Micro-mechanical properties of Song Dynasty tilestones based on nanoindentation tests and homogenization approach

Abstract

Song Dynasty tilestones are one type of ancient Chinese building materials. Studying their mechanical properties is of great significance for the design and development of restoration materials. It is a challenge to sample and perform traditional tests (ϕ50mm × 100mm) for the tilestone cultural relics. In this work, a combination of nanoindentation techniques and the homogenization calculation method based on the Mori–Tanaka model were used to determine the mechanical parameters of Song Dynasty tilestones. The study process involved several steps: (1) Using X-ray diffraction and scanning electron microscopy to examine the surface morphology and mineral composition of the tilestones. (2) Determining the mechanical parameters (i.e., the elastic modulus, hardness and fracture toughness) through nanoindentation tests. (3) Upgrading mechanical parameters from micro to meso-scale using the Mori–Tanaka model and comparing these with uniaxial compression test results. The result shows that the red tilestones and green tilestone are mainly composed of quartz, feldspar and mica. The average elastic modulus of the red tilestones and the green tilestones are 29.47 GPa and 30.21 GPa, respectively. Compared with the parameter result obtained by upscaling, the deviation rates of the red tilestones and green tilestones are 10.3% and 9.6%, respectively, which proves that the test method is reliable. The nanoindentation test and homogenization approach in this work provide the robust theoretical and practical basis for evaluating the mechanical strength of Song Dynasty tilestones.

Introduction

The splendid civilization of the Chinese nation has left behind many stone cultural relics that attract worldwide attention, such as the grottoes, stone carvings, steles, statues, and various buildings from the past dynasties. The stone cultural relics have important historical and cultural significance as well as research value [1,2,3]. As one of the ancient stone building materials, the tilestone has been used more than three thousand years. Song Dynasty tilestones are one of the representatives of ancient Chinese building materials and reflect the technical level and the aesthetic concept of architecture in the Song Dynasty [4].

At present, tilestone has been subjected to the combined effects of natural factors and human activities, resulting in varying degrees of damage, which seriously threaten their utilization and preservation [5]. In particular, stone cultural relics exposed to the natural environment have been subjected to long-term natural weathering, salt crystallization, air pollution, ultraviolet radiation, acid rain erosion, and gradually eroded and fell off until they are utterly destroyed. In order to protect historic buildings, many scholars have studied and analyzed the ancient brick-and-tile buildings. Li et al. analyzed the differences between the brick-and-tile buildings in the Qin and Han Dynasties and the Zhou Dynasty from a multidisciplinary perspective using integrated methods, such as archaeology, statistics, and theoretical analysis of ceramics technology [6]. Zhang systematically analyzed the brick-and-tile decoration in Han Dynasty, and the research results provided a material basis for in-depth exploration of the funeral system and production development of Jiangsu region in Han Dynasty [7]. The above research results provide valuable information for the comparison and study of tile-making techniques in different dynasties. In addition, some scholars also use modern testing techniques to analyze the structure and performance of tilestone. Zheng et al. used software to analyze the thermal performance of traditional empty bucket tile wall buildings [8]. Ren and Ji used 3D digital technology to reveal the decorative features and significance of the Matou wall and the door, they analyzed the concept of ecological energy saving in ancient Huizhou residential tilestone [9]. The above studies mainly focused on the tile combustion process, architectural structure and architectural culture. However, few studies focused on the mineral composition and mechanical properties of tilestone.

