#### Percolation Modeling

Theoretical and numerical modelings of self-healing materials are still in the initiation stage, which roughly stays in step with the experimental attempts in the field. Privman et al.50 focused the modeling program on time dependence of a gradual formation of damage (fatigue) and its manifestation in material composition. They formulated continuum rate equation for such a process and then carried out Monte Carlo simulations. Supposing *u*(*t*) stands for the fraction of material that is undamaged, *g*(*t*) the fraction of material consisting of healant-carrying capsules, *d*(*t*) the fraction of material that is damaged, and *b*(*t*) the fraction of material with broken capsules, the following expression can be given as:

- (5)

Here, the regime of small degree of degradation of the material is considered. At least for small times, *t*, *u*(*t*) ≈ 1, whereas *d*(*t*), *b*(*t*)*,* and *g*(*t*) are relatively small (*b*(0) = 0). For purposes of simple modeling, Privman et al.50 assumed that on average the capsules degrade with the rate, *P*, which is somewhat faster than the rate of degradation of the material itself due to its continuing use (fatigue), *p*, that is, *P* > *p*. The latter assumption was made to mimic the expected property that a significant amount of microcapsules embedded in the material may actually weaken its mechanical properties and, were it not for their healing effect, reduce its usable lifetime. Thus, by approximately taking

- (6)

*g*(*t*) can be yielded:

- (7)

For the fraction of the undamaged material, is written as:

- (8)

where *H*(*t*) is the healing efficiency:

- (9)

The healing efficiency is proportional to the fraction of capsules, as well as to the fraction of the damaged material, because this is where the healing process is effective. The latter will be approximated by *d*(*t*) ≈ 1−*u*(*t*), which allows for obtaining a closed equation for *u*(*t*). Accordingly, *H*(*t*) is expressed by:

- (10)

The healing efficiency is controlled by the parameter

- (11)

Although the formulated model is simple, it has the advantage of offering an exact solution.

- (12)

Equation 11 suggest that an important challenge in the design of self-healing materials will be to have the healing effect of most capsules cover volumes much larger than a capsule to compensate for a relatively small value of *g*(0), which is the fraction of the material volume initially occupied by the healant-filled capsules. As the healant cannot “decompress,” its healing action, after it spreads out and solidifies, should have a relatively long-range stress-relieving effect to prevent further crack growth over a large volume.

Considering that the above continuum modeling cannot address the details of morphological material properties and healant transport, Dementsov and Privman51 introduced conductance as a measure of self-healing material integrity. That is, conductivity is directly proportional to the local material “health.” By assuming square lattice (coordination number *z* = 4) bond percolation, the conductance, *C*(*t*), can be calculated through the bond-percolation mean-field formula:52

- (13)

The conductance is normalized to have *C*(0) = 1, and for simplicity the bond percolation probability is given by *u*(*t*), that is, the situation when the conductance of the healthy/healed material is maximal is considered, whereas, the other areas do not conduct at all. In the regime of a relatively low damage, which is likely the only one of practical interest, and also the one where the mean-filed expressions are accurate, we note that the conductance provides a convenient, proportional measure of the material degradation,

- (14)

where the constant *K* = *z*/(*z*−2) depends on the microscopic details of the material conductivity. Here *K* = 2, but in practical situations this parameter can be fitted from experimental data.

Results of Monte Carlo simulations51 have shown the general features of self-healing process. In particular, the onset of the material fatigue is delayed by developing a plateau-like time dependence of the material quality at initial times. In this low-damage regime, the changes in conductance, and likely in most other transport/response properties of the material that can be experimentally probed, measure the material quality degradation proportionally, whereas for larger damage at latter times, transport properties may undergo dramatic changes, such as the vanishing of the conductance, and they might not be good measure of the material integrity.

In a recent report, Dementsov and Privman53 extended the calculation of the conductance from 2D lattices to three-dimensional (3D). They demonstrated the competition between the healing capsules for three dimensions, where it is more profound than in their earlier-studied 2D systems. Even for low initial densities of the healing capsules, they interfere with each other and reduce each other's effective healing efficiency. Accordingly, it was suggested that not just the capsules density but their uniform dispersion in the medium (to avoid clumping) is of importance in designing self-healing materials.

