# Comparative Investigation on Small-Strain Stiffness Characteristics of Undisturbed and Compacted Highly Weathered Granites at Various Densities

Authors: Juntao Wang; Lingwei Kong; Junbiao Yan; Zhenhua Zhou
Journal: Geofluids
 (2022)

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## Abstract

To investigate the small-strain stiffness characteristics of highly weathered granite (HWG), a resonance column test system was used to conduct resonance column tests on highly weathered granite taken from the Lincang City, Yunnan Province, China. The effects of effective consolidation stress and structural changes is caused by remodeling process and initial dry density on the small-strain stiffness of HWG. Furthermore, the difference between the test data and other geotechnical materials were compared and analyzed. The test results show that the maximum dynamic shear modulusGmax of remodeled highly weathered granite samples is greater than that of undisturbed samples once the effective consolidation stress is smaller than 300 kPa, but the opposite result is observed once the effective consolidation stress exceeds 300 kPa, and the phenomenon was also explained from a microscopic perspective based on the scanning electron microscopy (SEM). The Gmax of remodeled highly weathered granite gradually increases with the increase in dry density and effective consolidation stress. At an identical effective consolidation stress, the dynamic shear modulus ratio G/Gmax−γ curves of remodeled highly weathered granites at different initial dry densities are nearly consistent. Additionally, according to the test data, the mathematical model of Gmax for remodeled highly weathered granite considering effective consolidation pressure and initial dry density was established and agrees well with the test results of practical engineering cases. The range of variation in G/Gmax was given by the research results, which were compared with the G/Gmax of weathered granite obtained from the existing research, and the results in this work can provide a valuable reference for analyzing dynamic stability of buildings in the engineering construction of weathered granite sites.


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## Body

## 1. Introduction

Granite occurs in the world broadly, and weathered granite results from in situ weathering process of granite, mainly consists of quartz, feldspar, and other clay minerals [1]. According to the different degrees of weathering, weathered granite is divided into moderately weathered granite, highly weathered granite (HWG), completely weathered granite, and granite residual soil. Due to its special genesis, weathered granite usually has a porous fissure structure, in which the particles have an irregular shape and large angles, and some particles are easy to break. In addition, it is prone to disintegration in case of water [2], which makes its physical and mechanical properties obviously different from clay and sandy soil. Compared with completely weathered granite and granite residual soil, highly weathered granite has a larger particle size and higher particle strength due to its relatively lower weathering degree, and less secondary cementing material in the fissure, which makes the structure of highly weathered granite more easily broken and loose, resulting in the difficulty in obtaining undisturbed samples of highly weathered granite [3], so there are fewer cases of research on highly weathered granite. There are a large number of structures and infrastructures in the weathered granite strata in the distribution zone of weathered granite, and the vibration waves generated by train vibration, earthquakes, and other dynamic loads are propagated in the weathered granite, which inevitably has certain effects on the weathered granite and the corresponding structures, so it is important to investigate the dynamic parameters of weathered granite and its dynamic characteristics. As far as the weathered granites with different degrees of weathering are concerned, their particle size composition and mineral weathering degree show obvious differences, so their dynamic properties also differ. Most of the existing studies involving the dynamic properties of weathered granites focus on residual soil, but a few focus on completely weathered granite. Ng et al. [4] investigated the maximum shear modulus Gmax of granite residual soils using suspension S-wave logging method and concluded that the test results are basically comparable with both the results of triaxial tests involving local displacement measurements and the predictions made using an empirical correlation (according to standard penetration test N values). Macari and Hoyos [5] used resonant column tests (RCT) to analyze the impacts of weathering and sample remodeling processes on the small-strain stiffness properties of granite residual soils in Hong Kong. Considering the effect of climate on granite residual soil, Yin et al. [6] investigated the influence of wet-dry cycle on the Gmax of granite residual soil by using the RCT and proposed the relationship curve of G/Gmax−γ based on the number of wet-dry cycles. Li et al. [7] investigated the dynamic properties of completely weathered granite under repeated loading by dynamic triaxial test and gave the formulas for the calculation of parameters including dynamic strength and dynamic modulus according to the experimental results. Niu et al. [3] determined the shear modulus as well as damping ratio of remodeled highly weathered granite using RCT apparatus and gave empirical ranges for dynamic parameters such as G/Gmax−γ curves. Summarizing the results of previous studies, one can see that the research on weathered granite mainly pays attention to the granite residual soil and completely weathered granite, but there are few studies on the dynamic responses of HWG, especially the relative research on the undisturbed HWG are rarely reported. In contrast, the buildings and structures of weathered granite strata are mostly located in undisturbed rock masses. Therefore, it is of great theoretical and engineering significance to study the dynamic properties of the undisturbed HWG.Generally, in the practical engineering, the analysis of dynamic responses at a small-strain amplitude under dynamic loads, such as train vibration and earthquake, are the focus of the study of force deformation of buildings and structures. When analyzing the dynamic response at a small-strain amplitude, the dynamic shear modulus is a key parameter. Many research results have shown that there are many factors influencing the small-strain dynamic characteristics of geotechnical materials, such as gradation [8], degree of saturation [9], consolidation pressure [10], number of wet-dry cycles [6], and stress history [11]. For geotechnical materials, dry density is one of the basic characteristics, and it can be equated to parameters such as initial compaction and initial pore ratio, and besides, the initial dry density also affects the small-strain stiffness characteristics. Liu, S. et al. [12] studied the effect of different initial relative compactness on the small-strain stiffness characteristics of calcareous sand and gave the recommended range of dynamic parameters for calcareous sand with different initial relative compactness. Feng and Sutter [13] analyzed the characteristics of the small-strain dynamic mechanical properties of rubber sand and investigated how dry density impacts the seismic performance of rubber sand. Liu, M. et al. [14] studied the influence of initial pore ratio on the small-strain stiffness of rubber sand at varying rubber contents and explained the role of different contents of rubber in rubber sand by mechanism analysis. Jie et al. [15] conducted RCT on loess at different initial dry densities in the unsaturated condition to study the impacts of dry density and degree of saturation on the small-strain dynamic properties, further monitored the variation in pore size of the samples during the wet-dry cycles, and concluded that the microporosity changes have significant effects on the changes of the maximum shear modulus of unsaturated loess. Summarizing the existing research results can reach a consensus that the initial dry density exerts a certain effect on the small-strain dynamic properties of geotechnical materials. The different initial dry densities have a huge impact on the strength and stiffness of remodeled weathered granite under large strain conditions [16], but how the initial dry density affects the stiffness properties of weathered granite under small-strain conditions remains unknown. There are few studies on the impacts of different initial dry densities on the dynamic responses of weathered granites, especially for HWGs.Due to the inhomogeneity and fragility of highly weathered granite, it is difficult to collect and prepared undisturbed samples. As a result, previous research usually use remodeled samples to substitute undisturbed samples to study the properties of the undisturbed HWG. However, whether the remodeled samples can reflect the true properties of undisturbed HWG has never been explored. Therefore, this work is aimed at studying the small-strain dynamic properties of undisturbed and remodeled HWG gotten from the Lincang City, Yunnan Province, China using RCT; the effect of the remodeling process on the dynamic shear modulus of HWG was explored and the corresponding mechanism was explained from a microscopic perspective as well. In addition, the influence of different initial dry densities on the small-strain stiffness properties of remodeled HWG is analyzed and relevant parameter calculation equations were proposed. The test data establish a good foundation for the design and analysis of buildings and structures in the HWG strata in the study area, and besides, the results in this work can also provide a good reference for other weathered granite areas.


## 2. Samples and Methods

### 2.1. Soil Characteristics

The HWG sample used in this study was taken from a slope in the Lincang City of Yunnan Province in China. The HWG at the sampling site is sandy and hand crushing would disturb its original structure. The standard penetration testN0 value of the research sample inverse calculated to the original surface was 96, and the samples were identified as HWG as per Code for Investigation of Geotechnical Engineering GB 50021 [17]. The parent granitic rock of the HWG is monzonite granite. The physical indexes of HWG were determined indoor based on the China National Standard for Geotechnical Testing Method GB/T 50123, as shown in Table 1 [18]. The sieving method was used to determine the grain size distribution, which are displayed in Figure 1 together with the other weathered granites from Hong Kong, Seoul, and Xinzhou. One can see that the maximal particle size of the HWG in this study exceeds 5 mm, among which sand grain occupy the greatest proportion, followed by gravel; however, the silt and clay contents are the least. According to the gain size distribution (GSD) parameters, the HWG in this work belongs to coarse-grained but well-graded sand. In addition, the GSD curve in this research is consistent with that of Lee and Coop [19] and Ng et al. [20] but is located significantly above that researched by Niu et al. [21]. The HWGs in this study were categorized as gravelly sand according to the unified soil classification system in Code for Investigation of Geotechnical Engineering GB 50021 [17]. In addition, different form the samples from Hong Kong [22], Seoul [19], and Xinzhou [21]; the HWG in the Lincang area has a significantly higher specific gravity.Table 1
Basic physical properties and mechanical indices of weathered granites (date from Lee and Coop [19], Niu et al. [21], and Ng and Chiu [22]).
SamplesGsρdgcm3e0IpClassificationLocationCuCcColorHWG2.7231.7860.51616.4Gravelly sandLincang6.21.15Yellow-brownLee and Coop [19]—1.713———SeoulNg and Chiu [22]2.61———Gravelly sandHong KongNiu et al.(2017) A2.602Xinzhou15.331.04Red-brownNiu et al.(2017) B2.636Xinzhou13.181.03Yellow-brownNiu et al.(2017) C2.609Xinzhou8.001.72GrayNote:Gs: specific gravity; ρd: natural dry density; e0: initial void ratio; and Ip: plasticity index.Figure 1
GSD of weathered granites (citation from Lee and Coop [19], Ng et al. [20], and Niu et al. [21]).


