Damage mechanics of interfacial media: Basic aspects, identification and application to delamination

Publisher Summary This chapter presents the development of a model to bridge between damage mechanics and delamination by including all the damage mechanisms in delamination analysis. For this, a damage meso-modeling that includes both inner layer damage mechanisms and interracial ones is used. Delamination often appears as the result of interactions among different damage mechanisms, such as fiber-breaking, transverse microcracking, and debonding of the adjacent layers themselves. At the meso-level, the laminate is described as a stacking sequence of inelastic and damageable homogeneous layers throughout the thickness and of damageable interlaminar interfaces. One limitation of the meso-modeling is that the fracture of the material is described by means of only two types of macrocracks: (1) delamination cracks within the interfaces and (2) cracks, orthogonal to the laminate, with each cracked layer being completely cracked in its thickness. The ideas and framework that govern the interface damage modeling are similar to those which are used for deriving the layer damage modeling.


INTRODUCTION
Our aim is to build a bridge between damage mechanics and delamination by including all the damage mechanisms in delamination analysis. Delamination often appears as the result of interactions between different damage mechanisms, such as fiber-breaking, transverse micro cracking and debonding of the adjacent layers themselves [1][2][3][4]. Thus a damage meso modeling, proposed in [5][6] and developed in [7][8][9], which includes both inner layer damage mechanisms and intetfacial ones is used At the meso-level, the laminate is described as a stacking sequence of inelastic and damageable homogeneous layers throughout the thickness and of damageable interlaminar interfaces. The single-layer model being identified, the aim is to determine the properties of any structures regarding delamination by knowing only a few characteristics of the interface.
The word interface denotes here a physical yet two-dimensional medium. At the present applications only concem static loading without buckling.
The single layer model and its identification, including damage (such as fiber-breaking, transverse cracking and deterioration of the fiber-matrix bond ) and inelasticity, were 1 previously developed [5],[1 0]. The interlaminar interface is a two-dimensional entity which ensures traction and displacement transfer from one ply to another. Its mechanical behaviour depends on the angles between the fibers of the two adjacent layers.
Here we pay special attention to the basic aspects of the interlaminar interface model: defmition, debonding and sliding effects modeling, qualitative connection with "micro information" and questions concerning identification. Therefore results given in [8] These compatisons require numerical tools in order to simulate, up to failure, the behaviour of any stacking sequence. The use of classical damage modeling for the simulation of failure has led to many theoretical and numerical difficulties which are well-understood at the present time [19]. The solution which is used for laminates, and more generally for composites, is based on the meso-model concept [ 6 ]. The physical meaning of this concept is that the state of damage is uniform in each meso-constituent. For example, the damage state is uniform throughout the thickness of each single layer. To be able to perfmm a complete analysis of the delamination process in all cases, damage models with delay effects are introduced for the in plane direction. These models should be, at least theoretically, combined with a dynamic analysis of the structure [8].

MESO-MODELING CONCEPT FOR COMPOSITE LAMINATES
Let us recall that delamination often appears as an interaction between fiber-breaking, transverse micro-cracking and debonding of adjacent layers itself. In order to take these mechanisms into account, the first issue is the scale at which they are modeled. For laminates three different scales may easily be defined, the micro scale of individual fiber, the meso scale associated with the thickness of the elementary ply and the macro scale which is the structural one. Due to the low thickness ( 1/10 of mm) of the elementary ply and the kinematics of the detetioration inside of the ply (fiber oriented ) it is possible and of interest to derive a material model at the mesoscale. The one proposed in [5][6] is defmed by two meso-constituents: -a single layer -an intetface which is a mechanical smface connecting two adjacent layers and depending on the relative orientation of their fibers. Let us recall that, in order to be able to perform a complete analysis of the delamination process in all cases, damage models with delay effects are introduced for the in-plane direction. These model should be, at least theoretically, combined with a dynamic analysis of the structure [8].

Interface
One limitation of the proposed meso-modeling is that the fracture of the material is described by means of only two types of macrocracks: -delamination cracks within the inte1faces -cracks, orthogonal to the laminate. with each cracked layer being completely cracked in its thickness.
Let us re.:all that the single layer model and its identification. including damage such as fiber-breaking and transverse micro-na.:king as well as inelastic effects were previously developed in l5].l!O]. In paragraph 3 the interface model is detailed.

