^{1}

^{1}

^{2}

^{1}

^{2}

The features of the pores in pervious concrete have a great influence on the mechanical property of the pervious concrete. In this study, a finite element model (FEM) with random pores has been used to simulate the mechanical characteristics of pervious concrete. First, pervious concrete specimens with two different porosities were prepared in the laboratory. Then, the specimens were cut and the pores features were extracted based on the images of cross-sections. Thereafter, the ellipse and roundness were introduced into a simulation model to describe the characteristics of pores including the size, area, shape, and inclined angle which have been developed randomly in the FEM using the Monte Carlo method. In the simulation model, the aggregate and cement paste have been simplified into a composite material, and the method of determining the composite modulus of aggregate-cement paste is discussed. The simulation results show that the shape and distribution of the pores have an obvious influence on the internal stress of the pervious concrete and the pore features can be considered reasonable for the validity of the simulations. In addition, the composite modulus of the aggregate-cement paste can be determined by comparing the simulation and test results. The recommended composite modulus in the pervious concrete simulation model is much lower than that of common impervious concrete.

Pervious concrete is material that has been used to favor the permeability of surfaces. The most singular characteristic that favors the selection of this special type of concrete is its interconnected porosity. There are many earlier studies on the pore feature and numerical simulations of pervious concrete. They found that there is a close correlation between porosity, permeability, and mechanical properties of pervious concrete. In the studies of Debnath and Sarkar [

There are two main methods to determine the pore structure of pervious concrete. One is to use the nondestructive test method by combining tomography and digital image processing [

Pervious concrete is a multiphase material. Both the experimental and numerical analyses were used to investigate mechanical properties of pervious concrete [

The above literature review shows that the method of random distribution aggregate model was usually used in the earlier studies. But the cement paste layers between the aggregates were difficult to define for pervious concrete because of the small thickness. In addition, there are a large number of macropores in pervious concrete which should not be ignored. In this study, using the random pore distribution method, a different way has been used to develop a numerical model. First, the pore image data of pervious concrete have been extracted from the cut specimen of the pervious concrete. Then, the pore features have been analyzed by digital image processing technology. Based on the pores characteristics, the random generation of pores has been programmed in the development of the pervious concrete simulation model.

In accordance with the requirements of the Chinese specification CJJT135-2009 [

Cement quality parameters.

Test items | Consistency (mm) | Initial setting time (min) | Final setting time (min) | Compressive strength (MPa) | |
---|---|---|---|---|---|

3 days | 28 days | ||||

Material parameters | 28 | 150 | 250 | 24.7 | 47.2 |

Two sizes of aggregate have been used in this study which was obtained from the local quarries in Shandong province of China. The two sizes are D1: 4.75–9.5 mm and D2: 9.5–13.2 mm basalt (Figure

Coarse aggregate: (a) particle size 4.75 mm∼7.5 mm; (b) particle size 9.5 mm∼13.2 mm.

Coarse aggregate properties.

Aggregate | Bulk density (kg/m^{3}) | Apparent density (kg/m^{3}) | Void in mineral aggregate (%) |
---|---|---|---|

D1 | 1643 | 2819 | 40.32 |

D2 | 1611 | 2855 | 41.35 |

D1 (80%): D2 (20%) | 1622 | 2830 | 41.17 |

Related studies have shown that the porosity of pervious concrete meeting both strength and permeability is generally between 20% and 30%. Thus, 22% and 28% have been selected as the designed porosity in the test, representing the lower and higher porosities, respectively. The two types of specimens have been labeled as 1# and 2#. According to the designed porosities, the amounts of raw materials per unit volume have been determined, as shown in Table

Compositions of pervious concrete.

Specimens group | Design porosity (%) | Coarse aggregate (kg/m^{3}) | Cement (kg/m^{3}) | Water (kg/m^{3}) |
---|---|---|---|---|

1# | 22 | 1631.7 | 417.2 | 125.1 |

2# | 28 | 1631.7 | 319.6 | 95.9 |

Six 100 mm cubic specimens for each designed porosity were prepared as follows. The quantities of material according to the composition in Table

Pervious concrete specimen preparation: (a) mixing; (b) compact; (c) molded specimens; (d) curing.