Meanwhile, it is more difficult to sample because of the protection standards of cultural relics and the preciousness of cultural relics. Therefore, an efficient and simple method to obtain the mechanical properties of the stone relics is urgently needed. In recent years, Computed Tomography (CT) technology and acoustic wave test have been used to detect the mechanical properties of stone relics and restoration materials [10,11,12]. Selçuk used ultrasonic diffraction technology to measure the rock cracks and crack depth, and proved the accuracy and reliability of the technology [13]. Maher focused on the use of Multi-Detector Computed Tomography (MDCT) to study an ancient Egyptian falcon hollow-cast bronze coffin, and the overall geometry of the coffin was obtained and all casting process components were revealed [14]. However, the non-destructive testing method has strict requirements on the test environment and samples, and the test process is easily affected, resulting in inaccurate test results. Nanoindentation technique is one of the rapidly developing techniques for surface mechanical testing in recent years. It has the advantages of being non-destructive, high-resolution, and easy to do, etc. The key mechanical parameters, such as hardness, elastic modulus, plastic strain and fatigue strength of different materials can be measured by nanoindentation test. In particular, nanoindentation testing methods greatly reduce the requirements for the quality and size of rock samples. For stone cultural relics which are difficult to sample, continuous mechanical testing can be carried out with a small number of samples, and then mechanical parameters of stone cultural relics can be obtained. Li and Shi et al. analyzed the mechanical properties of shale at different scales by nanoindentation test [15, 16]. Multi-scale structure models of mudstone were constructed from macroscopic, mesoscopic, and microscopic levels to obtain mechanical distribution rules of rock materials at different scales [17, 18]. The elastic modulus of the sandstone was determined by nanoindentation techniques, and the meso-heterogeneity of the sandstone was proven [19, 20]. The above research indicates that the mechanical parameters of rock mass at the microscopic level can be obtained by the nanoindentation test, which provides a train of thought for studying the law of rock deterioration. However, different from the natural rock mass, it is difficult to sample the tilestone cultural relics, which makes it impossible to carry out conventional mechanical tests. Thus, it is urgent to obtain the mineral composition and mechanical parameters of the tilestone cultural relics, so as to provide a basis for evaluating the damage state and restoration of ancient tilestone buildings.

In this work, the tilestone were prepared from the Shuntianmen Site, Kaifeng. The mechanical properties, including elastic modulus, hardness and fracture toughness of tilestones, were tested using grid nanoindentation. With the adoption of X-Ray Diffraction (XRD) and Scanning Electron Microscope (SEM), the material surface morphology and mineral composition in indentation area were quantitatively analyzed. Based on the results of nanoindentation test, the mechanical parameter at micro-scale were determined. The Mori–Tanaka model was used to obtain the mechanical parameters from micro-scale to macro-scale. In order to verify the applicability of nanoindentation technique for tilestone cultural relics and the accuracy of the model, the results from the uniaxial compression test were compared with those from the nanoindentation test. The work is a great significance for studying the mechanical properties of the Song Dynasty tilestone at micro-scale. Meanwhile, the mechanical properties of tilestone cultural relics are obtained by this method, which provides scientific guidance for the preparation of restoration materials and the restoration of tilestone cultural relics.

Materials and methods

Nanoindentation principle

In the nanoindentation test, the mechanical properties of material are measured by applying pressure to the surface of material using an indenter with a specific shape and mechanical parameters. The process of nanoindentation test includes three stages: loading, holding load and unloading. In the loading stage, the elastic deformation is transformed into elastoplastic deformation. In the holding load stage, the holding time of 5 s was set to eliminate the hysteresis effect of loading. In the unloading stage, the elastic deformation is restored, while the plastic deformation causes the material to form an indentation. Indentation profiles before and after unloading are shown in Fig. 1. P is the load of the indenter, mN; h_max is the maximum depth, nm; h_f is the residual depth of the indenter after unloading, nm; h_c is the contact depth, nm.

Fig. 1

Indentation profiles before and after unloading [17]

The load displacement curve are shown in Fig. 2. P_max is the load corresponding to the maximum depth, mN; U_e is the elastic energy; U_p is the plastic energy; U_c is the fracture energy.

