Experimental characterization of damage spreading and quasi-brittle failure
Quasi-brittle failure concerns a large variety of materials such as concrete, rocks, woods, fiberglass… It consists in a gradual deterioration of the material, which eventually leads to its abrupt failure. This degradation of the material properties is attributed to the progressive development of irreversible defects such as microcracks or voids, also called damage, while failure corresponds to the formation of a macroscopic crack that goes from one side of the sample to the other. The precursory microdefects can be defined as small flaws generally of similar dimensions as the characteristic mi-crostructural feature of the material . As they interact through the elastic field, their growth gives rise to complex collective behaviors that render the prediction of the material response a challenging task [8, 55, 88]. A collection of microscopic damage mechanisms such as crack face friction [64, 91], crack kinking [52, 98], pore collapse [32, 7] and others can be involved in the deterioration process. Yet, a remarkable feature of quasi-brittle failure is that even though multiple scales and dissipative mechanisms are involved during the failure process, common behaviors emerge. In this section, we attempt to give an overview of the common features observed experimentally and discuss the associ-ated challenges to understand them.
At the specimen scale: Average mechanical response of quasi-brittle materials
The irreversible defects formed during the loading result in a noticeable stiffness reduction of the altered material. The elastic moduli evolve as damage develops, which leads to a progressive decrease in load bearing capabilities of the structure. This has been observed for example upon the cycling loading of graphite/epoxy laminate , or triaxial testing of granite . The decrease of the Young’s modulus and increase of Poisson’s ratio obtained by Heap et al.  on Etna basalt is shown in Fig.1.1a as a function of the number of applied cycles of an increasing-amplitude stress-cycling experiment. However, direct relationship between the moduli and damage itself is difficult to obtain. Indeed, experimental measures of damage remain a complicated task, as direct determination of the micro-defects is often associated with destructive techniques to study the material microstructure. As reported in [20, 67], different authors obtained indirect measures by considering the remaining life of a cycling loading, the stiffness reduction, density or resistivity variations. Thus, damage is usually quantified indirectly through its effect on the material properties.
Since elastic moduli set the relationship between stress and strain, progressive stiffness degradation leads to strongly non-linear macroscopic response of quasi-brittle materials. This has been largely explored experimentally. A characteristic force-displacement behavior obtained by Fortin et al.  for a triaxial test performed on a Bleurswiller sandstone is shown in Fig. 1.1b. It can be separated in three phases: The response is first linear, the loading being too small to nucleate new microcracks or activate the preexisting defaults. After some critical loading, a non-linear response is observed, associated with the nucleation and stable growth of micro-defects before the ultimate strength is reached (peak load). If the system is controlled in strain, this phase is often followed by a progressive strain softening prior to an abrupt failure. Finally, one might observe a plateau regime taking place after the formation of a macroscopic crack or fault that corresponds to the sliding of one part of the sample on the other.
However, noticeable variations in the behavior are worth mentioning. At the macroscopic scale, failure is associated with the separation of the sample in two pieces by a macrocrack and is very often preceded by the formation of a localized band where most of the damage activity accumu-lates. The orientation of this band depends on the type of loading considered. For example, for the triaxial test of Fortin et al., an angle of about 45◦ was measured, whereas the macrocrack formed in a prismatic spruce wood sample compressed in parallel to the fibers was perpendicular to the loading direction, as shown on Fig. 1.2 (image reproduced from ).
Failure and strength predictions are fundamental and long-standing  issue in material science and structural engineering. A general problem when performing laboratory experiments is the change of size of the tested specimens. For testing in the laboratory rock or concrete structures, size rescaling is very often unavoidable, but the sample size appears to greatly control its strength. Hence, a quantitative understanding of the effect of the specimen size on quasi-brittle strength is an important matter. Many systematic characterizations of this phenomenon were performed, as for example [10, 18, 39, 95, 110], but an extended range of length scales is generally difficult to achieve since large scale experiments are arduous to realize.
