The effects of mismatch fracture properties in encapsulation-based self-healing concrete using cohesive-zone model

Abstract

Finite element analysis is developed to simulate the breakage of capsule in capsule-based self-healing concrete. A 2D circular capsule with different core-shell thickness ratios embedded in the mortar matrix is analyzed numerically along with their interfacial transition zone. Zero-thickness cohesive elements are pre-inserted into solid elements to represent potential cracks. This study focuses on the effects of mismatch fracture properties, namely fracture strength and energy, between capsule and mortar matrix into the breakage likelihood of the capsule. The extensive simulations of 2D specimens under uniaxial tension were carried out to investigate the key features on the fracture patterns of the capsule and produce the fracture maps as the results. The developed fracture maps of capsules present a simple but valuable tool to assist the experimentalists in designing appropriate capsule materials for self-healing concrete.

Keywords

self-healing concrete / interfacial zone / capsule materials / cohesive elements / fracture maps

Introduction

Particles-dispersed type is one of the heterogeneous materials which particles are embedded in matrix material. This combination is expected to enhance its utility especially for its fracture characteristic, such as ceramic particles embedded in metallic matrix for strengthening purpose [¹,²] and dispersing metallic particles inside ceramic matrix for enhancing its toughness [³]. There are so many variation novel methods to study fracture in materials can be found in the literatures. For example, finite element-based [⁴–⁸], extended finite element-based [⁹–¹²], cracking elements method [¹³], strong discontinuity embedded approach [¹⁴–¹⁶], multiscale method [¹⁷], meshfree [¹⁸–²²], remeshing technique [²³], phase-field [²⁴–²⁹], screened Poisson [³⁰], peridynamics [³¹,³²], and cohesive zone model [³³,³⁴]. Some alternatives techniques which are easy to implement in finite element model (FEM) which start at one higher level in the hierarchy of models have been developed recently [³⁵–³⁷].

Over the last few decades, some researchers have developed analytical studies to investigate the interaction between particles and inclusions [³⁸–⁴¹]. Numerical method-based techniques to investigate the shape irregularities of particles [⁴²,⁴³], interfacial layer between particle and matrix [⁴⁴], and pre-existing of particles law [⁴⁵] have also been developed. They focused only on mismatch of elastic properties from particle and matrix, without considering fracture of particle. Nevertheless, despite from stiffness of particle is very high, fracture in particle might happen instead of deflecting incoming crack as can be observed from experimental works [⁴⁶,⁴⁷]. Therefore, the main objective of this study is to investigate the mismatch fracture properties to predict the interaction between crack and particle. This objective is not only significant to observe interaction between crack and particle at microscale, but also to predict macroscopic fracture characteristic of particulate system. The strength of this material depends on bonding behavior between particle and matrix. The lower strength of bonding led to decreasing material capacity [⁴⁸] and enhancement of fracture toughness from material might not be achieved. The weak interfacial zone would trigger debonding phenomena instead of fracture in the particle as can be observed in some experimental works [⁴⁹,⁵⁰]. Whereas in some special cases, where matrix and particle are brittle, weak interfacial bonding is more favored in facilitating effective mechanism for energy to dissipate [⁵¹]. The advantage of dispersing particles inside matrix to elevate life-cycle capacity against failure has been adopted by some researchers in capsules-based self-healing concrete [⁵²,⁵³]. The capsules contain healing fluids are dispersed in mortar matrix during concrete mixing. Upon loading, an appearing crack propagates toward the capsule and breaks it then activating healing mechanism in the area of damage. This approach is claimed to be the most beneficial healing mechanism which is able to reach in every corner of the materials [⁵⁴]. Encapsulation-based self-healing concrete has been quite popular research topic these days [⁵⁵–⁶¹]. Mauludin et al. [⁶²] developed fracture model of capsule-based self-healing concrete at mesoscale level using cohesive elements. The capsules are made of Poly Methyl Methacrylate (PMMA) with different core-shell thickness ratios. They employed an efficient packing algorithm to generate randomly microstructures in the mortar matrix and subsequently inserted cohesive elements into each of element interfaces to represent the potential cracks. Gilabert et al. [⁶³] investigated the interaction of capsule interface embedded in homogeneous mortar matrix. They also studied debonding behavior from capsule with different thickness. Strong bonding between capsule and matrix will cause rupture of capsule shell and trigger the healing mechanism, whereas the interface cohesion and voids will give disadvantage to this system [⁶⁴]. However, this ideal bonding condition is not always observed in the experiment [⁶⁵]. Succeeding for releasing fluid into the area of damage will determine efficiency level of healing process. This situation only can be achieved when an incoming crack hits a capsule and breaks it. Numerical models to simulate interaction of a crack into an interface with or without the presence of flaws on the particles have been developed by some researchers [⁶⁶–⁶⁹]. The effects of interfacial transition zone strength between capsule’s wall and mortar matrix on the fracture of capsule has been recently investigated by Mauludin and Oucif [⁷⁰,⁷¹]. They studied the fracture mechanism of capsule with varying fracture properties of the interface while all elastic properties were assumed constant during simulation. The comprehensive simulation which considered all relevant factors, such as matrix cracking and fracture of capsule is needed. The simulation should take into account not only mismatch of elastic properties, but also fracture properties between these phases. The papers which discussed about the interaction between an incoming crack and capsule with considering mismatch properties are really hard to find. The effects of mismatch in elastic and fracture properties on breakage of capsule are investigated in this study. The crack pattern mechanism on each combination both of elastic and fracture parameters of capsule and matrix is observed and classified. Cohesive zone model in terms of cohesive element is used to simulate complex crack mechanisms which exist from multiple initiations. The zero thickness cohesive elements are embedded throughout initial meshes to represent all potential cracks inside materials. The significance of this study is to present a quick reference for experimentalists to examine and design of capsule-based self-healing concrete with proper combination of matrix and capsule properties. The polymeric capsule known for brittleness and can break even with minor stress [⁷²] is used with different shell thickness. The objective of this paper is to study the effect of mismatch in elastic and fracture properties on breakage of capsule and to study the effect of capsule shell thickness on breakage of capsule.