Fig. 2

Load–displacement curve obtained from nanoindentation [16]

The Oliver-Pharr method is widely used to analyze the nanoindentation loading and unloading curves [21]. According to the Oliver-Pharr method, the unloading curve of residual displacement is fitted to a power function by Eq. (1):

$$P = m\left( {h - h_{{\text{f}}} } \right)^{n}$$

(1)

where, m and n are the fitting parameters; S is the contact stiffness (the slope at the top of the unloading curve), which can be obtained by Eq. (2):

$${\text{S = }}\left( {\frac{{{\text{dP}}}}{{{\text{dh}}}}} \right)_{{{\text{h = h}}_{\max } }}$$

(2)

According to the relevant elastic contact theory, h_c can be obtained by Eq. (3):

$$h_{{\text{c}}} = h_{{{\text{max}}}} - \frac{{\varepsilon P_{{{\text{max}}}} }}{S}$$

(3)

where, ε the geometric constant for Berkovich indenter (ε = 0.7268) [22].

The indentation hardness and reduced modulus can be calculated through the following two equations:

$$H = \frac{{P_{{{\text{max}}}} }}{{A_{{\text{c}}} }}$$

(4)

$${\text{E}}_{{\text{k}}} = \frac{S\sqrt \pi }{{{2}\delta \sqrt {A_{{\text{c}}} } }}$$

(5)

where, H is the hardness, GPa; E_k is the reduced modulus, GPa; δ the geometric constant for Berkovich indenter (δ = 1.034). A_c is the projected contact area, nm². For Berkovich indenter, A_c = 24.5h_c². The elasticity modulus of sample can be obtained by Eq. (6):

$$E_{{\text{T}}} = \frac{{{1} - \mu_{T}^{{2}} }}{{\frac{{1}}{{E_{{\text{k}}} }} - \frac{{{1} - \mu_{{\text{i}}}^{{2}} }}{{E_{{\text{i}}} }}}}$$

(6)

where, E_T and E_i are the elastic modulus of the sample and the indenter, respectively, GPa; μ_T and μ_i are the Poisson’s ratio of sample and the indenter, respectively. In this work, the Poisson's ratio of red tilestone and green tilestone are 0.29 and 0.28, and they are obtained through uniaxial compression test [23]. The Poisson ratio of minerals (quart, feldspar and mica) are 0.2, 0.25 and 0.3, respectively, and they are obtained through an approximate assumption [24,25,26]. The indenter is diamond material in the nanoindentation test, and μ_i and E_i is 0.07 and 1140 GPa [27].

Fracture toughness is defined as a measure of a material's ability to resist crack growth and is commonly used to assess the strength and stability of defective materials [28]. In this study, fracture toughness can be calculated using the energy analysis method [29, 30]. The total energy (U_t) is composed of elastic energy (U_e), plastic energy (U_p) and fracture energy (U_c) in Eq. (7).

$$U_{{\text{t}}} = U_{{\text{e}}} + U_{{\text{p}}} + U_{{\text{c}}}$$

(7)

The relation between pure plasticity (U_p) and total energy (U_t) can be obtained by Eq. (8):

$$\frac{{U_{{\text{p}}} }}{{U_{{\text{t}}} }} = {1 - }\left[ {\frac{{{1 - 3}\left( {\frac{{h_{{\text{f}}} }}{{h_{{{\text{max}}}} }}} \right)^{{2}} { + 2}\left( {\frac{{h_{{\text{f}}} }}{{h_{{{\text{max}}}} }}} \right)^{{3}} }}{{{1 - }\left( {\frac{{h_{{\text{f}}} }}{{h_{{{\text{max}}}} }}} \right)^{{2}} }}} \right]$$

(8)

The total energy (U_t) can be calculated by Eq. (9):

$$U_{{\text{t}}} = \int_{{0}}^{{h_{{{\text{max}}}} }} {p_{{\text{c}}} } {\text{d}}h_{{\text{c}}} = \frac{{p_{{{\text{max}}}} h_{{{\text{max}}}} }}{{3}}$$

(9)

where, P_c is the required load which makes the indenter reach the contact depth (h_c). To calculate the fracture toughness, the critical energy release rate (G_c) needs to be obtained first by Eq. (10):

$$G_{{\text{c}}} = \frac{{\partial U_{{\text{c}}} }}{\partial A} = \frac{{U_{{\text{c}}} }}{{A_{{{\text{max}}}} }}$$

(10)

The (model I) fracture toughness (K_c) can be calculated by Eq. (11):

$$K_{{\text{c}}} = \sqrt {G_{{\text{c}}} E_{{\text{r}}} }$$

(11)

Site location and sample information

Site location

The Dongjingcheng Site from the Northern Song Dynasty is also known as the Shuntianmen Site, and it is located in Jinming District, Kaifeng City, Henan Province, China, as shown in Fig. 3.