At the scale of damage and microstructure: Temporal and spatial organization of precursors to failure
If the macroscopic response and elastic moduli allow observing the effect of defects for-mation on the elastic properties of the material, they do not provide any hint on the local damage processes. A useful tool is the use of acoustic emission (AE) measurements during experiments. As early as 1942, Obert et al.  realized that during the loading of rocks, noises were emitted, which they assumed related to cracking events. Indeed, during microcracking, acoustic waves, associated with dissipation processes, radiate in the material. This non-destructive technique consists in using piezoelectric sensors to record the acoustic waves emitted. Converted into an electric signal, the mea-surements are considered directly related to the local released elastic energy that takes place during microcracking [70, 29]. This technique allows characterizing both the temporal and spatial evolution of damage events, but also obtaining information on the microscopic mechanisms that take place. Considering some relation between the energy of the acoustic signal emitted and the energy dissi-pated into damage, one can study the evolution of the cumulated dissipated energy to characterize damage evolution, the distribution of the events intensity, their rate, duration and time separation or waiting time [22, 35, 42, 82]. Measurements performed with several receptors also allow to locate damage events and study damage spatial organization, in particular the phenomenon of localization, as discussed in the next paragraph. Finally, note that this approach can be made more quantitative since the different waveforms recorded can be used as indicators to characterize specific damage mechanisms, e.g. tensile, shear or implosive fracturing .
Spatial organization of the precursors during the transition from diffuse to localized damage Fortin et al. obtained projections of the AE hypocenters at different stages of the response of the triaxial test. Each cartography of Fig.1.3 corresponds to damage events taking place during given loading windows, indicated by the letters a-f on Fig. 1.1b. For low loadings in the non-linear part of the force-displacement response, and neglecting the peculiar behavior in the load application regions, damage events are randomly distributed in the core of the material (frame a). As the loading proceeds, damage starts to concentrate in the upper region (frames b-c). In the post-peak regime, damage events are localized: Hypocenters form a localization band that develops progressively (frame d). Finally, the abrupt failure is associated with the growth or sliding of the macroscopic band, as exemplified on frames e and f where all damage events take place in the new formed fault. Therefore, it seems that damage is first diffuse with a spatial distribution that is mainly dependent on the disorder distribution: Microcracks nucleate or grow at the weakest points of the sample. As the density of defects increases, progressive clustering is observed, leading to a complex organization of damage and the presence of a localized zone where macroscopic failure initiates. This transition has been characterized for example by the study of the change in entropy [35, 42] and diminution of fractal dimension of the damage clusters [71, 118], but a robust characterization of this localization process through the identification of a growing correlation length in the damage spatial distribution is still lacking.
Figure 1.3: AE hypocenter distributions represented as a projection on the(z, y) plane and obtained during a triaxial test at 10MP on Bleurswiller stone by , the loading intervals considered are indicated on Fig. 1.1b. Overlooking the regions where the load is applied, damage spreads uniformly in the material at small loading (a). As loading is increased, damage starts to organize and one can foresee the localization band forming (b)-(c). In the post-peak regime, enhanced localization is obtained (d), followed by material failure through sliding of the new created fault where damage concentrates exclusively in the band formed during localization (e)-(f).
Temporal organization: Intermittency and scale-free fluctuations
As seen before, damage during quasi-brittle failure exhibits a complex spatial organization. Knowing the spatial distribution of the dissipative events, one can extract some information on how far the system is from failure by considering the diffuse to localized damage transition. Once damage starts to localize, one can also predict the location of the macrocrack that leads to failure. In addition to that, information relevant for failure prediction can be obtained from the temporal evolution of these precursors. The cumulated signal recorded on the Bleurswiller sandstone is shown on Fig. 1.1b. Very few or no events take place in the initial linear part of the stress-strain curve. If the cumulated AE signal gradually increases for low loadings, a sharp raise is observed close to failure. In partic-ular, even through not exhibited here, rupture (taking place at a strain ∼ 1%) is associated with a surge in AE rate. The cumulated energy can be represented as a function of the distance to failure (Pc − P )/Pc, where Pc is the failure stress. This quantity ranges form one at the beginning of the experiment to zero at failure, and we could look at it as a control parameter within a critical phenom-ena description of quasi-brittle failure [1, 48]. Averaging over different experiments, the behavior obtained by Guarino et al.  when testing a planar sample of chipboard wood to which an effective tensile load is applied is shown in Fig. 1.4a. The power law behavior indicates a strong amplification of the average dissipated energy close to failure. In particular, the large increase close to failure that resembles a power law divergence argues for a critical transition interpretation. However, if there is a common agreement on the increase of the damage activity as failure is approached, its nature remains unclear as exponential laws have also been reported .
