distribution of springs. Such an in-terface, is defined as spring type. Goland and Reissner have noted that the thinness suggests considering constant stresses in the adhesive, and some years later, Gilibert and Rigolot [4] found a rational justification of this fact by means of the asymptotic expansion method, assuming that the thickness and the elastic properties of the adhe-sive had the same order of smallness ε. During the eighties and nineties, the relaxation of the perfect interface approximation was largely investigated, aiming principally to applicate these theories to composite materials with coated fibers or particles [5, 6], or in the case of decohesion and nucleation problems in cohesive zones [7, 8]. One of the first definition of imperfect interface was certainly due to Hashin and Benveniste [5, 6]. Particularly, Hashin concentrates his research in the case of composite material with
thin layer or coating enveloping its reinforcing constituents (fibers). Such an interfacial layer is generally referred to as interphase, and its presence can be due to chemical interaction between the constituents or it may be introduced by design aiming to im-prove the mechanical properties of the composite. Several investigations in literature, before and after the work of Hashin, modeled this kind of problem with the so-called three-phase-material theory. Such a description requires, obviously, the knowledge of the interphase properties. These constitutive informations are rarely available, pri-marily, because the interphase material properties are in situ properties which are not necessarily equal to those of the bulk material, and additionally, the interphase may vary within a transition zone from one constituent to another. Accordingly, in most cases, the interphase properties are unmeasurable. The Imperfect interface theory was formulated by Hashin [6, 9–11] in order to overcome these challenges. This alternative model was based to the main idea that if the interphase has significant effects on the overall response, then its properties must be significantly different from those of the constituents, in general, much more flexible. To this aim, the attempts for explicit modeling of the three-dimensional interphase are highly reduced by replacing it with a two-dimensional imperfect interface. In particular, within the Hashin imperfect in-terface model [6] the discontinuity in terms of displacements is allowed, instead, the continuity in terms of stresses, according local equilibrium, is preserved. Hashin, as Goland [3] before him, made the simplest assumption that the displacement disconti-nuity is linearly proportional to the traction vector:
σ(Ω1) nn = σnn(Ω2) = Dn[un] [un] = u(Ωn 1)− u(Ωn 2) σ(Ω1) ns = σns(Ω2) = Ds[us] [us] = u(Ωs 1)− u(Ωs 2) σ(Ω1) nt = σ (Ω2) nt = Dt[ut] [ut] = u(Ωt 1)− u (Ω2) t where σ(Ω1),(Ω2)
ij with i, j = n, s, t are the components of the interfacial stress vector of two deformable solids Ω1 and Ω2 with reference to a local orthogonal system of axes n, s, and t originating at some point on the interface, where n is the normal direction and s, t are tangential directions; [ui] and u(Ωi 1),(Ω2) with i = n, s, t are the components of displacement jump vector and displacement vector, respectively, both in normal and tangential directions. Dn and Dt,s are spring constant type material parameters, in normal and tangential directions, respectively, and they have dimension of stress divided by length. In the following, these parameters are referred as interface stiffnesses. It is worth to point out that infinite values of the interface stiffnesses imply vanishing of displacement jumps and therefore perfect interface conditions. At the other asymptotic limit, zero values of the stiffnesses imply vanishing of interface tractions and
Hashin, with his pioneering work, determined the effective elastic properties and thermal expansions coefficients both for unidirectional fiber composites with imperfect interfaces conditions [6] and for composites with spherical inclusions and particles [9, 10]. Moreover, he demonstrated that the three-phase-material approach was a special case of the imperfect interface theory. It is worth remarking that Hashin, as first, showed that the interface stiffnesses (he referred them as interface parameters) can be simply related to the interphase properties and geometry [6].
Hashin and Benveniste, independently, generalized the classical extremum principles of the theory of elasticity for composite bodies to the case of an imperfect interface described by linear relations between interface displacement jumps and tractions [5, 11]. In the work of B¨ovik [12], the idea to use a simple tool that is the Taylor expansion of the relevant physical fields in a thin interphase, combined with the use of surface differential operators on a curved surface, has been applied to achieve the representation of a thin interphase by an imperfect interface. The idea of a Taylor expansion was also adopted by Hashin to derive the spring-type interface model for soft elastic interfaces [10] and for interphases of arbitrary conductivity and elastic moduli [13]. More recently, Gu [14, 15] derived an imperfect interface model for coupled multifield phenomena (thermal conductivity, elasticity and piezoelectricity) by applying Taylor expansion to an arbitrarily curved thin interphase between two adjoining phases; he also introduced some new coordinate-free interfacial operators.
All the above cited imperfect interface models are derived by assuming an isotropic interphase.
In a quite recent work Benveniste [16], provided a generalization of the B¨ovik model to an arbitrarily curved three-dimensional thin anisotropic layer between two anisotropic media. Benveniste model is carried out in the setting of unsteady heat conduction and dynamic elasticity. The derived interface model consists of expressions for the jumps in physical fields, i. e. temperature and normal heat flux in conduction, and displacements and tractions in elasticity, across the interface.
Additionally, derivations of spring-type interface models by using asymptotic meth-ods, for different geometrical configurations, have been given, among other by Klarbring [17, 18] and Geymonat [19].
A much less studied imperfect interface condition is the one obtained starting from a stiff and thin interphase, the so called stiff interface (or equivalently hard interface). Differently from the soft case, the hard interface is characterized by a jump of the traction vector across the interface and by continuity of displacements. Benveniste
and Miloh [20], generalizing the study made by Caillerie [21] for curved interfaces, demonstrate that depending on its degree of stiffness with respect to the bodies in contact, a stiff thin interphase translates itself into a much richer class of imperfect interfaces than a soft interphase does. Within their study, a thin curved isotropic layer of constant thickness between two elastic isotropic media in a two-dimensional plane strain setting, is considered. The properties of the curved layer are allowed to vary in the tangential direction. It is shown that depending on the softness or stiffness of the interphase with respect to the neighboring media, as determined by the magnitude of the non-dimensional Lam´e parameters ¯λcand ¯µc, there exists seven distinct regimes of interface conditions according the following expressions:
¯ λi = ˜ λi εN µ¯i = ˜ µi εN
where ˜λc and ˜µc are non-dimensional constant Lam´e parameters of the material inter-phase, ε in the non-dimensional interphase thickness and N is a negative or positive integer or zero. Accordingly with the above definition these regimes may be distin-guished in: (a) vacuous contact type interface for N 6 −2, (b) spring type interface for N = −1, (c) ideal contact type interface for N = 0, (d) membrane type interface for N = 1, (e) inextensible membrane type interface for N = 2, (f) inextensible shell type interface for N = 3, (g) rigid contact type interface for N > 4. The first two conditions are characteristic of a soft interphase whereas the last four are characteristic of a stiff interphase. The cases (a), (c) and (g) are the classical ones: in case (a) the tractions vanish (debonding), in case (c) the displacements and tractions are continuous (perfect interface condition), and in case (g) the displacements vanish. Benveniste and Miloh [20], for the first time, derived the interface conditions for the hard cases (d), (e) and (f), by applying a formal asymptotic expansion for the displacements and stresses fields in the thin layer interphase.