Elastic modeling of the interface
The scheme which leads to the definition of the interface is classical for isotropic hi materials. The interlaminar connection is considered as a ply of matrix (denoted by Q) whose thickness (denoted by e). is small compared to the in-plane dimension whose characteristic length is denoted by L (figure 2.). where !l is the displacement field. Let us denote t the transposition, then: where: [!l] = u+ -.uis the difference of displacements between the upper and lower surfaces of Q. Thus, at the first order, the strain energy of Q is: where r is the area of the mid-plane of Q, and H is a (3,3) symmetric matrix. Let us denote by Q,l) the bisectors of the fiber directions. They are necessarily "orthotropic" directions of the interface, since a [8t. 8 2 ] interface is equivalent to a [8z,8t] interface. Then in the (Q,2.� axis, the elastic strain energy of the interface may be written as follows: The interlaminar connection is thus modeled as a two-dimensional entity which ensures stress and displacement transfers from one ply to another. Therefore, e being small, k0, kb, k6 have very high values and the interlaminar connection behaves, in elasticity, as a perfect bond: it ensures displacements and traction continuities. In the non-linear case, it ensures traction continuity only.

2 Interfacial damage indicators
The ideas and framework which govern the inte1face damage modeling are similar to those which are used for deriving the layer damage modeling [5], [10]. The effect of the deterioration of the interlaminar connection on its mechanical behaviour is taken into account by means of damage internal variables. The different damageable behaviour in "tension" and in "compression" are distinguished by splitting the strain energy into "tension-energy " and "compression-energy". More precisely we use the following expression, proposed in [5], of the energy per unit area: thus three internal damage indicators, associated with the three Fracture Mechanics modes are introduced.

Interfacial damage evolution laws
These evolution laws must satisfy the Clausius-Duheim inequality. Classically the damage energy release rates, associated with the dissipated energy ro, by damage and by unit area, are introduced: In what follows two type of damage evolution laws are described. The first type is based on the assumption that the evolution of the different damage indicators is strongly coupled and driven by a unique equivalent damage energy release rate. This type could be called "isotropic", even if the behaviour is the different mode are different. This is the type of model which has been used so far. The different versions mainly differ in the choice of the coupling.
The second class is inspired by our knowledge of the damage behaviour of the single layer. The in-plane behaviour of the layer is nearly brittle in normal tension and "ductile" in shear. Thus it can be assumed that the behaviour in mode I is btittle and that the behaviour in modes II and III is "ductile".

Isotropic dama&e evolution law
The following model, proposed in [14], [21], considers that the damage evolution is governed by means of an equivalent damage energy release rate of the following form: (2) this means that (i ) the evolution of the damage indicators are assumed to be coupled (as for single layers) (ii) the damage evolution depends (m ainly) on the maximal value of the equivalent damage energy release rate. y 1 , y 2 and a are material parameters. In terms of delamination modes, the first term is associated with the frrst opening mode, and the two others are associated with the second and third modes.
Compared to other damage evolution laws, used for example in [5][6][7][8][9], an enhanced coupling model, associated with the parameter a is proposed. The interest is to describe Fracture Mechanics failure loci which are quite general ( see paragraph 4.2).
A damage evolution law is then defined by the choice of a material function ro, such that : a simple case, used for application, is: where a critical value Yc and a threshold value Y0 are introduced. The high values of n case corresponds to brittle interface.
To summarize, the damage evolution law is defined by means of six intrinsic material parameters Y c.Y 0, y1, y2, a and n. It is shown in paragraph 4 that Y c. y 1 , y2 and a. are related to the critical energy release rates. 1l1e threshold Y 0 is introduced here in order to enlarge the possibility to describe both the creation of a delamination crack and its propagation. As regards the creation of a new delamination crack the significant parameters are Y 0, n and a .

Anisotropic dama!!e evolution law
Here the evolution of d is assumed to be govemed only by� and the evolution of d1 and d2 by a common shear damage energy release rate y12 with: two damage evolution laws are then defined: Consequently the damage evolution laws in mode I and modes II and III are decoupled.
These damage evolutions are always supposed to depend on the maximal value of the driving damage energy release rate.