The evolution of the cumulated energy suggests that either the size of the bursts or the event rate increases as damage progresses. To explore these two possibilities, the corresponding evolution of the bursts of dissipated energy obtained by Guarino et al. is represented as a function of the nor-malized applied pressure in Fig. 1.4b, where the cumulated measured energy is also indicated (dashed curve). Noticeably, damage events present a strong intermittency: the material exhibits phases were it deforms elastically, corresponding to silent periods were no AE is recorded, while short bursts of a broad range of amplitudes are observed in between. This behavior is often described as a crackling noise  and is observed for many physical systems, ranging from earthquakes faults dynam-ics  to the fracture of paper . A strong amplification is observed close to failure with bursts of increasing amplitude. An acceleration of the damage process takes place with an increase of the event rate .
Figure 1.4: Temporal behavior of AE events recorded by Guarino et al.  during the progressive damage of a chipboard wood up to failure at the pressure Pc: (a) Evolution of cumulated acoustic energy as a function of the distance to failure showing the diverging power law behavior obtained close to failure; (b) The corresponding rate of energy dissipated and cumulated energy (dashed curve) showing the intermittent crackling noise and acceleration of an amplified damage process as the sample is driven towards failure.
To characterize the amplitude of the energy bursts, we have to look at the statistics of avalanches in terms of probability density functions. The functions are usually obtained by con-sidering all bursts taking place during the experiments. They exhibit power law behaviors, of the form P (E) ∼ E−β, arguing that fluctuations at all scales take place. However, the universality of the power law exponent remains to be discussed as values ranging from 1.2 to 2 have been reported [27, 26, 35, 41, 42, 74, 82, 100]. Moreover, since the damage process amplifies as failure is approached, the acoustic signals might not have stationary statistical properties during the whole pro-cess of damage. As a result, computing the distributions at different distances to failure appears more appropriate. However, if some studies considered the lack of time invariance, the behavior obtained when considering different distances to failure remains under disagreement. Some studies report de-creasing exponent of the power law behavior values as failure is approached [2, 72], while the recent work of Baro´ et al.  reports a constant exponent close to 1.4, independently of the distance to fail-ure, as shown on Fig. 1.5 for a compression test on Vycor. A change of exponent is also reported as the loading condition is varied [2, 41]. Therefore, it seems that the general power law behavior, up to a cutoff, the origin of which is also unexplained, is common to all experiments whereas discrepancies remain on the actual value of exponents and their evolution as the material is driven towards failure. The lack of statistics may be responsible for the observed variations, as also the multitude of loading conditions and materials considered in the various studies. Finally, it should be mentioned that the linear relationship between the AE recorded signal and dissipated energy remains to be clarified as non-linear relations might actually relate them together.
Figure 1.5: Time invariant power law distributions of exponent 1.4 of energy bursts obtained by Baro´ et al.  using AE during a compression test on Vycor. The lack of time dependency is in contradiction with the behaviors of the cumulated and rate of dissipated energy, which exhibit a strong acceleration as failure is approached.