In the present thesis, two kind of imperfect interface conditions are essentially re-ferred to: the soft interphase case which brings to a spring-type linear (Chapter 3) and nonlinear interface (Chapter 4), and the hard interphase case (Chapter 4) which brings to a more general interface model that includes, as will be shown in the next section, the perfect interface conditions. In order to make an analogy with the Benveniste’s classifications, the cases with N = −1 and N = 0 will be analyzed. It is worth re-marking some differences between the hard interface case considered in this thesis and that defined by Benveniste and Miloh for N = 0. In fact in the thesis, the case N = 0, according formers papers [22–26] will be studied within the framework of the higher order theory. This choice, extensively detailed in the following, leads to the evidence
As a result, the perfect interface has been established to be a particular case of the hard interface condition at the zero-order [25], in what follows this evidence will be analytically derived within the asymptotic framework.
The imperfect interface models, object of the present thesis, are consistently de-rived by coupling a homogenization approach for microcracked media under the non-interacting approximation (NIA) [27–30], and arguments of asymptotic analysis [22–25]. Such a method, is defined imperfect interface approach. The thesis is organized as fol-lows: the first chapter is devoted to present and detail the framework of the imperfect interface approach; in Chapters 3 and 4 such an approach is enforced in order to derive several interface models; Chapter 5 is consecrated to the numerical applications and to the validation of these interface models; finally, Chapter 6 represents an enhancing and a theoretical application of the imperfect interface approach, herein employed to derive a non-conforming contact model.
Generalities
The imperfect interface models proposed in the thesis will be derived following the so defined imperfect interface approach, which is a method coupling asymptotic arguments and micromechanical homogenization theories. The principal ingredients of such an approach are extensively detailed in what follows.
2.1
Asymptotic expansion theory
A rigorous and mathematically elegant way to recover the governing equations of both soft and hard interfaces is represented by the use of the concepts of the asymptotic expansion method. The asymptotic expansions technique was already well-known in the seventies, when was applied to rigorously derive the plate and shell theories in both linear and nonlinear cases by Ciarlet [31–33]. This method was, immediately afterwards, developed by Sanchez-Palencia [34, 35] in order to derive the homogenized response of composite materials. This asymptotic homogenization technique is based on the choice of a geometrically small parameter (e.g., the size of the microstructure or the thickness of a thin layer) and on the expansion of the relevant physical fields (e.g., displacement, stress and strain) in a power series with respect to the chosen small parameter.
In the nineties, the asymptotic theory was successfully applied to recover the gov-erning equations of imperfect interface in solids mechanics [17–19, 36, 37].
In some circumstances, it is useful to avail of the so-called high order theory, i.e. to take into account for higher order terms in the asymptotic expansions of the rele-vant fields, in order to have more accurate interface governing equations. The most important cases are when the thickness ε of the bonding material has a no-negligible
dimension with respect to the characteristic lengths of the adherents, and when the adhesive material is stiffer than the adherents.
In literature is well-established that, if the material stiffness of the adhesive is comparable with the material stiffness of the adherents, then various mathematical approaches (asymptotic expansions [13, 16, 18–20, 38, 39], Γ−convergence techniques [21, 22, 37], energy methods [23–25]) can be used to obtain the model of perfect interface at the first (zero) order in the asymptotic expansion. More recently, some authors availing of the higher order theory, have demonstrated that at the next (one) order, it is obtained a non-local model of imperfect interface, due to the presence of tangential derivatives entering the interface equations [13, 22–25, 38].
The analysis of the regularity of the limit problems and of the singularities of the stress and displacement fields near the external boundary of the adhesive have been considered by Geymonat et al. [40]. Nevertheless, they have focused their studies only on the soft interface at the zero-order. On the contrary, the hard interface case and the higher order cases are already open challenges for the mathematicians.
The asymptotic technique adopted to derive the soft and the hard imperfect in-terface models presented in this thesis is the matching asymptotic expansions [35]. In this method, the derivation of the governing equations of the interface is performed by adopting the strong formulation of the equilibrium problem, i. e. by writing the classical compatibility, constitutive and equilibrium equations. In particular, the higher order terms of the asymptotic expansions are taken into account following the approach by Rizzoni et al. [25]. The zero-order terms are classical and well-established in the literature, on the contrary, the terms computed at the one-order have been recently derived both for the hard interfaces ([22–24]) and the soft interfaces ([25]).
2.1.1 Matched asymptotic expansion method
It is worth pointing out that the application of the asymptotic methods to obtain gov-erning equations of imperfect interface starting from thin layer problems in the elasticity framework, have a consistent mathematical background [6, 9, 10, 13, 21, 31–35]. Ould Khaoua among other, in his doctoral thesis [41], studied the elastic equilibrium prob-lemPεunder the hypothesis of small perturbations. The author demonstrates that the solution of the reference problem (i.e. with an elastic thin layer of thickness ε)Pε, that is expressed in terms of both stress and displacement fields (σε, uε), for ε → 0 admits a limit (σ, u) and that this limit solution is also the solution of the limit problem P (Pε → P for ε → 0). Additionally, Ould Khaoua [41] has found, as Hashin [6] be-fore, that the mechanical and geometrical characteristics of the layer (interphase) are retained in the interface stiffnesses of the soft interface governing equations.
The matched asymptotic expansion method [22–25, 42], adopted in this thesis, is detailed in the following paragraphs.
2.1.1.1 General notations
With reference to [25], let the problem general notations be detailed. A thin layer Bε with cross-section S and uniform small thickness ε 1 is considered, S being an open bounded set in R2 with a smooth boundary. In the following Bε and S will be called interphase and interface, respectively. The interphase lies between two bodies, named as adherents, occupying the reference configurations Ωε± ⊂ R3. In such a way, the interphase represents the adhesive joining the two bodies Ωε+ and Ωε−. Let Sε± be taken to denote the plane interfaces between the interphase and the adherents and let Ωε= Ωε±∪Sε±∪Bε denote the composite system comprising the interphase and the adherents.
