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Article

The Influence of Oscillatory Frequency on the Structural Breakup and Recovery of Coastal Mud

by
Hengye Gu
1,2,
Shaohua Wang
1,
Zhitong Su
1 and
Yuezhang Xia
1,3,*
1
Institute of Port, Coastal and Offshore Engineering, Ocean College, Zhejiang University, Zhoushan 316021, China
2
Hainan Institute of Zhejiang University, Sanya 572025, China
3
Guangxi Key Laboratory of Beibu Gulf Marine Resources, Environment and Sustainable Development, Beihai 536000, China
*
Author to whom correspondence should be addressed.
J. Mar. Sci. Eng. 2023, 11(11), 2073; https://doi.org/10.3390/jmse11112073
Submission received: 6 September 2023 / Revised: 25 October 2023 / Accepted: 25 October 2023 / Published: 30 October 2023
(This article belongs to the Section Ocean Engineering)
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Abstract

:
The structural breakup and recovery of coastal mud are closely related to wave propagation, mud transportation, and coastal morphology evolution. Due to the influence of climate, topography, and other factors, the wave frequency in marine environments is more variable than fixed. To investigate the mud structural breakup and recovery process under oscillatory shear loads with different frequencies, a series of oscillatory rheological experiments of the coastal mud collected from the tidal flats of Zhairuoshan Island, Zhejiang province, China, were carried out. The results revealed that the structural breakup of coastal mud had a two-step transition process. The fluidization occurs more rapidly at higher frequencies, but the influence of frequency on the two yield stresses is limited. In addition, frequency has a complex effect on the structural recovery of coastal mud. The normalized equilibrium storage modulus (G∞′/G0′) does not change monotonically with frequency. Moreover, the viscosity quickly approaches equilibrium when a shear load is applied. After that, when a low-frequency load is applied, G∞′/G0′ is no longer related to the pre-shear duration. However, when a high-frequency load is applied, G∞′/G0′ of the mud sample pre-sheared for 500 s is significantly larger than that of the sample pre-sheared to the minimum viscosity. This study is anticipated to provide reference and supplementary test data for understanding the interaction between waves of different frequencies and muddy seabed.

1. Introduction

In coastal environments, cohesive sediments usually exist in the form of coastal mud [1,2,3]. As a viscoelastic material, coastal mud contains clay, silt, sand, seawater, and a small amount of organic matter. The oscillatory shear loads created by waves result in wave–mud interactions [4,5]. Wave–mud interactions cause structural changes in coastal mud and affect mud transportation [6]. Under a large oscillatory shear load, the microstructure of coastal mud breaks up, and then the solid mud gradually transforms into fluid mud [7,8,9]. Fluid mud has low strength and high fluidity, and can be transported via external forces [8]. In contrast, under a very low shear load, the microstructure of the coastal mud gradually recovers, and the fluid mud will be dewatered to form a new muddy bed [10]. The structural change in coastal mud may cause numerous cohesive sediment problems, such as channel siltation [11], coastal morphology evolution [12,13], submarine mud flow [14], pollutant diffusion [11,15,16], and wave attenuation [17]. Additionally, as one of the characteristic elements of waves, frequency has an important influence on wave–mud interactions and mud structural changes [5]. Thus, accurately understanding the structural breakup and recovery process of coastal mud and the influence of oscillatory frequency on this process plays a crucial role in studying wave–mud interactions and is of great significance to coastal engineering design and marine hazards.
The rheological properties of natural mud can not only characterize its resistance to flow and deformation but also reflect its microstructure changes [1]. It is reported that the structural breakup of coastal mud is closely related to the rheological properties of coastal mud, especially the yield stress [18,19]. In recent years, a two-step transition process of coastal mud has been observed by researchers in rheological experiments [20,21,22,23]. With the increase in shear strength, the rheological behavior of coastal mud is classified into the following states: solid state, solid–fluidic state, and fluidic state. In addition, the two-step transition feature relates to the two critical shear stresses when coastal mud transits its state from the solid to the solid–fluidic state and from the solid–fluidic state to the fluidic state. This process is related to the destruction of the microstructure of coastal mud.
Thixotropy is one of the typical rheological properties of coastal mud. If the viscosity of a material changes with the shear duration, and the viscosity gradually recovers when the shear load is removed, the material can be stated as thixotropic [24,25]. Thixotropy has been identified as a crucial aspect related to the structural recovery process of coastal mud. In the past several decades, the thixotropic behavior of minerals and artificial suspensions has been studied in detail [26,27,28,29,30]. However, few studies have focused on the thixotropy of natural mud. Zhang et al. [31] observed a significantly lower storage modulus in disturbed mud compared to deposited mud when studying wave–mud interactions. Shakeel et al. [32] noted substantial differences in the storage modulus and yield stress between diluted and undisturbed mud, which may be attributable to the incomplete structural recovery of the mud. Yang et al. [18] found that the upward flow curve of mud was significantly different from the downward flow curve when investigating the thixotropy of coastal mud.
The structural breakup process of coastal mud is affected by numerous factors, such as water content, density, particle size, salinity, temperature, and pH. In the literature, correlations between these factors and rheological parameters have been extensively reported [33,34,35,36,37]. In the study of thixotropy, Shakeel et al. [19]. investigated the effects of pre-shear rate, pre-shear time, mud density, organic matter content, and geometry of the rheometer on the linear viscoelastic response of the mud collected from the Port of Hamburg. Nevertheless, research on the effect of oscillation frequency on the structural changes in coastal mud is very limited. In most literature, the frequency of oscillatory rheological experiments is a fixed value. However, the frequency of waves in marine environments typically ranges from 0.01 Hz to 10 Hz [38]. Thus, it is necessary to investigate the structural changes in coastal mud at different oscillatory frequencies.
While it is recognized that the oscillatory shear loads created by waves induce the structural breakup of coastal mud, knowledge about the structural recovery process of coastal mud after being pre-sheared is still limited [19,39]. In addition, frequency, as one of the wave characteristic elements, has not been well considered in studying the structural change in coastal mud. This prompted us to investigate the following: (i) the structural breakup and recovery process of coastal mud; and (ii) the effect of oscillatory frequency on the structural breakup and recovery of coastal mud. This study is anticipated to provide reference and supplementary test data for understanding the interaction between waves of different frequencies and a muddy seabed.