Modeling of capsule-crack interaction

Simulation set-up

The simulation set-up used in Ref. [⁷⁰] is adopted here in this study. 2D rectangular concrete specimens with a circular capsule placed in the center were modeled. The uniaxial tension loadings are applied at the top of specimens. The specimen dimensions are 50 mm × 25 mm and the fixed capsule’s diameter of 2 mm is used. The ratio of core-shell thickness 1:1 is considered in this preliminary simulation to ensure the resistant ability of capsule’s wall for fracture during mixing process [⁷²]. For simplicity, the capsule core in this study assumed to be solid elements. To simulate starting point of propagation crack, an edge-crack is modeled at the left side of specimen with fixed length 4 mm. The horizontal distance between the crack tip and the capsule is chosen large enough to ensure the influence of the particle on the crack driving force is suitably neglected. The geometry of model, boundary, and loading conditions are illustrated in Fig. 1. Vertical displacements were prescribed on the top of specimen. The Abaqus/Explicit with displacement-controlled algorithm was used to simulate the model with loading time t = 0.005 s, and all simulations were finished at a displacement d = 0.1 mm.

Cohesive zone model in Abaqus

Zero thickness cohesive element COH2D4 in Abaqus was built based on cohesive crack model. The cohesive zone is defined by the constitutive behavior of traction-separation laws as described in Fig. 2. It illustrates the interaction between a cracked surface and the constitutive relation. The bilinear cohesive law between traction t across the surface, and d, which is separation displacement on the cohesive surface, is adopted in Ref. [⁶²]. It requires at least two parameters, such as the fracture energy, which is the total area under the traction-separation curve:

(1)G_{\mathrm{n}\mathrm{f}}\mathrm{}{\displaystyle {=\int }_{\mathrm{0}}^{{\delta }_{\mathrm{n}\mathrm{f}}}}{t}_{\mathrm{n}}\left({\delta }_{\mathrm{n}}\right)\mathrm{d}{\delta }_{\mathrm{n}}=\mathrm{}{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{t}_{\mathrm{n}\mathrm{o}}{\delta }_{\mathrm{n}\mathrm{f}},

and the cohesive strength t_{\mathrm{n}\mathrm{o}}, {\delta }_{\mathrm{n}\mathrm{f}} is the separation displacement; and G_{\mathrm{n}\mathrm{f}} is the normal fracture energy.