### 2.2. Sample Preparation

Section2.1 indicates that the HWG in this investigation possesses a higher ratio of large grain size, and the grains greater than 0.075 mm account for 95.16 percent. Furthermore, the HWG has the common features of weathered granite, such as a loose structure, small interparticle cohesiveness, and fast disintegration in the presence of moisture [2]. Meanwhile, it contains a large amount of brittle minerals, including quartz and unweathered feldspar, so the weathered granites are characterized by hard particles and loose texture, which makes it hard to prepare undisturbed samples of weathered granite merely using typical methods such as knife cutting [3]. This research employs a circular computer numerical control (CNC) abrasive wire sawing machine to deal with weathered granites, and the obtained undisturbed sample is shown in Figure 2(a). One can clearly see that the sample remains complete and keeps a flat surface.Figure 2
(a) The undisturbed HWG sample; (b) the remolded HWG sample. (Figure2(a) from Wang et al. [26] Figure 5(a)).
(a)(b)To compare the effect of remolding and dry density on the small-strain stiffness properties of HWG, remolded samples of HWG were added, and the remolded samples were prepared using the five-layer moist tamping method. The photo of remolded HWG B sample is shown in Figure2(b). Comparison of Figures 2(a) and 2(b) show that the remolded sample is significantly different from the undisturbed sample. Unlike the undisturbed sample, in which the original structure can be seen, the structure of the remolded sample is completely destroyed. Overall, the sample is more uniform with fine particles covering the surface of coarse particles, and the color of the sample is brownish yellow.Prior to the RCT, the samples underwent vacuum saturation for 24 hours. Subsequently, the preliminarily saturated sample was then mounted in the resonance column, and the backpressure was gradually applied to the sample to enhance its saturation. At the same time, the Skempton B value was examined to ensure it exceeded 0.96. Then, the sample was then subjected to consolidation pressure. When the sample consolidation finished, the resonance column tests were performed.


### 2.3. Test Equipment and Test Program

To investigate the structural changes caused by remolding process, initial dry densities, and effective confining pressureσm′ on the small-strain behavior of HWGs, 50 RCTs were conducted. For each sample, ten effective confining stresses were applied. The samples and the specific schemes of the tests are provided in Table 2. The number of remodeled samples was four, numbered HWG R1, HWG R2, HWG R3, and HWG R4, denoting dry densities of 1.596, 1.672, 1.789, and 1.919 g/cm3, respectively. It should be noted that the dry density of the remodeled HWG R3 is consistent with that of the undisturbed sample. The dry densities of the remodeled sample HWG R1 and R2 are less than that of the undisturbed sample, and the dry density of the remodeled HWG R4 is higher than that of the undisturbed sample.Table 2
Experimental schemes of RCTs.
SamplesDry densitiesg/cm3Confining pressure (kPa)Backpressure (kPa)Effective confining pressure (kPa)Undisturbed HWG1.786150, 20010050, 100HWG R11.596300, 400200, 300HWG R21.672500, 600400, 500HWG R31.789700, 800600, 700HWG R41.919900, 1000800, 900The RCT apparatus used in this work was a Stokoe-type resonance column device. The top end of the sample was free. Its bottom end was fixed. The shear modulus can be measured continuously by the device in a strain of 0.001% to 0.1%.Figure3 also shows the specific procedure of sample loading: (1) the saturated sample is fixed on the pedestal of RCT apparatus; then, the inner chamber and the top cap are installed, respectively; (2) the device of the electromagnetic drive is installed, during which the verticality of the sample is checked; (3) the acceleration sensor, displacement sensor, and top fixing plate are installed and fixed, respectively; and (4) the outer chamber is fixed, and the sample is further saturated by the backpressure. Once cell pressure was applied on the sample, the backpressure control system monitored the backpressure volume. The consolidation under a given confining pressure was completed when the volumetric change rate of the backpressure control system is less than 1 mm3/min. When the consolidation process finished, the test began.Figure 3
Schematic view of the RCT apparatus. (From Wang et al. [26], Figure 6.)In the whole test process, a torsional excitation force was applied at the free end of the sample stage by stage. Subsequently, the sample was vibrated at various frequencies to determine the shear strain. Therefore, the values of dynamic shear modulusG corresponding to different effective consolidation stresses can be determined, according to the following equation:
(1)G=ρ2πfHβ2,where ρ denotes the density of the sample after consolidation, f means the resonance frequency, H refers to the height of the sample after consolidation, and β is the characteristic value of the frequency equation of torsional vibration.


## 2.1. Soil Characteristics

The HWG sample used in this study was taken from a slope in the Lincang City of Yunnan Province in China. The HWG at the sampling site is sandy and hand crushing would disturb its original structure. The standard penetration testN0 value of the research sample inverse calculated to the original surface was 96, and the samples were identified as HWG as per Code for Investigation of Geotechnical Engineering GB 50021 [17]. The parent granitic rock of the HWG is monzonite granite. The physical indexes of HWG were determined indoor based on the China National Standard for Geotechnical Testing Method GB/T 50123, as shown in Table 1 [18]. The sieving method was used to determine the grain size distribution, which are displayed in Figure 1 together with the other weathered granites from Hong Kong, Seoul, and Xinzhou. One can see that the maximal particle size of the HWG in this study exceeds 5 mm, among which sand grain occupy the greatest proportion, followed by gravel; however, the silt and clay contents are the least. According to the gain size distribution (GSD) parameters, the HWG in this work belongs to coarse-grained but well-graded sand. In addition, the GSD curve in this research is consistent with that of Lee and Coop [19] and Ng et al. [20] but is located significantly above that researched by Niu et al. [21]. The HWGs in this study were categorized as gravelly sand according to the unified soil classification system in Code for Investigation of Geotechnical Engineering GB 50021 [17]. In addition, different form the samples from Hong Kong [22], Seoul [19], and Xinzhou [21]; the HWG in the Lincang area has a significantly higher specific gravity.Table 1
Basic physical properties and mechanical indices of weathered granites (date from Lee and Coop [19], Niu et al. [21], and Ng and Chiu [22]).
SamplesGsρdgcm3e0IpClassificationLocationCuCcColorHWG2.7231.7860.51616.4Gravelly sandLincang6.21.15Yellow-brownLee and Coop [19]—1.713———SeoulNg and Chiu [22]2.61———Gravelly sandHong KongNiu et al.(2017) A2.602Xinzhou15.331.04Red-brownNiu et al.(2017) B2.636Xinzhou13.181.03Yellow-brownNiu et al.(2017) C2.609Xinzhou8.001.72GrayNote:Gs: specific gravity; ρd: natural dry density; e0: initial void ratio; and Ip: plasticity index.Figure 1
GSD of weathered granites (citation from Lee and Coop [19], Ng et al. [20], and Niu et al. [21]).


## 2.2. Sample Preparation

Section2.1 indicates that the HWG in this investigation possesses a higher ratio of large grain size, and the grains greater than 0.075 mm account for 95.16 percent. Furthermore, the HWG has the common features of weathered granite, such as a loose structure, small interparticle cohesiveness, and fast disintegration in the presence of moisture [2]. Meanwhile, it contains a large amount of brittle minerals, including quartz and unweathered feldspar, so the weathered granites are characterized by hard particles and loose texture, which makes it hard to prepare undisturbed samples of weathered granite merely using typical methods such as knife cutting [3]. This research employs a circular computer numerical control (CNC) abrasive wire sawing machine to deal with weathered granites, and the obtained undisturbed sample is shown in Figure 2(a). One can clearly see that the sample remains complete and keeps a flat surface.Figure 2
(a) The undisturbed HWG sample; (b) the remolded HWG sample. (Figure2(a) from Wang et al. [26] Figure 5(a)).
(a)(b)To compare the effect of remolding and dry density on the small-strain stiffness properties of HWG, remolded samples of HWG were added, and the remolded samples were prepared using the five-layer moist tamping method. The photo of remolded HWG B sample is shown in Figure2(b). Comparison of Figures 2(a) and 2(b) show that the remolded sample is significantly different from the undisturbed sample. Unlike the undisturbed sample, in which the original structure can be seen, the structure of the remolded sample is completely destroyed. Overall, the sample is more uniform with fine particles covering the surface of coarse particles, and the color of the sample is brownish yellow.Prior to the RCT, the samples underwent vacuum saturation for 24 hours. Subsequently, the preliminarily saturated sample was then mounted in the resonance column, and the backpressure was gradually applied to the sample to enhance its saturation. At the same time, the Skempton B value was examined to ensure it exceeded 0.96. Then, the sample was then subjected to consolidation pressure. When the sample consolidation finished, the resonance column tests were performed.