It is assumed that the adhesive and the adherents are perfectly bonded and thus, the continuity of the displacement and stress vector fields acrossSε± is ensured.
An orthonormal Cartesian basis (O, i1, i2, i3) is introduced and let (x1, x2, x3) be taken to denote the three coordinates of a particle. The origin lies at the center of the interphase midplane and the x3−axis runs perpendicular to the open bounded set S, as illustrated in Fig. 2.1.
The materials of the composite system are assumed to be homogeneous and linearly elastic and let A±, Bε be the fourth-rank elasticity tensors of the adherents and of the interphase, respectively. The tensors A±, Bε are assumed to be symmetric, with the minor and major symmetries, and positive definite. The adherents are subjected to a body force density f± : Ωε± 7→ R3 and to a surface force density g± : Γεg 7→ R3 on Γεg ⊂ (∂Ωε
+\Sε+) ∪ (∂Ωε−\Sε−). Body forces are neglected in the adhesive.
On Γεu = (∂Ωε+\Sε+) ∪ (∂Ωε−\Sε−) \ Γεg, homogeneous boundary conditions are prescribed:
uε = 0 on Γεu, (2.1)
where uε : Ωε 7→ R3 is the displacement field defined on Ωε. Γε
g, Γεu are assumed to be located far from the interphase, in particular, the external boundaries of the interphase Bε (∂S × (−ε
2, ε
2)) are assumed to be stress-free. The fields of the external forces are endowed with sufficient regularity to ensure the existence of equilibrium configuration.
2.1.1.2 Rescaling process
The rescaling phase in the asymptotic process represents a mathematical construct [33], not a physically-based configuration of the studied composed system. This standard
Ωε Ω-ε ε 2 - ε 2 S ε S-ε Bε Γε Γ ε Ω+ Ω -+ 1 ⁄ 2 -1 ⁄ 2 S + S -B Γ g Γu Ω+ Ω -S Γg Γu g g S S
(a)
(b)
(c)
x
3x
1,x
2z
1,z
2x
3x
1,x
2z
3Fig. 2.1: Asymptotic procedure - Synoptic sketch of three steps performed in the matched asymptotic expansion approach: (a) the reference configuration with the ε-thick interphase; (b) the rescaled configuration; (c) the final configuration with the zero-thick interface.
step is used in the asymptotic expansion method, in order to eliminate the dependency on the small parameter ε of the integration domains. This rescaling procedure can be seen as a change of spatial variables ˆp : (x1, x2, x3) → (z1, z2, z3) in the interphase [33]:
z1= x1, z2= x2, z3= x3 ε (2.2) resulting ∂ ∂z1 = ∂ ∂x1 , ∂ ∂z2 = ∂ ∂x2 , ∂ ∂z3 = ε ∂ ∂x3 (2.3) Moreover, in the adherents the following change of variables ¯p : (x1, x2, x3) → (z1, z2, z3) is also introduced:
z1 = x1, z2 = x2, z3 = x3± 1
2(1 − ε) (2.4)
where the plus (minus) sign applies whenever x ∈ Ωε+ (x ∈ Ωε−), with ∂ ∂z1 = ∂ ∂x1 , ∂ ∂z2 = ∂ ∂x2 , ∂ ∂z3 = ∂ ∂x3 (2.5) After the change of variables (2.2), the interphase occupies the domain
B = {(z1, z2, z3) ∈ R3 : (z1, z2) ∈S, |z3| < 1
2} (2.6)
and the adherents occupy the domains Ω±= Ωε±±12(1−ε)i3, as shown in Fig. 2.1. The sets S± = {(z1, z2, z3) ∈ R3 : (z1, z2) ∈S, z3 = ±12} are taken to denote the interfaces betweenB and Ω± and Ω = Ω+∪ Ω−∪B ∪ S+∪S− is the rescaled configuration of the composite body. Lastly, Γu and Γg indicates the images of Γεu and Γεg under the change of variables, and ¯f± := f±◦ ¯p−1 and ¯g± := g±◦ ¯p−1 the rescaled external forces.
2.1.1.3 Concerning kinematics
Following the approach proposed by [22, 25], let focus on the kinematics of the elastic problem. After taking ˆuε = uε◦ ˆp−1 and ¯uε = uε◦ ¯p−1 to denote the displacement fields from the rescaled adhesive and adherents, respectively, the asymptotic expansions of the displacement fields with respect to the small parameter ε take the form:
uε(x1, x2, x3) = u0+ εu1+ ε2u2+ o(ε2) (2.7a) ˆ
uε(z1, z2, z3) = ˆu0+ εˆu1+ ε2uˆ2+ o(ε2) (2.7b) ¯
uε(z1, z2, z3) = ¯u0+ ε¯u1+ ε2u¯2+ o(ε2) (2.7c)
Interphase: The displacement gradient tensor of the field ˆuε in the rescaled inter-phase is computed as:
ˆ H = ε−1 " 0 uˆ0 α,3 0 uˆ03,3 # + " ˆ u0 α,β uˆ1α,3 ˆ u03,β uˆ13,3 # + ε " ˆ u1 α,β uˆ2α,3 ˆ u13,β uˆ23,3 # + O(ε2) (2.8)
where α = 1, 2, so that the strain tensor can be obtained as:
e(ˆuε) = ε−1ˆe−1+ ˆe0+ εˆe1+ O(ε2) (2.9) with: ˆ e−1 = 0 1 2uˆ 0 α,3 1 2uˆ 0 α,3 uˆ03,3 = Sym(ˆu 0 ,3⊗ i3) (2.10) ˆ ek= Sym(ˆukα,β) 1 2(ˆu k 3,α+ ˆuk+1α,3 ) 1 2(ˆu k 3,α+ ˆuk+1α,3 ) uˆk+13,3 = Sym(ˆu k
,1⊗i1+ ˆuk,2⊗i2+ ˆuk+1,3 ⊗i3) (2.11)
where Sym(·) gives the symmetric part of the enclosed tensor and k = 0, 1, and ⊗ is the dyadic product between vectors such as: (a ⊗ b)ij = aibj. Moreover, the following notation for derivatives is adopted: f,j denoting the partial derivative of f with respect to zj.
