BRIEF RESEARCH REPORT
published: 28 September 2021
doi: 10.3389/fphy.2021.740993
Normalized Shear Modulus and
Damping Ratio of Soil–Rock Mixtures
With Different Volumetric Block
Proportions
Shengnian Wang 1*, Xinqun Gao 1, Honglei Hui 1, Wei Ma 2, Chong Shi 3 and Peng Zhang 1*
1
College of Transportation Science and Engineering, Nanjing Tech University, Nanjing, China, 2State Key Laboratory of Frozen
Soil Engineering, Northwest Institute of Eco-Environment and Resources, Chinese Academy of Sciences, Lanzhou, China,
3
Institute of Geotechnical Engineering, Hohai University, Nanjing, China
Edited by:
Wanqing Shen,
Université de Lille, France
Reviewed by:
Lanlan Yang,
Jiangnan University, China
Kai Wu,
Southeast University, China
*Correspondence:
Shengnian Wang
shengnian.wang@njtech.edu.cn
Peng Zhang
zhangpeng-mail@njtech.edu.cn
Specialty section:
This article was submitted to
Interdisciplinary Physics,
a section of the journal
Frontiers in Physics
Received: 14 July 2021
Accepted: 13 August 2021
Published: 28 September 2021
Citation:
Wang S, Gao X, Hui H, Ma W, Shi C
and Zhang P (2021) Normalized Shear
Modulus and Damping Ratio of
Soil–Rock Mixtures With Different
Volumetric Block Proportions.
Front. Phys. 9:740993.
doi: 10.3389/fphy.2021.740993
Frontiers in Physics | www.frontiersin.org
The volume fraction of rock blocks plays a particularly significant role in static/dynamic
shear behaviors of soil–rock mixtures (SRM). Large-scale cyclic triaxial tests for SRM with
different volumetric block proportions (VBPs) were performed at different confining
pressures to investigate the reduction of dynamic shear modulus (G) and the increase
of damping ratio (λ). Results indicate that VBP has a significant effect on the dynamic
behaviors of SRM. The higher VBP is more likely to result in a gentler reduction of G and a
faster increase of λ. The variations of dynamic shear modulus ratio (G/G0) and normalized
damping ratio (λnor) fall within relatively narrow bands but are very different with gravelly
soils and sands due to VBP with particle size larger than 2 mm. The G/G0 and λnor can be
characterized by empirical functions about normalized shear strain amplitude (γnor).
Keywords: soil–rock mixture, shear modulus, damping ratio, volumetric block proportion, dynamic properties
INTRODUCTION
Soil–rock mixtures (SRMs) are inhomogeneous and heterogeneous geological materials widely used
in infrastructure projects such as road foundations, earth dams, and slopes [1]. However, since their
mechanical and physical properties are significantly determined by the interaction and properties of
rock blocks, SRM belongs to neither the category of soil nor rock [2]. Great difficulties in parameter
determination represent a significant challenge in engineering practice and thus have attracted the
attention of scholars from home and abroad.
Soil dynamic parameters, including dynamic shear modulus (G), dynamic shear modulus ratio
(G/G0, G0 was initial shear modulus), and damping ratio (λ) from small to large shear strain
amplitude (ca), are significant indexes for the seismic design and stability analysis of structures and
geo-structures subjected to seismic/repeated loading. Seed et al. [3], Rollins et al. [4], and Hardin and
Kalinski [5] examined the G and λ of gravelly soils by cyclic triaxial tests. However, only a few tests
were performed on gravel and gravelly soils due to the large size of the testing apparatus required. Lin
et al. [6] pointed out that the large proportion of gravels and the unusual gap grading were the causes
for the differing behavior of gravelly deposits. Tanaka [7] found that the G/G0-ca relation of an
undisturbed soil could be approximately described by that of the reconstituted soil sample. Araei
et al. [8] believed that the dynamic characteristics of reconstituted gravelly soils were significantly
affected by gravel content, confining pressure, and loading frequency. Wang et al. [9] further
indicated that the dynamic soil properties showed significant changes when VBP was higher than
30%. Alhassan and VandenBerge [10] proposed a better distinguish between the behavior of
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Dynamic Properties of Soil–Rock Mixtures
FIGURE 1 | G and λ of SRM with different VBPs and CPs.
compacted sand and gravel soils in normalized dynamic shear
modulus and damping ratio with shear strain. Zhang et al. [11]
studied the dynamic properties and damage evolution of silty
soils with different volumetric block proportions (VBPs). Xu et al.