It has been well shown that quasi-brittle failure is preceded by a complex evolution of pre-cursory damage events. The use of AE allowed clarifying the heterogeneous materials response. Generally, it can be divided in four identified stages: The material first deforms elastically before microcracks nucleate and grow, leading to a nonlinear macroscopic response as the material stiff-ness deteriorates. During this phase, damage events are randomly distributed in space, microcracks forming in the weakest points of the material or extending the existing micro-defects. Their varying amplitudes remain rather low in magnitude and the crackling noise is characterized by large time separation between events. As the system evolves towards failure, damage progressively organizes, forming clusters of non-trivial fractal dimension. This results from the increased density of microc-racks in the system: As damage takes place somewhere in the system, the resulting stress variations in the neighborhood of the microcrack overlap with other microcracks, leading to complex interac-tions and cascade processes. This complex collective behavior of microcracks is well revealed by an increased AE activity, with a strong amplification of the damage burst size as the system gets close to failure. Finally, in the region where damage has localized, a macrocrack is formed. The system enters in another regime where the mechanics of the material is entirely governs by the behavior of the newly formed macrocrack or fault, similarly to crack propagation problems in brittle materials.
Moreover, the study of the temporal behavior of the precursors revealed interesting features such as a possible diverging dissipated energy at failure, and the presence of scale-free fluctuations, both indicated by power law behaviors. However, the lack of statistics does not allow for a clear assessment of the behaviors and power law exponent values. In particular, the change in behavior as failure is approached is not well established and calls for further experimental investigations. Finally, if the importance of the coaction of both the inherent disorder of the material and microcracks inter-actions is now well established, a better determination on the role of each component is still needed to understand the observed behavior. The common behaviors reported in quasi-brittle fracture ex-periments raise the following questions: Can we rationalize the temporal and spatial behavior of the precursors? How can we use their statistical properties to understand damage spreading and localiza-tion? What is the influence of the stress redistributions on damage evolution? More importantly, can we predict damage localization and failure? These questions led to numerous model descriptions that we classify in two main classes in the next section.
The theoretical description of quasi-brittle failure is a long standing problem that has moti-vated a great number of researches, leading to the elaboration of various theoretical approaches. Yet, two main classes can be identified: Continuum damage mechanics and statistical models. The former consists in a continuous description of damage at a coarse scale through internal variables. Damage reflects the level of microcracking and affects the local mechanical behavior of the material through the degradation of the elastic moduli. At all time damage can be tracked and thermodynamical con-cepts allow for a rigorous description of the failure process. However, damage evolution remains described phenomenologically, the response is that of a statistically homogeneous material. A major problem in these approaches is the strong mesh dependency observed numerically, leading to a zero total dissipated energy at failure. The introduction of localization limiters, in particular through the use of non-local damage theory, has allowed overcoming this issue. However, non-locality is intro-duced heuristically, and physically motivated arguments to justify these models are still lacking.
On the other hand, the framework of statistical models relies on the central role played by microstructural heterogeneities. The solids are described by an array of discrete bonds with disor-dered properties interacting through an explicit law of force redistribution or implicitly through, e.g. elasticity laws. If these models allow capturing the main features of quasi-brittle failure in disordered materials, they still lack thermodynamics consistency and are hence limited to qualitative predictions of material failure behaviors. In this section, we briefly describe each approach, from which concepts are borrowed to build our model.
At the continuum scale: Damage mechanics
The concept of damage mechanics was first introduced in the context of creep rupture by Kachanov  who described the effect of gradual degradation of the material on its elastic proper-ties through a continuous local damage variable. This concept was later generalized, as for example in [20, 60, 66] in particular through the theory of irreversible thermodynamics. Continuum damage mechanics (CDM) based models, consist in introducing one or more continuous internal damage vari-ables in the equations predicting the mechanical fields. This local damage parameter is representative, at a coarse scale, of the damage density constituted by micro-defects on a representative volume el-ement (RVE). The constitutive equations of damage evolution are formulated using the theory of irreversible thermodynamics combined with phenomenological or micromechanics considerations. It allows the description of the inelastic response of the damaged material as well as damage growth and localization.
Table of contents :
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
1.2 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
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
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
3.2 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
4 Damage spreading toward failure: Statistics of fluctuations
4.1 Statistics of fluctuations in damage models with positive load redistributions ( = 0) 64
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
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