The volumetric method has been used to calculate the void content and the connected porosity of the prepared specimens. Firstly, the mass of the cured pervious concrete specimen in water was measured. Then, the specimens were placed in a drying box and the mass was measured after drying. Finally, the bulk volume of specimen was measured by a vernier caliper. Using equations (1)–(3), the total void ratio and the connected void ratio have been determined:_{t} is theoretical density of the pervious concrete materials; _{0} is the void content; _{e} is the connective porosity; _{c} is the mass ratio of cement to coarse aggregate; _{a} is the apparent density of coarse aggregate; _{c} is the apparent density of cement; _{s} is the gross bulk density of pervious concrete; _{2} is the dry mass of the specimen; _{1} is the mass of the specimen in water; _{w} is the density of water; and

The cubic specimens were cut vertically and a high-definition digital camera was used to obtain the cross-sectional image. In the process of image acquisition, the quality of digital image was often affected by some noise, such as the camera quality and lighting. In order to improve the image quality, the cut section image needs to be enhanced. The original image was therefore processed by the software PSCC based on the smoothing and sharpening algorithm in the spatial domain. Taking the specimen with the 22% designed porosity as an example, the test specimen was vertically cut into a 100 mm square. The original image and the enhanced image are shown in Figure

Digital image of cut section: (a) original 2D digital image; (b) enhanced 2D digital image.

To obtain the pore information from the enhanced image, the steps are as follows:

Select a reasonable threshold to segment the enhanced image.

Compare the threshold with the gray value of pixels in the image: if the gray value of the pixel area is higher than the threshold, the area is marked 1; otherwise it is marked 0.

Import the binarized image into the software Image-Pro Plus, and set the scale parameters to calculate the planar pore characteristics in the digital image.

The binarized image and planar pore characteristic parameters are shown in Figure

Planar pores extraction from cut section image: (a) binarized image; (b) pores feature parameters extraction.

The pores structural characteristics mainly include three aspects: porosity, pore size, and pore shape.

A single pore area can be obtained by counting the number of pixels in the pore. All the pore areas can be obtained by calculating the sum of all the pore pixels in the section image. The planar void ratio is the ratio of all the pore areas to the cross-sectional area. Because the unit in the software Image-Pro Plus is pixels, used to describe the digital image, it is necessary to convert the pore pixel data to an area unit. By adjusting the digital image to 1000 × 1000 pixels, the pore area in mm^{2} can be calculated as follows:

To facilitate the description of the pore morphology, the pores are simplified to a circular and an elliptical shape as described by the equivalent diameter and roundness, respectively. The equivalent diameter (_{e}) of a single pore is calculated as follows:

The pore shape of an ellipse is defined by the roundness which is the ratio between the long and short axis of an equivalent ellipse (Figure

Schematic diagram of pore shape.

To analyze the influence of pores on the mechanical properties of pervious concrete, the aggregate-cement is assumed to be a homogeneous material in the simulation model and the pores are randomly distributed among aggregate-cement materials. The pore characteristics, such as size, shape, and porosity, are determined according to the actual data from the image analysis data. The steps for developing a FEM of pervious concrete are as follows:

Analyze the pore structural characteristic including the equivalent diameter and long axis and short axis radius of an elliptical pore and the pore area. Further, the pores are divided into several groups based on the equivalent diameter.

The pores are randomly generated for every group, and the size, position, and angle of the pores are determined randomly within the set range.

The effectiveness need to be verified when one pore is generated. If the new pore overlapped with an existing pore, the newly generated pore is invalid and deleted. This process of pore generation is repeated until the new pore did not intersect with any of the existing pores. When the total areas of the generated pores reach a predetermined value, the process of pore generation is stopped.

All the pores areas are subtracted from the model region by the Boolean operation in the finite element software Ansys to generate a two-dimensional model with random pores.

Apply the boundary conditions, material properties, and load conditions to solve.

Taking the cross-sections of the specimens with 15% and 20% porosities as examples, the random circular pores and the random elliptical pores generated by the finite element model are, respectively, shown in Figures

Finite element model with planar porosity of 15%: (a) random circular pores; (b) random ellipse pores.

Finite element model with planar porosity of 20%: (a) random circular pores; (b) random ellipse pores.

