Computational modeling of fracture in capsule-based self-healing concrete: A 3D study

Abstract

We present a three-dimensional (3D) numerical model to investigate complex fracture behavior using cohesive elements. An efficient packing algorithm is employed to create the mesoscale model of heterogeneous capsule-based self-healing concrete. Spherical aggregates are used and directly generated from specified size distributions with different volume fractions. Spherical capsules are also used and created based on a particular diameter, and wall thickness. Bilinear traction-separation laws of cohesive elements along the boundaries of the mortar matrix, aggregates, capsules, and their interfaces are pre-inserted to simulate crack initiation and propagation. These pre-inserted cohesive elements are also applied into the initial meshes of solid elements to account for fracture in the mortar matrix. Different realizations are carried out and statistically analyzed. The proposed model provides an effective tool for predicting the complex fracture response of capsule-based self-healing concrete at the meso-scale.

Graphical abstract

Keywords

3D fracture / self-healing concrete / spherical / cohesive elements / heterogeneous

1 Introduction

Cracks in concrete must be carefully monitored and periodically repaired to ensure the durability and safety of structures. The annual cost related to concrete repair works is high and it is estimated to be about half of the initial construction costs [1]. Hence, using self-healing concrete is highly beneficial for reducing maintenance and replacement costs. It allows early repair of damage which cannot be easily accessed. The underlying mechanical behavior of capsule-based self-healing concrete, which determines the structural performance and reliability, is determined by its composition, heterogeneity, and loading history. Therefore, a better understanding and prediction of the fracture process in self-healing concrete, by experiments and numerical modeling, is of major importance. However, it still poses a major challenge.

The first production of self-healing cement can be traced back to 1990 [2] where fibers and sealant were combined to reduce cracking inside the cement matrix by releasing chemical compounds from fibers into the cement matrix. Experiments on self-healing in polymer materials have also been conducted under fatigue loading [3]. The researchers used micro-capsules, filled with healing agents, embedded in substantive materials to simulate a self-healing process. It was expected that an incoming crack would hit the capsule and break it, so the healing liquid could be released into the damaged area, initiating the healing process, as shown in Fig. 1.

Compared to the development of numerical models in concrete in general, researches in numerical modeling of self-healing concrete are infrequently found in Refs. [4–11]. First attempts to simulate cracking and healing in composite materials were conducted by White et al. [12]. Two analytical models for encapsulated particles determining the healing capability of materials were proposed by Zemskov et al. [13]. They calculated the probability of cracks hitting an encapsulated particle with different combinations of crack length, particle size, and inter-particle distance. Mookhoek et al. [14] developed a model to study the effects of elongated capsules on the healing efficiency. A self-healing concrete model at mesoscale level was treated as a three or four phases substance consisting of aggregates, the mortar matrix, capsules and their interfaces. The mechanical response of self-healing concrete at the macroscale strongly depends on its meso-structure. Hence, numerical 24 modeling of self-healing concrete at the meso-scale becomes significant and was used in this study.

Some scholars studied the possibility of capsules being impacted by cracks both in 2D and 3D. Lv and Chen [15] studied these possibilities for different volume fractions and capsule shell thickness [16]. A dosage analysis of self-healing agents required for capsules combined with probability theory was performed in Ref. [17]. Recent experimental studies have shown that the bonding between the capsule shell and mortar matrix is not perfect [18,19] and can influence the possibility of capsules being hit by the crack, which is important for an efficient healing capability. The strong interfacial cohesion between the capsule and mortar matrix is needed in order to force the capsule to break, as reported by Ref. [20]. Mauludin and Oucif [21,22] developed a 2D fracture model but studied only a single capsule embedded in concrete. They investigated the effect of the interfacial strength of a capsule shell on crack propagation and trajectory, showing that only a strong interfacial bonding of capsule shell and its surroundings can break the capsule shell. They also developed a 2D model of fracture in heterogeneous capsule-based self-healing concrete at the meso-scale in Ref. [23], investigating the influence of different volume fractions and capsule thickness on the cracking behavior.

However, a full 3D model might be necessary for reliable predictions, which was the focus of this study. The 3D results were compared to 2D results of our former studies. Most capsules were composed of Poly Methyl Methacrylate-PMMA since they break easily even under small deformations and have good resistance in mixing process compared to other materials such as polystyrene (PS), poly lactic acid (PLA), and glass [19]. The healing agent inside the capsule was modeled as solid element, hence the liquid flow into the crack area, its polymerization, and healing efficiency was not studied in this work. To represent the complex fracture process in multi-phases medium, cohesive interface elements (CIEs) were pre-inserted into the discretization both at the interfaces and the mortar-matrix. The objectives of this study were: i) to simulate the complex fracture process in 3D and ii) to quantify the influence of different volume fraction of capsules on the strength and fracture energy.