Adherents: The displacement gradient tensor of the field ¯uε in the adherents is computed as: ¯ H = " ¯ u0 α,β u¯0α,3 ¯ u03,β u¯03,3 # + ε " ¯ u1 α,β u¯1α,3 ¯ u13,β u¯13,3 # + O(ε2) (2.12)
so that the strain tensor can be obtained as: e(¯uε) = ε−1¯e−1+ ¯e0+ ε¯e1+ O(ε2) (2.13) with: ¯ e−1 = 0 (2.14) ¯ ek= Sym(¯ukα,β) 1 2(¯u k 3,α+ ¯ukα,3) 1 2(¯u k 3,α+ ¯ukα,3) u¯k3,3 = Sym(¯u k ,1⊗ i1+ ¯uk,2⊗ i2+ ¯uk,3⊗ i3) (2.15) with k = 0, 1. 2.1.1.4 Concerning equilibrium
With reference to the work by [22, 25], the stress fields in the rescaled adhesive and adherents, ˆσε = σ◦ ˆp−1and ¯σε = σ◦ ¯p−1respectively, can be represented as asymptotic expansions: σε= σ0+ εσ1+ O(ε2) (2.16a) ˆ σε= ˆσ0+ ε ˆσ1+ O(ε2) (2.16b) ¯ σε= ¯σ0+ ε ¯σ1+ O(ε2) (2.16c)
Equilibrium equations in the interphase: As body forces are neglected in the
adhesive, the equilibrium equation is:
div ˆσε= 0 (2.17)
Substituting the representation form (2.16b) into the equilibrium equation (2.17) and using (2.3), it becomes:
0 = σˆiα,αε + ε−1σˆi3,3ε
= ε−1ˆσi3,30 + ˆσiα,α0 + ˆσ1i3,3+ εˆσiα,α1 + O(ε) (2.18) where α = 1, 2. Eq. (2.18) has to be satisfied for any value of ε, leading to:
ˆ
σ0i3,3 = 0 (2.19)
ˆ
σi1,10 + ˆσi2,20 + ˆσ1i3,3 = 0 (2.20)
Eq.(2.19) shows that ˆσ0
i3is not dependent on z3 in the adhesive, and thus it can be written:
ˆσ0i3 = 0 (2.21)
where [·] denotes the jump between z3= 12 and z3 = −12.
In view of (2.21), Eq. (2.20), for i = 3, can be rewritten in the integrated form
[ˆσ331 ] = −ˆσ13,10 − ˆσ023,2 (2.22)
Equilibrium equations in the adherents: The equilibrium equation in the adher-ents is:
div ¯σε+ ¯f = 0 (2.23)
Substituting the representation form (2.16c) into the equilibrium Eq. (2.23) and taking into account that it has to be satisfied for any value of ε, it leads to:
div ¯σ0+ ¯f = 0 (2.24)
div ¯σ1 = 0 (2.25)
2.1.1.5 Matching external and internal expansions
Due to the perfect bonding assumption betweenBε and Ωε±, the continuity conditions at Sε
± for the fields uε and σε lead to matching relationships between external and internal expansions [22, 25]. In particular, in terms of displacements the following relationship have to be satisfied:
uε(xα, ± ε 2) = ˆu ε(z α, ± 1 2) = ¯u ε(z α, ± 1 2) (2.26)
where xα := (x1, x2), zα:= (z1, z2) ∈S. Expanding the displacement in the adherent, uε, in Taylor series along the x3−direction and taking into account the asymptotic expansion (2.7a), it results:
uε(xα, ± ε 2) = u ε(x α, 0±) ± ε 2u ε ,3(xα, 0±) + · · · = u0(xα, 0±) + εu1(xα, 0±) ± ε 2u 0 ,3(xα, 0±) + · · · (2.27) Substituting Eqs. (2.7b) and (2.7c) together with formula (2.27) into continuity con-dition (2.26), it holds true:
u0(xα, 0±) + εu1(xα, 0±) ± ε 2u 0 ,3(xα, 0±) + · · · = uˆ0(zα, ± 1 2) + εˆu 1(z α, ± 1 2) + · · · = u¯0(zα, ± 1 2) + ε¯u 1(z α, ± 1 2) + · · · (2.28)
After identifying the terms in the same powers of ε, Eq. (2.28) gives: u0(xα, 0±) = uˆ0(zα, ± 1 2) = ¯u 0(z α, ± 1 2) (2.29) u1(xα, 0±) ± 1 2u 0 ,3(xα, 0±) = uˆ1(zα, ± 1 2) = ¯u 1(z α, ± 1 2) (2.30)
Following a similar analysis for the stress vector, analogous results are obtained [22, 25]: σi30(xα, 0±) = ˆσi30(zα, ± 1 2) = ¯σ 0 i3(zα, ± 1 2) (2.31) σi31(xα, 0±) ± 1 2σ 0 i3,3(xα, 0±) = ˆσi31(zα, ± 1 2) = ¯σ 1 i3(zα, ± 1 2) (2.32) for i = 1, 2, 3.
Using the above results, it is possible to rewrite Eqs. (2.21) and (2.22) in the following form:
[[σ0i3]] = 0, i = 1, 2, 3 [[σ133]] = −σ0
13,1− σ023,2− hhσ33,30 ii (2.33)
where [[f ]] := f (xα, 0+) − f (xα, 0−) is taken to denote the jump across the surface S of a generic function f defined on the limit configuration obtained as ε → 0, as schematically outlined in Fig. 2.1, while it is set hhf ii := 12(f (xα, 0+) + f (xα, 0−)).
It is worth to point out that all the equations written so far are independent of the constitutive behavior of the material.
2.1.1.6 Concerning constitutive equations
The specific constitutive behavior of the materials is now introduced [22, 25]. In partic-ular, the linearly elastic constitutive laws for the adherents and the interphase, relating the stress with the strain, are given by the equations:
¯
σε= A±(e(¯uε)) (2.34a)
ˆ
σε= Bε(e(ˆuε)) (2.34b)
where A±ijkl, Bijklε are the elasticity tensor of the adherents and of the interphase, re-spectively.
It is worth pointing out that in order to achieve the interface law via this asymp-totic approach, the only assumption on the constitutive behavior of constituents, to do necessarily, is that of linear elastic materials. Thereby, no assumption is herein made on the anisotropy of both constituents and on their soundness.
In what follows, within the framework of the imperfect interface approach it has been shown that it is possible to account for different interphase anisotropy conditions and for damage phenomena in the interphase.
In the following section, reference is made to the analysis of the interphase behavior, detailing both the soft and the hard interphase cases.