[12] investigated the dynamic properties of sand–gravel mixtures
with varying contents of gravel, loading cycles, and loading
frequencies. Ye et al. [13] summarized the empirical models of
dynamic shear modulus and the range of reduction curves for
coarse-grained soils. It is not a stretch to infer that VBPs will
significantly affect the dynamic properties of SRM. This study
conducted undrained cyclic triaxial tests on SRM with different
VBPs and confining pressures (CPs) to investigate their dynamic
characteristic evolution. The relationships of G and λ with shear
strain amplitude (ca) were discussed in detail.
vacuum extractor until Skempton’s B achieved 0.95. The axial
strain amplitude during loading process increased from 1 × 10–5
to 1 × 10–2 level by level. The specimen was reconsolidated at the
same confining pressure for 15 min at least to achieve an effective
pore water pressure dissipation as each strain loading level ended.
The loading frequency was 0.5 Hz, and the loading cycle was 5.
RESULTS AND DISCUSSION
Figure 1A shows the G of SRM with different VBPs and CPs. The
G of SRM exhibits degradation with ca on the whole. High CP
always results in larger G values for SRM with the same VBP. The
more significant deviation in CP, the more tremendous difference
in the G of SRM. The G of SRM at low CP illustrates a slight
discrepancy of G regardless of VBP. However, when CP increases,
the difference in the G of SRM increases with VBP. These
variations should be due to specimens’ compaction. High VBP
may cause significant overhead phenomena in SRM. The higher
CP can compact SRM specimens effectively, thereby resulting in
denser structures with higher stiffness. The G0 shows that both
VBP and CP can positively impact soils’ fundamental stiffness.
Figure 1B shows the λ of SRM with different VBPs and CPs.
The λ of SRM increases with ca and shows a zonal distribution on
the whole. The whole growth process of λ can divide into the
initial stage (ca < 0.01%), the growth stage (0.01% ≤ ca ≤ 1.0%),
and the stable stage (ca > 1.0%). High VBP always results in λ
growing earlier in the initial stage. The growth rate of λ is
approximately the same at the growth stage, no matter what
VBP is. Results of SRM with the same VBP show that high CP can
narrow the distribution of λ.
Figure 2A presents the envelope curves of G/G0 of SRM with
different VBP and CP, which exhibits a similar behavior
identified by previous researchers. The G/G0 falls within a
relatively narrow band at the ca of 5 × 10–4% but has almost
no effect on the curve shape of G/G0 in ca less than the order of
10−4%. In 1972, Hardin and Drnevich [14] proposed a hyperbolic
shear modulus reduction function for soils. Darendeli [15]
EXPERIMENTS
The Large-Scale Cyclic Triaxial System (GCTS HCA-300) was
used to study the dynamic behavior of SRM. SRM samples were
prepared with fine-grained soil and rock blocks. The fine-grained
soil with particle size less than 2 mm was a residual soil formed in
Quaternary, collected at Nanjing Tech University’s campus. The
dry density of this soil was 1.63 g/cm3. The natural water content
was 21.58%. The maximum dry density was 1.91 g/cm3, and the
optimum moisture content was 12%. Considering that the sizes of
cylindrical samples were 100 mm in diameter and 200 mm in
height, the maximum diameter of rock blocks was limited to
20 mm to avoid the size effect. The natural density of dry rock
blocks with particle sizes ranging from 2 to 20 mm was 2.36 g/
cm3. Since the aerial phenomenon of rock blocks would result in
significant difficulties in packing SRM samples into the mold
when VBP was higher than 60%, only five VBP (0/15/30/45/60,
%) combined with four CP (100/200/400/800, kPa) were
considered in undrained cyclic triaxial tests. Given the water
absorption of rock blocks, the extra 5% of water was mixed into
SRM to avoid rock blocks absorbing water from the fine-grained
soil. When SRM specimens were ready, they were saturated by a
Frontiers in Physics | www.frontiersin.org
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Wang et al.