2. Materials and Methods

2.1. Materials

The coastal mud samples were collected from the tidal flats of Zhairuoshan Island in Zhejiang Province, China (Figure 1a,b). Ten mud samples were collected at each sampling location using a gravity-based grab sampler to ensure the repeatability of the experiments. The collected mud samples were sent to the laboratory located on the Zhoushan Campus of Zhejiang University. The coastal mud samples were first dried at 105 °C for 24 h and then sieved through a 0.5 mm mesh to remove impurities. It is noteworthy that the samples had the same rheological properties after being dried at different temperatures. The dried samples were then mixed with two amounts of deionized water to obtain two experimental samples (Zhoushan#1 and Zhoushan#2). The particle size distribution of the experimental samples was measured using static light scattering techniques (Figure 1c). The density of the experimental samples was measured using the core cutter method. The detailed physical characteristics of the experimental samples are shown in Table 1.

2.2. Equipment

In this study, the rheological experiments were conducted using the rotational rheometer DHR-I equipped with a parallel-plate fixture. The parallel-plate geometry can produce constant shear loads on the materials and has been successfully used in previous research [14,40,41]. The geometry has a diameter of 25 mm and a 1 mm gap between the upper and lower plates. The minimum torque in dynamic rheological tests is 20 nN∙m. To eliminate the influence of temperature, the temperature was controlled at 25 °C throughout all the tests. Each test was conducted three times to exclude the influence of shear histories and check the repeatability of the measurements.

2.3. Measurement Method

2.3.1. Structural Breakup Tests

To investigate the structural breakup of the coastal mud under oscillatory shear loads, the experimental methodologies performed in this study are structured as follows:
(1) Initially, a resting time of 3 min was allowed for the Zhoushan#1 sample to ensure the reproducible state of the sample before shearing.
(2) An amplitude sweep test was carried out on the Zhoushan#1 sample. In the test, the sample was applied to sinusoidal shear strain, which increases logarithmically from 10−2 to 103%, and oscillatory frequency was considered 0.25, 0.5, 1, 2, and 4 Hz.
(3) To further investigate the effect of oscillatory frequency on the rheological properties of the Zhoushan#1 sample, frequency sweep tests were conducted. In this test, the frequency was varied from 0.1 to 4.1 Hz, and the strain amplitude was considered as 0.03 (solid state), 3 (the beginning period of the solid–fluidic state), 50 (the late period of the solid–fluidic state), and 500% (fluidic state).
The basis for selecting the above frequency range is as follows. On the one hand, the frequency of waves in marine environments typically ranges from 0.01 Hz to 10 Hz [38], and the range of the applied frequency (0.1–4.1 Hz) covers the frequency of most coastal waves. On the other hand, the rotational rheometer DHR-I cannot produce constant shear loads on the sample when the oscillatory frequency is higher than 4 Hz or lower than 0.1 Hz in the structural breakup tests.
In all tests, eight measuring points were set in each decade on a log scale with a measuring time of 15 s for each point, and the rheological parameters, such as stress amplitude, complex viscosity, phase angle, storage modulus, and loss modulus, were obtained. The technology flow and protocol of the experiments are given in Figure 2.

2.3.2. Structural Recovery Tests

To investigate the structural recovery of the coastal mud after being disturbed by oscillatory shear loads, the experimental methodologies performed in this study are structured as follows:
(1) Similar to the structural breakup test, a resting time of 3 min was given to the Zhoushan#2 sample before all the tests. To determine the initial storage modulus (G0′) before being pre-sheared with oscillatory shear loads, oscillatory time sweep tests were carried out within the linear viscoelastic region of the coastal mud (Figure 3a). Moreover, preliminary amplitude sweep tests were carried out to estimate the linear viscoelastic region (Figure 3b).
(2) A pre-shearing step was then carried out to destroy the microstructure of the Zhoushan#2 mud sample. According to the literature overview that we performed, the coastal mud shows a “two-step” transition process. There are two yield stresses, namely, static yield stress and fluidic yield stress, which are the critical stresses when the coastal mud transitions from the solid to the solid–fluidic state and from the solid–fluidic state to the fluidic state, respectively. Considering that coastal mud behaves as a solid material when subjected to shear loads lower than the static yield stress, the resulting damage to its structures is limited. Furthermore, considering the frequency of waves in marine environments typically ranges from 0.01 Hz to 10 Hz, two types of shear loads, where the mud was in a solid–fluidic state (40 Pa) and a fluidic state (100 Pa), combined with eight frequencies of 0.1 Hz, 0.25 Hz, 0.5 Hz, 1 Hz, 2 Hz, 4 Hz, 8 Hz, and 10 Hz, were chosen to investigate the structural recovery process of the Zhoushan#2 sample and the relationship between oscillatory frequency and structural recovery of the sample. The pre-shearing step in the test was executed for 500 s. It was observed that the viscosity could reach a constant value before the end of the pre-shearing step (see Section 3.2.2). This implies that the mud sample has reached a steady-state structure. In addition, the pre-shear time was adjusted to the time when the sample reaches the minimum value of its viscosity to investigate the effect of pre-shear duration.
(3) Oscillatory time sweep tests were carried out within the linear viscoelastic region to recover the structure of the coastal mud [42,43,44]. The linear viscoelastic region after the pre-shearing step was also determined using the amplitude sweep test (Figure 3b). The structural recovery step was executed for 800 s, which was long enough for the mud sample to achieve an equilibrium structure according to prior research [19].
The technology flow and protocol of the experiments are given in Figure 4.