2.1.1.7 Internal/interphase analysis
Recalling the seven-regimes distinguish made by Benveniste and Miloh [20] (see Section 1), basically two of these typologies of interphase are considered in the present work of thesis. The first interphase type, called soft interphase, is defined as an interphase material whose elastic properties are linearly rescaled with respect to the interphase thickness ε. The second type, referred as hard interphase, is characterized by elastic
moduli, which, on the contrary, do not depend on the thickness ε. The matched
asymptotic expansion method applied to soft and hard interphases gives rise to soft and hard interface laws, respectively.
These two cases are relevant for the development of the interface laws classically used in technical problems. Moreover, models of perfect and imperfect interfaces, which are currently used in finite element codes, are known to arise from the hard and the soft interface conditions expanded at the first (zero) order [16, 17, 20–22]. The interface laws at the higher order, both in the soft and in the hard cases, are object of recent studies [25] which are recalled in the following.
Soft interphase analysis: Assuming that the interphase is soft, let the interphase elasticity tensor Bε be defined as [25]:
Bε= εB (2.35)
where tensor B does not depend on ε. Referring to Voigt notation rule, its components can be expressed as:
Kkijl:= Bijkl (2.36)
Taking into account relations (2.9), (2.16b) and (2.35), the stress-strain law (2.34b) takes the following form:
ˆ
σ0+ ε ˆσ1 = B(ˆe−1+ εˆe0) + o(ε) (2.37)
As Eq. (2.37) is true for any value of ε, the following expressions are derived: ˆ
σ0 = B(ˆe−1) (2.38a)
ˆ
Substituting Eq. (2.36) into Eq. (2.38a) it results: ˆ
σij0 = Bijklˆekl−1 = Kkijleˆ−1kl (2.39) and using Eq. (2.10), it follows that:
ˆ
σ0ij = K3juˆ0,3 (2.40)
for j = 1, 2, 3. Integrating Eq. (2.40) written for j = 3, with respect to z3, it results: ˆ
σ0i3= K33 ˆu0
(2.41) which represents the classical law for a soft interface at the zero-order.
Recalling a recent study by Rizzoni et al. [25], it is possible to formulate the soft interface law at the one-order. Accordingly, by substituting the expression (2.36) into (2.38b) and by using formula (2.11) written for k = 0, one has:
ˆ
σ1ij = K1juˆ0,1+ K2juˆ0,2+ K3juˆ1,3 (2.42) for j = 1, 2, 3. Moreover, by taking into account formula (2.40), written for j = 1, 2, the equilibrium equation (2.20) explicitly becomes:
(K31uˆ0,3),1+ (K32uˆ0,3),2+ ( ˆσ1i3),3= 0 (2.43) and thus, integrating with respect to z3 between −12 and 12, it gives:
ˆσ1i3 = −K31 ˆu0
,1− K 32 ˆu0
,2 (2.44)
It is worth remarking that the stress components ˆσi30 (with i = 1, 2, 3) are indepen-dent of z3, because of the Eq. (2.19). Consequently, taking into account Eq. (2.40) written for j = 3, the derivatives ˆu0i,3 are also independent of z3; thus, the displacement components ˆu0
i are a linear functions of z3. Therefore, Eq. (2.44) reveals that the stress components ˆσ1
i3, with i = 1, 2, 3, are linear functions in z3, allowing to write the following representation form for the stress components:
ˆ
σ1i3= ˆσ1i3 z3+ h ˆσ1i3i (2.45)
where hf i(zα) := 12 f (zα,12) + f (zα, −12). Substituting equation (2.42) written for j = 3 into expression (2.45) and integrating with respect to z3 it yields:
h ˆσ1i3i = Kα3hˆu0i,α+ K33 ˆu1
where the sum over α = 1, 2 is implied. Combining equations (2.44), (2.45) and (2.46), it results: ˆ σ1(zα, ± 1 2)i3 = K 33[ˆu1](z α) + 1 2(K α3∓ K3α)ˆu0 ,α(zα, 1 2) + 1 2(K α3± K3α)ˆu0 ,α(zα, − 1 2) (2.47)
The soft interface laws at zero-order and at one-order, expressed by Eqs. (2.41) and (2.47) respectively, have to be formulated in their final form in terms of the stresses and displacements fields in the final configuration (see Fig. 2.1-(c)). To this aim, using the matching relations (2.29)-(2.32), the final formulations of the soft interface laws at zero-order and at one-order, respectively, are the following [25]:
σ0(·, 0)i3 = K33[[u0]], (2.48)
σ1(·, 0±)i3 = K33([[u1]] + hhu0,3ii) + 1 2(K α3∓ K3α)u0 ,α(·, 0+) +1 2(K α3± K3α)u0 ,α(·, 0−) ∓ 1 2σ 0 ,3(·, 0±)i3 (2.49)
where the symbol (·) represents the coordinates (x1, x2) in a generic point of the sys-tem Ω+∪ Ω− in the final configuration. In detail, Eq. (2.48) represents the classical spring-type interface law, derived from an interphase characterized by a finite stiffness. Moreover, Eq. (2.49) allows to evaluate the stress vector at the higher (one) order, highlighting that the stress vector σ1(·, 0±)i3 depends not only on displacement jump at one-order but also on the displacement and stress fields evaluated at the zero-order and their derivatives.
In order to have a complete expression of the effective stress field in the reference configuration (see Fig. 2.1-(a)), Eqs. (2.16b) and (2.7c) must be substituted in Eqs. (2.48) and (2.49). Finally, it results:
σε(·, 0±)i3 ≈ K33[[uε]] + ε K33hhuε ,3ii +1 2(K α3∓ K3α)uε ,α(·, 0+) +1 2(K α3± K3α)uε ,α(·, 0−) ∓ 1 2σ ε ,3(·, 0±)i3 (2.50) It is worth remarking that Eq. (2.50) improves the classic interface law at zero-order by linearly linking the stress vector and the relative displacement via a higher order term, involving the in-plane first derivatives of the displacement. Moreover, (2.50) allows to clearly quantify the error committed in the interface constitutive equation by modeling a ε-thick layer with a soft interface law at the zero-order (first right-side
term in Eq. (2.50)). In particular, if the in-plane gradient of displacement and/or the out-of-plane gradient of stress are relevant, the additive term in Eq. (2.50)) depending on ε can not be neglected in the interface constitutive law.
Hard interphase analysis: For a hard interphase, the elasticity tensor Bε takes the following form [22, 25, 26]:
Bε= B (2.51)
where the tensor B does not depend on ε, and Kjl is still taken to denote the matrices such that Kkijl := Bijkl (Voigt notation).