Dynamic Properties of Soil–Rock Mixtures
FIGURE 2 | Envelope curves of G/G0 and λ of SRM with different VBPs and CPs.
FIGURE 3 | G/G0 and normalized λ of SRM with γnor.
further proposed a modified hyperbolic model based on testing of
sand and gravel samples as
α
G/G0 1/1 + ca /cr ,
but gentler. This find also highlights that the envelope curves of
G/G0 for gravelly soils are below that of SRM.
Figure 2B shows the envelope curves of normalized λ (λnor) of
SRM with different VBP and CP, following the empirical model
proposed by Chen et al. [16].
(1)
where cr is the reference shear strain and is determined as the
shear strain when G/G0 0.5; α is the regression parameter. The
G/G0 of SRM thus can be characterized by following this
hyperbolic model. It can also be found that when VBP in
SRM is higher than 45%, the G/G0 curve begins to overlap
with the bounds proposed by Rollins et al. [4]. When VBP in
SRM is less than 45%, the G/G0 curve almost does not overlap
anymore, especially in ca between 0.01 and 0.1%. Seed et al. [3]
indicated that the curve characteristics of G/G0 for natural gravel
deposits with VBP of 92% at least were similar to that of sandy soil
Frontiers in Physics | www.frontiersin.org
λnor λ0 (1 − G/G0 )β /(λmax − λmin ),
(2)
where λmin and λmax are the minimum and maximum damping
ratios, λ0 and β are regression parameters related to soil
properties. Similarly, the variation of λnor falls within a
relatively narrow band at the ca of 5 × 10–3% and has almost
no effect on the shape of λnor with ca < 10–3%. The reason why the
envelope curves of λnor for SRM is lower than gravelly soils
examined by Seed et al. [3] and Rollins et al. [4] may be that the
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Dynamic Properties of Soil–Rock Mixtures
soil with high VBP is harder to compact but more likely to form a
loose structure in low CP when subjected to cyclic loadings,
thereby resulting in a faster increase of λnor.
Figure 3 illustrates the G/G0 and λnor of SRM with normalized
shear strain amplitude (cnor ca/cr). Both of them are falling within
the relatively narrow bands. Namely, the G/G0 and λnor of SRM with
cnor are not sensitive to VBP and CP. According to the modified
hyperbolic model as Eq. 1 proposed by Darendeli [15], the nonlinear
relationship of G/G0 with cnor for SRM was fitted. It can be found
that this modified hyperbolic model is also appropriate to SRM with
an excellent correlation coefficient up to 0.9879.
Taking Eq. 1 into Eq. 2 yields the following:
β
λnor cαnor /1 + cαnor .
VBP and CP. This study’s results can provide a valuable reference
to understand the dynamic response characteristics and energy
dissipation mechanisms of SRM and other similar geo-materials.
DATA AVAILABILITY STATEMENT
The original contributions presented in the study are included in
the article/Supplementary Material; further inquiries can be
directed to the corresponding authors.
AUTHOR CONTRIBUTIONS
(3)
Funding acquisition and formal writing of the work, SW and XG;
investigation and data analysis of the work, HH; review and
editing of the work, WM, CS, and PZ. All authors have read and
agreed to the published version of the manuscript.
The fitting result of λnor shows that it has a high correlation of
0.9789 with cnor. So the empirical correlation can also be used to
characterize the changes in the λ of SRM under cyclic loadings.
CONCLUSION
FUNDING
Cyclic triaxial tests were conducted to understand the dynamic
properties of SRM further. Outcomes indicate that both VBP and
CP can positively impact the fundamental stiffness (G0) and λ of
SRM. High VBP in coarse-grained soils is more likely to result in a
gentler reduction in G/G0 and a faster increase of λnor. Both G/G0
and λnor of SRM can be evaluated by functions of cnor regardless of
This work was supported by the National Natural Science
Foundation of China (No. 41902282), the Natural Science
Foundation of Jiangsu Province (No. BK20171006), and the
State Key Laboratory of Frozen Soil Engineering (No.
SKLFSE201809).
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Conflict of Interest: The authors declare that the research was conducted in the
absence of any commercial or financial relationships that could be construed as a
potential conflict of interest.
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