#### Continuum and Molecular-Level Modeling of Fatigue Crack Retardation

To model the behavior of self-healing polymeric system under more realistic loading environments, Geubelle and coworkers54 constructed a numerical approach to describe the fatigue crack retardation in the self-healing polymer developed by White et al.,47 which contained microencapsulated dicyclopentadiene and Grubbs' catalyst. Before review of their model, experimental observation of fatigue crack propagation and healing induced-reduction in crack growth rate in the composites55 is summarized hereinafter.

It has been found that for successful *in situ* self-healing, the healing agent released into the crack plane must have enough time to polymerize. If the crack growth rate is too fast, little or no healing will occur. These observations can be explained in terms of a competition between two time scales. The development of healing efficiency in this material system is as follows.56 After an initial dwell period of about 30 min during which no appreciable healing is measured, the healing efficiency increases rapidly, tapering off to a maximum healing efficiency after about 10 h, showing the presence of a characteristic time for healing denoted hereafter by τ_{heal}. On the other hand, retardation and arrest of the propagating crack in the self-healing material is influenced by the healing kinetics, the cyclic loading applied on the structure and the fracture properties of the material determine the crack growth rate. This fact introduces in the self-healing system another time scale τ_{crack}, the characteristic time for crack propagation. This time scale may increase or decrease with crack length depending on whether the crack growth is unstable or stable, respectively. For higher loading levels, no crack retardation is possible as τ_{crack} is substantially less than τ_{heal}, thus allowing little or no healing. If the load level is decreased, the crack propagation rate in the material also reduces, thereby increasing τ_{crack} and enabling the crack retardation to take place. The presence of a rest period during which the loading is not applied but healing takes place further facilitates crack healing. Finally, for much lower load levels, τ_{heal} becomes considerably smaller than τ_{crack}, and the crack retards or gets completely arrested.

Crack retardation in self-healing polymers is mainly controlled by the following two effects: crack bridging (or adhesive) effect associated with the adhesion of the healing agent to the crack flanks, and crack closure effect associated with the solid wedge formed by the deposited polymer behind the crack. Geubelle and coworkers54 combined the cohesive modeling for fatigue crack propagation57 and a contact algorithm to enforce crack closure. The healing kinetics of the self-healing system was incorporated by introducing along the fracture plane, a state variable representing the evolving degree of cure of the healing agent. The atomic scale processes during the cure of the healing agent were modeled using a coarse-grain molecular dynamics model specifically developed for this purpose. The core part of their modeling is as follows.

The cohesive model of an inserted wedge in the wake of an advancing fatigue crack gives the rate of change of the cohesive stiffness, *k*_{coh}, which describes an irreversible unloading–reloading path that accounts for the fatigue-induced irreversibility:57

- (15)

where *N*_{f} denotes the number of loading cycle experienced by the material point as the onset of failure, ψ and β are material parameters describing the fatigue-induced degradation of cohesive properties, and is the rate of change of the normal separation. When a wedge of thickness Δ_{n}* is inserted behind the crack tip, the crack faces experience a contact force whenever Δ_{n} − Δ_{n}* ≤ 0.

As soon as the healing reaction is triggered, the monomer released behind the crack tip polymerizes and creates a solid wedge in the wake of the crack. The wedge thickness, δ*, and the degree of cure, α, can be correlated through an explicit linear form based on the molecular-level simulations:

- (16)

where Δ_{n}* denotes the maximum thickness that can be achieved by the wedge, and α_{threshold} the threshold value of curing degree (below which the polymer acts like a liquid, and above which it responds stiffly).

As alluded to earlier, the key issue that needs to be captured when modeling the fatigue failure of the self-healing composite is the competition between how quickly the crack propagates and how fast it can be healed. The healing and failure processes are each characterized by their own time scales. For the cure kinetics process, the time scale is simply given by:

- (17)

where *B* is the rate constant depending on temperature and the catalyst concentration. The other time scale, τ_{crack}, can be expressed as the time needed for the fatigue crack to propagate a distance equal to the active cohesive zone length, *l*_{c}, located ahead of the crack front:

- (18)

where *da/dN* denotes the crack advance per cycle and ω the loading frequency. The cohesive zone length is related to the failure properties of the material.