3. Results

3.1. Structural Breakup Process

3.1.1. The Two-Step Transition Process

Strain amplitude sweep tests can display the two-step transition process of coastal mud. The variation of stress amplitude (τ0), storage modulus (G′), and loss modulus (G″) of the Zhoushan#1 mud sample with strain amplitude (ε0) is displayed as rheological curves (Figure 5a), whereas complex viscosity (η0) and phase angle (δ) are shown in Figure 5b. Depending on the characteristics of the rheological curve, the transition process of the sample could be classified into the following three states:
  • Solid state (ε0 < 0.24%): At a strain amplitude of ε0 < 0.03%, the stress amplitude increases linearly from 0.39 Pa to 1.12 Pa with the increase in strain amplitude, and the Zhoushan#1 mud sample shows a storage modulus of 3763.21–3779.40 Pa, a loss modulus of 976.26–981.99 Pa, a complex viscosity of 614.25–615.85 Pa∙s, and a phase angle of 14.69–14.74°. This shows that, at the beginning of the solid state, the storage modulus, loss modulus, complex viscosity, and phase angle all remain almost unchanged, and the value of G′ is one order of magnitude larger than that of G″. These phenomena indicate that the sample is in the linear region, and the oscillatory shear loads only cause elastic deformation. At a strain amplitude of ε0 > 0.03%, τ0 increases nonlinearly to a local maximum τ0 = 2.86 Pa, which is stated as static yield stress τos (superscript denotes static yield and subscript denotes oscillatory shear). G′, G″, η0, and δ change from 3763.21 Pa to 1079.45 Pa, 976.26 Pa to 547.92 Pa, 614.25 Pa∙s to 192.67 Pa∙s, and 14.74° to 26.91°, respectively, by increasing the strain amplitude. These phenomena indicate that the sample exhibits nonlinear viscoelastic behavior and that plastic deformation occurs. At the end of the state, the Zhoushan#1 sample yields for the first time and undergoes the transition from a solid state to a solid–fluidic state.
  • Solid–fluidic state (0.24 < ε0 < 236.83%): At a strain amplitude of ε0 < 13.31%, the stress amplitude remains unchanged: τ0 = 2.86 Pa. Then, the stress amplitude increases to fluidic yield stress τof = 17.09 Pa (the superscript denotes fluidic yield and the subscript denotes oscillatory shear) gradually. In this state, with the increase in strain amplitude, G′, G″, η0, and δ change from 1079.45 Pa to 2.04 Pa, 547.92 Pa to 6.92 Pa, 192.67 Pa∙s to 1.15 Pa∙s, and 26.91° to 73.67°, respectively. It is noted that G′ decreases faster than G″ and is lower than G″ eventually. Moreover, the relative sizes of G′ and G″ reverse at the critical strain ε0 = 0.65%, defined as modulus crossover [45,46]. G′ > G″, the sample exhibits solid characteristics; G′ < G″, the sample exhibits fluid characteristics. At the end of the state, the Zhoushan#1 sample yields again and undergoes the transition from the solid–fluidic state to the fluidic state.
  • Fluidic state (ε0 > 236.83%): In this state, the stress amplitude continues to rise. Moreover, with the increase in ε0, the values of G′, G″, and η0 decrease from 2.04 Pa to 0.43 Pa, 6.92 Pa to 2.64 Pa, and 1.15 Pa∙s to 0.42 Pa∙s, respectively. G″ is dominant, and the Zhoushan#1 mud sample behaves like fluids.

3.1.2. Amplitude Sweep Tests at Five Different Frequencies

It can be observed that larger frequencies result in a larger stress amplitude, a larger storage modulus, and a lower viscosity in general (Figure 6 and Table 2). For loss modulus, larger frequencies lead to lower values in the solid state but larger values in the solid–fluidic and fluidic states.
In the solid state, G′, G″, and τ0 all change slightly with frequency at the same strain amplitude, and the influence of frequency is limited (Figure 6 and Table 2). In late solid–fluidic and fluidic states, the influence of frequency is relatively larger. G′, G″, and τ0 all exhibit a significant increasing trend with the increase in frequency at the same strain amplitude. In addition, it is worth noting that the stress amplitude of the sample subjected to low-frequency oscillatory shear loads is significantly higher than that of the sample subjected to high-frequency oscillatory shear loads at the same strain amplitude in the early solid–fluidic state, which is inconsistent with the relationship between stress amplitude and oscillatory frequency in general (symbol of the ellipse in Figure 6a).
The fluidic critical strain (εof) of the sample subjected to high-frequency oscillatory shear load is lower than that of the sample subjected to low-frequency oscillatory shear load, and the sample subjected to high-frequency load has a narrower solid–fluidic state (Figure 6 and Table 3). In addition, it is worth noting that the influence of frequency on the values of static yield stress (symbol of the triangle in Figure 6a) and fluidic yield stress (symbol of the rectangle in Figure 6b) is limited.