2 Heterogeneous capsules-based concrete

2.1 Distribution of aggregates and capsules

The Fuller curve is often used to define the aggregate distribution in concrete [24], which is categorized based on the sieve analysis into different numbers of segments. In this study, we adopted the aggregate size distributions from Hirsch [25] summarized in Table 1. The cut-off aggregates size of 2.36 mm was adopted and concrete was assumed to consist of mortar, aggregates, capsule walls and cores. The coarse aggregates in normal concrete usually constitute about 30%–50% of the total volume of the concrete.

Based on images produced by X-ray photography, the coarse aggregates shapes were found to be circular and elliptical in 2D or spherical and ellipsoidal in 3D [26]. Du et al. [27] reported that the shape of the aggregates has no significant influence on the macroscopic behavior of concrete. For the sake of simplicity, we therefore assumed spherical shapes for the aggregates. We also generated 2D models for comparison assuming circular aggregates. Experiments show that the agitation rates determine the size and the thickness of the capsules. The diameter range of capsules commonly range from 1 to 8 mm [19,28] and we present results for a diameter of 2.0 mm in this paper, whereas the core-wall thickness ratio of 1 was applied to take account of capsules’ strong resistance during the mixing process [19].

2.2 Microstructures generation

The packing algorithm to generate the meso-structure in concrete is adopted from our previous works [23,29]. The main purpose of the procedure is to generate aggregates and capsules repeatedly in random manner until the target area or volume is achieved.

A series of meso-scale models with dimensions of 25 mm × 25 mm in 2D and 25 mm × 25 mm × 25 mm in 3D were generated; see Figs. 2 and 3, respectively. An aggregate volume fraction of 30%, see Table 1, was assumed constant along with the variation capsule volume of fraction. The minimum space allowed between the boundary of the concrete model and the edge of the aggregates or capsules was 0.5 mm.

3 Cohesive zone model

A bilinear cohesive zone model (CZM) as illustrated in Fig. 4 was used [30]; G_c denotes the critical energy release rate, t_n^0 is the cohesive strength, and δ_f is the ‘maximum’ crack opening/sliding when the cohesive tractions have decayed to zero. This model is readily available in ABAQUS, which we used for all computational studies.

It is based on an effective relative crack opening δ_m, which depends on the normal crack opening (δ_n) and two shear components (δ_s), and (δ_t) defined in a local coordinate system:

(1){\delta }_{\text{m}}=\sqrt{{\delta }_{\text{n}}^2+{\delta }_{\text{s}}^2+{\delta }_{\text{t}}^2},

(2)

where ⟨⟩ is the Macaulay bracket. A damage variable D is introduced to describe the bilinear traction-separation law, where the damage evolution is expressed as

(3)D=\frac{{\delta }_{\text{mf}}\left({\delta }_{\text{m,max}}-{\delta }_{\text{m0}}\right)}{{\delta }_{\text{m,max}}\left({\delta }_{\text{mf}}-{\delta }_{\text{m0}}\right)},

where δ_mf indicating the effective relative displacements at point of failure; δ_m,max, denotes the maximum relative displacement attained throughout the loading history, and δ_m0 is the effective relative displacement at initial point of damage. More details about the cohesive traction model can be found in the ABAQUS manual.

The concrete samples were meshed with 3-node triangular solid elements (CPS3) for our 2D models (used for comparison to 3D results) and 4-node tetrahedra solid elements (C3D4) in 3D. To represent cracks, 4-node (COH2D4) and 6-node (COH3D6) cohesive elements were pre-inserted into the initial generated element interfaces in 2D and 3D, respectively. The post-processing was done by incorporating a Python script into ABAQUS batch processing, as in our previous study [23].

Seven sets of CIEs were automatically generated, namely, aggregates (CIE-AA), mortar matrix (CIE-MM), capsule shells (CIE-SS), capsule cores (CIE-CC), the interface of aggregates and mortar matrix (CIE-AM), the interface of capsule cores and shells (CIE-CS), and the interface of capsule shells and mortar matrix (CIE-SM). The interface between capsule cores and shells were assumed to exist based on SEM testing reported in Ref. [31].