## 2.3. Test Equipment and Test Program

To investigate the structural changes caused by remolding process, initial dry densities, and effective confining pressureσm′ on the small-strain behavior of HWGs, 50 RCTs were conducted. For each sample, ten effective confining stresses were applied. The samples and the specific schemes of the tests are provided in Table 2. The number of remodeled samples was four, numbered HWG R1, HWG R2, HWG R3, and HWG R4, denoting dry densities of 1.596, 1.672, 1.789, and 1.919 g/cm3, respectively. It should be noted that the dry density of the remodeled HWG R3 is consistent with that of the undisturbed sample. The dry densities of the remodeled sample HWG R1 and R2 are less than that of the undisturbed sample, and the dry density of the remodeled HWG R4 is higher than that of the undisturbed sample.Table 2
Experimental schemes of RCTs.
SamplesDry densitiesg/cm3Confining pressure (kPa)Backpressure (kPa)Effective confining pressure (kPa)Undisturbed HWG1.786150, 20010050, 100HWG R11.596300, 400200, 300HWG R21.672500, 600400, 500HWG R31.789700, 800600, 700HWG R41.919900, 1000800, 900The RCT apparatus used in this work was a Stokoe-type resonance column device. The top end of the sample was free. Its bottom end was fixed. The shear modulus can be measured continuously by the device in a strain of 0.001% to 0.1%.Figure3 also shows the specific procedure of sample loading: (1) the saturated sample is fixed on the pedestal of RCT apparatus; then, the inner chamber and the top cap are installed, respectively; (2) the device of the electromagnetic drive is installed, during which the verticality of the sample is checked; (3) the acceleration sensor, displacement sensor, and top fixing plate are installed and fixed, respectively; and (4) the outer chamber is fixed, and the sample is further saturated by the backpressure. Once cell pressure was applied on the sample, the backpressure control system monitored the backpressure volume. The consolidation under a given confining pressure was completed when the volumetric change rate of the backpressure control system is less than 1 mm3/min. When the consolidation process finished, the test began.Figure 3
Schematic view of the RCT apparatus. (From Wang et al. [26], Figure 6.)In the whole test process, a torsional excitation force was applied at the free end of the sample stage by stage. Subsequently, the sample was vibrated at various frequencies to determine the shear strain. Therefore, the values of dynamic shear modulusG corresponding to different effective consolidation stresses can be determined, according to the following equation:
(1)G=ρ2πfHβ2,where ρ denotes the density of the sample after consolidation, f means the resonance frequency, H refers to the height of the sample after consolidation, and β is the characteristic value of the frequency equation of torsional vibration.


## 3. Test Results and Analysis

### 3.1. Variation in Dynamic Shear Modulus at Different Effective Consolidation Stresses

Figure4 displays the change trend of dynamic shear modulus (G) with shear strain (γ) at various effective consolidation pressures, from which it is found that all samples at different effective consolidation stresses follow the same change trend. Specifically, the G of each specimen exhibits a decreasing trend as shear strain increases. Basically, the soil stiffness manifests less decay of G when the shear strain is minimal (0.001%). Once γ is greater than 0.001%, G becomes more strain-dependent, and the decay gradient of G increases with an increase in shear strain. The behavior is more visible in the same sample at larger effective consolidated stresses. When consolidation stress proceeds to increase, the decay of G is more significant. This indicates that as shear strain increases, the correlation between stress and strain of HWG evolves from linear to nonlinear gradually, which is also evidenced by other geotechnical media [10, 23].Figure 4
Relationship betweenG and γ: (a) undisturbed HWG; (b) HWG R1; (c) HWG R2; (d) HWG R3; and (e) HWG R4. (Data points halved.)
(a)(b)(c)(d)(e)Furthermore, with an increase in the effective consolidation stress, all samples’ curves are elevated higher. This indicates that at the same shear strain,G increases with consolidation stress. Similar behavior has been found in residual soil [5], sandy soil [8], and sedimentary soil [24]. In fact, during the isotropic consolidation, the effective consolidation pressure narrows down the interparticle pore space, so the particles become more compact, and the sample’s capacity to withstand dynamic shear deformation is enhanced dramatically. The sample structure becomes more compact when the effective consolidation stress increases. Meanwhile, the intergranular contact area is enlarged, and the shear wave’s propagation velocity is easier to get through the sample.Figure5 presents the change in G with γ of HWGs at various initial dry densities under the same consolidation stress. It can be found that even at the same effective confining stress, the difference in initial dry density causes these curves to show varying characteristics. Specifically, at an identical strain, the G increases with increasing dry density and consolidation pressure. Some interesting phenomena were observed that the curves of HWG R1 and R2 samples almost overlapped at low consolidation pressures (50, 100, and 200 kPa). In contrast, the G of HWG R3 and R4 samples is significantly larger than that of the HWG R1 and R2 samples, and the higher the dry density, the higher the G. Therefore, a conclusion can be drawn that at an identical shear strain and consolidation stress, higher initial dry densities can produce a higher G, while the decay rate of G is greater with the further increase in shear strain.Figure 5
Variation inG with γ at various effective consolidation stresses: (a) 50 kPa; (b) 100 kPa; (c) 200 kPa; (d) 400 kPa; and (e) 800 kPa. (Data points halved.)
(a)(b)(c)(d)(e)


### 3.2. Maximum Dynamic Shear Modulus of HWG

The maximum dynamic shear modulusGmax occurs once γ approaches the minimal value [25]. To get Gmax, the curve fitting method is commonly adopted. To match the experimental data in this research, the Stokoe formula [23] was employed as follows:
(2)G=Gmax1+γ/γrc,where γr denotes the reference shear strain when G/Gmax is 50%; c is the fitting curvature parameter. Table 3 displays the fitting parameters and Figure 5 gives some fitting curves. It can be found that these fitting curves agree well with the test data, and the regression analysis coefficients R2 of all fitted curves were greater than 0.99.Table 3
Fitting parameters of experimental data at various dry densities.
SampleParameterEffective consolidation pressure50 kPa100 kPa200 kPa300 kPa400 kPa500 kPa600 kPa700 kPa800 kPa900 kPaHWGGmax58.3390.37133.96190.47243.11280.44312.19340.99367.67405.34γr0.02340.03440.04280.05460.05570.05760.06160.06550.06870.0682c0.68570.70380.70670.69680.72130.75980.76320.79250.77710.7541R20.99870.99830.99900.99850.99930.99960.99950.99910.99980.9998HWG R1Gmax60.7287.49121.68162.92192.95230.37269.99305.07339.47365.97γr0.04750.05830.08660.09650.10880.12340.11890.12110.12810.1359c0.54910.55980.62900.63030.67370.68220.73970.74210.73790.7157R20.99420.99800.99830.99910.99830.99660.99880.99990.99910.9974HWG R2Gmax63.8392.01128.56174.70212.88246.27284.52309.81339.85367.01γr0.03240.05440.09610.09330.09920.10940.10970.12190.12900.1334c0.50110.58210.54070.61370.67060.67930.70100.72090.72740.7471R20.99140.99570.99870.99770.99920.99790.99920.99820.99850.9990HWG R3Gmax88.66126.10174.16220.80257.68297.91323.93349.41372.76401.15γr0.02580.04060.06050.07130.08340.88880.10150.10900.11400.1143c0.60240.69350.69500.71880.75070.78340.74000.76650.78340.8372R20.99770.99750.99730.99820.99880.99930.99920.99920.99960.9994HWG R4Gmax74.08109.10153.38199.96234.23269.96296.96328.90350.93381.68γr0.04780.05100.06910.07180.08180.08900.09790.10540.11570.1147c0.69260.69740.73180.70540.77670.78140.80780.78160.78310.8119R20.99850.99840.99860.99690.99840.99900.99820.99680.99870.9980


#### 3.2.1. Analysis of the Effect of the Consolidation Stress on theGmax of HWG

Figure6 summarizes the results of the Gmax for all samples. It can be observed that the Gmax of each sample increases when the consolidation pressure increases. This is mainly because with the increase in effective consolidation pressure, the particles move or roll to make the particles reorganize, resulting in a gradual decrease in the pore ratio. The specimen is more compact and the intergranular contact area becomes larger; the shear wave propagation velocity in the sample is enhanced. This causes the resonant frequency to increase when the effective consolidation stress is enhanced, so the Gmax increases as well. In general, for the remodeled sample, a larger initial dry density makes a smaller pore ratio, so the contact area between the soil particles becomes larger. Therefore, the shear wave propagates in the specimen faster and leads to a greater Gmax at the same consolidation pressure.Figure 6
Relationship betweenGmax and σm′.


#### 3.2.2. Analysis of Effect of the Sample Structure on theGmax of HWG

Besides, a comparison was drawn between theGmax of HWG R3 and the results gotten by the undisturbed HWG sample (Figure 7(a)). From the Gmax−σm′ curve, one can see that once effective consolidation stress is less than 300 kPa, and Gmax of HWG R3 is greater than that of the undisturbed HWG sample at the same effective consolidation pressure. The opposite phenomenon was found when the effective consolidation pressure was higher than 300 kPa. Figure 7(a) illustrates the fact that the HWG R3 is more resistant to deformation than the undisturbed HWG sample at low effective consolidation pressures (<300 kPa). In contrast, the opposite phenomenon occurs at high effective consolidation pressures (>300 kPa). The same phenomenon that at low consolidation pressures the Gmax of remolded sample was greater compared with the undisturbed sample has been also observed in granite residual soils gotten at shallow depths [5].Figure 7
(a)Gmax−σ′ curves for the undisturbed HWG sample and the remolded HWG R3 sample; (b) compression curves of the undisturbed HWG sample and the remolded HWG R3 sample for comparison.
(a)(b)These differences can be explained, in part, by the structural differences between the two samples. The original structure of the sample was destroyed in the process of remolding, and the order of the particle size was chaotic. As the grains are disordered, a greater force between the grains is required to obtain the same dry density. As a result, the shear wave propagates faster in the sample and has a higher resonance frequency than that in the undisturbed sample, so theGmax is larger. When consolidation stress is enhanced, the pore volume begins to contract, the intergranular contact of particles becomes tighter, and the shear wave propagates faster, so the shear modulus increases gradually. Figure 7(b) manifests that the void ratio of the undisturbed sample varies more considerably than that of the remolded sample because the original structure of the undisturbed sample (the interlocking structure) is more conducive to the consolidation and drainage of the sample compared with the disorderly structure of the remolded sample. Therefore, the structure of the undisturbed sample is more compact, which facilitates wave velocity propagation. In the cases of the identical consolidation stress variation, the Gmax increment of the undisturbed sample is larger compared with the remolded sample. Therefore, the Gmax of the undisturbed sample is greater than that of the remolded sample at a larger consolidation pressure.