3.1.3. Frequency Sweep Tests

Frequency sweep tests were carried out to further study the effect of frequency on the structural breakup of coastal mud. In the solid state (Figure 7a), the storage modulus, loss modulus, and stress amplitude are independent of frequency: τ0 = 1.28 Pa, G′ = 4137 Pa, and G″ = 925 Pa. The value of G′ is one order larger than that of G″. These phenomena indicate the linear viscoelastic behavior of the mud sample [10,37]. At the beginning of the solid–fluidic state (Figure 7b), the Zhoushan#1 mud sample under oscillatory shear loads with different frequencies has similar values of stress amplitude, storage modulus, and loss modulus that range from 3.50 Pa to 4.31 Pa, 1142.24–1199.23 Pa, and 611.18–659.98 Pa, respectively. The value of G′ is still higher than that of G″, indicating that elasticity still dominates. In the late solid–fluidic state (Figure 7c), with the increase in frequency, G′, G″, and τ0 increase from 0.90 Pa to 4.45 Pa, 0.40 Pa to 3.78 Pa, and 1.36 Pa to 8.06 Pa, respectively. The value of G′ is already lower than that of G″, indicating that viscosity dominates. In the fluidic state (Figure 7d), G′, G″, and τ0 increase from 16.50 Pa to 30.76 Pa, 0.40 Pa to 3.38 Pa, and 3.00 Pa to 5.15 Pa, respectively. In this state, the microstructure of the mud sample has been completely destroyed. The results of the frequency sweep tests are consistent with those of the amplitude sweep tests, as shown in Figure 6, which indicate that frequency has an important influence on the structural breakup of coastal mud.

3.2. Structural Recovery Process

3.2.1. Effect of Oscillatory Frequency in the Case of the Same Pre-Shear Duration

As shown in Figure 8, the time evolution of the normalized storage modulus G′/G0′ exhibits a similar pattern in the structural recovery stage, regardless of the magnitude of the frequency. G′/G0′ shows a rapid increase at the beginning of the stage, followed by a progressively slow increase over time until it ultimately reaches an equilibrium state. Moreover, Figure 8 shows obvious differences in G′/G0′ when the Zhoushan#2 sample was pre-sheared at different oscillatory frequencies, suggesting that the frequency significantly affects the structural recovery process of coastal mud.
To further analyze the structural recovery process of coastal mud after pre-shearing by external loads with different frequencies, the equilibrium storage modulus G∞′ for the structural recovery of the Zhoushan#2 mud sample was estimated by fitting the experimental data to a stretched exponential function [47,48]. The time evolution of the storage modulus of the material after being pre-sheared with oscillatory shear loads is as follows:
G ′ G 0 ′ = G i ′ G 0 ′ + G ∞ ′ G 0 ′ − G i ′ G 0 ′ 1 − exp − t t r
where G0′ is the initial storage modulus, Gi′ is the storage modulus right after the pre-shearing stage, G∞′ is the equilibrium storage modulus, and tr is the characteristic time of the sample, which denotes the time required for G′ of the sample to reach 63% of G∞′.
However, as shown in Figure 9a, after fitting some of the experimental data shown in Figure 8 with the above equation, we found that the experimental data at some frequencies, such as 4 Hz, are well fitted: R2 = 0.99036, while the fitting curve at other frequencies, such as 0.1 Hz, has a low correlation coefficient (R2 = 0.95386). In addition, we found that the value of G′/G0′ almost remains unchanged after recovering for more than 750 s (Figure 8 and Figure 9a). The structure recovery test, with a recovery time of more than 800 s, also confirmed this phenomenon. It can be suggested that the value of G′/G0′ after 800 s of structural recovery is close to the value of G∞′/G0′. Therefore, the storage modulus after 800 s of structural recovery was selected as the equilibrium storage modulus (G∞′) in this study.
G∞′/G0′ displays a decreasing trend with increasing frequency until a critical frequency (F′) and is followed by a significant increase for higher frequencies (Figure 9b). The difference in critical frequency is relatively significant. The sample under the oscillatory shear load of 40 Pa shows F′ = 1 Hz, whereas the sample under the oscillatory shear load of 100 Pa shows F′ = 2 Hz. It should be noted that for low frequencies F = 0.1, 0.25, 0.5, and 1 Hz, a slight decrease in G∞′/G0′ with frequency was obtained at 40 Pa. However, G∞′/G0′ changes significantly with frequency under the oscillatory shear load of 100 Pa (Figure 9b and Table 4). This implies that the low-frequency oscillatory shear load lower than the fluidic yield stress has little effect on the structural recovery of the Zhoushan#2 sample, while the oscillatory shear load larger than the fluidic yield stress has a significant effect on the structural recovery of the sample. The variation trend of G∞′/G0′ with pre-shear frequency in the test is very similar to the variation trend of G∞′/G0′ with pre-shear rate observed by Shakeel et al. [19].

3.2.2. Effect of Oscillatory Frequency in the Case of Different Pre-Shear Duration

Figure 10a illustrates the time evolution of the viscosity of the Zhoushan#2 sample with varying pre-shear duration at 40 Pa and 2 Hz in the pre-shear step. After pre-shearing for 200 s, the viscosity values with different pre-shear times are almost the same: η0 = 0.55 Pa∙s, indicating that the microstructure of the mud sample has reached equilibrium. In the recovery step, the values of G′/G0′ of the Zhoushan#2 mud sample with different pre-shear durations are very close after recovering for the same time (Figure 10b).
It is evident that the Zhoushan#2 mud sample with different pre-shear durations has similar values of Gi′/G0′ and G∞′/G0′ that range from 0.032 to 0.034 and 0.759–0.778, respectively (Figure 11). It is consistent with the prior study when investigating the structural recovery process of the screen-printing silver pastes [49]. The rapid convergence of viscosity to equilibrium indicates the rapid disintegration of the microstructure of coastal mud under the oscillatory shear load, and the increase in shear duration does not appear to have a significant effect on the extent of damage to the internal structures of the mud in this stage. This is evident from similar Gi′/G0′ and G∞′/G0′ values observed at different shear durations, suggesting that the shear duration has a limited influence on the structural recovery of the mud.
However, we found that the viscosity of the Zhoushan#2 sample would increase rather than remain unchanged after reaching the minimum value under the high-frequency oscillatory shear load in the pre-shearing step (Figure 12a). As shown in Figure 12b–f, the value of G′/G0′ remains almost unchanged after recovering for more than 750 s, even if the pre-shear time is less than 500 s. For the frequencies of 4 and 8 Hz, when the oscillatory shear load (τ0 = 40 Pa) is lower than the fluidic yield stress, the values of G′/G0′ with different pre-shear durations can reach a similar equilibrium value after recovering for 800 s (Figure 12b,c, and Table 4). However, as the oscillatory shear load and frequency increase, the pre-shear duration has a significant effect on the structural recovery of the mud sample. G∞′/G0′ of the sample that had been pre-sheared for 500 s is significantly higher than that of the sample that had been pre-sheared to the minimum viscosity (Figure 12d–f and Table 4). Moreover, with the increase in frequency and oscillatory shear load, the difference of G∞′/G0′ of the Zhoushan#2 mud sample at the above two different pre-shear durations increases.

