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## At the microstructure scale: Statistical models of quasi-brittle fracture

To capture the characteristic intermittency and spatial organization of damage, other ap-proaches borrowing concepts issued from out-of-equilibrium statistical physics have been proposed. In these discrete models, the material is described by an array of discrete bonds or elements. Het-erogeneities are introduced in the system via a statistical distribution of some properties, like failure threshold, or sometimes initial defects such as vacancies. During mechanical loading, the damage evolution is inferred from force balance in the network. The introduction of loads or energy redis-tributions, either explicitly or implicitly, allows to reproduce the complex interplay between disorder and microcracks interactions as the system is loaded up to failure.

The most widely known model is probably the fiber bundle model (FBM). In its simpler formulation, see [93] for a review, it consists in a 1D bundle of brittle fibers having randomly dis-tributed failure strengths. As the external force is quasistatically increased, the weakest elements fail and the force carried by the broken fibers is redistributed among all intact fibers (global load sharing redistribution rule). The redistribution allows to trigger further breaking of fibers and the number of failures taking place between two equilibrium states allows defining an avalanche size. This sim-plified approach to quasi-brittle fracture allows to capture the temporal intermittency and power law distributions of avalanche sizes observed experimentally, as well as study of the evolution of the sys-tem up to its failure.

Moreover, the simplified framework allows for extensive analytical calculations. The criti-cal force at failure has been determined [92] as also the temporal behavior of the precursors to failure. Indeed, a power law decrease of exponent 5/2 [59] is established for the distribution of avalanches sizes when considering all damage events. If one considers a restricted loading frame, for example the last 10 percent of avalanches prior to failure, two power law behaviors emerge: For small avalanches, an exponent 3/2 is determined while for larger ones, the exponent 5/2 dominates [94]. The transi-tion avalanche size can be considered as a cutoff parameter, which is shown to follow a power law increase with the distance to failure, clarifying the nature of the transition to failure. These results are essential as more complicated models usually do not allow for analytical determinations. Hence these behaviors serve as references for comparisons with that obtained numerically or from other the-oretical descriptions.

### Our approach: A mesoscopic description of damage spreading in heteroge- neous media

We chose to benefit from both approaches by using some aspects of non-local continuum damage model together with other aspects used in statistical approaches. We consider a continuous non-local damage model formulated in a thermodynamical framework. The damage growth derives from the comparison of the damage driving force and the damage resistance. The former, the non-local rate of energy restitution is calculated from the free elastic energy. The latter can be viewed as its thermodynamic counterpart, the rate of dissipated energy. This last parameter is chosen as an increas-ing function of damage, representative of the material hardening. The non-locality is introduced via a non-local damage parameter in the stiffness expression. The expression of the interaction function is motivated by the recent study of Demery´ et al. [Demery´ et al.] who calculated the exact expression of the non-local redistribution function in the context of weakly disordered elasto-damageable solids. Moreover, to capture the characteristic fluctuations during damage spreading, we introduce material heterogeneities at a mesoscopic scale through a random distribution of damage resistance.

#### Non-local damage model description of a heterogeneous material

Before detailing our model, we provide a brief general description of our approach. We build our model on classical concepts of continuum damage mechanics. Yet, instead of considering homogeneous material properties, we introduce spatial variations in the material resistance over a

typical scale ξh. To do so we consider media with heterogeneous fields of damage energy. Within this approach, material heterogeneities are introduced at a mesoscopic scale, intermediary between the microscale at which microfracturing processes take place and the macroscale at which the material can be seen as homogeneous. Moreover, the material is characterized by its damage field which rep-resents in each material point the density of formed microcracks within an elementary volume δx3. In principle, addressing numerically damage evolution within such a heterogeneous material would require to mesh the material at a fine scale δx ξh with respect to the heterogeneity size. However, we will see that using a coarser discretization δx = ξh does not affect damage evolution, so that in the following, material heterogeneity and material representative element will be confounded.

Since our main focus is the effect of material heterogeneities on damage spreading within the material, we will consider simple loading conditions. In this first part of the thesis, materials will be loaded uniformly in tension, by the application of a displacement on the upper side of the specimen. During loading, damage will increase and affect the strength of an element via its stiffness, which decreases with damage. A central feature of our model is a non-local interaction function that allows to describe how stresses are redistributed after each damage event. In this study, a main objective is to understand how the nature of these interactions affects damage spreading and localization in heterogeneous solids.

**Energy based damage criterion**

We consider a system of ND elements, where D is the dimension of the system Σ and is equal to 1 or 2, with periodic boundary conditions. Elements are distributed equidistantly in space and indexed by the position ~x of their center in the system of size LD. As discussed the in introduction, these elements coincide with the typical heterogeneities of the material of size ξh. As a result, the total material size is effectively NDξhD. The material is clamped between two rigid plates, as schematically represented in 1D in Fig. 2.1a where each element is represented by a spring: The bottom plate is maintained fixed whereas a uniform macroscopic displacement Δ, perpendicular to the x-axis in 1D and to the (x, y)-plane in 2D, is applied to the upper plate. Due to these clamping conditions, the microscopic displacement is homogeneous, equal to the macroscopic one, such that the macroscopic force writes as Z Z F = σ(~x)d~x = Δ k[d(~x)]d~x (2.1).

where σ is the stress field and k[d(~x)] = σ(~x)/Δ the local stiffness of an element which depends on its damage level d(~x). This scalar damage parameter ranges from zero when the element is intact to one when it is fully broken. For quasi-brittle fracture, it relates to the density of microcracks formed within the elementary volume represented by this element. On a broader perspective, it quantifies the degree of damage and the associated loss of stiffness of the elementary element. The variation of the elastic constant with damage is a central feature of continuum damage models [66]. However, as done in non-local damage models [9], we introduce here a dependency on the non-local parameter, d.