#### 3.2.3. Analysis of Effect of Dry Density on the Dynamic Characteristics of the Remolded HWG

From Section3.2.1, it is apparent that the effective consolidation pressure exerts a significant impact on the Gmax of HWG. Similar conclusions have been reported by scholars according to abundant test results for soil and sand, and various mathematical models for predicting soil Gmax have been proposed based on this. Huang et al. [8] indicated that the Gmax of sand can be obtained by the following equation, which is modified from the famous ‘Hardin formula’ [27].
(3)Gmax=K∙pa∙σm′pan,where pa=100kPa; σm′ refers to effective confining stress; and K and n are the fitting parameters. Defining G0=K∙pa, Equation (3) can be rewritten as
(4)Gmax=G0∙σm′pan,where G0 is the Gmax when the effective consolidation stress becomes 100 kPa. Equation (4) indicates that Gmax reaches zero once effective confining stress becomes zero, but this cannot comply with a real scenario because in fact, the Gmax of the soil must not be 0 MPa even if the effective stress becomes zero. Therefore, Equation (4) is modified as follows to satisfy the real phenomenon.
(5)Gmax=G0∙1+σm′Pan.Equation (4) is, in fact, a monotonic function that progressively reduces to Equation (4) if effective pressure increases stage by stage.Figure8(a) exhibits the fitting parameters of Gmax at different initial dry densities and the fitting curve. The values of R2 exceed 0.99, denoting a good fit of data. In addition, G0 and n increases and decreases, respectively, with an increase in initial dry density. The variation in n can be attributed to the fact that since the sample at a lower initial dry density has a higher porosity, its skeleton is easier to be compressed when subjected to consolidation stress.Figure 8
(a)Gmax−σ′ curves for remodeled samples at different initial dry densities; (b) variation of G0 and n at initial dry densities of the remolded HWG.
(a)(b)Based on the correlation between fitting parameters and initial dry density in Figure8(a), G0, n, and initial dry density showed a linear relationship. When these fitting parameters were compared with the initial dry density as shown in Figure 8(b), it indicates these fitting parameters present a good linear relationship with the initial dry density. Substituting the linear equations for G0 and n in Equation (5), Equation (5) can be shown as
(6)Gmax=A∙ρd0+B∙1+σm′paC∙ρd0+D,where A and B refer to fitting parameters for G0; C and D denote fitting parameters for n; ρd0 denotes the initial dry density; pa is the atmospheric pressure (pa=100kPa in this paper).The parameters ofG0 and n in the Lincang area were derived according to regression analysis. The linear fitting formula in Figure 8(b) presented a good correlation between G0, n, and initial dry density ρd0 (R2 are greater than 0.98). Finally, Equation (6) becomes the following form:
(7)Gmax=78.6646ρd0−73.77481+σm′pa−0.5585ρd0+1.8160.Formula (7) can be directly used to calculate the Gmax of the remolded HWGs at varying consolidation stresses and dry densities.Figure9 depicts the distribution of the experimental values of Gmax of the HWG samples at each initial dry density and the predicted values obtained by using the Equation (7). Figure 9 indicates that for the HWG samples at each initial dry density and effective consolidation pressure, the predicted and experimental values of Gmax obtained by the modified model have a good agreement, with the basic relative error not exceeding 5% at most points. This means that the prediction model based on dry density can well take into account the impacts of consolidation stress and initial dry density, which is a good reference value for the analysis of the dynamic response of related sites.Figure 9
Relationships of measuredGmax and predicted Gmax using the model modified by the initial dry density.


### 3.3. Decay Trends ofGmax of HWG

#### 3.3.1. Analysis of the Impact of Effective Consolidation Stress on the Decay Trends ofGmax

TheGmax characterizes the decay properties of the dynamic shear modulus as Gmax develops, and it is a key input parameter related to the soil dynamics. The test data were standardized using Gmax for ease of analysis. Transforming Equation (2) to yield the following equation, which can reflect the stiffness attenuation properties of the material.
(8)GGmax=11+γ/γrc.Figure10 displays the dynamic shear modulus ratio values. Table 3 lists the fitting parameters. When consolidation stress increases, γr and c of all samples basically increases accordingly, which means for all samples, the stiffness decay gradient declines with the improvement of consolidation stress. This may be due to the fact that as consolidation pressure increases, the pore ratio of the sample falls, the particles interact more tightly, thus increasing the resistance to deformation. As a result, the G may be kept at higher shear strain amplitudes.Figure 10
Normalized stiffnessG/Gmax−γ decay curves: (a) undisturbed HWG; (b) HWG R1; (c) HWG R2; (d) HWG R3; and (e) HWG R4. (Data points halved.)
(a)(b)(c)(d)(e)


#### 3.3.2. Analysis of the Effect of the Remodeled Process on the Decay Characteristics ofGmax

To compare the impact of the remodeling process on the small-strain stiffness characteristics of the weathered granite, Figure11 draws a comparison between the stiffness attenuation of the remolded HWG R3 and that of the undisturbed HWG sample. One can see that the at each effective consolidation pressure, shear modulus values of the remolded HWG R3 are all distributed above the G/Gmax values of the undisturbed HWG. The results illustrate that with increasing dynamic shear strain, the G/Gmax of remolded HWG R3 presents basically a lower decay rate compared with the undisturbed HWG sample. Besides, with an increase in dynamic shear strain, the G/Gmax of the remolded HWG R3 decreases at a smaller rate and a narrower change interval. Similar results were observed in loess [28] and highly structured clay [29]. This phenomenon might be explained as follows. The undisturbed sample has natural defects such as initial fissuring. The deformation could magnify these defects such as a larger width of fissuring, which influences the shear wave propagation. However, the soil experiences a structural breakdown during the manual remolding and recompaction. Hence, a new soil structure is formed, which is more uniform and has fewer cracks and other defects than the undisturbed sample. The shear wave propagation in the remolded sample is less sensitive to deformation than that in the undisturbed sample. Therefore, the stiffness’ decay of the remolded sample is less than that of the undisturbed sample at an identical shear strain.Figure 11
Normalized stiffnessG/Gmax−γ attenuation curves for the undisturbed HWG sample and the remolded HWG R3 sample at varying consolidation stresses: (a) 100 kPa; (b) 200 kPa; (c) 400 kPa; and (d) 800 kPa.
(a)(b)(c)(d)


#### 3.3.3. Analysis of Impact of Initial Dry Density on the Decay Characteristics ofGmax

Figure12 displays the change characteristics of G/Gmax−γ curve of the remodeled HWG samples at varying initial dry densities at the identical consolidation pressure, which shows that the decay curves of the HWG samples at varying initial dry densities at an effective consolidation pressure of 50 kPa have some differences, showing a certain dispersion. However, after the effective increase in consolidation pressure (>100 kPa), the differences between the G/Gmax−γ curves of HWGs at different initial dry densities are very small and can be basically considered consistent, which can be described by the same normalized curve. This indicates that although the initial structural properties and structural strength of the HWG samples at different initial dry densities have some differences at low effective consolidation pressures due to minor differences in the remolding process, which in turn leads to some minor differences in the G/Gmax−γ curves, the effective consolidation pressure weakens such differences and eventually eliminates the impact of initial dry density onG/Gmax−γ curves when consolidation pressure is elevated.Figure 12
NormalizedG/Gmax−γ decay curves for remolded samples at varying initial dry densities at different effective consolidation stresses: (a) 50 kPa; (b) 100 kPa; (c) 200 kPa; (d) 400 kPa; and (e) 800 kPa. (Data points halved.)
(a)(b)(c)(d)(e)