The effective relative displacements {\delta }_{\mathrm{m}} can be defined after calculating all corresponding displacements both in normal ({\delta }_{\mathrm{n}}) and tangential directions ({\delta }_{\mathrm{s}}):

(2){\delta }_{\mathrm{m}}=\mathrm{}\sqrt{{\langle {\delta }_{\mathrm{n}}\rangle }^{\mathrm{2}}+{\delta }_{\mathrm{s}}^{\mathrm{2}}},

where \langle \rangle refers to the Macaulay bracket and

(3)

The scalar index D is used to determine level of damage and it is given by

(4)D={\displaystyle \frac{{\delta }_{\mathrm{m},\mathrm{f}}({\delta }_{\mathrm{m},\max }-{\delta }_{\mathrm{m},\mathrm{o}})}{{\delta }_{\mathrm{m},\max }({\delta }_{\mathrm{m},\mathrm{f}}-{\delta }_{\mathrm{m},\mathrm{o}})}},

where d_m,f is the effective crack opening at complete failure, d_m,max is the maximum crack opening obtained during the loading history, and d_m,o is the effective crack opening when the damage is initiated. In this study, the cohesive zone model in the specimen is assumed to be damaged when a quadratic function involving the ratio of nominal stress reaches a value of one

(5){\left\{{\displaystyle \frac{\langle t_{\mathrm{n}}\rangle }{t_{\mathrm{n}\mathrm{0}}}}\right\}}^{\mathrm{2}}+{\left\{{\displaystyle \frac{t_{\mathrm{s}}}{t_{\mathrm{s}\mathrm{0}}}}\right\}}^{\mathrm{2}}=1,

where t_{\mathrm{n}} and t_{\mathrm{s}} refer to the corresponding traction both in normal and tangential directions, respectively.

Mesh discretization

All samples in this simulation are initially meshed with 3-node solid elements (CPS3). The mesh with element length of 1 mm is used in this study as illustrated in Fig. 3(a). The 4-node zero thickness cohesive elements are inserted into initial element meshes to represent potential cracks, such as inside the mortar matrix (COH_MM), inside capsule shell (COH_SS), between capsule core and capsule shell (COH_CS), and between capsule shell and mortar matrix (COH_SM). Since the flow process of healing agent into the crack area is not addressed in this study, the capsule core in this simulation is assumed to be solid and cracks are not allowed to initiate in the capsule core. The Python script combined with Abaqus batch processing technique is used to generate duplicate nodes and insert cohesive element automatically at each element interface. The typical discretization of the model after insertion of cohesive elements around the sample can be observed in Fig. 3(b).

Parametric studies

In capsule-based self-healing material, the fracture of capsule is highly expected. Once the capsule is fractured, it will release the healing agent to the area of damage and sealed the crack. The main objective of the present analysis is to study the effect of mismatch in elastic and fracture properties on the crack propagation. To do this, the extensive parametric studies are carried out considering that the elastic and fracture properties of capsule are varied relative to the fracture properties of the mortar matrix, such as the elastic modulus, E_c/E_m, the fracture strength, t_c/t_m, and the fracture energy, G_c/G_m. All values of parametric simulations and the range considered in the present study are summarized in Table 1. The subscript m and c indicates for mortar matrix and capsule, respectively. The quantity ratio given by E_c to E_m, t_c to t_m, and G_c to G_m are hereafter referred to as elastic ratio, strength ratio, and toughness ratio, respectively. For the sake of simplicity, the Poisson’s ratios of the mortar matrix and the capsule, n_m and n_c, are kept constant in all simulations, with n_m = 0.2 and n_c = 0.3 as found in Ref. [⁷³]. The bonding between the capsule and the mortar matrix is assumed to be perfect, with applying the large value of fracture strength and fracture energy into the interfacial zone, such that the separation at the capsule-mortar matrix interface is restrained. The shell thickness ratio of 1:1 is used in these preliminary simulations to investigate the effects of mismatch fracture properties on fractured capsule with the variation of elastic ratios between the mortar matrix and the capsule.