The compressive strength of the specimens was tested at the age of 28 days in accordance with ASTM C39-15a and Chinese specification CJJT135-2009. The loading rate is 1 mm/min and the compressive force was applied until the specimen displayed a well-defined fracture pattern. During the compressive test, the load magnitude and deformation were recorded at the same time by a control computer (Figure

Compressive test control system: (a) compression test machine; (b) compression test control computer.

In Table

Volume parameters of specimens.

Specimen number | Design porosity (%) | Void content (%) | Average void content (%) | Connective porosity (%) | Average connective porosity (%) | Specimen volume (mm^{3}) | |
---|---|---|---|---|---|---|---|

1# | 1-1 | 22 | 22.0 | 21.8 | 16.8 | 16.6 | 1015 |

1-2 | 22.3 | 16.3 | 1012 | ||||

1–3 | 21.5 | 16.9 | 1014 | ||||

1–4 | 22.6 | 17.1 | 1013 | ||||

1–5 | 21.0 | 16.5 | 1005 | ||||

1–6 | 21.5 | 16.2 | 1008 | ||||

2# | 2-1 | 28 | 27.0 | 27.3 | 25.2 | 25.5 | 1010 |

2-2 | 27.6 | 25.8 | 1012 | ||||

2-3 | 27.3 | 25.7 | 1012 | ||||

2–4 | 26.9 | 25.0 | 1010 | ||||

2–5 | 27.4 | 25.5 | 1009 | ||||

2–6 | 27.6 | 25.8 | 1007 |

According to the pore data extracted from the section image of the pervious concrete, both the equivalent diameter and the roundness of the pores have been calculated and analyzed. The results show that the equivalent diameter ranges from 0 to 10 mm for specimen 1# and ranges from 0 to 13 mm for specimen 2# (Figure

The number of pores with different equivalent diameter: (a) specimen 1#; (b) specimen 2#.

In Figure

The number of pores with different roundness: (a) specimen 1#; (b) specimen 2#.

For the subsequent simulation models, the pores have been divided into five groups based on the equivalent diameter (_{e}), i.e., tiny pores (_{e} ≤ 1.0 mm), small pores (_{e} = 1.0～2.5 mm), medium pores (_{e} = 2.5～5.0 mm), large pores (_{e} = 5.0～9.0 mm), and super pores (_{e} ≥ 9.0 mm). Accordingly, the lengths of long axis and short axis of the pores have been extracted and sorted, as shown in Tables

Statistical results of pores feature for specimen 1#.

Parameter | Range | ||||
---|---|---|---|---|---|

Tiny pores | Small pores | Medium pores | Large pores | Super pores | |

Equivalent diameter (mm) | <1.0 | 1–2.5 | 2.5–5 | 5–8 | >9 |

Long axis range (mm) | 0.15–1.9 | 1.2–3.6 | 3–9 | 7.6–13 | 12.6–20 |

Short axis range (mm) | 0.12–0.75 | 0.74–2 | 1.3–3.4 | 2–8 | 5.4–9.6 |

Planar porosity (%) | 0.07 | 1.2 | 3.05 | 5.8 | 4.1 |

Statistical results of pores feature for specimen 2#.

Parameter | Range | ||||
---|---|---|---|---|---|

Tiny pores | Small pores | Medium pores | Large pores | Super pores | |

Equivalent diameter (mm) | <1.0 | 1–2.5 | 2.5–5 | 5–8 | >9 |

Long axis range (mm) | 0.15–1.02 | 1.3–3.8 | 3–8.4 | 5.8–13 | 12–18 |

Short axis range (mm) | 0.1–0.75 | 0.7–1.8 | 1.4–4.4 | 1.9–7 | 5–8.4 |

Planar porosity (%) | 0.01 | 0.6 | 5.5 | 9.4 | 6.1 |

To verify the random pore models, the size, area, and number of pores in each group have been recorded by a program in the process of pores generation. The recorded statistical data have been used to verify whether the porosity in the FEM is consistent with reality. The comparisons are shown in Table

Porosity comparison between FEM and reality.