Taking into account relations (2.9) and (2.16b), the stress-strain equation (2.34b) takes the following form:
ˆ
σ0+ ε ˆσ1 = B(ε−1ˆe−1+ ˆe0+ εˆe1) + o(ε) (2.52) As Eq. (2.52) is true for any value of ε, the following conditions are derived:
0 = B(ˆe−1) (2.53a)
ˆ
σ0 = B(ˆe0) (2.53b)
Taking into account Eq. (2.10) and the positive definiteness of the tensor B, relation (2.53a) gives:
ˆ
u0,3= 0 ⇒ [ˆu0] = 0 (2.54)
which corresponds to the kinematics of the perfect interface. Substituting Eq. (2.11) written for k = 0 into (2.53b) one has:
ˆ
σ0ij = K1juˆ0,1+ K2juˆ0,2+ K3juˆ1,3 (2.55) for j = 1, 2, 3. Integrating Eq. (2.55) written for j = 3, with respect to z3, it results:
[ˆu1] = (K33)−1σˆ0i3− Kα3uˆ0,α
(2.56) Recalling the Eq. (2.55) ( written for j = 1, 2), equilibrium equation (Eq. (2.20)) explicitly becomes:
(K11uˆ0,1+ K21uˆ0,2+ K31uˆ1,3),1+ (K12uˆ0,1+ K22uˆ0,2+ K32uˆ1,3),2+ ( ˆσ1i3),3= 0 (2.57) and thus, integrating with respect to z3 between −1/2 and 1/2 and using (2.56), it gives: ˆσ1i3 = − Kαβuˆ0,β− K3α[ˆu1] ,α = − Kαβuˆ0,β− K3α(K33)−1 ˆ σ0i3− Kβ3uˆ0,β ,α (2.58)
with the Greek indexes (α, β = 1, 2) are related, as usual, to the in-plane (x1, x2) quantities.
It is worth noting that in Eq. (2.58) higher order effects occur and they are related to the appearance of in-plane derivatives, which are usually neglected in the classical first (zero) order theories of interfaces [22, 25, 26]. These new terms are related to second-order derivatives and as a consequence, indirectly, to the curvature of the deformed interface. By non-neglecting these terms it is possible to model a membrane effect in the adhesive [25].
In the hard case also, it is possible to derive a final form of the interface laws in terms of the stresses and displacements fields in the final configuration (Fig. 2.1-(c)). Using matching relations (2.29), (2.30) and (2.31), (2.32) the interface laws, calculated both at zero-order and at one-order, can be rewritten as follows [25, 26]:
[[u0]] = 0 (2.59) [[u1]] = −(K33)−1σ0i3− Kα3u0,α − hhu0 ,3ii (2.60) [[σ0 i3]] = 0 (2.61) [[σ1 i3]] = − Kαβu0,β+ K3α(K33)−1 σ0i3− Kβ3u0,β ,α− hhσ 0 ,3 i3ii (2.62) Eqs. (2.59) and (2.61) represent the classical perfect interface law characterized by the continuity of the displacement and stress vector fields [20]. Additionally, Eqs. (2.60) and (2.62) are imperfect interface conditions, allowing jumps in the displacement and in the stress vector fields at the higher (one) order across the interfaceS [25]. Moreover, Eqs. (2.60) and (2.62) highlight that these jumps depend on the displacement and the stress fields at the zero-order and on their first and second order derivatives [26].
As done in the soft case, the constitutive law for the hard interface written in terms of displacement jumps and stresses in the reference configuration (Fig. 2.1-(a)) can be derived (with reference to [25, 26]). By considering the expansions (2.16a) and (2.7a) combined with Eqs. (2.59), (2.60), (2.61) and (2.62). The obtained imperfect interface laws reads as:
[[uε]] ≈ −ε(K33)−1σεi3+ Kα3uε,α − hhuε,3ii (2.63) [[σε i3]] ≈ ε − Kαβuε,β+ K3α(K33)−1 σεi3− Kβ3uε,β ,α −hhσε,3 i3ii (2.64) For the sake of completeness about the imperfect interface obtained via asymp-totic methods, it is worth remarking that Rizzoni et al. [25] formulated an imperfect interface analysis via an asymptotic method based on energy minimization. Their ap-proach, based on the weak formulation of the equilibrium problem is useful to ensure
the soundness of the method based on the matched asymptotic expansions. Moreover, they demonstrate that both approaches lead to the same governing equations of the imperfect interface, both at first (zero) order and second (one) order. As it is well-established, the weak formulation is the development basis of numerical procedures, such as finite element method. Consecutively, these latter can be used to perform nu-merical analyses aimed at evaluating the influence of the higher order effects in the overall response of the interface, especially in the hard case, as done by Dumont at al. [26]. For further details on the energetic asymptotic method, which is not considered in the context of this thesis, refer to former papers [23, 25].
The matched asymptotic expansions formulation, detailed in this section, has been used to derive the imperfect interface models proposed within the framework of this thesis and fully detailed in the Chapters 3 and 4.
Finally, it is worth anticipating that in Chapter 4 the proposed asymptotic pro-cedure is generalized within the context of the finite strain theory [90, 92] in order to derive an imperfect interface model able to take into account large deformations occurring at the interface level.
2.2
Homogenization for microcracked media
The imperfect interface models presented in this thesis are derived starting from a heterogeneous interphase, comprising an elastic material in which a microstructure is embedded. This section focuses on this kind of interphase materials, particularly, the adopted technique to achieve their effective elastic moduli is detailed. The homogeniza-tion of a heterogeneous material, is classically understood, from a mathematical point of view, as an asymptotic technique [34, 35], nevertheless, the homogenization method considered, within the framework of this thesis, follows a micromechanical approach [43, 44].
In detail, the considered microstructure consists in a random or non-random micro-cracks distribution embedded into the elastic interphase material. This assumption is made in order to derive an interface model able to take into account for damage. In what follows (Chapters 3 and 4), it will be shown that different interface models de-rive starting from different assumptions on interphase material and on the microcracks structure.
As it is well-established [34], the homogenization techniques for microcracked media have been, in the most cases, derived from more general formulations for dry pores (or voids) of various shapes, e. g. spherical or ellipsoidal [10, 45–47]. For this reason, the
homogenization theories herein considered, which has been discussed in the following, can be extended to the case of dry pores [28, 48–50].
