The following four steps are required to establish the geometric model of mineralized precipitate using the RPGPA (Yin et al. 2011; Yang et al. 2012).

The first step is to determine the population of particles of different calcium carbonate crystal phases in the precipitate. The particles of calcite, aragonite and vaterite have the shape of block, prism and sphere, respectively, according to the scanning electron microscopy observations (SeifanBerenjian 2019; Zehner et al. 2021). Therefore, in the two-dimensional model of precipitate, squares, rectangles with aspect ratio of 2:1, and circles are used to simulate calcite, aragonite and vaterite particles. After the total packing area is determined, the particle numbers of calcite, aragonite and vaterite can be determined according to their volume proportion in the precipitate.

The second step is to generate and pack different crystalline particles. A packing area is given according to the microbeam notch size. To avoid the conflict between the particles and the packed particles in the packing process, and improve the packing efficiency, all the particles are reduced according to a certain proportion, and then randomly packed in the packing area. Their sizes are recovered until the packing process is over.

The third step is to oscillate to equilibrium. To eliminate the overlap or intrusion between particles that may occur after particle size recovery, a large stiffness is assigned to the boundary or wall of the packing area and particles to simulate the randomly mixing process. When there is overlap or intrusion between particles, and between particles and walls, their large repulsive force will make the overlap or intrusion disappear to ensure that particles are reasonably distributed in the packing area.

The fourth step is to establish the repaired microbeam model. The generated precipitate model is filled into the notch in the microbeam to establish the geometric model of repaired microbeam.

In general, in particle reinforced composites, the matrix material and its interface with the reinforced particles have lower strength than the particles. Therefore, under the action of load, damage and crack initiate in the matrix material and its interface between the reinforced particles. In precipitates, the particles of calcite, aragonite and vaterite are the reinforcement, while intergranular materials containing a large number of voids, impurity materials and amorphous calcium carbonate are the matrix. The CZM (Yang et al. 2012; Zhang et al. 2016, 2018) is used to describe the damage and cracking behavior occurring inside the matrix, between the matrix and the reinforced particles, and between the matrix and the repaired material during loading.

The CZM assumes that the material interface which is likely to be damaged and cracked is bonded by a virtual bonding surface. When damage occurs, the virtual bonding surface opens. A bilinear constitutive model is used to describe the relationship between the tractive force t acting on the virtual bonding surface and its opening displacement δ (Wang et al. 2015). The damage D of the virtual bonding surface in the softening stage is defined as(Gad et al. 2021)

$$ D = \frac \left( } \right)}} - \delta _ } \right)}} $$

(1)

in which, \(\:_\) is the opening displacement when damage initiates in the virtual bonding surface, and \(\:_\) is the opening displacement when the virtual bonding surface is completely cracked. When D = 0, no damage occurs in the virtual bonding surface. However, when D = 1, the virtual bonding surface is completely cracked. The fracture energy is the energy consumed when a unit of material breaks. It can be expressed as

$$ G = \frac\delta _ t_ ~~~~ $$

(2)

in which, \(\:_\) is the strength of the virtual bonding surface.

In order to analyze the deformation and failure of the repaired microbeams under three-point bending loading, the following material parameters must be determined first: elastic modulus E, Poisson ratio ν and density ρ of calcite, aragonite, vaterite, precipitate matrix and cement mortar; tensile strength t0, initial stiffness kn and fracture energy GF of precipitate matrix and cement mortar; and tensile strength t0, initial stiffness kn and fracture energy GF of the interfaces of precipitate matrix with calcite, aragonite, vaterite particles and cement mortar.

Luo et al. (2022) and Agbaje et al. (2017) gave the elastic modulus of 73.2 GPa for single crystal calcite and 140 GPa for single crystal aragonite, respectively. Sevcík et al. (2018) gave the elastic modulus of vaterite 31–58 GPa. Xu et al. (2020) gave the density of calcite 2.71 g/cm3, the density of aragonite 2.95 g/cm3 and the density of vaterite 2.64 g/cm3. Chen et al. (2001) and Hearmon (1979) found that the Poisson ratio of calcium carbonate crystals was about 0.31. Wang et al. (2024) gave the elastic modulus, Poisson ratio and density of cement mortar 26 GPa, 0.2 and 2.2 g/cm3, respectively. Zhang et al. (2016) gave the tensile strength of cement mortar 3.1 MPa and the fracture energy of 380 N/m. With reference to the abovementioned literatures, the main material parameters used in this paper are shown in Table 1. It must be pointed out that the fracture parameters of the precipitate matrix and its interface with the cement mortar and different crystal phase particles are first given by referring to some references (Gorna et al. 2008; Freeman et al. 2023; Bruno 2019), and then appropriately adjusted according to the comparison between the calculations and experiments.