Fig. 3

The location of Shuntianmen site

It adjacent to the Royal Garden-Qionglin Garden and Jinming Pool. The Site is a square area measuring 160m in length and 100m in width. In the Site, many stone artifacts and ancient building materials of different periods are excavated. These cultural relics and ancient building materials have important reference value for the study of contemporary culture.

Samples preparation

Red tilestones and green tilestones were chosen to conduct the test from the Shuntianmen Site in Kaifeng. These tilestones were cut into cubes (1 cm × 1 cm × 0.5 cm) by cutting machine, and then the samples were solidified by the epoxy resin. These tilestones were polished by sandpaper with 800 mesh, 1000 mesh, 1500 mesh, 2000 mesh and 3000 mesh [31]. These tilestones were polished by using an oily metallographic suspension with particle sizes of 3μm, 1μm and 0.3μm to obtain a complete sample. In this study, four sets of red tilestone samples and four sets of green tilestone samples were provided, as shown in Fig. 4.

Fig. 4

Tilestone samples prepared for a nanoindentation test

Test equipment

The nanoindentation test were performed on a Nanoindenter (Agilent Technologies G200), as shown in Fig. 5.

Fig. 5

G200 Nanoindenter

The Nanoindenter with a load resolution of 50 nN, in which a displacement resolution of less than 0.01 nm in the z direction and 1μm in the x and y direction. Moreover, the minimum pressing depth is greater than 500 μm, and the range of the effective working area is 100 mm × 100 mm [32]. The samples were tested by dynamic loading method, and the maximum depth was set to 2000 nm. In this work, the grid dot matrix was used. Thirty indentation points on the surface of red tilestones and green tilestones were set and numbered. The size of the point matrix was 100 μm × 125 μm, and the interval of measuring points was 25μm, as shown in Fig. 6.

Fig. 6

Gridding dot matrix diagram

Test scheme

To solve the problem that the mechanical parameters of stone cultural relics are difficult to determine, a combination of nanoindentation techniques and the homogenization calculation method were used to determine the mechanical parameters of Song Dynasty tilestones in this work. The test scheme was divided into three steps.

• (1) Using the XRD and SEM to examine the surface morphology and mineral composition of the tilestones.

• (2) Determining the mechanical parameters (i.e., the elastic modulus, hardness and fracture toughness) through nanoindentation tests.

• (3) Upgrading mechanical parameters from micro-scale to meso-scale using the Mori–Tanaka model and comparing these with uniaxial compression test results.

Results and discussion

Mineral composition and micro-structure of tilestones

Mineral composition of tilestone was obtained from the standard PDF card. The XRD pattern of red tilestone and green tilestone are shown in Fig. 7. The mass fraction of red tilestone and green tilestone are shown in Tables 1 and 2.

Fig. 7

The XRD pattern

Table 1 Quantitative results of mineral components in red tilestone

Table 2 Quantitative results of mineral components in green tilestone

The result shows that the red tilestones and green tilestone are mainly composed of quartz, feldspar and mica. The mass fraction of quartz in red tilestones and green tilestones is 42.9% and 45.1%, respectively. The mass fraction of mica in red tilestones and green tilestones is 27.6% and 28.2%, respectively. The mass fraction of feldspar in red tilestones and green tilestones is 20.8% and 16.3%, respectively. At the same time, there are a few minerals such as illite and chlorite in red tilestone and green tilestone. The above result shows that the mineral types in red tilestone and green tilestone are similar, but the mass fraction of mineral is different. In the productive process of tilestones, the raw materials (clay and sand) and combustion temperature of red tilestones and green tilestones are the same, the combustion time and cooling method are different [33, 34].