Accordingly, Geubelle and coworkers54 conducted numerical simulations. The study of the effect of different loading and healing parameters shows a good qualitative agreement between experimental observations and simulation results.

#### Continuum Damage and Healing Mechanics

Barbero et al.58 studied damage and healing behavior of fiber reinforced polymer composites. The microcracks were taken as distributed damages, which proved to be able to initiate the release of healing agent from microcapsules embedded in composites' matrix. Similarly, healing processes can be considered opposite to damage. Therefore, continuum damage mechanics can be applied. Barbero et al. firstly generalized continuum damage mechanics including healing processes and then proposed continuum damage-healing mechanics. Furthermore, the theory was used to establish a specific model for fiber-reinforced polymer-matrix composites experiencing damage, plasticity, and healing. The mesoscale constitutive model was developed in a consistent thermodynamic framework that automatically satisfies the thermodynamic restrictions. The degradation and healing evolution variables were obtained by introducing proper dissipation potentials motivated by physically based assumptions.

Recently, Barbero et al.59 simplified their model. A damage/healing model where both effects were described in a single thermodynamic space was presented, which consists of three main ingredients the damage variable, the free-energy potential, and the damage evolution equations. Some of the main issues are reviewed as follows.

Damage represents distributed, irreversible phenomena that cause stiffness and strength reductions. When distributed damage controls the mechanical behavior, many materials, including polymer-matrix composites, usually exhibit a quasibrittle macroscopic behavior. For example, experimental observations on polymer-matrix composite before failure show a continuous distribution of microcracks in the matrix. During loading, the total energy of the system is dissipated mainly into new surface formation, whereas a minor fraction is used to nucleate existing microcracks. In this context, damage can be represented by a second-order tensor *D*. As healing is related to the extent of repair of distributed damage, it is also represented by a second-order diagonal tensor *H* with principal directions aligned with those of the damage tensor *D*. The principal values *h*_{1}, *h*_{2,}*h*_{3} represent the area recovery normal to the principal directions, which are aligned with the material directions *x*_{1}, *x*_{2}, *x*_{3} in the material coordinate system. It is postulated that healing tensor, *h*_{i}, is proportional to the damage tensor, *d*_{i}:

- (19)

where the proportionality constant, η_{i}, is the efficiency of the healing system. The principal values of the healed-damage tensor are given by:

- (20)

For fiber composites, the self-healing system is incapable of healing fiber damage, which results in η_{1} = 0. Once the healing agent is released, it travels by capillary action and penetrates all the microcracks regardless of orientation, and thus η_{2} = η_{3} = η.

For identification, all that is required is experimental determination of the healing efficiency as a function of damage. Under shear loading *G*_{12}, the amount of damage *d*_{1} in the fiber direction is negligible when compared to the amount of damage *d*_{2} transverse to the fibers. Therefore, the change in the (unloading) shear modulus due to both damage and healing is given by

- (22)

and the healing efficiency can be calculated as

- (23)

Taking into account that the induced damage *d*_{2} is a function of the applied strain, it is possible to represent the efficiency as a function of damage with a polynomial as:

- (24)

Shear tests of composite laminates were performed. The results indicated that the computational model tracks the damaging stress-strain behavior very well. Additional tests on the samples not used in the parameter identification were also carried out to verify the predictive capabilities of the model. It was observed that it indeed possesses certain predictability as expected.

#### Discrete Element Modeling and Numerical Study

So far, most theoretical works dealing with self-healing materials are based on continuum approaches. They act on a coarse grained level and the materials must be (or are assumed to be) sufficiently homogeneous on that coarse grained level. In contrast, many-particle simulations like the discrete element model (DEM)60, 61 complement experiments on the scale of small “representative volume elements” (RVEs). They allow detailed insight into the kinematics and dynamics of the samples as all information about all particles and contacts is available at all times. DEM requires only the contact forces and torques as the basic input to solve the equations of motion for all particles in such systems. Furthermore, the macroscopic material properties, such as, among others, elastic moduli, cohesion, friction, yield strength, dilatancy, or anisotropy can be measured from such RVE tests.