4 Inelastic effects coupled with damage
Although their identification has not yet been investigated, in-plane inelastic effects, due to friction, certainly exist. Moreover, one may assume that the level ofinterfacial damage modifies the interfacial inelastic properties, as in the layer case [10]. Following [8], these inelastic aspects can be taken into account by introducing an inelastic patt of the displacement discontinuities [U 1 ] P , [U2] P . Damage and inelasticity coupling is then modeled by defining the following effective quantities: Considering the previous relations it appears that no friction is assumed in the normal direction. A plasticity like-model which makes use of these effective quantities is then defined: f defines the elastic domain, a1 and a2 are two material parameters and R(p) is the effective hardening function. A Coulomb like-effect is taken into account by means of the normal effective stress cr�. Within the framework of standard mate1ials one obtains: where: Due to the difficulty of the identification we propose, in a first approach, to chose:

Extension: damage model with delay effects
In order to get, in all cases, a consistent model for the description of the rupture a variant of the previous damage model, introducing delay effects [6] is introduced. It as to be combined, in the principle, with a dynamical analysis of the structure. This variant ensures the physical following properties: -a variation of the d1iving force Y does not lead to an instantaneous variation of d -the damage rate is bounded More precisely the rate of the damage indicator is defined by: In many practical situations, a model without delay effect is sufficient This is the case, in particular, for 2D problems where the crack is described by a line. Let us define the location of the crack by d= l. In order to explicitly integrate the equation (4) , a steady-state situation is assumed. This means that the process zone (defined as the area where d is between 0 and 1) is u·anslated from L'1a over time L'1 T (Fig . 4 ).

Figure. 4 Diagram of a steady-state delamination process
In this case one obtains: In such a steady state process Gc(a) reaches its stabilised value at the propagation denoted by og. This leads to: and using relation (5) the energy release rate is split into three tenns: A mixed mode situation is defined by the mode coupling ratios en and em. such that: Xct1 = en Xct and Xd 2 = em Xct The previous considerations do not depend on the damage evolution law. To go further the type of damage evolution law must be specified. An example of an isotropic damage evolution law is treated below.
In that case relation (9) becomes: Gn = en G r and Gm = em Gr and, making use of relations (3), (7) and (9' ), one obtains: and thus for pure mode situations: Substituting y 1 and y2 in relation (12) and using (9) into account, one obtains: Relations (12) and (13) show that the only significant parameters of the Damage Evolution law are, Yc, y1 , y2 and a. From equation (19) it appears that a governs the shape of the failure locus in mixed-mode.
There exist few experimental results concerning mixed-mode crack propagation. In  In the first situation inner layer damage mechanisms may be activated leading to an apparent energy release rate different from the local interfacial one. In that case, a non-linear damage analysis should be performed.
R curve-like phenomena appear when the size of the non linear domain is comparable to that of the specimen. This happens, for example, in the problem of fiber-bridging. In that case also it appears to be more interesting to use Damage Mechanics rather than LEFM.  In order to clruify this point, two computations have been pe1formed. The first one uses an interface whose behaviour was identified with the (0°/0') test (i. e. Gel= .45N/mm), and the second one uses an interface identified with the (900/90') test (i.e. Gel= 1.05N/mm). where (Gg)a pp is the apparent critical energy release rate obtained by means of the compliance formulae ( denoted here by C), P� is the total dissipated power inside the layer.

Influence of inner-layer damai:e mechanisms
In this case, the last te1m is thus equal to . 6 N/mm.

R-curve phenomena: example of fiber-bridging
The existence of fiber-bridging gives rise to R-curve like phenomenon. This type of phenomenon has long been the subject of intensive research [22][23][24]. The main difference, which occurs, for example in the case of fiber-bridging, is that the size of the equivalent process zone is very large. In [25], one can fmd a "large scale model" which is, in fact, a damage model. In [26], it is proposed to use an inte1face damage model (of the same type as the one presented in this paper) associated with a Griffith criterion at the tip of the fiber bridge. For the elastic moduli of the interface it appears that the specific length of the interface is, in this case, connected to an average length of the part of the fibers involved in the bridge.
This would lead to a value for the rigidity of the interface that is much lower than the usual one. The influence of n is also easy to understand. The lower the value of n, the larger the area which is "significantly" damaged. In order to get an idea of the relative importance of these coefficients on propagation, the case of a D.C. B. specimen with an initial delaminated area of 25 mm was treated. Using the relation d= 1 for the definition of the crack, the most influential parameter is interface stiffness. But this influence is relatively small.
In fact, in the case of fiber-bridging, one problem is to define the value of the length of the crack. Expeiimentally, this value corresponds to the tip of the fiber-bridge.
Consequently a part of the interface specimen ahead of the crack tip is contributing to its stiffness. A simple way to take this aspect into account is to defme the crack tip using a value de of the damage lower than I. In tigure 7 a result obtained with this value fixed at . 5