Feng et al. (2021) observed by scanning electron microscope (SEM) that all precipitate particles were calcite and the average particle size was about 2.4 μm in the repaired mortar samples using MICP. Zehner et al. (2020) measured the volume content of particles in the precipitate to be about 45%. In this paper, we assume that there is a notch with size of 20 μm×20 μm in the bottom midspan of a unit thickness microbeam with a dimension of 160 μm×40 μm. The MICP technique is used to repair the microbeam. It is assumed that the precipitate particles are all calcite and the particle size is 2.4 μm. The geometric model of precipitate is established by using the RPGPA. Solid triangular elements are used to discretize the grid, and the CZM elements (Zhang et al. 2016) are inserted into the cement mortar in the midspan region of the microbeam, the precipitate matrix and its interface with the particles and the cement mortar. The finite element models of the perfect microbeam, notched microbeam and repaired microbeam are established, as shown in Fig. 1. Taking into account the balance between calculation accuracy and time cost (Yin et al. 2011), a mesh size of 0.8 μm is selected in the midspan region of the microbeam, while a mesh size of 1.5 μm is selected in the region away from the midspan.

Finite element models of perfect microbeam, notched microbeam and repaired microbeam

The displacement constraint is applied to the two ends of the microbeam bottom with 70 μm distance from the midspan. The horizontal and vertical displacement constraints are applied to the left constraint point, but only the horizontal displacement constraint is applied to the right constraint point. The deformation and failure process of the microbeam under three-point bending load is simulated by applying a monotonically increasing concentrated load to the top of the microbeam midspan. Figure 2a shows the load-displacement curves of perfect, notched, and repaired microbeams. It can be seen that, compared with the perfect microbeam, the notched microbeam decreases significantly in the initial stiffness and ultimate load of the ascending stage, but the displacement corresponding to the ultimate load does not change significantly. After repaired using the MICP technique, compared with the notched microbeam, although the initial stiffness does not change significantly, the ultimate load is significantly increased, and the displacement corresponding to the ultimate load is also significantly increased. In general, the ultimate load can be used to characterize the bearing capacity of the microbeam, so it indicates that the bearing capacity of the notched microbeam is recovered to a certain extent.

Load-displacement curves from numerical simulations

In order to eliminate the influence of particle distribution and crystal orientation on the calculation results of repaired microbeams, for a given crystal particle combination, the RPGPA is used to randomly create 10 sample models with different particle distributions and crystal orientations. Figure 2b shows the envelope of their load-displacement curves. It can be seen that all curves basically coincide in the ascending and descending stages, which indicates that the particle distribution and crystal orientation have very little influence on the calculation results. Therefore, the influence of particle distribution and crystal orientation is not considered in the subsequent analysis and discussion. Yin et al. (2012) also concluded that the aggregate distribution had little effect on the mechanical properties of asphalt mixture.

Nuaklong et al. (2023) evaluated the self-repairing effect of shrinkage crack in mortar containing microencapsulated bacterial spores by the permeability and compressive strength. In this study, the repairing effect of the notched microbeam through the MICP technique is evaluated by the bearing capacity and fracture dissipation energy. The recovery rate of bearing capacity can be obtained by the following expression.

$$\:_=\frac_-_}_-_}$$

(3)

where, fc, fd, and fb are the ultimate loads of the perfect, notched, and repaired microbeams under three-point bending loading, respectively. Similarly, the recovery rate of the fracture dissipation energy can be obtained by

$$\:_=\frac_-_}_-_}$$

(4)

where, Gc, Gd, and Gb are the fracture dissipation energy of the perfect, notched, and repaired microbeams, respectively.

The calculation shows that 10 sample models have the average ultimate load of 0.000964 N, and the average recovery rate of 20.94%. Feng et al. (2021) carried out an experimental study on repairing concrete beams by MICP technique, and found that the average recovery rate of the ultimate load of beams was about 20.41%, which is only 2.52% difference from our result. Chen et al. (2021), Zheng et al. (2021), Bayati et al. (2023) and Fu et al. (2024) all carried out experimental studies on repairing cement mortar beams with MICP technique, and the average recovery rates of beam ultimate loads obtained are 21.16%, 5.30%, 10.05% and 20.40%, respectively. The results given by Chen et al. (2021) and Fu et al. (2024) are also very close to our result, but the results by Zheng et al. (2021) and Bayati et al. (2023) are much different from our result. Combined with the SEM images, it is found that the crystal type and particle size in precipitate are significantly different from our models. Considering that the crystal type and particle size of the precipitate can significantly affect the mechanical properties of the precipitate (Xu et al. 2020; Cheng et al. 2013), this should be the main reason for the difference between the above experimental results and our result.