## 3.1. Variation in Dynamic Shear Modulus at Different Effective Consolidation Stresses

Figure4 displays the change trend of dynamic shear modulus (G) with shear strain (γ) at various effective consolidation pressures, from which it is found that all samples at different effective consolidation stresses follow the same change trend. Specifically, the G of each specimen exhibits a decreasing trend as shear strain increases. Basically, the soil stiffness manifests less decay of G when the shear strain is minimal (0.001%). Once γ is greater than 0.001%, G becomes more strain-dependent, and the decay gradient of G increases with an increase in shear strain. The behavior is more visible in the same sample at larger effective consolidated stresses. When consolidation stress proceeds to increase, the decay of G is more significant. This indicates that as shear strain increases, the correlation between stress and strain of HWG evolves from linear to nonlinear gradually, which is also evidenced by other geotechnical media [10, 23].Figure 4
Relationship betweenG and γ: (a) undisturbed HWG; (b) HWG R1; (c) HWG R2; (d) HWG R3; and (e) HWG R4. (Data points halved.)
(a)(b)(c)(d)(e)Furthermore, with an increase in the effective consolidation stress, all samples’ curves are elevated higher. This indicates that at the same shear strain,G increases with consolidation stress. Similar behavior has been found in residual soil [5], sandy soil [8], and sedimentary soil [24]. In fact, during the isotropic consolidation, the effective consolidation pressure narrows down the interparticle pore space, so the particles become more compact, and the sample’s capacity to withstand dynamic shear deformation is enhanced dramatically. The sample structure becomes more compact when the effective consolidation stress increases. Meanwhile, the intergranular contact area is enlarged, and the shear wave’s propagation velocity is easier to get through the sample.Figure5 presents the change in G with γ of HWGs at various initial dry densities under the same consolidation stress. It can be found that even at the same effective confining stress, the difference in initial dry density causes these curves to show varying characteristics. Specifically, at an identical strain, the G increases with increasing dry density and consolidation pressure. Some interesting phenomena were observed that the curves of HWG R1 and R2 samples almost overlapped at low consolidation pressures (50, 100, and 200 kPa). In contrast, the G of HWG R3 and R4 samples is significantly larger than that of the HWG R1 and R2 samples, and the higher the dry density, the higher the G. Therefore, a conclusion can be drawn that at an identical shear strain and consolidation stress, higher initial dry densities can produce a higher G, while the decay rate of G is greater with the further increase in shear strain.Figure 5
Variation inG with γ at various effective consolidation stresses: (a) 50 kPa; (b) 100 kPa; (c) 200 kPa; (d) 400 kPa; and (e) 800 kPa. (Data points halved.)
(a)(b)(c)(d)(e)


## 3.2. Maximum Dynamic Shear Modulus of HWG

The maximum dynamic shear modulusGmax occurs once γ approaches the minimal value [25]. To get Gmax, the curve fitting method is commonly adopted. To match the experimental data in this research, the Stokoe formula [23] was employed as follows:
(2)G=Gmax1+γ/γrc,where γr denotes the reference shear strain when G/Gmax is 50%; c is the fitting curvature parameter. Table 3 displays the fitting parameters and Figure 5 gives some fitting curves. It can be found that these fitting curves agree well with the test data, and the regression analysis coefficients R2 of all fitted curves were greater than 0.99.Table 3
Fitting parameters of experimental data at various dry densities.
SampleParameterEffective consolidation pressure50 kPa100 kPa200 kPa300 kPa400 kPa500 kPa600 kPa700 kPa800 kPa900 kPaHWGGmax58.3390.37133.96190.47243.11280.44312.19340.99367.67405.34γr0.02340.03440.04280.05460.05570.05760.06160.06550.06870.0682c0.68570.70380.70670.69680.72130.75980.76320.79250.77710.7541R20.99870.99830.99900.99850.99930.99960.99950.99910.99980.9998HWG R1Gmax60.7287.49121.68162.92192.95230.37269.99305.07339.47365.97γr0.04750.05830.08660.09650.10880.12340.11890.12110.12810.1359c0.54910.55980.62900.63030.67370.68220.73970.74210.73790.7157R20.99420.99800.99830.99910.99830.99660.99880.99990.99910.9974HWG R2Gmax63.8392.01128.56174.70212.88246.27284.52309.81339.85367.01γr0.03240.05440.09610.09330.09920.10940.10970.12190.12900.1334c0.50110.58210.54070.61370.67060.67930.70100.72090.72740.7471R20.99140.99570.99870.99770.99920.99790.99920.99820.99850.9990HWG R3Gmax88.66126.10174.16220.80257.68297.91323.93349.41372.76401.15γr0.02580.04060.06050.07130.08340.88880.10150.10900.11400.1143c0.60240.69350.69500.71880.75070.78340.74000.76650.78340.8372R20.99770.99750.99730.99820.99880.99930.99920.99920.99960.9994HWG R4Gmax74.08109.10153.38199.96234.23269.96296.96328.90350.93381.68γr0.04780.05100.06910.07180.08180.08900.09790.10540.11570.1147c0.69260.69740.73180.70540.77670.78140.80780.78160.78310.8119R20.99850.99840.99860.99690.99840.99900.99820.99680.99870.9980


### 3.2.1. Analysis of the Effect of the Consolidation Stress on theGmax of HWG

Figure6 summarizes the results of the Gmax for all samples. It can be observed that the Gmax of each sample increases when the consolidation pressure increases. This is mainly because with the increase in effective consolidation pressure, the particles move or roll to make the particles reorganize, resulting in a gradual decrease in the pore ratio. The specimen is more compact and the intergranular contact area becomes larger; the shear wave propagation velocity in the sample is enhanced. This causes the resonant frequency to increase when the effective consolidation stress is enhanced, so the Gmax increases as well. In general, for the remodeled sample, a larger initial dry density makes a smaller pore ratio, so the contact area between the soil particles becomes larger. Therefore, the shear wave propagates in the specimen faster and leads to a greater Gmax at the same consolidation pressure.Figure 6
Relationship betweenGmax and σm′.


### 3.2.2. Analysis of Effect of the Sample Structure on theGmax of HWG

Besides, a comparison was drawn between theGmax of HWG R3 and the results gotten by the undisturbed HWG sample (Figure 7(a)). From the Gmax−σm′ curve, one can see that once effective consolidation stress is less than 300 kPa, and Gmax of HWG R3 is greater than that of the undisturbed HWG sample at the same effective consolidation pressure. The opposite phenomenon was found when the effective consolidation pressure was higher than 300 kPa. Figure 7(a) illustrates the fact that the HWG R3 is more resistant to deformation than the undisturbed HWG sample at low effective consolidation pressures (<300 kPa). In contrast, the opposite phenomenon occurs at high effective consolidation pressures (>300 kPa). The same phenomenon that at low consolidation pressures the Gmax of remolded sample was greater compared with the undisturbed sample has been also observed in granite residual soils gotten at shallow depths [5].Figure 7
(a)Gmax−σ′ curves for the undisturbed HWG sample and the remolded HWG R3 sample; (b) compression curves of the undisturbed HWG sample and the remolded HWG R3 sample for comparison.
(a)(b)These differences can be explained, in part, by the structural differences between the two samples. The original structure of the sample was destroyed in the process of remolding, and the order of the particle size was chaotic. As the grains are disordered, a greater force between the grains is required to obtain the same dry density. As a result, the shear wave propagates faster in the sample and has a higher resonance frequency than that in the undisturbed sample, so theGmax is larger. When consolidation stress is enhanced, the pore volume begins to contract, the intergranular contact of particles becomes tighter, and the shear wave propagates faster, so the shear modulus increases gradually. Figure 7(b) manifests that the void ratio of the undisturbed sample varies more considerably than that of the remolded sample because the original structure of the undisturbed sample (the interlocking structure) is more conducive to the consolidation and drainage of the sample compared with the disorderly structure of the remolded sample. Therefore, the structure of the undisturbed sample is more compact, which facilitates wave velocity propagation. In the cases of the identical consolidation stress variation, the Gmax increment of the undisturbed sample is larger compared with the remolded sample. Therefore, the Gmax of the undisturbed sample is greater than that of the remolded sample at a larger consolidation pressure.


### 3.2.3. Analysis of Effect of Dry Density on the Dynamic Characteristics of the Remolded HWG

From Section3.2.1, it is apparent that the effective consolidation pressure exerts a significant impact on the Gmax of HWG. Similar conclusions have been reported by scholars according to abundant test results for soil and sand, and various mathematical models for predicting soil Gmax have been proposed based on this. Huang et al. [8] indicated that the Gmax of sand can be obtained by the following equation, which is modified from the famous ‘Hardin formula’ [27].
(3)Gmax=K∙pa∙σm′pan,where pa=100kPa; σm′ refers to effective confining stress; and K and n are the fitting parameters. Defining G0=K∙pa, Equation (3) can be rewritten as
(4)Gmax=G0∙σm′pan,where G0 is the Gmax when the effective consolidation stress becomes 100 kPa. Equation (4) indicates that Gmax reaches zero once effective confining stress becomes zero, but this cannot comply with a real scenario because in fact, the Gmax of the soil must not be 0 MPa even if the effective stress becomes zero. Therefore, Equation (4) is modified as follows to satisfy the real phenomenon.
(5)Gmax=G0∙1+σm′Pan.Equation (4) is, in fact, a monotonic function that progressively reduces to Equation (4) if effective pressure increases stage by stage.Figure8(a) exhibits the fitting parameters of Gmax at different initial dry densities and the fitting curve. The values of R2 exceed 0.99, denoting a good fit of data. In addition, G0 and n increases and decreases, respectively, with an increase in initial dry density. The variation in n can be attributed to the fact that since the sample at a lower initial dry density has a higher porosity, its skeleton is easier to be compressed when subjected to consolidation stress.Figure 8
(a)Gmax−σ′ curves for remodeled samples at different initial dry densities; (b) variation of G0 and n at initial dry densities of the remolded HWG.
(a)(b)Based on the correlation between fitting parameters and initial dry density in Figure8(a), G0, n, and initial dry density showed a linear relationship. When these fitting parameters were compared with the initial dry density as shown in Figure 8(b), it indicates these fitting parameters present a good linear relationship with the initial dry density. Substituting the linear equations for G0 and n in Equation (5), Equation (5) can be shown as
(6)Gmax=A∙ρd0+B∙1+σm′paC∙ρd0+D,where A and B refer to fitting parameters for G0; C and D denote fitting parameters for n; ρd0 denotes the initial dry density; pa is the atmospheric pressure (pa=100kPa in this paper).The parameters ofG0 and n in the Lincang area were derived according to regression analysis. The linear fitting formula in Figure 8(b) presented a good correlation between G0, n, and initial dry density ρd0 (R2 are greater than 0.98). Finally, Equation (6) becomes the following form:
(7)Gmax=78.6646ρd0−73.77481+σm′pa−0.5585ρd0+1.8160.Formula (7) can be directly used to calculate the Gmax of the remolded HWGs at varying consolidation stresses and dry densities.Figure9 depicts the distribution of the experimental values of Gmax of the HWG samples at each initial dry density and the predicted values obtained by using the Equation (7). Figure 9 indicates that for the HWG samples at each initial dry density and effective consolidation pressure, the predicted and experimental values of Gmax obtained by the modified model have a good agreement, with the basic relative error not exceeding 5% at most points. This means that the prediction model based on dry density can well take into account the impacts of consolidation stress and initial dry density, which is a good reference value for the analysis of the dynamic response of related sites.Figure 9
Relationships of measuredGmax and predicted Gmax using the model modified by the initial dry density.