4. Discussion

4.1. Structural Breakup Analysis

With increasing oscillatory shear load, coastal mud exhibits a two-step transition process (Figure 13a), which may be described via the microstructure of coastal mud. As shown in Figure 13b, the initial structure of the coastal mud before the structural breakup is composed of many aggregates and chains. Aggregates are composed of cohesive sediment particles, while chains represent interparticle electrostatic bonding, direct contact of particles, and biological effects [50]. In the linear region, shear forces less than the strength of the initial structure could not break the chains between the aggregates, and could only cause elastic deformation. In the late solid state, the chains between the aggregates are partly destroyed, resulting in plastic deformation and nonlinear viscoelastic behavior. In the solid–fluidic state, the initial structure gradually breaks into smaller-size aggregates, resulting in a reduction in structural strength and the occurrence of modulus crossover. The behavior of coastal mud gradually changes from the solid state (elasticity dominant) to the fluidic state (viscosity dominant). Beyond the fluidic yield stress, the aggregates are fully destroyed, and the flow units become individual particles [37], resulting in the rapid development of deformation. The two yield stresses correspond to the destruction of chains between aggregates and between particles, respectively.
In the solid state, the viscoelastic solid coastal mud has a stable microstructure and is not sensitive to the oscillation frequency. With the increase in oscillatory shear strength, viscosity gradually dominates, and the coastal mud behaves like a viscous fluid. Thus, the influence of frequency on the rheological properties of coastal mud is mainly exhibited in late solid–fluidic and fluidic states.
In addition, the Zhoushan#1 mud sample subjected to low-frequency oscillatory shear loads has a larger stress amplitude at the beginning of the solid–fluidic state. A possible explanation for this phenomenon is that the high-frequency oscillatory shear load may lead to a sudden collapse of the microstructure, resulting in a sudden loss of the stress amplitude τ0 when the sample yields for the first time.
The values of static and fluidic yield stresses of coastal mud are related to physical characteristics such as density [33], ion concentration [51], water content [52,53], mineral composition [54,55,56] and organic matter content [57]. However, the effect of oscillatory frequency on the values of two yield stresses is limited. Moreover, compared with the sample subjected to low-frequency oscillatory shear loads, the sample subjected to high-frequency oscillatory shear loads has narrower solid–fluidic states and transforms into fluids more rapidly. The oscillatory frequency corresponds to the number of times the shear load is applied per unit time. In the amplitude sweep tests, the oscillatory shear load increased logarithmically rather than remaining unchanged. For such shear loads, the effect of oscillatory frequency on the structural breakup of coastal mud is mainly exhibited in the transition process rather than the yield stresses. Therefore, it is better to use the fluidic yield stress to define fluidizations than the critical strain amplitude [5,58].

4.2. Structural Recovery Analysis

In the structural recovery tests, the input oscillatory shear load is constant. For such shear loads, the oscillatory frequency can significantly affect the structural change of coastal mud. With the increase in oscillatory frequency, the variation of the normalized equilibrium storage modulus (G∞′/G0′) is not monotonous. From the analysis of Section 4.1, when the oscillatory shear load is lower than the fluidic yield stress, the initial structure (Figure 14a) breaks into small-size aggregates. After pre-shearing, it is assumed that the small-size aggregates can constitute a weaker microstructure (Figure 14b). In the recovery step, over time, some strength will be recovered due to the reconstruction of the aggregates (Figure 14f). Despite this, the stability of the microstructure after the structural recovery stage is lower than that of the initial structure, implying G∞′/G0′ < 1. However, at a high-frequency shear load, the initial structure will be broken into smaller-size aggregates and individual sediment particles, even if the pre-shear load is lower than the fluidic yield stress (Figure 14c). In the recovery step, it is expected that aggregates, particles, and chains can form a more stable microstructure (Figure 14g). For the high frequencies of 4, 8, and 10 Hz, the normalized equilibrium storage modulus G∞′/G0′ is then even larger than 1. This phenomenon can be attributed to the “densification” of the coastal mud after its internal structures are broken into smaller units when the mud is sheared using an external load, which was observed by Van Den Tempel [59].
The initial structure is completely broken into individual sediment particles when the oscillatory shear load is greater than the fluidic yield stress (Figure 14e). In the recovery step, it is expected that particles and chains could also form a better microstructure (Figure 14i). For the high frequencies of 8 and 10 Hz, G∞′/G0′ is also larger than 1 but lower than that at 40 Pa. The reason is that a greater number of particles are surrounded by the pore water, and these particles become suspended in the pore water, resulting in difficulties in forming internal structures and a reduced ability to resist external loads [11,60]. On the other hand, even if the oscillatory shear load is greater than the fluidic yield stress, the slow oscillatory shear load caused by low frequency cannot completely destroy the interconnected network in the mud sample. As a result, the aggregates cannot be completely divided into individual particles (Figure 14d). In the recovery step, these aggregates and particles will form a better structure than the structure shown in Figure 14f (Figure 14h).
In addition, it is worth noting that the structural recovery begins in the pre-shearing step when the sample is subjected to high-frequency oscillatory shear loads. The structural recovery in the pre-shearing step leads to a higher normalized equilibrium storage modulus (G∞′/G0′) of the coastal mud after the structural recovery step.