Simulation results

Effect of mismatch elastic and fracture properties

The present simulations to study the effect of mismatch elastic and fracture properties on the crack propagation are reported in this section. The first part of this section provides an overview of parametric study with highlighting different parameters used and also describes the particular crack patterns. The results of the parametric study will be summarized in the last section in term of fracture maps to distinguish the mechanism of fracture.

Crack patterns

The location of incoming crack is assumed to be located directly in front of the capsule, so there is no crack offset assumed in all simulations. The relative elastic and fracture properties ratio between the matrix and the capsule define the actual path of crack as it approaches into the capsule as shown in Fig. 4. Figures with E_c/E_m = 2 correspond to the stiffer capsule (Figs. 4(a) and 4(e)), whereas figures with E_c/E_m = 1/7 correspond to softer capsule (Figs. 4(d) and 4(f)). The other relative ratio of fracture strength and toughness are varied correspondingly as described in the Figures. The capsule is initially undamaged and has a good bonding with the surrounding mortar matrix.

All crack patterns illustrated in Fig. 4 are initially the same until it reaches a distance around to the diameter of the capsule. It can be seen that capsule fracture is observed in three cases (Figs. 4(a), 4(b), and 4(d)) whereas crack deflection occurs in other cases (Figs. 4(c), 4(e), and 4(f)). Using the same elastic ratio between the capsule and the mortar matrix, the incoming crack deflects along the capsule when the strength of the capsule is larger than the strength of the mortar matrix whereas the capsule fracture occurs when the strength of the capsule is less than the strength of the mortar matrix as shown in Figs. 4(f) and 4(d), 4(c) and 4(b), 4(e) and 4(a), respectively. Nonetheless, the mismatch in elastic properties between the capsule and the mortar matrix does not guarantee capsule fracture despite higher stiffness of the capsule.

Fracture maps

The effect of mismatch in elastic properties is investigated by considering three different elastic ratios, namely, E_c/E_m = 2, 1, and 1/7. In all simulations, the incoming crack has no initial offset from the center of the capsule (the crack lies directly in front of the center of the capsule). The results of these extensive simulations are summarized in the form of a fracture map, as illustrated in Fig. 5. The corresponding curve for each elastic properties ratios 2, 1, and 1/7, separates the areas of the fracture strength ratio t_c/t_m and the fracture energy ratio G_c/G_m for which the capsule would be either fractured or not because of an incoming crack, as shown in Figs. 5(a), 5(b) and 5(c), respectively.

The comparison of corresponding regions between areas of capsule fracture and crack deflection from all elastic ratios can be seen in Fig. 6. As can be observed in Fig. 6, the regions of capsule fracture correspond to the left and bottom sides of the particular curve whereas the regions of crack deflection correspond to the right and top sides of the particular curve. A large number of simulations are extremely needed to define the curves shown in Fig. 6 with identifying not only the particular strength and energy ratios but also the transition phenomenon between two cases (capsule fracture and crack deflection) in each mismatch of elastic properties.

In general, the trend of capsule fracture is identified by a decrease of capsule stiffness whereas crack deflection is favored with an increase of capsule stiffness. This finding is in good agreement with the established conclusion found in the literature which indicates that an incoming crack will be attracted to a softer particle meanwhile a stiffer particle will cause a crack to be deflected far away from the particle. However, as shown in Fig. 6, the behavior of capsule fracture is strongly dependent on the mismatch in fracture properties, especially when using softer capsule (e.g., E_c/E_m = 1/7) compared to the one with stiffer capsule (e.g., E_c/E_m = 2). The interesting part that can be pointed out from fracture map shown in Fig. 6 is that capsule fracture occurs in softer capsule, even if the capsule strength is higher than the mortar matrix strength up to a factor of 2 or even more, given the energy ratio of capsule is adequately low. Regarding to stiffer capsule, capsule fracture is prevented if the capsule strength is higher than the mortar matrix strength by a factor of 1.1, regardless of lower fracture energy of the capsule.

