1 Introduction

Acoustic emission (AE) is relatively high-frequency (i.e. typically non-audible) elastic stress waves that propagate through materials surrounding the generation source. AE is generated when a material undergoes irreversible changes in its internal structure, which can result from crack formation, ageing, temperature gradients, or external mechanical forces. AE has had successful application in a range of domains to detect, locate, and characterise damage or deformation, for example, structural health monitoring, quality control, system feedback, and process monitoring [2, 19, 62]. AE monitoring offers the potential to sense particle-scale interactions that lead to macro-scale responses of granular materials; however, AE is seldom used in geotechnical engineering because of limited understanding of the fundamental links between particle-scale mechanisms and AE generation, and hence, there are challenges in the interpretation of measured AE.

Fundamental laboratory studies on the AE behaviour of soils carried out in the 1970s, 1980s, and 1990s (e.g.) [25, 26, 41, 45, 54, 55] led to the following qualitative conclusions: well-graded soils generate more AE than uniformly graded soils; angular particles generate more AE than rounded particles; AE amplitude increases with particle size; higher imposed stresses generate greater AE activity; and AE activity increases with imposed strain rate. More recent laboratory studies have advanced this early work to quantify empirical relationships between AE activity and imposed stress level, strain rate, initial relative density, stress history, and particle size and shape (e.g. [30,31,32, 38, 46, 47, 51]). In addition, AE activity during bond breakages of cemented soils has been investigated [28].

The focus of AE research in geotechnical engineering over the past 60 years has been to quantify links between measured AE and soil strength and deformation behaviour to enable interpretation of the field performance of geotechnical infrastructure assets. Recent advances have been made in the interpretation of soil/structure interaction behaviour from AE measurements using physical modelling and field experiments. For slope monitoring applications, relationships between measured AE and slope deformation behaviour have been quantified, enabling early warning of accelerating slope movements and failure [13, 40, 48,19,20,, 49]. For pile monitoring applications, relationships between measured AE and load–displacement behaviour are developed [34,35,36]. For buried pipe monitoring, relationships between AE activity and pipe displacement in uplift and differential ground motion experiments have been established [50, 52].

Despite these significant developments in AE monitoring in the laboratory and field, there remains a paucity of understanding of the links between fundamental mechanisms at the particulate scale that lead to AE activity at the macro-scale, which is measured in laboratory experiments and geotechnical systems in the field. Addressing this gap in knowledge is the focus of the present study. A suite of AE generation mechanisms in soil bodies and soil/structure systems has been proposed: particle–particle interactions; soil–structure interface friction; particle contact network rearrangement (e.g. release of contact stress and stress redistribution as interlock is overcome and regained); degradation at particle asperities; particle crushing; and structure surface degradation [25, 38, 47, 51].

The objective of this study was to establish links between particulate-scale energies and AE activity measured at the macro-scale in experiments. To achieve this, a programme of 3D DEM simulations was performed on granular soil/steel structure interfaces for comparison with the experimental results obtained by Smith et al. [51] for AE generation during sand/steel interface shearing. Energy partitions during the programme of DEM simulations were monitored and compared with AE measured in the physical tests. DEM was selected because it can capture the fundamental particulate-scale energies, whereas previous studies that have attempted to simulate AE generation in particulate materials have used fibre bundle models, which assume the mechanism of AE generation is the release of strain energy during stick–slip (e.g. [38, 39]). New understanding of the links between particulate-scale energies and AE generation will enable improved interpretation of AE measurements made in the laboratory and field and will underpin the development of theoretical and numerical approaches to model and predict AE behaviour in particulate materials.

2 DEM simulations of soil/structure interaction

2.1 Experimental measurements

Smith et al. [51] performed a programme of 14 large direct shear tests on Leighton Buzzard sand/steel interfaces with AE measurement. These tests comprised one-dimensional (1D) compression, load–unload–reload (LUR) compression, direct shear at constant displacement rate (CDR) and accelerating displacement rate (ADR), and shearing displacement cycles (SDC). The results showed that AE generation was influenced by the normal effective stress, mobilised shearing resistance and shearing velocity. In addition, compression-induced AE activity was negligible until the current stress conditions exceeded the maximum that had been experienced in the past: that is, granular soil/steel systems experience the Kaiser effect in compression. The Kaiser effect is a phenomenon, observed in rock mechanics [27], whereby AE activity is negligible until the current stress level exceeds the previous maximum, that is, AE activity is principally generated during plastic deformation and is influenced by stress history. The results from these experiments were used in this study to: (1) calibrate the mechanical response of the DEM simulations and (2) establish links between the measured AE in experiments with energy dissipation in DEM. In the experiments performed by Smith et al. [51], a Wille Geotechnik automatic large (300 × 300 mm constant cross-sectional area) direct shear apparatus (ADS-300) was used and configured for soil/steel interaction testing with an AE sensor coupled to the steel plate as shown in Fig. 1. Ring-down counts (RDC) are the AE units presented in the experimental results in Smith et al. [51]. RDC per unit time are the number of times the AE waveform crosses a programmable threshold level (set to 0·01 V) within a predefined time interval and are a measure of the AE signal energy.