## 3.2.1. Analysis of the Effect of the Consolidation Stress on theGmax of HWG

Figure6 summarizes the results of the Gmax for all samples. It can be observed that the Gmax of each sample increases when the consolidation pressure increases. This is mainly because with the increase in effective consolidation pressure, the particles move or roll to make the particles reorganize, resulting in a gradual decrease in the pore ratio. The specimen is more compact and the intergranular contact area becomes larger; the shear wave propagation velocity in the sample is enhanced. This causes the resonant frequency to increase when the effective consolidation stress is enhanced, so the Gmax increases as well. In general, for the remodeled sample, a larger initial dry density makes a smaller pore ratio, so the contact area between the soil particles becomes larger. Therefore, the shear wave propagates in the specimen faster and leads to a greater Gmax at the same consolidation pressure.Figure 6
Relationship betweenGmax and σm′.


## 3.2.2. Analysis of Effect of the Sample Structure on theGmax of HWG

Besides, a comparison was drawn between theGmax of HWG R3 and the results gotten by the undisturbed HWG sample (Figure 7(a)). From the Gmax−σm′ curve, one can see that once effective consolidation stress is less than 300 kPa, and Gmax of HWG R3 is greater than that of the undisturbed HWG sample at the same effective consolidation pressure. The opposite phenomenon was found when the effective consolidation pressure was higher than 300 kPa. Figure 7(a) illustrates the fact that the HWG R3 is more resistant to deformation than the undisturbed HWG sample at low effective consolidation pressures (<300 kPa). In contrast, the opposite phenomenon occurs at high effective consolidation pressures (>300 kPa). The same phenomenon that at low consolidation pressures the Gmax of remolded sample was greater compared with the undisturbed sample has been also observed in granite residual soils gotten at shallow depths [5].Figure 7
(a)Gmax−σ′ curves for the undisturbed HWG sample and the remolded HWG R3 sample; (b) compression curves of the undisturbed HWG sample and the remolded HWG R3 sample for comparison.
(a)(b)These differences can be explained, in part, by the structural differences between the two samples. The original structure of the sample was destroyed in the process of remolding, and the order of the particle size was chaotic. As the grains are disordered, a greater force between the grains is required to obtain the same dry density. As a result, the shear wave propagates faster in the sample and has a higher resonance frequency than that in the undisturbed sample, so theGmax is larger. When consolidation stress is enhanced, the pore volume begins to contract, the intergranular contact of particles becomes tighter, and the shear wave propagates faster, so the shear modulus increases gradually. Figure 7(b) manifests that the void ratio of the undisturbed sample varies more considerably than that of the remolded sample because the original structure of the undisturbed sample (the interlocking structure) is more conducive to the consolidation and drainage of the sample compared with the disorderly structure of the remolded sample. Therefore, the structure of the undisturbed sample is more compact, which facilitates wave velocity propagation. In the cases of the identical consolidation stress variation, the Gmax increment of the undisturbed sample is larger compared with the remolded sample. Therefore, the Gmax of the undisturbed sample is greater than that of the remolded sample at a larger consolidation pressure.


## 3.2.3. Analysis of Effect of Dry Density on the Dynamic Characteristics of the Remolded HWG

From Section3.2.1, it is apparent that the effective consolidation pressure exerts a significant impact on the Gmax of HWG. Similar conclusions have been reported by scholars according to abundant test results for soil and sand, and various mathematical models for predicting soil Gmax have been proposed based on this. Huang et al. [8] indicated that the Gmax of sand can be obtained by the following equation, which is modified from the famous ‘Hardin formula’ [27].
(3)Gmax=K∙pa∙σm′pan,where pa=100kPa; σm′ refers to effective confining stress; and K and n are the fitting parameters. Defining G0=K∙pa, Equation (3) can be rewritten as
(4)Gmax=G0∙σm′pan,where G0 is the Gmax when the effective consolidation stress becomes 100 kPa. Equation (4) indicates that Gmax reaches zero once effective confining stress becomes zero, but this cannot comply with a real scenario because in fact, the Gmax of the soil must not be 0 MPa even if the effective stress becomes zero. Therefore, Equation (4) is modified as follows to satisfy the real phenomenon.
(5)Gmax=G0∙1+σm′Pan.Equation (4) is, in fact, a monotonic function that progressively reduces to Equation (4) if effective pressure increases stage by stage.Figure8(a) exhibits the fitting parameters of Gmax at different initial dry densities and the fitting curve. The values of R2 exceed 0.99, denoting a good fit of data. In addition, G0 and n increases and decreases, respectively, with an increase in initial dry density. The variation in n can be attributed to the fact that since the sample at a lower initial dry density has a higher porosity, its skeleton is easier to be compressed when subjected to consolidation stress.Figure 8
(a)Gmax−σ′ curves for remodeled samples at different initial dry densities; (b) variation of G0 and n at initial dry densities of the remolded HWG.
(a)(b)Based on the correlation between fitting parameters and initial dry density in Figure8(a), G0, n, and initial dry density showed a linear relationship. When these fitting parameters were compared with the initial dry density as shown in Figure 8(b), it indicates these fitting parameters present a good linear relationship with the initial dry density. Substituting the linear equations for G0 and n in Equation (5), Equation (5) can be shown as
(6)Gmax=A∙ρd0+B∙1+σm′paC∙ρd0+D,where A and B refer to fitting parameters for G0; C and D denote fitting parameters for n; ρd0 denotes the initial dry density; pa is the atmospheric pressure (pa=100kPa in this paper).The parameters ofG0 and n in the Lincang area were derived according to regression analysis. The linear fitting formula in Figure 8(b) presented a good correlation between G0, n, and initial dry density ρd0 (R2 are greater than 0.98). Finally, Equation (6) becomes the following form:
(7)Gmax=78.6646ρd0−73.77481+σm′pa−0.5585ρd0+1.8160.Formula (7) can be directly used to calculate the Gmax of the remolded HWGs at varying consolidation stresses and dry densities.Figure9 depicts the distribution of the experimental values of Gmax of the HWG samples at each initial dry density and the predicted values obtained by using the Equation (7). Figure 9 indicates that for the HWG samples at each initial dry density and effective consolidation pressure, the predicted and experimental values of Gmax obtained by the modified model have a good agreement, with the basic relative error not exceeding 5% at most points. This means that the prediction model based on dry density can well take into account the impacts of consolidation stress and initial dry density, which is a good reference value for the analysis of the dynamic response of related sites.Figure 9
Relationships of measuredGmax and predicted Gmax using the model modified by the initial dry density.


## 3.3. Decay Trends ofGmax of HWG

### 3.3.1. Analysis of the Impact of Effective Consolidation Stress on the Decay Trends ofGmax

TheGmax characterizes the decay properties of the dynamic shear modulus as Gmax develops, and it is a key input parameter related to the soil dynamics. The test data were standardized using Gmax for ease of analysis. Transforming Equation (2) to yield the following equation, which can reflect the stiffness attenuation properties of the material.
(8)GGmax=11+γ/γrc.Figure10 displays the dynamic shear modulus ratio values. Table 3 lists the fitting parameters. When consolidation stress increases, γr and c of all samples basically increases accordingly, which means for all samples, the stiffness decay gradient declines with the improvement of consolidation stress. This may be due to the fact that as consolidation pressure increases, the pore ratio of the sample falls, the particles interact more tightly, thus increasing the resistance to deformation. As a result, the G may be kept at higher shear strain amplitudes.Figure 10
Normalized stiffnessG/Gmax−γ decay curves: (a) undisturbed HWG; (b) HWG R1; (c) HWG R2; (d) HWG R3; and (e) HWG R4. (Data points halved.)
(a)(b)(c)(d)(e)