Herbst and Luding62 presented a model for self-healing in particulate materials based on a piecewise linear contact model for elastoplastic, adhesive, viscous, and frictional particle–particle interactions. The contact model included a memory variable, that is, the contact laws were history dependent. The proposed self-healing model admitted to set the damage detection sensitivity and rate as well as the strength of the healed contacts. It could therefore be a model for a material including a healing agent in, for example, capsules.

A typical self-healing process consists of locally detection of damage and initiation of local healing when damage is detected. Accordingly, the following self-healing scheme that is compatible with (but not limited to) the contact model was introduced: (i) detect damage, (ii) heal detected damage, (iii) run simulation for a time τ_{D}, and (iv) go to (i). This procedure allowed to set the damage detection rate 1/τ_{D} at which damage is detected (and subsequent self-healing is activated), and to specify the healing sensitivity and the healing strength separately.

The model parameters have been systematically studied most prominently by examining the stress–strain response of the samples and additionally by monitoring the fraction of healed contacts. It was found that low damage detection sensitivity led to unsatisfactory self-healing as too few contacts were healed. On the other hand, very high damage detection sensitivity resulted in healing of contacts that were not critical. Healed contacts within the framework of this model, once healed cannot be healed again when they became critical again in the future. The healing intensity was a nonlinear function of the adhesion parameter *k*_{SH}. It saturated for large *k*_{SH} and set the strength of the healed contacts. With increasing healing intensity, the material can sustain larger stresses and failed at higher strains. As a function of *k*_{SH} the material strength showed strong fluctuations for large *k*_{SH}, that is, in the regime where the healing intensity was almost constant. These fluctuations were due to local contact instabilities. For very large healing intensities, that is, for very large adhesive strength of the healed contacts, a contact rapidly failed when the tensile limit was reached, resembling a local, brittle failure at the contact level. Thus, moderate values for the adhesive constant after healing, *k*_{SH}, led to the best healing results. The fraction of healed contacts increased a little with the healing intensity, which corresponded to the final strength of the (solidified) healing agent, that is, the strength of the final bonding.

Based on a RVE, in which either spherical or elongated capsules were uniformly dispersed, Mookhoek et al.63 conducted a numerical study of the release of healing agent for liquid-based self-healing systems. In their model, the capsules did not overlap nor were in direct contact with each other. In the case of elongated capsules, a cylindrical shape terminated by a half sphere at either end was assumed. A cubic RVE was defined using periodic boundary conditions. To obtain a RVE with a desired capsule volume, capsules were sequentially positioned in this RVE using a random number generator for the position of centre of gravity of each capsule. Accordingly, the released healing agent volume, *V*_{released}, per crack area was given by:

- (25)

- (26)

where *A* denotes the cross-sectional area of the RVE with volume *V*, φ is the capsules volume fraction, *r* is the spherical capsule radius, *V*_{cap} is the single capsule volume, and *AR* is the aspect ratio of cylindrical microcapsules.

Equation 25 shows that with decreasing microcapsule size the total amount of liquid released decreases rapidly, which agrees with the estimation of Rule et al.64 Small, for example, nanosized, spherical microcapsules were suggested only to be used for healing of small scale damage (such as interfacial debonding) and not for enhancing the healing efficiency in general. Comparison between the results from eq 25 and eq 26 indicated that for a given capsule volume and a given capsule volume fraction, more healing agent can be delivered to a crack surface by the cylindrical capsules. Therefore, a strong improvement in the healing potential was predicted for elongated capsules. The spatial orientation of such elongated microcapsules also had a significant effect on the healing potential. Considerable reductions of capsule loading concentrations for a particular healing efficiency should be considered to preserve the material intrinsic properties to a larger extend.