2.2.1 Recall on the Eshelby’s problem
Let start with a brief recall of the Eshelby’s problem, which establishes the general background of the homogenization methods for heterogeneous media in linear poroe-lasticity. The original problem treated by Eshelby in his 1957 paper [43] concerns an elastic inclusion in an infinite elastic homogeneous medium, herein one refers to the formulation of Dormieux et al. [51] in the case of microporoelasticity, also reported in his thesis by Nguyen [52]. Let identify a representative elementary volume (REV) Ω of boundary ∂Ω, comprising of a solid phase Ωs of boundary ∂Ωs and of the spherical pores (or elliptic cracks) which occupy the complementary domain Ωpof boundary ∂Ωp. The pores are supposed to be filled with a fluid which induces a hydrostatic pressure p. For the sake of simplicity, let assume that the pores do not intersect the REV boundary ∂Ω. Moreover, let define the volume fraction φ of pores, as:
φ = Vp V
where Vp and V are the volume of the domain Ωp and the volume of the overall domain Ω, respectively. Let assume the REV Ω be subjected to a constant macroscopic strain tensor E, as a result the displacement field along the boundary ∂Ω is : u(¯x) = E.¯x with ¯x ∈ ∂Ω. The relation between the average strain ¯ over the domain Ω and the macroscopic strain E reads as:
¯ = hiΩ = 1 V Z ∂Ω (n ⊗su) dS = E (2.65)
where ⊗s is the symmetric tensorial product between vectors defined as a ⊗s b = 1
2(a ⊗ b + b ⊗ a) and h•iΩ= 1 V
R
Ω(•) dΩ is the average operator over the solid domain Ω of volume V . This relationship points out that the local strain fields in the REV domain can be considered as the overlap of a constant field given by the imposed boundary conditions and of a perturbation depending on the local elastic properties and on the interaction between the phases. The microscopic stress field is defined as:
σ(x) = Cs: A : E − Cs: A∗p (2.66)
where Cs is the local elasticity tensor of the solid phase; A and A∗ are the fourth rank and the second rank localization tensor in terms of strain, respectively. As a result, the
average stress hσi over the domain Ω is: Σ = hσiΩ= 1 V Z Ω σ(x) dΩ = 1 V Z Ωs σ(x) dΩ + Z Ωp −pI dΩ ! = (1 − φ)hσis− φpI (2.67) where hσis = V1 s R
Ωsσ(x) dΩ is the average stress over the solid domain Ωs. Introducing
(2.66) in (2.67), it is possible to re-write the macroscopic stress Σ as follows [51]:
Σ = hCijkl: Apqkh: ¯khi = ˜C : E − pBb (2.68)
where Bb is the Biot tensor.
In the case of dry pores (p = 0), Eq. 2.68 becomes:
Σ = ˜C : E (2.69)
where the macroscopic elasticity tensor ˜C is given by the following relation:
˜
C = (1 − φ)Cs: hAis= Cs(I − φhAip) (2.70)
where hAip is the average localization tensor in the dry pores.
The Eshelby’s problem defines the Hill tensor P0 in the case of parallel ellipsoidal pores [51], as:
P0 = SE : C−10 = SE : S0 (2.71)
where S0and C0 are respectively the compliance and the elasticity tensors of the infinite medium. SE is a fourth rank tensor, called Eshelby’s tensor, which depends on the ellipsoidal shape, on the orientation of the pores and on C0. It worth to remark that:
hAip = [I − SE]−1 (2.72)
If the solid matrix is made of an isotropic material, C0 assumes the following form:
C0= 3k0J + 2µ0K (2.73)
where k0 and µ0 are the bulk modulus and the shear modulus of the infinite medium, respectively; J = 13δij⊗ δij and K = I − J are the projection tensors on the spaces of the fourth rank isotropic tensors and deviatoric tensors, respectively. For further details on Eshelby homogenization one can refer to Dormieux’s book [51].
2.2.2 Non-interacting approximation (NIA)
In the present thesis one focuses on the non-interaction approximation (NIA) to the problem of effective properties whereby interactions between inhomogeneities are ne-glected. The reasons of this choice are manifold, and they are briefly detailed in what follows (the reader can be found more details in the works of Sevostianov and Kachanov [30, 49, 53–56]). The NIA constitutes, surely, the simplest approach to the problem of determinate the effective properties of a heterogeneous material that allows one to focus on the effect of inhomogeneity shapes -in particular, of mixtures of diverse shapes-without interference of the interaction effects.
In the NIA micromechanical approach the quantities directly related to the inhomo-geneities are proportional to a proper microstructural parameter, which gives a measure of the microcracks (or pores) density and which choice is not so trivial as extensively discussed in [27, 28, 30]. It is worth to anticipate that such parameters do not, generally, reduce to volume fractions.
The distinction between the NIA and the dilute limit approximation (or dilute con-centration) needs to be clarified [53]. These two assumptions are often treated as syn-onyms; on the contrary, they are quite different. Substantially, the small concentration assumption automatically imply to neglect the interactions between inhomogeneities, the inverse is not necessarily true. Thereby, it is worth noting that the NIA predic-tions may remain sufficiently accurate at substantial concentrapredic-tions [30]. It has been estimated that until 15 ÷ 20% of the concentration parameter, dilute limit and NIA give comparable results in terms of effective elastic properties [53] (see Fig. 2.3). If the concentration parameter is small then, of course, interactions are weak (provided mutual positions of inhomogeneities are random). However, the effect of interactions on the overall properties may remain weak at finite, even substantial, concentrations, principally for two reasons. Firstly, interactions can be weak, even in the local sense, at finite concentrations: at distances of one pore radius, field perturbations due to the pore become negligible. Secondly, local interactions may produce opposite effects of field amplification or field shielding, so that the overall effect of interactions remains weak, due to their reciprocal override. In other words, smallness of the geometrical pa-rameter (ratio of the inhomogeneity size to spacing between them), or the dilute limit, is a more restrictive assumption than the non-interaction approximation [30]. Several comparisons with experimental data are made by Sevostianov and coauthors in order to establish a range of concentration parameter values for NIA homogenization. In details, in [57] a comparison of the effective elastic stiffnesses of short-fiber reinforced thermo-plastics, calculated using NIA, with experimental data is carried out, highlighting a
good accuracy of the prediction estimates until 40% of the concentration parameter. Moreover, direct measurements of stiffness in closed cell aluminum foam are compared with NIA predictions by [58], obtaining a very good agreement until 75 ÷ 85% of the porosity parameter.
Referring to Fig. 2.2, it is important noting that the formulation within the frame-work of the NIA, strictly speaking, predict only the initial slope (∂E/∂φ)φ=0where E is the effective physical constant of interest and φ is the parameter of concentration. This slope may correspond to more than one formulation, for example, to E(φ) = E0+φ E0(0) or to E(φ) = [E0− φ E0(0)]−1. One of them may remain accurate at substantially larger range of concentrations: the linearization, i.e. the dilute limit, may substantially reduce this range. Therefore, one should not automatically linearize the NIA results.