Effect of capsule shell thickness

This section presents the simulation results to investigate the effect of capsule shell thickness on the fracture of capsule. An overview of parametric studies with highlighting different parameters will be explained at the first part along with the particular crack patterns. The fracture maps as results of the parametric studies will be created and summarized in the last section to differentiate the fracture mechanism of the capsule.

Crack patterns

All simulations presented here are based on the capsule core-shell thickness ratios taken from Ref. [⁶²] as shown in Fig. 7. All crack patterns as illustrated in Fig. 8, which initiate from left to right, are distinguished from capsule thickness with different fracture properties ratios between the capsule and the mortar matrix. The elastic modulus ratios of 1/7 between the capsule and the mortar matrix was used in all simulations [⁶²] regardless the capsule shell thickness. There are four different shell thickness cases considered in this study, namely Ratio 1:1 (Figs. 8(a) and 8(b)), Ratio 5:1 (Figs. 8(c) and 8(d)), Ratio 10:1 (Figs. 8(e) and 8(f)), and Ratio 15:1 (Figs. 8(g) and 8(h)). Capsule fractures in all thickness are observed when there is no mismatch in fracture properties between the capsule and the mortar matrix (Figs. 8(a) and 8(g)) and when the mismatch ratio in fracture properties up to 1.5 were used (Figs. 8(c) and 8(e)). When the fracture strength of the capsule is twice larger than the mortar matrix (Figs. 8(d), 8(f), and 8(h)), the incoming crack deflects the path along the capsule and could not penetrate into the shell whereas in other case, capsule fracture occurs when the small toughness ratio between the capsule and the mortar matrix was applied even though the largest thickness of capsule was used (Fig. 8(b)). Nonetheless, the mismatch in fracture energy (toughness) has significant effect on capsule fracture despite larger thickness of the capsule.

Fracture maps

The effect of capsule shell thickness is investigated by considering four different shell thickness ratios, namely, Ratio 1, 5, 10, and 15. In all presented simulations, the approaching crack has no initial offset from the center of the capsule (the crack lies directly in front of the center of the capsule). The results of these extensive simulations are summarized in the form of a fracture map which compares corresponding regions between areas of capsule fracture and crack deflection from all thickness ratios as shown in Fig. 9. As can be seen in Fig. 9, the regions of capsule fracture correspond to the left and bottom sides of the particular curve whereas the regions of crack deflection correspond to the right and top sides of the particular curve. A large number of simulations are significantly needed to determine the curves with identifying not only the particular strength and energy ratios but also the transition phenomenon between two cases (capsule fracture and crack deflection) in each thickness ratio. Consequently, it can be concluded from the fracture map (Fig. 9) that the capsule shell thickness has a significant influence on the capsule fracture, but only in combination with the fracture properties (fracture strength and energy).

Conclusions

Numerical models of capsule-based self-healing concrete have been developed to analyze the effect of mismatch in elastic and fracture properties on fractured capsule. The cohesive zone model is used to simulate fracture process under uniaxial tension. The effects of mismatch in elastic and fracture properties along with different capsule shell thickness on capsule fracture are evaluated. Complex interactions between the capsule and the incoming crack exist (crack nucleation and propagation) and can be observed in term of crack patterns and fracture maps. The main conclusions based on the obtained results can be drawn as follow.

1) The path of an incoming crack is strongly dependent upon combination of the elastic and the fracture properties.

2) The mismatch in fracture strength between the capsule and the mortar matrix was found to be the significant aspect on capsule fracture when there is sufficient bonding strength between the capsule and the mortar matrix.

3) The mismatch in fracture energy (toughness) also plays significant role in defining the crack trajectory despite higher stiffness of the capsule.

4) Given the same elastic modulus ratios between the capsule and the mortar matrix, capsule shell thickness plays an important factor in determining the fracture path around the capsule.

5) Given the same elastic modulus ratios, the smaller thickness of capsule will likely be fractured compared to the larger ones.

For design purposes in capsule-based self-healing concrete, it is crucial to select an appropriate combination of mechanical properties between the capsule and the mortar matrix. This study shows that a high bonding strength between the capsules and the mortar matrix is significantly required while fracture strength of the capsules should be sufficiently small. With this criterion, a capsule-based self-healing mechanism can be triggered successfully by an approaching crack.

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