The red tilestone and the green tilestone at the scale of 5μm and the scale of 400nm were observed by SEM, as shown in Figs. 8 and 9. The blue part represents the pores in the figure.

Fig. 8

Mineral and pore distribution in red tilestone

Fig. 9

Mineral and pore distribution in green tilestone

From the Fig. 8, the internal pores of the red tilestone are blocky and unevenly distributed. The internal pore diameter of the red tilestone is large, which leads to the loading discontinuity easily in the test. From the Fig. 9, the internal pores of the green tilestone are linearly and concentratedly distributed, and the pore diameter is small. In the productive process of tilestones, the high temperature causes the internal expansion of tilestone and the formation of pores. The red tilestone adopts the method of cooling in air, the internal pore cannot contract rapidly. The green tilestone adopts the method of cooling in water, the external temperature dropped rapidly, the internal pores contracted rapidly.

Results of nanoindentation test

The load–displacement curves at different indentation points of red tilestone and green tilestone were obtained by the nanoindentation test to study the deformation and mechanical parameters of tilestone, as shown in Fig. 10.

Fig. 10

Load–displacement curves

The peak load can reflect the mechanical strength of each indentation point. The peak load of load–displacement curve of red tilestone is concentrated between 40 mN-70 mN. The average peak load is 57 mN, and the maximum peak load is 113 mN, as shown in Fig. 10a. The peak load of load–displacement curve of green tilestone is concentrated between 50 mN-80 mN, the average peak load is 64mN and the maximum peak load is 225 mN, as shown in Fig. 10b. The above data shows that the mechanical strength of green tilestones is higher than red tilestones. When the indentation depth is constant, the indenter passes through high strength minerals and the peak load will increase.

The load–displacement curves of two indentation points (No. 12 and No. 27) of green tilestone were selected to analyze the deformation process of the indentation point under different loads, as shown in Fig. 11.

Fig. 11

Load–displacement curves at different indentation points

There are three stages during the loading process, elastic stage, elastoplastic stage and plastic stage. The deformation of No.12 indentation point changes from elastic to elastoplastic when the displacement is 516 nm, and changes from elastoplastic to plastic phase when the displacement reaches 2000 nm. For the No.27 indentation point, the deformation changes from elastic to elastoplastic when the displacement is 573 nm, and changes from elastoplastic to plastic phase when the displacement reaches 2100 nm. The result shows that the deformation process of different indentation points is different. During the loading process, the slope of the load–displacement curve increases rapidly in the elastic and elastoplastic stages, and decreases in the plastic stage. When the yield strength of mineral is the same as the applied external load, elastic deformation will occur. When the applied external load exceeds the yield strength of minerals, plastic failure will occur.

Meanwhile, the load–displacement curves of red tilestone and green tilestone appear obvious ‘pop-in’ during loading stage, as shown in Fig. 12. Tilestone is a heterogeneous material with uneven distribution of pores and minerals. When the indenter contacts with micro-cracks, pores, or interfaces between hard material and soft materials, the indenter moves rapidly and jumps in a short time with the increase of load.

Fig. 12

Load–displacement curves with pore concentration and normal pore distribution

The mechanical parameters of tilestone, such as hardness, elastic modulus and fracture toughness can were calculated by Eqs. (1)–(11) and the load–displacement curve. The averaging method was used to analyze test data in this work [35] and the results were calculated, as shown in Table 3.

Table 3 Mechanical parameters from nanoindentation

In terms of the red tilestone, the average elastic modulus, hardness and fracture toughness is 29.47 GPa, 2.2 GPa and 1.28 MPa·m^1/2, respectively. The standard deviation is 6.4 GPa, 0.41 GPa and 0.32 MPa·m^1/2, respectively. In terms of the green tilestone, the average elastic modulus, hardness and fracture toughness is 30.21 GPa, 2.59 GPa and 1.76 MPa·m^1/2, respectively. The standard deviation is 6.54 GPa, 0.37 GPa and 0.32 MPa·m^1/2, respectively. At micro-scale, as the indentation points are on different minerals, the mechanical parameters tested would fluctuate somewhat. The result shows that the average mechanical parameters of green tilestone are larger than red tilestone, and directly reflects that the overall strength of green tilestone is higher than red tilestone. The reason for this result is that the combustion time and cooling method are different [33].