5. Conclusions

The mud structural breakup and recovery process after being sheared by oscillatory shear load was investigated by a series of rheological experiments on coastal mud samples collected from the tidal flats of Zhairuoshan Island in China. The conclusions are summarized as follows:
(1) Structural breakup process: Coastal mud undergoes a process from a solid state (elastic effect dominates) to a fluidic state (viscous effect dominates) under oscillatory shear loads. This process is related to the structural breakup of coastal mud. When the sample yields for the first time, the initial structure is broken into small-size aggregates. When the sample yields for the second time, the aggregates are broken into individual sediment particles.
(2) The effect of frequency on the structural breakup of coastal mud: Oscillatory frequency has little effect on the structural breakup process of coastal mud in the solid state but has a very important influence on the other two states. At larger frequencies, the fluidic yield point appears at lower strain amplitudes, and the solid–fluidic state is narrower. The fluidization occurs more rapidly. However, the influence of frequency on the values of two yield stresses is limited.
(3) Structural recovery process: The time evolution of the normalized storage modulus exhibits a similar pattern in the structural recovery stage, regardless of the magnitude of the frequency. The normalized storage modulus shows a rapid increase at the beginning of the stage, followed by a progressively slower increase over time until it ultimately reaches an equilibrium state.
(4) The effect of frequency on the structural recovery of coastal mud: The normalized equilibrium storage modulus G∞′/G0′ displays a decreasing trend with increasing frequency until a critical frequency and is followed by a significant increase for higher frequencies. The low-frequency oscillatory shear load lower than the fluidic yield stress has little effect on the structural recovery of the Zhoushan#2 sample, while the oscillatory shear load larger than the fluidic yield stress has a significant effect on the structural recovery of the sample. In addition, when high-frequency shear loads are applied, G∞′/G0′ of the mud sample being pre-sheared for 500 s is significantly larger than that of the sample being pre-sheared to the minimum viscosity.