### 3.3.2. Analysis of the Effect of the Remodeled Process on the Decay Characteristics ofGmax

To compare the impact of the remodeling process on the small-strain stiffness characteristics of the weathered granite, Figure11 draws a comparison between the stiffness attenuation of the remolded HWG R3 and that of the undisturbed HWG sample. One can see that the at each effective consolidation pressure, shear modulus values of the remolded HWG R3 are all distributed above the G/Gmax values of the undisturbed HWG. The results illustrate that with increasing dynamic shear strain, the G/Gmax of remolded HWG R3 presents basically a lower decay rate compared with the undisturbed HWG sample. Besides, with an increase in dynamic shear strain, the G/Gmax of the remolded HWG R3 decreases at a smaller rate and a narrower change interval. Similar results were observed in loess [28] and highly structured clay [29]. This phenomenon might be explained as follows. The undisturbed sample has natural defects such as initial fissuring. The deformation could magnify these defects such as a larger width of fissuring, which influences the shear wave propagation. However, the soil experiences a structural breakdown during the manual remolding and recompaction. Hence, a new soil structure is formed, which is more uniform and has fewer cracks and other defects than the undisturbed sample. The shear wave propagation in the remolded sample is less sensitive to deformation than that in the undisturbed sample. Therefore, the stiffness’ decay of the remolded sample is less than that of the undisturbed sample at an identical shear strain.Figure 11
Normalized stiffnessG/Gmax−γ attenuation curves for the undisturbed HWG sample and the remolded HWG R3 sample at varying consolidation stresses: (a) 100 kPa; (b) 200 kPa; (c) 400 kPa; and (d) 800 kPa.
(a)(b)(c)(d)


### 3.3.3. Analysis of Impact of Initial Dry Density on the Decay Characteristics ofGmax

Figure12 displays the change characteristics of G/Gmax−γ curve of the remodeled HWG samples at varying initial dry densities at the identical consolidation pressure, which shows that the decay curves of the HWG samples at varying initial dry densities at an effective consolidation pressure of 50 kPa have some differences, showing a certain dispersion. However, after the effective increase in consolidation pressure (>100 kPa), the differences between the G/Gmax−γ curves of HWGs at different initial dry densities are very small and can be basically considered consistent, which can be described by the same normalized curve. This indicates that although the initial structural properties and structural strength of the HWG samples at different initial dry densities have some differences at low effective consolidation pressures due to minor differences in the remolding process, which in turn leads to some minor differences in the G/Gmax−γ curves, the effective consolidation pressure weakens such differences and eventually eliminates the impact of initial dry density onG/Gmax−γ curves when consolidation pressure is elevated.Figure 12
NormalizedG/Gmax−γ decay curves for remolded samples at varying initial dry densities at different effective consolidation stresses: (a) 50 kPa; (b) 100 kPa; (c) 200 kPa; (d) 400 kPa; and (e) 800 kPa. (Data points halved.)
(a)(b)(c)(d)(e)


## 3.3.1. Analysis of the Impact of Effective Consolidation Stress on the Decay Trends ofGmax

TheGmax characterizes the decay properties of the dynamic shear modulus as Gmax develops, and it is a key input parameter related to the soil dynamics. The test data were standardized using Gmax for ease of analysis. Transforming Equation (2) to yield the following equation, which can reflect the stiffness attenuation properties of the material.
(8)GGmax=11+γ/γrc.Figure10 displays the dynamic shear modulus ratio values. Table 3 lists the fitting parameters. When consolidation stress increases, γr and c of all samples basically increases accordingly, which means for all samples, the stiffness decay gradient declines with the improvement of consolidation stress. This may be due to the fact that as consolidation pressure increases, the pore ratio of the sample falls, the particles interact more tightly, thus increasing the resistance to deformation. As a result, the G may be kept at higher shear strain amplitudes.Figure 10
Normalized stiffnessG/Gmax−γ decay curves: (a) undisturbed HWG; (b) HWG R1; (c) HWG R2; (d) HWG R3; and (e) HWG R4. (Data points halved.)
(a)(b)(c)(d)(e)


## 3.3.2. Analysis of the Effect of the Remodeled Process on the Decay Characteristics ofGmax

To compare the impact of the remodeling process on the small-strain stiffness characteristics of the weathered granite, Figure11 draws a comparison between the stiffness attenuation of the remolded HWG R3 and that of the undisturbed HWG sample. One can see that the at each effective consolidation pressure, shear modulus values of the remolded HWG R3 are all distributed above the G/Gmax values of the undisturbed HWG. The results illustrate that with increasing dynamic shear strain, the G/Gmax of remolded HWG R3 presents basically a lower decay rate compared with the undisturbed HWG sample. Besides, with an increase in dynamic shear strain, the G/Gmax of the remolded HWG R3 decreases at a smaller rate and a narrower change interval. Similar results were observed in loess [28] and highly structured clay [29]. This phenomenon might be explained as follows. The undisturbed sample has natural defects such as initial fissuring. The deformation could magnify these defects such as a larger width of fissuring, which influences the shear wave propagation. However, the soil experiences a structural breakdown during the manual remolding and recompaction. Hence, a new soil structure is formed, which is more uniform and has fewer cracks and other defects than the undisturbed sample. The shear wave propagation in the remolded sample is less sensitive to deformation than that in the undisturbed sample. Therefore, the stiffness’ decay of the remolded sample is less than that of the undisturbed sample at an identical shear strain.Figure 11
Normalized stiffnessG/Gmax−γ attenuation curves for the undisturbed HWG sample and the remolded HWG R3 sample at varying consolidation stresses: (a) 100 kPa; (b) 200 kPa; (c) 400 kPa; and (d) 800 kPa.
(a)(b)(c)(d)


## 3.3.3. Analysis of Impact of Initial Dry Density on the Decay Characteristics ofGmax

Figure12 displays the change characteristics of G/Gmax−γ curve of the remodeled HWG samples at varying initial dry densities at the identical consolidation pressure, which shows that the decay curves of the HWG samples at varying initial dry densities at an effective consolidation pressure of 50 kPa have some differences, showing a certain dispersion. However, after the effective increase in consolidation pressure (>100 kPa), the differences between the G/Gmax−γ curves of HWGs at different initial dry densities are very small and can be basically considered consistent, which can be described by the same normalized curve. This indicates that although the initial structural properties and structural strength of the HWG samples at different initial dry densities have some differences at low effective consolidation pressures due to minor differences in the remolding process, which in turn leads to some minor differences in the G/Gmax−γ curves, the effective consolidation pressure weakens such differences and eventually eliminates the impact of initial dry density onG/Gmax−γ curves when consolidation pressure is elevated.Figure 12
NormalizedG/Gmax−γ decay curves for remolded samples at varying initial dry densities at different effective consolidation stresses: (a) 50 kPa; (b) 100 kPa; (c) 200 kPa; (d) 400 kPa; and (e) 800 kPa. (Data points halved.)
(a)(b)(c)(d)(e)


## 4. Discussion

### 4.1. The Variations of the Dynamic Characteristics in the Remodeling Process Based on the Microstructure

To observe the effects of the remodeling process and consolidation stress on the microstructure of HWG, HWG samples were tested using scanning electron microscopy (SEM), as presented in Figure13. The undisturbed HWG without consolidation, as illustrated in Figure 13(a), has an embedded locking structure. The distribution of fractures between the particles is notable, with a large width. The orientation of main fissures is unorganized, and the majority of fissures intersect each other. In addition, almost no cementitious agents are observed, but the shapes of adjacent particles are well-matched, which indicates that the intergranular fractures are caused by weathered mineral loss. It is worth noting that some of the quartz grains have flakes of extensively worn primary minerals between them, and these flake structures accumulate in face-to-face contact. For remodeled HWG without consolidation, as shown in Figure 13(c), the soil experiences a structural breakdown during the manual remolding and recompaction. Hence, a new soil skeleton is formed, and the disposition, orientation, and structure of the soil are completely different. In terms of the mineral structure characteristics, the fine-grained minerals of the unconsolidated remodeled sample are attached on the surface of coarse grains which mainly show point-point contact structure and part of their contact in the point-line form, with the irregular arrangement between particles (Figure 12(c)). Unlike the particle interlocking structure in undisturbed samples, the remolded sample particles are randomly oriented and overlap. Besides, the pores of the remodeled sample are mostly isolated pores and secondary pores, with various morphologies and diameters, and some millimeter-level pores can be observed. In the unconsolidated state, the contact area between the particles of the remodeled sample is larger than that of the undisturbed sample, the shear wave propagates more easily in the remolded sample, and the Gmax of the remodeled sample is larger than that of the undisturbed sample.Figure 13
SEM photos of HWG and HWB R3: (a) original state for undisturbed HWG (magnification=200×); (b) effective consolidation pressure 900 kPa for undisturbed HWG (magnification=200×); (c) the natural state for remolded HWG R3 (magnification=50×); (d) effective consolidation pressure 900 kPa for remolded HWG R3 (magnification=50×). (Figures 13(a) and 13(b) from Wang et al. [26] Figure 13).
(a)(b)(c)(d)The photographs of the samples after consolidation show that both samples are structurally compacted. For these samples that have undergone resonance column testing, the undisturbed HWG has a more compact structure, its intergranular fissures are occlusive, as shown in Figure13(b). Therefore, after experiencing the consolidation process, the undisturbed HWG has a tighter structure, thus making shear waves propagate more quickly in the sample, increasing the sample’s shear modulus. The undisturbed samples mainly show the closure of open fissures, while the remolded samples mainly show a significant reduction in the diameter of the pores lapped between the particles. After consolidation at the effective consolidation pressure (900 kPa) (Figure 12(d)), the pores of the remolded samples are still partially harder to close, and thus macroscopically exhibit less volume change compared with the undisturbed samples. The undisturbed sample is denser than the remolded sample and the shear wave propagates more easily in the undisturbed sample. The Gmax of the remolded weathered granite at this consolidation pressure is smaller than that of the undisturbed weathered granite.