To obtain the micro-mechanical parameters of different minerals and achieve the upscaling from micro-scale to macro-scale, the internal minerals of tilestone were analyzed by Energy Dispersive Spectrometer (EDS) and the mineral mechanical parameters were determined by the nanoindentation test, as shown Table 4 [36, 37].

Table 4 Mechanical parameters of minerals in red tilestone and green tilestone

The result shows that the elastic modulus of quartz is 94 GPa and 95 GPa for red tilestone and green tilestone. The elastic modulus of mica is 47 GPa and 46 GPa for the red tilestone and green tilestone. The elastic modulus of feldspar is 52 GPa and 50 GPa for the red tilestone and the green tilestone. The elastic modulus of the internal mineral for the red tilestone and green tilestone is roughly the same. The results show that the difference of combustion time and cooling method will not lead to the change of mechanical strength of minerals in red tilestone and green tilestone. The main reason is that the crystal structure of quartz, mica and other minerals are stable, and the mechanical properties do not change at the high temperature.

Mechanical parameters based on upscaling method

The mechanical parameter is calculated by the nanoindentation test at the micro-scale and cannot reflect the macroscopic mechanical properties of the tilestone. In this study, an upscaling method based on Mori–Tanaka model was used to analyze the mechanical parameters of tilestone from micro-scale to macro-scale [38, 39]. Mori–Tanaka model is a mature theory, which has been studied for decades and is widely used to predict the effective stiffness of various composite materials [40, 41].

In this study, the tilestone was approximately regarded as a complex composed of quartz, feldspar, mica and pores. The other minerals were chlorite, calcite and illite, which had similar mechanical properties to mica. Therefore, other minerals were classified as mica in the homogenization calculation. In the homogenization calculation process, the mica particles and the pore structure were homogenized into uniform mica matrix because the pore concentrated near mica. Then, the quartz, feldspar and mica matrix were homogenized into uniform materials equivalent to actual tilestone [42, 43]. The mineral homogenization procedure are shown in Fig. 13.

Fig. 13

Minerals homogenization procedure [42]

For the red tilestone, the volume fraction of mica, quartz and feldspar is f₀ = 36%, f₁ = 43%, f₂ = 21%, respectively. For the green tilestone, the volume fractions of mica, quartz and feldspar is f₀ = 38%, f₁ = 46%, f₂ = 16%, respectively. The porosity of red tilestone and green tilestone is 18.4% and 21.7%, respectively. Then, the mica matrix porosity (φ) of red tilestone and green tilestone is 55.6% and 52.7%, respectively. The shear modulus and bulk modulus of mica matrix can be calculated by bringing the sample porosity and indentation test results into the Eqs. (12)–(13), as follows.

$$k_{{0}} { = }\frac{{{4}\left( {{1} - \varphi } \right)k_{s} \mu_{s} }}{{{4}\mu_{s} + {3}\varphi k_{s} }}$$

(12)

$$\mu_{{0}} { = }\frac{{\left( {{1} - \varphi } \right)\mu_{s} }}{{{1} + {6}\varphi \frac{{k_{s} + {2}\mu_{s} }}{{{9}k_{s} + {8}\mu_{s} }}}}$$

(13)

where, k₀ and μ₀ is the shear modulus and bulk modulus of mica matrix, GPa. φ is mica porosity. k_s and μ_s is the shear modulus and bulk modulus of mica particle, GPa [44].