Author Contributions

Conceptualization, H.G. and S.W.; methodology, H.G., S.W. and Y.X.; validation, H.G.; formal analysis, H.G. and Z.S.; investigation, H.G.; resources, Y.X.; data curation, H.G.; writing—original draft preparation, H.G.; writing—review and editing, Y.X.; visualization, H.G.; supervision, S.W. and Y.X.; project administration, Y.X.; funding acquisition, Y.X. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the National Natural Science Foundation of China: No. 42076177; Guangxi Key Laboratory of Beibu Gulf Marine Resources, Environment and Sustainable Development: No. MRESD-2022-B05.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. (a) Geographical location of Zhairuoshan Island; (b) sampling locations in Zhairuoshan Island; (c) particle size distribution of the experimental samples.
Figure 1. (a) Geographical location of Zhairuoshan Island; (b) sampling locations in Zhairuoshan Island; (c) particle size distribution of the experimental samples.
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Figure 2. (a) The technology flow of the structural breakup tests; (b) the resting step; (c) amplitude sweep tests; and (d) frequency sweep tests.
Figure 2. (a) The technology flow of the structural breakup tests; (b) the resting step; (c) amplitude sweep tests; and (d) frequency sweep tests.
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Figure 3. (a) The initial storage modulus (G0′) before the breakup of the structure is determined using the oscillation time sweep test; (b) stress amplitude sweep tests to determine the linear viscoelastic region of the sheared and unsheared mud sample. The linear viscoelastic region is represented with dotted lines and arrows.
Figure 3. (a) The initial storage modulus (G0′) before the breakup of the structure is determined using the oscillation time sweep test; (b) stress amplitude sweep tests to determine the linear viscoelastic region of the sheared and unsheared mud sample. The linear viscoelastic region is represented with dotted lines and arrows.
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Figure 4. (a) The technology flow of the structural recovery tests; (b) the experimental protocol to investigate the structural recovery of the coastal mud after being sheared with the oscillatory shear load; (c) the schematic diagram of the change in storage modulus (G′) at each stage. G0′ is the storage modulus before structural breakup, Gi′ is the storage modulus right after pre-shearing, and G∞′ is the equilibrium storage modulus as t→∞.
Figure 4. (a) The technology flow of the structural recovery tests; (b) the experimental protocol to investigate the structural recovery of the coastal mud after being sheared with the oscillatory shear load; (c) the schematic diagram of the change in storage modulus (G′) at each stage. G0′ is the storage modulus before structural breakup, Gi′ is the storage modulus right after pre-shearing, and G∞′ is the equilibrium storage modulus as t→∞.
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Figure 5. Measured results of the Zhoushan#1 sample at a fixed frequency of 1 Hz in the amplitude sweep test. (a) Stress amplitude (τ0), storage modulus (G′), and loss modulus (G″); (b) complex viscosity (η0) and phase angle (δ). The black triangle and square represent the static yield stress (τos) and fluidic yield stress (τof), respectively. The orange circle represents the modulus crossover.
Figure 5. Measured results of the Zhoushan#1 sample at a fixed frequency of 1 Hz in the amplitude sweep test. (a) Stress amplitude (τ0), storage modulus (G′), and loss modulus (G″); (b) complex viscosity (η0) and phase angle (δ). The black triangle and square represent the static yield stress (τos) and fluidic yield stress (τof), respectively. The orange circle represents the modulus crossover.
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Figure 6. Measured results of the Zhoushan#1 sample at five frequencies of 0.25, 0.5, 1, 2, and 4 Hz in the amplitude sweep test. (a) Stress amplitude; (b) complex viscosity; (c) storage modulus; (d) loss modulus. The triangle symbol denotes static yield stress τos and the rectangle denotes fluidic yield stress τof. The ellipse symbol indicates the difference in stress amplitude between the sample with high frequency and the sample with low frequency at the beginning of the solid–fluidic state.
Figure 6. Measured results of the Zhoushan#1 sample at five frequencies of 0.25, 0.5, 1, 2, and 4 Hz in the amplitude sweep test. (a) Stress amplitude; (b) complex viscosity; (c) storage modulus; (d) loss modulus. The triangle symbol denotes static yield stress τos and the rectangle denotes fluidic yield stress τof. The ellipse symbol indicates the difference in stress amplitude between the sample with high frequency and the sample with low frequency at the beginning of the solid–fluidic state.
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Figure 7. Measured results of the Zhoushan#1 sample at strain amplitudes of (a) 0.03, (b) 3, (c) 50, and (d) 500% in the frequency sweep test.
Figure 7. Measured results of the Zhoushan#1 sample at strain amplitudes of (a) 0.03, (b) 3, (c) 50, and (d) 500% in the frequency sweep test.
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Figure 8. The normalized storage modulus G′/G0′ of the Zhoushan#2 sample as a function of time after being pre-sheared with oscillatory shear loads with different frequencies: (a) pre-shear load τ = 40 Pa and (b) pre-shear load τ = 100 Pa.
Figure 8. The normalized storage modulus G′/G0′ of the Zhoushan#2 sample as a function of time after being pre-sheared with oscillatory shear loads with different frequencies: (a) pre-shear load τ = 40 Pa and (b) pre-shear load τ = 100 Pa.
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Figure 9. (a) The normalized storage modulus G′/G0′ of the Zhoushan#2 sample as a function of time after being pre-sheared with oscillatory shear loads with a magnitude of 100 Pa and frequencies of 0.1, 1, and 4 Hz. The solid lines in the figure represent the fitting curves. (b) The normalized equilibrium storage modulus G∞′/G0′ of the Zhoushan#2 sample as a function of the frequency.
Figure 9. (a) The normalized storage modulus G′/G0′ of the Zhoushan#2 sample as a function of time after being pre-sheared with oscillatory shear loads with a magnitude of 100 Pa and frequencies of 0.1, 1, and 4 Hz. The solid lines in the figure represent the fitting curves. (b) The normalized equilibrium storage modulus G∞′/G0′ of the Zhoushan#2 sample as a function of the frequency.
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Figure 10. (a) The time evolution of the viscosity of the Zhoushan#2 mud sample with varying pre-shear duration at 40 Pa, 2 Hz; (b) the time evolution of the normalized storage modulus of the Zhoushan#2 sample after being pre-sheared with varying pre-shear duration at 40 Pa, 2 Hz.
Figure 10. (a) The time evolution of the viscosity of the Zhoushan#2 mud sample with varying pre-shear duration at 40 Pa, 2 Hz; (b) the time evolution of the normalized storage modulus of the Zhoushan#2 sample after being pre-sheared with varying pre-shear duration at 40 Pa, 2 Hz.
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Figure 11. The normalized storage modulus right after the pre-shear stage Gi′/G0′ and normalized equilibrium storage modulus G∞′/G0′ of the Zhoushan#2 sample as a function of the pre-shear duration at 40 Pa and 2 Hz.
Figure 11. The normalized storage modulus right after the pre-shear stage Gi′/G0′ and normalized equilibrium storage modulus G∞′/G0′ of the Zhoushan#2 sample as a function of the pre-shear duration at 40 Pa and 2 Hz.