### 4.2. Recommended Range of the Dynamic Parameters for HWG

As Figure14 shows, this work categorized the experimental data and determined the change range of stiffness parameters of HWG in the Lincang area, whereby giving parameter selections for the dynamic response analysis of HWG strata in the Lincang area. On this basis, the test data from other geomaterials were introduced as well to facilitate comparative studies. Figure 14 manifests the decay curve of G for undisturbed, and remolded HWG presents a similar changing tendency but with different decay rates. Existing studies indicate the undisturbed soil is more sensitive to consolidation pressure and shear strain, resulting in greater attenuation rate. The resonant column small-strain (10−6~10−3) test reveals that the weathered granite exhibits almost similar dynamic responses with the normal sand samples [30]. The decay rate of G/Gmax of HWGs in this study is significantly below the ceiling limit value of the sand data provided by Seed and Idriss [10]. The most data points of HWG in this study are scattered above lower limiting value of the compacted HWG data range provided by Niu et al. [3], while some data points are overstep the ceiling limit value of the compacted HWG data range provided by Niu et al. [3]. Furthermore, all data point of the HWG in this study are distributed below the ceiling limit value of the sand data coverage provided by Yuan et al. [30]. As the Figure 14 shows that the lower limit data of the sand coincided with the ceiling limit value of the compacted HWG data provided by Niu et al. [3], and the results of Lincang HWG found in the two data ranges. Furthermore, the range of experimental findings of granite residual soil examined by Liu, X. et al. [31] is similar to the range of remolded HWG supplied by Niu et al. [3]. When the strain exceeds 10-4, the data range of HWG in this study progressive change, almost covering the data range of sand provided by Seed and Idriss [10] and Yuan et al. [30], and it has little overlap within the data recommended by Niu et al. [3], indicating that the stiffness decay of weathered granite in different areas has disparity. Although the attenuation trend of the stiffness of weathered granite in different locations is consistent, the decay rate varies greatly. If the test data of other locations are mechanically used to the Lincang area, there will be a huge discrepancy. Compared with the data range recommended by Niu et al. [3] and Liu, X. et al. [31], the HWGs in the Lincang area have a lower stiffness decay rate and a wider variation range. As a result, the HWGs in the Lincang area are more opposed to distortion than those in other regions and are more appropriate for engineering construction.Figure 14
Variation in normalized dynamic shear modulus with shear strain. Data source: Niu et al. [3], Seed and Idriss [10], Yuan et al. [30], and Liu, X. et al. [31].


## 4.1. The Variations of the Dynamic Characteristics in the Remodeling Process Based on the Microstructure

To observe the effects of the remodeling process and consolidation stress on the microstructure of HWG, HWG samples were tested using scanning electron microscopy (SEM), as presented in Figure13. The undisturbed HWG without consolidation, as illustrated in Figure 13(a), has an embedded locking structure. The distribution of fractures between the particles is notable, with a large width. The orientation of main fissures is unorganized, and the majority of fissures intersect each other. In addition, almost no cementitious agents are observed, but the shapes of adjacent particles are well-matched, which indicates that the intergranular fractures are caused by weathered mineral loss. It is worth noting that some of the quartz grains have flakes of extensively worn primary minerals between them, and these flake structures accumulate in face-to-face contact. For remodeled HWG without consolidation, as shown in Figure 13(c), the soil experiences a structural breakdown during the manual remolding and recompaction. Hence, a new soil skeleton is formed, and the disposition, orientation, and structure of the soil are completely different. In terms of the mineral structure characteristics, the fine-grained minerals of the unconsolidated remodeled sample are attached on the surface of coarse grains which mainly show point-point contact structure and part of their contact in the point-line form, with the irregular arrangement between particles (Figure 12(c)). Unlike the particle interlocking structure in undisturbed samples, the remolded sample particles are randomly oriented and overlap. Besides, the pores of the remodeled sample are mostly isolated pores and secondary pores, with various morphologies and diameters, and some millimeter-level pores can be observed. In the unconsolidated state, the contact area between the particles of the remodeled sample is larger than that of the undisturbed sample, the shear wave propagates more easily in the remolded sample, and the Gmax of the remodeled sample is larger than that of the undisturbed sample.Figure 13
SEM photos of HWG and HWB R3: (a) original state for undisturbed HWG (magnification=200×); (b) effective consolidation pressure 900 kPa for undisturbed HWG (magnification=200×); (c) the natural state for remolded HWG R3 (magnification=50×); (d) effective consolidation pressure 900 kPa for remolded HWG R3 (magnification=50×). (Figures 13(a) and 13(b) from Wang et al. [26] Figure 13).
(a)(b)(c)(d)The photographs of the samples after consolidation show that both samples are structurally compacted. For these samples that have undergone resonance column testing, the undisturbed HWG has a more compact structure, its intergranular fissures are occlusive, as shown in Figure13(b). Therefore, after experiencing the consolidation process, the undisturbed HWG has a tighter structure, thus making shear waves propagate more quickly in the sample, increasing the sample’s shear modulus. The undisturbed samples mainly show the closure of open fissures, while the remolded samples mainly show a significant reduction in the diameter of the pores lapped between the particles. After consolidation at the effective consolidation pressure (900 kPa) (Figure 12(d)), the pores of the remolded samples are still partially harder to close, and thus macroscopically exhibit less volume change compared with the undisturbed samples. The undisturbed sample is denser than the remolded sample and the shear wave propagates more easily in the undisturbed sample. The Gmax of the remolded weathered granite at this consolidation pressure is smaller than that of the undisturbed weathered granite.


## 4.2. Recommended Range of the Dynamic Parameters for HWG

As Figure14 shows, this work categorized the experimental data and determined the change range of stiffness parameters of HWG in the Lincang area, whereby giving parameter selections for the dynamic response analysis of HWG strata in the Lincang area. On this basis, the test data from other geomaterials were introduced as well to facilitate comparative studies. Figure 14 manifests the decay curve of G for undisturbed, and remolded HWG presents a similar changing tendency but with different decay rates. Existing studies indicate the undisturbed soil is more sensitive to consolidation pressure and shear strain, resulting in greater attenuation rate. The resonant column small-strain (10−6~10−3) test reveals that the weathered granite exhibits almost similar dynamic responses with the normal sand samples [30]. The decay rate of G/Gmax of HWGs in this study is significantly below the ceiling limit value of the sand data provided by Seed and Idriss [10]. The most data points of HWG in this study are scattered above lower limiting value of the compacted HWG data range provided by Niu et al. [3], while some data points are overstep the ceiling limit value of the compacted HWG data range provided by Niu et al. [3]. Furthermore, all data point of the HWG in this study are distributed below the ceiling limit value of the sand data coverage provided by Yuan et al. [30]. As the Figure 14 shows that the lower limit data of the sand coincided with the ceiling limit value of the compacted HWG data provided by Niu et al. [3], and the results of Lincang HWG found in the two data ranges. Furthermore, the range of experimental findings of granite residual soil examined by Liu, X. et al. [31] is similar to the range of remolded HWG supplied by Niu et al. [3]. When the strain exceeds 10-4, the data range of HWG in this study progressive change, almost covering the data range of sand provided by Seed and Idriss [10] and Yuan et al. [30], and it has little overlap within the data recommended by Niu et al. [3], indicating that the stiffness decay of weathered granite in different areas has disparity. Although the attenuation trend of the stiffness of weathered granite in different locations is consistent, the decay rate varies greatly. If the test data of other locations are mechanically used to the Lincang area, there will be a huge discrepancy. Compared with the data range recommended by Niu et al. [3] and Liu, X. et al. [31], the HWGs in the Lincang area have a lower stiffness decay rate and a wider variation range. As a result, the HWGs in the Lincang area are more opposed to distortion than those in other regions and are more appropriate for engineering construction.Figure 14
Variation in normalized dynamic shear modulus with shear strain. Data source: Niu et al. [3], Seed and Idriss [10], Yuan et al. [30], and Liu, X. et al. [31].


## 5. Conclusions

Taking the HWG in Lincang area as the research object, this work conducted the RCTs on undisturbed HWG and the remodeled HWG at varying initial dry densities. The following conclusions can be drawn:(1)
The small-strain stiffness of undisturbed HWG shows an identical trend with the remodeled samples, i.e., the dynamic shear modulus is mainly characterized by highly nonlinear characteristics. Using the Stokoe equation can fit well theG−γ curve of HWG in Lincang area(2)
Structural changes obviously influence the small-strain stiffness of HWGs. Once consolidation stress <300 kPa,Gmax of remodeled HWG samples are larger compared with undisturbed samples; however, when consolidation pressure exceeds 300 kPa, Gmax of undisturbed HWG samples are greater than that of the remodeled sample. At an identical consolidation stress, G/Gmax−γ curve of undisturbed remodeled HWGs at the same shear strain is always below than that of remodeled HWG samples, and the decay rate of the G/Gmax−γ curve is larger for the undisturbed sample compared with remodeled samples(3)
TheGmax of the remodeled weathered granite at a lower initial dry density responses to the consolidation stress more intensely. According to the test data, the fitting curves for Gmax of remodeled HWG samples were given, and the fitting parameters, such as G0 and n, are linearly related with initial dry density. Additionally, based on the test results, the mathematical model of Gmax of remodeled HWG considering effective consolidation pressure and initial dry density was established and agreed well with test results in practical engineering cases, which can provide a reference and basis for the research on dynamic behaviours of weathered granite areas. Meanwhile, the G/Gmax−γ curves of remodeled HWGs at different initial dry densities at an identical effective consolidation stress are nearly consistent



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*Source: 3935815-2022-10-22.xml*