4 Numerical simulations

4.1 Description of the models

Let us consider 2D concrete specimens assuming plane strain conditions as well as 3D specimens under uniaxial tension as illustrated in Fig. 5. Their dimensions are 25 mm × 25 mm for 2D specimens and 25 mm × 25 mm × 25 mm for 3D specimens [32]. A uniformly distributed horizontal displacement was applied on the right surface of the specimen. Vertical displacements of nodes at the lower left edge of the model were fixed, whereas the other nodes were free. All material properties used in our simulations can be found in Table 2 and were adopted from [16,19,29,32–34]. Aggregates have a higher strength than the mortar matrix and their interfaces, and experimental evidence has suggested no cracks inside the aggregates. The healing liquid inside the capsule was modeled as linear elastic solid. Therefore, its Poisson’s ratio should be close to 0.5. The capsule shell properties composed of urea formaldehyde were adopted from recent experimental data and assumed to have perfect bonding [33,34]. Fracture properties were taken from other recent experimental studies [16,19,32]. For simplicity and due to lack of information, the shear behavior of fracture is assumed to be equal to the one in normal direction.

4.2 Mesh dependency study

We first determined the influence of the discretization on the results—for three different meshes differing in element length, namely coarse mesh (L_e = 2.0 mm), medium mesh (L_e = 1.5 mm), and fine mesh (L_e = 1.0 mm) as illustrated in Fig. 6. The coarse mesh had 318704 solid elements and 636263 CIEs, the medium mesh 354651 solid elements and 687313 CIEs, and the fine mesh 422854 solid elements and 841401 CIEs. Figures 7 and 8 show very similar results for the medium and the fine mesh. The stress value was obtained by dividing the total horizontal reaction forces with the cross-sectional area of the specimen. The medium and fine meshes exhibited similar fracture patterns where only one single macrocrack occurred at the final stage of fracture. On the other hand, the coarse mesh showed two macro-cracks (Figs. 8(a)−8(c)). For the subsequent analysis, discretizations with an element length ofL_e = 1.25 mm were employed. Figure 9 depicts exemplary stress-displacement curves for 4 different realizations. Similar results were obtained for other realizations.

4.3 Volume fraction of capsules

The effect of volume fraction of capsules on the mechanical behavior of capsules-based self-healing concrete was investigated. Three different volume fractions were considered here: 2%, 5%, and 10% while the volume fraction of aggregates was kept constant at 30%. Figure 10 shows the associated models. For each volume fraction, 50 random samples were modeled to ensure statistical convergence.

4.4 Tensile behavior

Figures 11 and 12 show the curves of mean stress and mean dissipation energy for both 2D and 3D numerical simulation with a capsule volume fraction of 5% in both cases. The two curves in Fig. 11 are qualitatively similar but show significant quantitative differences, which is probably due to the different dimensions (2D versus 3D).

We believe the increases in the ultimate stress and fracture dissipation energy and decrease in standard deviation were due to thickness effects and less smooth fracture surfaces in the 3D model (see Section 4.5). Furthermore, in the softening region (see point Y at Fig. 13), the standard deviation for the 3D samples (0.52 MPa) was larger than for the 2D samples (0.42 MPa).

The mean ultimate stress obtained by the 2D and 3D models were 2.75 and 3.50 MPa, respectively, which was a 27.3% difference. Similarly, the mean dissipation energy predicted by 2D and 3D models were 41.5 and 57.5 mJ, respectively, which was a difference of 38.6% (Fig. 12). This higher value was attributed to the different post-localization behavior. Figure 13 shows the standard variation in the stress versus the displacement both 2D and 3D samples. The standard deviation at the ultimate stress from 2D modeling was 0.095 MPa, whereas for 3D modeling it was 0.045 MPa (see point X at Fig. 13). The results suggest the use of a 3D heterogeneous model for the capsules-based self-healing concrete. Therefore, the subsequent results presented in this study are based only on 3D simulations.

To evaluate the effect of the capsule volume fraction on the mechanical behavior of capsules-based self-healing concrete, three different volume fractions were considered here: 2%, 5%, and 10%. The stress-displacement curves can be found in Fig. 14 and Fig. 15, while Fig. 16 depicts the dissipated energy versus the displacement. Upper and lower bounds of the peak stress for 50 realizations and different capsule volume fractions are illustrated in Fig. 15.

The highest mean stress was obtained around 3.58 MPa for specimens with volume fraction of capsules of 0%, followed by capsules with volume fractions of 2%, 5%, and 10% which had mean stress of 3.53, 3.52, and 3.50 MPa, respectively. The more capsules added into the samples, the lower was the strength of the samples, whereas the dissipation energy increased. The initial elastic stiffness of the samples decreased with increasing capsule volume fraction.