Fig. 1
figure 1

Schematic diagram of the Wille Geotechnik automatic large (300 × 300 mm cross-sectional area) direct shear apparatus (ADS-300) configured for soil/steel interaction testing with an AE sensor coupled to the upper surface of the steel plate (used in experiments by and modified after Smith et al. [16,50,])

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2.2 Simulation methodology

The DEM simulations of soil/steel structure interaction were performed using the commercial code PFC3D [22], which adopts the discrete element method proposed by [11]. The model of the direct shear apparatus was prepared by generating six rigid planar walls, all of which have three degrees of freedom so that a servo-mechanism can be used to apply or maintain a target stress. The top and bottom walls have dimensions 120 mm × 120 mm and the lateral walls have dimensions 120 mm × 30 mm, forming a box with an internal cuboid space with dimensions 100 mm × 100 mm × 25 mm (i.e. the DEM model had a length scale factor of 1/3 compared to the experiment), as shown in Fig. 2. Because soil particle size and size distribution influence AE behaviour, the simulated particles were generated to replicate the particle size distribution of the Leighton Buzzard sand (LBS) used in the experiments by Smith et al. [51] (Fig. 3). Spherical particles were used with calibrated particle interaction parameters to ensure the DEM soil followed trends in behaviour comparable to LBS in the experiments. Table 1 provides a summary of the geometry of the direct shear box and soil particles used in the simulations in this study and in the experiments carried out by [51].

Fig. 2
figure 2

DEM model configuration for direct shear of soil/structure interaction

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Fig. 3
figure 3

Particle size distributions of the LBS used by Smith et al. [51] in direct shear experiments and the DEM soil used in simulations in this study

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Table 1 Geometry details of the experiments and DEM simulations
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Each DEM simulation comprised sample preparation to a target initial void ratio, one-dimensional compression to a target normal stress, and then shearing under constant normal stress. Uniform and relatively dense DEM samples were achieved by applying isotropic compression to 1 kPa with zero interparticle friction. The interparticle friction and particle–wall friction coefficients were subsequently increased to 0.42 and 0.30, respectively. This procedure, which has been widely used in DEM to obtain medium dense and dense samples (e.g. [43, 56]), led to initial void ratios that were equivalent to an initial relative density of approximately 75% for the LBS used in the experiments. The normal or isotropic stresses were imposed using the servo-mechanism until a tolerance (defined as \({\text{(required}}\;{\text{ stress}} - {\text{target}}\; {\text{stress}}/{\text{target}}\;{\text{ stress}} < 0.001\)) was reached. After equilibrium was established, one-dimensional (1D) compression was applied to a target normal stress.

The 1D compression stage was applied by moving the top wall downwards at a constant velocity of 0.5 mm/s until the target normal stress was achieved. To reduce the stress variance at the onset of wall movement, the velocity was increased in increments until the target velocity was reached and then kept constant. Load–unload–reload compression tests were performed in specific simulations by applying downward–upward–downward velocity cycles to the top wall in the z-direction (the cartesian coordinate system is adopted, as shown in Fig. 2) while keeping the lateral and base walls fixed.

The shearing stage was performed by applying a constant or accelerating shear velocity in the positive y-direction to the base wall (i.e. steel plate) while keeping lateral walls fixed and maintaining constant normal stress with the servo-mechanism. In this study, the horizontal shear stress at the soil–steel plate interface (i.e. at the bottom of the soil sample) is defined as the shear stress, and the displacement of the base wall (i.e. steel plate) is defined as the shear displacement, which is consistent with the shear stress and displacement measurements in experiments conducted by Smith et al. [51]. Shear displacement cycles were performed in specific experiments by applying positive and negative shear velocity cycles to the base wall in the y-direction. Table 2 provides a summary of the simulations performed in this study.