The quartz and feldspar can be approximately regarded as dense and uniform material. The shear modulus and bulk modulus of minerals (quartz, feldspar and mica) can be calculated by Eqs. (14)–(15), as follows.

$$k_{{\text{r}}} { = }\frac{{E_{{\text{r}}} }}{{{2}\left( {{1 + }\nu_{{\text{r}}} } \right)}}$$

(14)

$$\mu_{{\text{r}}} { = }\frac{{E_{{\text{r}}} }}{{{3}\left( {{1 - 2}\nu_{{\text{r}}} } \right)}}$$

(15)

where, r = 0, 1, 2, s is mica matrix, quart, feldspar, mica particle; k_r and μ_r is the shear modulus and bulk modulus of minerals, GPa; E_r is the elastic modulus of minerals, GPa, which can be obtained by nanoindentation test and EDS mapping; ν_r is the Poisson’s ratio of minerals. Therefore, the shear modulus and bulk modulus of the tilestone can be calculated based on Mori–Tanaka model by Eqs. (16) - (17), as follows.

$$K_{{{\text{hom}}}} { = }\left( {\sum\nolimits_{{r = {0}}} {f_{r} \frac{{k_{r} }}{{{3}k_{r} + {4}\mu_{{0}} }}} } \right)\left( {\sum\nolimits_{{r{ = 0}}} {\frac{{f_{r} }}{{{3}k_{r} + {4}\mu_{{0}} }}} } \right)^{{ - 1}}$$

(16)

$$G_{{{\text{hom}}}} { = }\frac{{\sum\nolimits_{{r = {0}}} {f_{r} \frac{{\mu_{r} }}{{\mu_{{0}} \left( {{9}k_{{0}} + {8}\mu_{{0}} } \right) + {6}\mu_{r} \left( {k_{{0}} + {2}\mu_{{0}} } \right)}}} }}{{\sum\nolimits_{{r = {0}}} {\frac{{f_{r} }}{{\mu_{{0}} \left( {{9}k_{{0}} + {8}\mu_{{0}} } \right) + {6}\mu_{r} \left( {k_{{0}} + {2}\mu_{{0}} } \right)}}} }}$$

(17)

where, f_r represent the volume fraction of minerals. K_hom and G_hom represent the shear modulus and bulk modulus of the tilestone, GPa [44].

Based on the above equation, the bulk modulus (K_hom^h) and shear modulus (G_hom^h) of red tilestone is 9.286 GPa and 13.09 GPa. The bulk modulus (K_hom^q) and shear modulus (G_hom^q) of green tilestone is 9.327 GPa and 13.677 GPa. Therefore, the elastic modulus of the tilestone at the meso-scale can be calculated by Eq. (18).

$$E_{\hom } { = }\frac{{{9}K_{\hom } G_{\hom } }}{{{3}K_{\hom } { + }G_{\hom } }}$$

(18)

where, E_hom represent elastic modulus of the tilestone in meso-scale, GPa. The elastic modulus of red tilestone and green tilestone is 26.72 GPa and 27.56 GPa by Eq. (17). Compared with the parameter results obtained by nanoindentation test, the deviation rates of red tilestone and green tilestone were 10.3% and 9.6%.

Uniaxial compression test of samples

The uniaxial compression test of red tilestone and green tilestone was carried out to verify the rationality of nanoindentation test and homogenization approach based on the Mori–Tanaka model. The test sample were prepared into cylinders with 25 mm in diameter and 50 mm in height, as shown in Fig. 14. Three samples of green tilestone were prepared, G1, G2 and G3, respectively and three samples of red tilestone were prepared, R1, R2 and R3, respectively.

Fig. 14

The sample of uniaxial compression test

The uniaxial compression stress–strain curve of samples are shown in Fig. 15. The result shows that the difference of peak stress and maximum strain between red tilestone and green tilestone is not obvious, indicating that the difference of macroscopic mechanical strength between the two kinds of tilestone is small. In addition, the slope of stress–strain curve of green tilestone is larger than that of red tilestone during loading, which is related to the cooling method of two tilestones. The green tilestone adopts the method of cooling in water, which on the one hand prevents the divalent iron ions in the green tilestone from being oxidized, and on the other hand makes the internal pores contracted rapidly, the brittleness is enhanced, and the slope of stress–strain curve is larger.