E
Φ
NIA dilute limitFig. 2.2: Comparison among NIA and dilute limit behavior - Scheme of the path of a generic elastic moduli E(φ) with respect the parameter of concentration φ under the NIA and the dilute limit assumption.
The NIA constitutes the basic building block for various approximate schemes that account for interactions by placing non-interacting inhomogeneities into some sort of ef-fective environment, basically one can distinguish between efef-fective matrix approaches and effective field approaches.
In the framework of effective matrix schemes, each inhomogeneity, treated as an isolated one, is placed into certain homogeneous material that have different properties with respect to the bulk matrix. In what follows the principal schemes, developed in literature within this approach, are briefly outlined. In the self-consistent scheme, the homogeneous material possesses the (unknown) effective properties. This scheme was first employed, probably, by Clausius [59] and then developed in works of Bruggeman [60] for effective conductivity, by Kr¨oner [61] for elastic properties of polycrystals, by Hill and Budiansky [62] for effective elastic properties of matrix composites, and by Budiansky and O’Connell [63] for cracked materials. The generalized self-consistent
scheme first proposed by Kerner [64] and then developed by Christensen and Lo [65] differs from the self-consistent one in introducing an intermediate layer between the inhomogeneity and the virgin matrix material. Explicit formulas for this scheme have been obtained only in cases of 3-D spherical or 2-D circular inhomogeneities. The differ-ential scheme can be considered as an infinitesimal version of the self-consistent scheme. It was first proposed by Bruggeman [60] and was further developed by McLaughlin [66], and Zimmerman [67]. It introduces inhomogeneities in small increments until the final volume fraction is reached. On each increment, a set of non-interacting inhomogeneities is added to the homogenized material with the properties determined by the previously embedded inhomogeneities. As pointed out by McLaughlin [66], the total concentra-tion of inhomogeneities introduced to the matrix does not add up to the final volume concentration since certain fraction of “new” inhomogeneities falls into places already occupied by the “old” ones.
The above-mentioned schemes in the effective matrix framework, all have the short-coming that in anisotropic case, they need the type of the overall anisotropy and its orientation to be hypothesized a priori.
Within the framework of the effective field schemes each inhomogeneity, also in this case treated as a single one, is placed into the unaltered matrix material. The interactions between the inhomogeneities are accounted for by assuming that the iso-lated inhomogeneity is subjected to a field that differs from the remotely applied one. In the simplest version, the Mori–Tanaka’s scheme, this field is taken as average over the matrix and it is assumed to be the same for all inhomogeneities. In more advanced Kanaun–Levin’s method [68], the effective field can reflect the statistics of mutual po-sitions of inhomogeneities; for multi-phase composites, it may be different for different sets of inhomogeneities [69]. In the Maxwell’s scheme [70] the far field produced by the considered set of inhomogeneities is equated to the far field produced by a fictitious do-main of certain shape that possesses the (unknown) effective properties. It is worth to note that the choice of the mentioned shape is non-trivial. Maxwell scheme, originally developed for electrical conductivity of a matrix containing spherical inhomogeneities, has been extended to the elastic properties of a material containing either randomly oriented [71] or parallel [72] ellipsoidal inhomogeneities of identical aspect ratios. Addi-tionally, this scheme has been, recently, discussed by Sevostianov and Kachanov [53, 54] and by Levin et al. [73].
Both effective matrix and effective field schemes aim to obtain the so-called property contribution tensor of the inhomogeneity. Under the assumptions of linear elasticity and small perturbation, the determination of such a tensor is a crucial issue. Note that the summation over it gives the change in the effective properties due to the presence
of inhomogeneities.
Within the framework of determination of the effective elastic properties and under the assumptions of linear elasticity and small perturbation, the property contribution tensor of an inhomogeneity is a crucial issue: summation over it gives the change in the effective properties due to the presence of inhomogeneities. The inhomogeneity contribution tensor, generally, depends on the physical constants of the matrix and the inhomogeneity, and on the shape and orientation of the latter. Let the average strain, over the RVE Ω of volume V comprising the volume of pores Vp, be represented as the following sum:
= S0 : σ0+ ∆ (2.74)
where S0 is the compliance tensor of the matrix. The traction vector on ∂Ω have the form t|∂Ω = σ0 · n where σ0 is a constant tensor also called far-field, or remotely applied, stress [45], and n is the positive-outer normal. In absence of inhomogeneities, it would have been uniform in Ω. The material is assumed to be linear elastic, hence the additional strain (average over Ω) due to inhomogeneities is a linear function of the applied stress: ∆ = V p V H : σ 0 (2.75) where H is a fourth-rank compliance contribution tensor of the inhomogeneity. In the case of multiple inhomogeneities, the additional compliance due to their presence is given by their sum as follows [55]:
∆S = 1
V X
k
VkpHk (2.76)
This kind of formulation is also referred to as stress-based approach. It is often ad-vantageous to formulate the problem of effective elastic properties in terms of the complementary elastic potential f (σ) -a quadratic function of stress- such that the effective compliances Sijkl are given by differentiation: ij = Sijklσkl = ∂f /∂σij. The main advantage of using this formulation is that the symmetrization imposed by the potential often leads to simplifications that may otherwise be overlooked. In this case, one represents the elastic potential in stress f (σ) = 12σ : S : σ as a sum-representation similar to the one used by Eshelby [44]:
f = f0+ ∆f (2.77)
where f0 is the potential of the bulk material and in the case of the isotropic matrix of mechanical properties E0 and ν0, is given by
f0 = 1 + ν0 2E0 σijσji− ν0 2E0 (σkk)2 (2.78)
![Fig. 2.3: Comparison between homogenization techniques for microcracked media - The effective modulus E N (ρ) obtained in NIA stress-based (see Chapter 3) and strain-based approaches [80] is compared with those obtained in dilute limit scheme (DL), Mori-Ta](https://thumb-eu.123doks.com/thumbv2/123doknet/14505124.719971/50.892.279.682.211.477/comparison-homogenization-techniques-microcracked-effective-obtained-chapter-approaches.webp)
![Fig. 3.4: REA with a linear crack - Sketch of the ε-thick representative elementary area (REA) taken into account in the 2-d homogenization process, according the brick/mortar interface model by Rekik and Lebon [87, 88].](https://thumb-eu.123doks.com/thumbv2/123doknet/14505124.719971/63.892.162.669.213.455/sketch-representative-elementary-account-homogenization-process-according-interface.webp)