**Failure behavior of the heterogeneous system: numerical observations**

In this section we focus first on 1D systems while 2D systems, that display similar behaviors, are left for Sec. 3.2.3. We study the response of a material made of heterogeneities with unstable failure properties using a negative stiffness parameter a = −0.3. The results are here shown for a hardening parameter value = 8 but the system behavior is the same as long as a and are verifying the conditions described in Table 2.1. We consider several realization of the disorder, the intensity of which (g) ranges from 0.001 to 0.2. The influence of the internal length parameter, with `0 = 5 or 10, and shape parameter is investigated.

**Critical damage at failure: global energy minimization**

We now explore an alternative approach to predict the localization and failure thresholds of the elasto-damageable materials studied here. In the previous section, we performed a stability analysis of the damaged medium based on the distribution of local damage driving force. This allowed us to determine the unstable mode qc over which damage field heterogeneities develop and the critical damage level at which this mode starts to grow. Assuming that the damage distribution remains close to the homogeneous state, solution of the equation Y (d0) = Yc(d0), even after damage localization, we have also determined the critical damage level at which the full system fails. Despite this rather rough approximation, our predictions provided a correct estimate of the failure threshold.

However, in an attempt to provide a better assessment of the failure threshold, we propose here a complementary approach based on the minimization of the total energy of the damaged media. Here, we take advantage of our damage model formulation based on energy consideration. The approach consists in assuming that, at a given applied displacement , the damage field takes a certain shape, either given by Eq. (3.12) or Eq. (3.19), and determine the set of solutions (d0,d0) for which the energy is minimum. Only the admissible solutions are considered: both d0 and d0 must be positive, increasing functions of , otherwise the solution is not taken into account. Unlike for the linear stability analysis where the homogeneous solution (2.19) is used, no relationship is here assumed between the applied loading and the homogeneous contribution of the damage field. As will be shown in this section, the same localization criterion is obtained, whereas the failure prediction shows a better agreement with the numerical results. For the sake of simplicity, to perform this minimization we limit our study to the case of 1D systems.

**Before localization: Prediction of the homogeneous material evolution and its loss of stability**

The energy of the system corresponds to the total energy of Eq. (2.4) where we neglect the term of external work. This amounts to seek for the damage distribution d(x) that minimizes the energy stored in the material in elastic and damage energy at some fixed loading condition. Using d(x) = d0+d0 ˜(q0)cos(q0x),

**Table of contents :**

Table of contents

Introduction

**1 Context and motivation **

1.1 Experimental characterization of damage spreading and quasi-brittle failure

1.1.1 At the specimen scale: Average mechanical response of quasi-brittle materials

1.1.2 At the scale of damage and microstructure: Temporal and spatial organization of precursors to failure

1.1.3 Conclusions

1.2 Damage models

1.2.1 At the continuum scale: Damage mechanics

1.2.2 At the microstructure scale: Statistical models of quasi-brittle fracture

1.2.3 Our approach: A mesoscopic description of damage spreading in heterogeneous media

1.3 Conclusions

**2 Model formulation and limit cases analysis **

2.1 Non-local damage model description of a heterogeneous material

2.1.1 Energy based damage criterion

2.1.2 Material parameters

2.1.3 Interaction function

2.1.4 Numerical integration

2.2 On the relevance of the combination of interactions and disorder

2.2.1 Interacting homogeneous system response

2.2.2 Non-interacting heterogeneous system response

2.3 Conclusions

**3 Damage localization and failure predictions of the quasi-brittle material **

3.1 Failure behavior of the heterogeneous system: numerical observations

3.1.1 Macroscopic response

3.1.2 Deviation from the homogeneous system response and failure damage thresholds

3.1.3 Damage field spatial organization

3.1.4 Conclusions

3.2 Critical loadings prediction: linear stability analysis

3.2.1 Stability analysis of the homogeneous damage states

3.2.2 Application to 1D media

3.2.3 Application to 2D media

3.3 Critical damage at failure: global energy minimization

3.3.1 Before localization: Prediction of the homogeneous material evolution and its loss of stability

3.3.2 After localization: Failure prediction

3.4 Conclusions

**4 Damage spreading toward failure: Statistics of fluctuations **

4.1 Statistics of fluctuations in damage models with positive load redistributions ( = 0)

4.1.1 Avalanche definition

4.1.2 Temporal behavior: avalanche size evolution

4.1.3 Spatial structure of the avalanches

4.1.4 Spatial organization of the cumulated damage field

4.2 Statistics of fluctuations in damage models with finite wavelength at localization (> c)

4.3 Theoretical analysis of the damage growth fluctuations: A disordered elastic interface analogy

4.3.1 Analogy between damage growth and driven disordered elastic interface

4.3.2 Linearized evolution equation for the damage growth

4.3.3 Exponents prediction

4.4 Conclusions

**5 Comparison with experiments: Compression of a 2D array of hollow soft cylinders **

5.1 Experimental setup

5.2 Mechanical behavior

5.2.1 Macroscopic Force-Displacement response

5.2.2 Intermittent dynamics of the compaction process: Avalanches definitions

5.3 Characterization and prediction of localization

5.3.1 Determination of the localization threshold from the deformation field

5.3.2 Analytical investigation of localization

5.4 Temporal fluctuations study

5.5 Conclusions

Conclusions

**Bibliography **