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Figure 12. (a) The time evolution of the viscosity of the Zhoushan#2 sample when being pre-sheared with 40 Pa and 100 Pa at 10 Hz with varying pre-shear duration; (b–f) the time evolution of the normalized storage modulus of the Zhoushan#2 sample after being pre-sheared with 40 Pa and 100 Pa with varying pre-shear duration: (b) 40 Pa, 4 Hz; (c) 40 Pa, 8 Hz; (d) 40 Pa, 10 Hz; (e) 100 Pa, 8 Hz; and (f) 100 Pa, 10 Hz. Additionally, “500 s” represents the pre-shearing time of 500 s, and “Min” represents the time required for pre-shearing to the lowest viscosity.
Figure 12. (a) The time evolution of the viscosity of the Zhoushan#2 sample when being pre-sheared with 40 Pa and 100 Pa at 10 Hz with varying pre-shear duration; (b–f) the time evolution of the normalized storage modulus of the Zhoushan#2 sample after being pre-sheared with 40 Pa and 100 Pa with varying pre-shear duration: (b) 40 Pa, 4 Hz; (c) 40 Pa, 8 Hz; (d) 40 Pa, 10 Hz; (e) 100 Pa, 8 Hz; and (f) 100 Pa, 10 Hz. Additionally, “500 s” represents the pre-shearing time of 500 s, and “Min” represents the time required for pre-shearing to the lowest viscosity.
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Figure 13. (a) The stress amplitude evolution with increasing strain amplitude of the Zhoushan#1 mud sample. (b) Schematic diagram of the structural breakup of coastal mud. The black circle and triangle represent the static yield stress and fluidic yield stress, respectively. Yellow circles represent the sediment particles, green curves represent electrostatic bonding between the sediment particles, and the blue background represents the water environment.
Figure 13. (a) The stress amplitude evolution with increasing strain amplitude of the Zhoushan#1 mud sample. (b) Schematic diagram of the structural breakup of coastal mud. The black circle and triangle represent the static yield stress and fluidic yield stress, respectively. Yellow circles represent the sediment particles, green curves represent electrostatic bonding between the sediment particles, and the blue background represents the water environment.
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Figure 14. Schematic diagram of structural breakup and recovery at different pre-shear loads and frequencies. Resting stage: (a) Initial structure. The structures in the pre-shearing stage: (b) Low frequency and low shear strength; (c) high frequency and low shear strength; (d) low frequency and large shear strength; (e) high frequency and large shear strength. The structures in the recovery stage: (f) Low frequency and low shear strength; (g) high frequency and low shear strength; (h) low frequency and large shear strength; (i) high frequency and large shear strength. Yellow circles represent the sediment particles, green curves represent electrostatic bonding between the sediment particles, and the blue background represents the water environment.
Figure 14. Schematic diagram of structural breakup and recovery at different pre-shear loads and frequencies. Resting stage: (a) Initial structure. The structures in the pre-shearing stage: (b) Low frequency and low shear strength; (c) high frequency and low shear strength; (d) low frequency and large shear strength; (e) high frequency and large shear strength. The structures in the recovery stage: (f) Low frequency and low shear strength; (g) high frequency and low shear strength; (h) low frequency and large shear strength; (i) high frequency and large shear strength. Yellow circles represent the sediment particles, green curves represent electrostatic bonding between the sediment particles, and the blue background represents the water environment.
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Table 1. The physical characteristics of the experimental samples.
Table 1. The physical characteristics of the experimental samples.
Sample IDSampling LocationD10
(μm)		D50
(μm)		D90
(μm)		Water Content
(%)		Density
(kg /m³)
Zhoushan#1Sampling location11.454.5824.1080.001531.50
Zhoushan#2Sampling location22.429.8631.1075.001554.60
Table 2. Values of the stress amplitude (τ0), storage modulus (G′), loss modulus (G″), and complex viscosity (η0) of the Zhoushan#1 mud sample subjected to oscillatory shear loads with different frequencies at the strain amplitude (ε0) of 0.03, 1, 75, and 1000%.
Table 2. Values of the stress amplitude (τ0), storage modulus (G′), loss modulus (G″), and complex viscosity (η0) of the Zhoushan#1 mud sample subjected to oscillatory shear loads with different frequencies at the strain amplitude (ε0) of 0.03, 1, 75, and 1000%.
ε0 (%)Rheological StateF (Hz)τ0 (Pa)G′ (Pa)G″ (Pa)η0 (Pa∙s)
0.03 Solid state0.250.973247.14950.821869.32
0.50.973423.80930.91988.49
11.123585.27903.58570.78
21.193687.14871.05299.72
41.403970.89787.51161.75
1Early solid–fluidic state0.253.00246.11187.16196.84
0.52.89212.83193.5591.57
12.54154.56206.8041.09
22.44132.22209.7019.73
42.51167.52201.6110.43
75Late solid–fluidic state0.251.860.202.151.65
0.53.530.744.671.50
15.010.946.611.06
26.671.848.690.71
410.665.7712.990.57
1000Fluidic state0.2518.760.141.841.18
0.524.900.292.450.79
126.810.432.640.43
227.210.942.560.22
428.161.452.990.13
Note: F is the input oscillatory frequency.
Table 3. Values of the yield stresses and critical strains of the sample subjected to an oscillatory shear load with different frequencies.
Table 3. Values of the yield stresses and critical strains of the sample subjected to an oscillatory shear load with different frequencies.
Frequency (Hz)τos (Pa)εos (%)τof (Pa)εof (%)W (%)
0.253.0300.28018.7591014.7501014.47
0.53.1330.31121.757417.571417.26
12.9600.31520.850313.666313.351
23.0110.31219.511235.912235.600
43.0730.31619.647177.536177.220
Note: τos and τof are the static yield stress and fluidic yield stress of the Zhoushan#1 sample under oscillatory shear load, respectively. εos and εof are the static critical strain and fluidic critical strain of the Zhoushan#1 sample under oscillatory shear load, respectively. W represents the width of the solid–fluidic stage: W = εof − εos.
Table 4. Values of G∞′/G0′ of the Zhoushan#2 sample after being pre-sheared using two oscillatory shear loads with two pre-shear durations.
Table 4. Values of G∞′/G0′ of the Zhoushan#2 sample after being pre-sheared using two oscillatory shear loads with two pre-shear durations.
Sample IDτ0 (Pa)F (Hz)Pre-Shear DurationG∞′/G0′V
Zhoushan#2400.1500 s0.668/
0.250.634
0.50.568
10.529
20.778
4500 s1.3550.009
Min1.346
8500 s1.7770.042
Min1.819
10500 s1.9250.315
Min1.610
1000.1500 s1.314/
0.251.091
0.51.055
10.748
20.405
40.925
8500 s1.4610.437
Min1.024
10500 s1.8690.848
Min1.021
Note: “500 s” represents the pre-shearing time of 500 s, and “Min” represents the time required for pre-shearing to the lowest viscosity. F is the input oscillatory frequency. V represents the difference of G∞′/G0′ at the above two pre-shear durations.
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MDPI and ACS Style

Gu, H.; Wang, S.; Su, Z.; Xia, Y. The Influence of Oscillatory Frequency on the Structural Breakup and Recovery of Coastal Mud. J. Mar. Sci. Eng. 2023, 11, 2073. https://doi.org/10.3390/jmse11112073

AMA Style

Gu H, Wang S, Su Z, Xia Y. The Influence of Oscillatory Frequency on the Structural Breakup and Recovery of Coastal Mud. Journal of Marine Science and Engineering. 2023; 11(11):2073. https://doi.org/10.3390/jmse11112073

Chicago/Turabian Style

Gu, Hengye, Shaohua Wang, Zhitong Su, and Yuezhang Xia. 2023. "The Influence of Oscillatory Frequency on the Structural Breakup and Recovery of Coastal Mud" Journal of Marine Science and Engineering 11, no. 11: 2073. https://doi.org/10.3390/jmse11112073

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