Parameter pores | Range | |||||
---|---|---|---|---|---|---|

Tiny pores | Small pores | Medium pores | Large pores | Super pores | ||

Specimen 1# | Actual porosity (%) | 0.07 | 1.2 | 3.05 | 5.8 | 4.1 |

Circular pore model (%) | 0.07 | 1.12 | 3.07 | 5.95 | 4.10 | |

Ellipse pore model (%) | 0.07 | 1.14 | 3.14 | 6.01 | 4.11 | |

Specimen 2# | Actual porosity (%) | 0.01 | 0.6 | 5.5 | 9.4 | 6.1 |

Circular pore model (%) | 0.01 | 0.60 | 5.53 | 9.57 | 6.51 | |

Ellipse pore model (%) | 0.01 | 0.62 | 5.6 | 9.61 | 6.6 |

Stress cloud in simulation model for specimen 1#: (a) random circular pores model; (b) random ellipse pores model.

Stress cloud in simulation model for specimen 2#: (a) random circular pores model; (b) random ellipse pores model.

Variation of stress-strain curve of simulation model with composite modulus of aggregate-cement paste material: (a) specimen 1#; (b) specimen 2#.

^{−3} and 3.5 × 10^{−3} for the elliptical pore model and the circular pore model, respectively. This may be caused by the random distribution of the angle and the roundness of the elliptical pores, which can easily result in stress concentration surrounding the pores, especially around the flat pores. So, it is necessary to consider the pore shape in the simulation model of pervious concrete. Certainly, the pore features should be evaluated fully based on the actual specimens before developing the simulation model.

Effect of pores shape on the simulation results: (a) specimen 1#; (b) specimen 2#.

With the increase in the loading force, an increase of deformation was measured during the compressive test in the laboratory. Figure

Compressive property of pervious concrete: (a) specimen 1#; (b) specimen 2#.

In pervious concrete, aggregates are bonded by cement paste. The modulus of aggregate-cement paste is affected by the aggregate modulus, the cement content, and the bond strength of cement paste. The composite modulus reflects the composite mechanical property of an aggregate-cement paste material. To determine a reasonable modulus, the composite modulus varies widely from 3,000 to 35,000 MPa, and the compressive stress of 10 MPa is applied on top surface of the simulation models. The relationships between the vertical strain and the composite modulus have been calculated and the results are shown in Figure

Composite modulus of aggregate-cement paste determination in simulation model: (a) specimen 1#; (b) specimen 2#.

In this study, a random pore FEM has been developed to simulate the mechanical property of pervious concrete. In the simulation model, the influence of pore characteristics, including porosity, shape, and pore size, has been considered and the determination of the composite modulus of the aggregate-cement paste has also been discussed. The conclusions are as follows:

The random pore FEM can be used to simulate the real pore features, including the size, shape, and random distribution. In addition, the aggregate and cement paste has been simplified into one composite material which can significantly improve the efficiency in the model meshing, calculations, and result extracting.

The distribution and size of pores have an important effect on the stress distribution of pervious concrete. If the distance between bigger pores is small, the stress around the pores may be high. For instance, when the horizontal distance between two big pores is small, the compressive stress between the pores is very large. In another case, when the vertical distance between two big pores is small, the tensile stress between the pores is very high. In these cases, microcracks or damage may occur around the pores.

In the random pore FEM, ellipse and roundness have been used to characterize the features of real pores. The method can describe the shape and size of actual pores. In particular, the inclined angle of pores can be also represented by adjusting the degree of long axis or short axis of the ellipse. The simulation results show that the stress and strain in the random elliptical pore model are larger than those in the random circular pore model. This is induced by the difference in the stress concentration around circular and elliptical pores.

The composite modulus of aggregate-cement paste is an important parameter in the simulation model, which affects the accuracy of simulation results. By comparing the simulation results and the test data, a reasonable composition modulus for pervious concrete has been obtained. In this study, the composite modulus of aggregate-cement paste varies from 4,000 MPa to 6,000 MPa for the pervious concrete with porosity of 20%∼30%. It is noticed that the value of composite modulus is remarkably lower than that of common cement concrete.

The data supporting the conclusions of the present study can be obtained from the corresponding author.

The authors declare that they have no conflicts of interest.

J.S. and J.L. conceptualized the study and were involved in original draft preparation; J.S. contributed to methodology, performed formal analysis, and was involved in project administration; J.L. was responsible for software and resources; J.L. and F.L. were involved in validation, performed data curation, and contributed to funding acquisition; F.L. contributed to investigation.

This research was funded by “A Project of Shandong Province Higher Educational Science and Technology Program,” grant number J17KA213.