For comparison, we also computed the tensile strength using the rule of mixtures (ROM) although it tended to exceed the original tensile strength of FRP rebars [35]. The tensile strength of capsule-based self-healing concrete can be 138 predicted by a linear combination from tensile properties of matrix (m), aggregates (a), and capsules (c) as follows:

(4){\sigma }_{\text{CSHC}}={\sigma }_{\text{m}}{V}_{\text{m}}+{\sigma }_{\text{a}}{V}_{\text{a}}+{\sigma }_{\text{c}}{V}_{\text{c}},

where σ_CSHC, σ_m, σ_a, and σ_c indicate the tensile strength of capsule-based self-healing concrete, the mortar matrix, aggregates, and capsules, respectively (MPa); V_m, V_a, and V_c are the associated volume fractions, respectively (%). According to Eq. (4), substituting the volume fractions of capsules ranging from 2%, 5%, 10% and tensile properties from Table 2, the predicted tensile strengths of capsule-based self-healing concrete were 3.14, 3.35, and 3.7 MPa, respectively. These ROM predictions were in good agreement with the numerical results.

The mean dissipation energy over the displacement for different volume fraction of capsules––2%, 5%, and 10%––can be seen in Fig. 16. The dissipated energy increased with increasing capsule volume fraction. The dissipated energy increased by 13.2% (from 53 to 60 mJ) when the capsule volume fraction was increased by 8%. The higher micro-cracks density indicated that more dissipation energy was needed when more capsules are embedded in the samples.

Our numerical results agreed well with experimental observations from Wang et al. [36], Kanellopoulos et al. [37], and Lv et al. [38], using micro-encapsulated bacterial spores, pig-gelatine microcapsules carrying sodium silicate, and poly (phenol-formaldehyde-PF) microcapsules, respectively. They showed that there was a significant decrease in the strength of capsule-based materials compared to pristine concrete. Note also that the capsules-based models were more ductile when more capsules were added into the samples (Fig. 14). Similar observations were reported in recent experimental studies [38].

4.5 Crack patterns

Figure 17 shows the microcrack evolution during the loading stages. The red color indicates the ‘active’ (opened) cohesive elements. Note that the deformations are magnified with a factor of 15 to better visualize the microcracks. In most cases, the final fracture pattern was dominated by a single main macrocrack which evolved from coalescence of the microcracks between two closest and largest aggregates at the top and the bottom of the samples. The fracture evolution inside the 3D specimen is presented in Fig. 18. At an early loading stage, a number of microcracks nucleated at the interfaces between the aggregates and mortar matrix. Some of them were located near the capsules, as can be seen in Fig. 18(a). With increasing loading, the microcracks grew continuously––mainly between two closest and largest aggregates––before forming a macroscopic crack.

Some of the micro-cracks grew toward the capsules breaking their shells as indicated in Fig. 18(b). The cohesive elements shown in red color in Figs. 17 and 18 indicate a damage value of D ≥ 0.95; D = 1 is complete failure. The fracture surface of the final model is visualized in Fig. 19; in this plot the cohesive elements have been removed.

The breakage of the capsule shell caused by an incoming crack at an early stage is crucial in self-healing concrete. Once the capsule shell is broken, the healing liquid inside the shell will flow into the damaged area and seal the crack to restore its capacity. Figure 20 shows the fracture of the capsule shells. The red cohesive elements on the capsule shells represent a damage index higher than 0.95.

The distinct difference in the more complex fracture pattern between 2D and 3D models discussed in the previous section is the reason behind the discrepancies in the dissipated energy and load-deflection curves. 3D samples presumably predict the fracture patterns more realistically, especially in the thickness direction compared to 2D samples. Hence, these simulation results emphasize the importance of 3D modeling of self-healing concrete at the meso-scale.

5 Conclusions

Encapsulation-based self-healing concrete with randomly distributed spherical aggregates and capsules were investigated through extensive 3D numerical simulations. Capsules-based self-healing concrete were considered as multi-phase material consisting of aggregates, capsule shells, capsule core and their interfaces. The peak stress and dissipated energy versus the applied displacement were monitored. Pre-inserted cohesive elements were employed to simulate potential crack patterns. For comparison, we also carried out 2D simulations. The effect of the capsule volume fraction on the macroscopic uniaxial tensile strength, dissipated energy, and stiffness of self-healing concrete was investigated. The following conclusions can be summarized from our numerical studies.

1) 3D modeling has a pronounced influence not only on macroscopic mechanical properties but also on the crack patterns, compared to associated 2D models. The 3D model results in a higher ultimate stress and a lower standard deviation in the post-localization region compared to the results of the 2D model.

2) The macroscopic response of self-healing concrete is strongly dependent on the amount (per volume) of capsules.

3) When the amount (per volume) of aggregates remains constant, the tensile strength decreases with increasing capsule volume ratio.

4) For a constant aggregate volume fraction, the dissipated energy in tension increases with increasing capsule volume ratio.

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