Fig. 15

Uniaxial compression stress–strain curve of samples

The elastic modulus of G1, G2 and G3 is 28.38 GPa, 27.22 GPa and 27.05 GPa, respectively, and the average elastic modulus is 27.55 GPa. The elastic modulus of R1, R2 and R3 is 27.79 GPa, 26.79 GPa and 25.97 GPa, respectively, and the average elastic modulus is 26.85 GPa. In this test, the uniaxial compression test sample and the standard sample are different, so the uniaxial compression test results need to be corrected and the correction coefficient is 0.883 [45]. The modified elastic modulus of red tilestone and green tilestone is 24.78 GPa and 25.43 GPa.

Comparison and analysis of multi-scale parameters

The results of the nanoindentation test, upscaling method, and uniaxial compression test, correspond to the results of the micro-scale, meso-scale, and macro-scale, respectively. The results of mechanical parameter at multi-scale are shown in Table 5 and Fig. 16.

Table 5 Three methods elastic modulus results

Fig. 16

Mechanical parameters of red tilestone and green tilestone at multiple scales

The result shows that the calculated values from the upscaling model have a good agreement with those obtained by uniaxial compression test and nanoindentation test. However, the calculated results by the averaging approach of nanoindentation data are higher in values. The reasons are as follows: On the one hand, the test sites of nanoindentation are relatively intact parts with small defects, avoiding areas such as micro-holes and weak interfaces. As a result, the measured mineral elastic modulus value is higher than the actual value, and the equivalent elastic modulus obtained by homogenization method is significantly higher than the measured value of uniaxial compression test. On the other hand, the mechanical properties of the mineral grain interface are generally weaker than those of the mineral itself, and there is a weak interlayer on the meso-scale. Therefore, the weak interlayer between minerals is ignored, which leads to the equivalent elastic modulus obtained by the homogenization method higher than the measured value of uniaxial compression test.

Conclusions

In this paper, the restoration and protection of the tilestone architectural is setting at the Shuntianmen Site. The mechanical parameters of Song Dynasty tilestones were determined using a combination of nanoindentation techniques and the homogenization calculation method based on the Mori–Tanaka model. The application of nanoindentation technique was evaluated in the protection of tilestone cultural relics. The conclusions are as follows.

• (1) With the adoption of XRD and SEM, the mineral composition and micro-structure distribution of Song Dynasty tilestones was investigated. The main mineral components of red tilestone and green tilestone are quartz, feldspar and mica. The mass fraction of quartz, feldspar and mica is 42.9%, 20.8%, 27.6% in red tilestone, respectively. The mass fraction of quartz, feldspar and mica is 45.1%, 16.3%, 28.2% in green tilestone, respectively. The reason for this result is that the combustion time and cooling mode are different.

• (2) The variation law of mechanical property of Song Dynasty tilestones was revealed by the nanoindentation test. The average peak load of red tilestone is 57 mN, and the maximum peak load is 113 mN. The average peak load of green tilestone is 64 mN, and the maximum peak load is 225 mN. During the loading process, when contacting with micro-cracks, pores, or interface between hard material and soft materials, the indenter moves rapidly.

• (3) The mechanical parameters of tilestone, such as hardness, elastic modulus and fracture toughness were calculated by the nanoindentation test. In terms of the red tilestone, the average elastic modulus, hardness and fracture toughness is 29.47 GPa, 2.2 GPa and 1.28 MPa·m^1/2, respectively. In terms of the green tilestone, the average elastic modulus, hardness and fracture toughness is 30.21 GPa, 2.59 GPa and 1.76 MPa·m^1/2, respectively. The overall strength of the green tilestone is higher than that of the red tilestone.

• (4) The elastic modulus of red tilestone and green tilestone calculated by the upscaling method based on the Mori–Tanaka model is 26.72 GPa and 27.56 GPa, respectively. The elastic modulus of red tilestone and green tilestone is 24.78 GPa and 25.43 GPa by uniaxial compression test. The results of mechanical parameters at micro-scale, meso-scale, and macro-scale are compared and analyzed, which shows high consistency. Meanwhile, it is verified that the nanoindentation technique is used to study the mechanical property of tilestone cultural relics.