INITIATION PREDICTION
The study of the creation of a new delamination crack is often investigated by means of Edge Delamination Tension specimens [27]. In such a case Fracture Mechanics is not well adapted. The reason is that the energy release rate tends to zero with the size of the crack.
Moreover, the analysis of stress singularities does not allow a simple comparison of the state of different interfaces. In fact the type and the exponent of the singularity depend on the interface. Thus, for the forecast of the onset of a edge delamination crack, computational methods of elastic edge effects [28][29][30]associated with criteria [16] are used. These criteria are often based upon the average of inter laminar stresses on a specific distance from the free edge. Nevertheless delamination does not always occur where stresses are maximal.
Moreover the specific distance from the free edge does not seem to be an intrinsic parameter when geometry and stacking sequence vary.
In addition, delamination, especially at its onset, appears to result from an intricate interaction between inner layer damage mechanisms and the deterioration of the inter laminar interface itself [3][4]. In that case its seems adequate to use the previously defmed meso modeling for the layer and the interface. This was done in [11] for the prediction of initiation and propagation of delamination and damages around initially circular holes. In this type of situation, classical approaches are clearly insufficient, and damage mechanics appears today to be, perhaps, the only way to deal with such a problem.
In order to emphasise the interest of Damage Mechanics of Interface for the prediction of initiation, let us consider the case where damage phenomena are supposed to be located only on the interface. The delamination analysis is carried out as a post-processor of an elastic laminate shell computation. A specialised software (Edge Damage Analysis) was developed where the layers, modeled as elastic, are connected by damageable interlarninar interfaces [12], [31]. EDT specimens under tension and compression were simulated. In such cases the numerical problem is set in a strip perpendicular to the edge. This type of problem has been studied in a similar way in [32].
The simulations are compared with experimental results, in the case of a TI00-5208 material [16][17] for mode I and mode I &II delamination ( Fig. 8 and 9 ).
In mode I cases, delamination occurs on the mid-plane interface. The values of the longitudinal su·ain at the onset of delamination are compared. The same value of Y c was used for the six cases which have been u·eated. Even they were nearly pure mode I cases, the state of stress is very different from one case to the other. It is thus surprising that only one parameter allowed for obtaining such a similarity ( a maximum relative error of about 10%, figure 8). A tentative explanation is that, in each case, it is the same interface [±0'1 submitted to the same mode of loading which is prone to delamination. In mixed mode cases the values of the coupling coefficient must also be specified. It was assumed that y 1 = y 2 with an estimated value of .2. The comparison is still encouraging and the location of the onset of delamination was con·ectly predicted (Fig. 9). Nevertheless the results are less satisfactory than in pure mode I since an average relative error of 15 % was obtained. A tentative explanation is that the same values of the material characteristics of the    The load history is shown in figure 10.

CONCLUSION
A Meso-Damage modeling of laminates whose aim is to determine the properties of any structures with regards to delamination, by knowing only a few characteristics of the interface has been detailed. Examples show that this approach is promising for the prediction of delamination in various circumstances [11][12][13]. Nevertheless, one of the ftrst challenges will be to build a good interface model, i. e. a model which allows for predicting, using a single set of parameters, both the creation and the propagation of a delamination crack on a large range.
Difficulties are also encountered when trying to get a complete identification of the interlaminar interfacial model. The goal will be then to define canonical tests which allow a complete and precise identification of the interlaminar interface model. Progress can be made by comparing experiments and tests in the case of the creation and beginning of propagation of a new delamination crack.
We also want to insist on the necessity of including more physics in the modeling of the inte1face. In fact, a more precise description of the interlaminar connection with respect to its constitution and geomeuical characteristics could lead to a more precise link between meso and micro-parameters.