Table 2 Summary of the simulation programme performed in this study
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2.3 Contact model

A programme of preliminary simulations was performed with two contact models: (1) the linear model based on [11], comprising linear elastic normal and shear stiffness, normal and shear dashpots with viscosity, and frictional shear, and (2) the rolling resistance linear model based on [1, 11, 58], which is a modified linear model that incorporates torque to resist rolling particle motion. The results demonstrated that rolling motion and hence rolling friction were negligible in comparison with frictional slip for the mechanism under investigation, that is, sand deforming against a smooth steel plate (i.e. minimal interlocking at the interface). For example, the incremental frictional slip energy was between four and six orders of magnitude greater than rolling friction energy. Moreover, the linear model significantly improved the mechanical response (e.g. shear stress and volumetric strain) calibration compared to the rolling resistance linear model. The linear model was used for the main simulation programme because rolling was shown to be insignificant for the mechanism under investigation.

2.4 Calibration of parameters

Table 3 provides a summary of the DEM model parameters used in this study. Initial values for parameters were selected from the literature to represent the real soil and interface, for example, LBS particle density, LBS interparticle friction and LBS/steel interface friction from [47, 51, 58], and stiffness and damping values recommended in [42]. An iterative process followed whereby the mechanical stress–strain response of the DEM model outputs were compared with the experimental results in Smith et al.[51], and values for parameters were systematically modified to improve the match in mechanical behaviour between the simulation and the experiment.

Table 3 Summary of DEM parameters used in this study
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Spherical particles were adopted over real particle shapes in this study for a series of reasons: computational efficiency; lack of consensus in the literature on the influence of particle shape on AE generation (e.g.) [25, 47, 46]; all experiments used the same soil and hence particle shape was kept constant between tests; the mechanism was sand deforming against a smooth steel plate with negligible interlocking; and friction (particle–particle and particle-boundary) parameters were adopted to represent the real LBS and interface behaviour.

2.5 Energy calculations

The goal of this study was to establish links between energy dissipation processes at the particulate scale in DEM with AE activity at the macro-scale in experiments. To achieve this, energy partitions were traced in each simulation. The increment of external energy applied to the system, \({\textit d}{\mathrm{W}}_{\mathrm{external}}\), is the sum of two components (Eq. (1)): the increment of boundary energy (i.e. work done transmitted to the system across its boundaries, which was generated by the motion of the rigid walls in this study), \({dW}_{\mathrm{boundary}}\); and body work, which is generated by gravitational acceleration, \({dW}_{\mathrm{body}}\):

$${d}{\mathrm{W}}_{\mathrm{external}}={dW}_{\mathrm{boundary}}+{dW}_{\mathrm{body}}$$
(1)

The external energy, \({\textit d}{\mathrm{W}}_{\mathrm{external}}\), is converted to components of internal energy in the DEM simulation: change in elastic strain energy stored in linear springs, \({dE}_{\mathrm{strain}}\); change in translational kinetic energy (particle motion), \({dE}_{\mathrm{kinetic}}\); energy lost in frictional sliding when particle contact frictional strength is exceeded, \({dE}_{\mathrm{friction}}\); and energy lost in damping due to the rearrangement of particles, \({dE}_{\mathrm{damping}}\) [4, 8, 20, 24, 42, 57, 59]. Applying the first law of thermodynamics to these components results in Eq. (2):

$${\textit d}{\mathrm{W}}_{\mathrm{external}}={dW}_{\mathrm{boundary}}+{dW}_{\mathrm{body}}={dE}_{\mathrm{strain}}+{dE}_{\mathrm{kinetic}}+{dE}_{\mathrm{friction}}+{dE}_{\mathrm{damping}}$$
(2)

For any loading or unloading increment, \({dE}_{\mathrm{friction}}\) and \({dE}_{\mathrm{damping}}\) must be positive by definition, hence they can be combined as plastic dissipated energy:

$${dW}_{\mathrm{dissipation}}={dE}_{\mathrm{friction}}+{dE}_{\mathrm{damping}}$$
(3)

This energy dissipation term, \({dW}_{\mathrm{dissipation}}\), is analogous to ‘plastic energy’ used by others [57, 20]. Frictional slip and damping energy are irrecoverable (i.e. plastic) and hence dissipated from the system. Energy converted to sound (i.e. AE) must be dissipated (lost) from the system and hence this energy dissipation (or plastic energy) that combines frictional slip and damping is compared with AE in this study. This is superficially analogous to irrecoverable (plastic) work in conventional soil plasticity. Particle crushing and damage are excluded from the DEM simulations because of the relatively low normal stresses and observations in the experiments (including sample particle size distributions before and after experiments) revealed negligible particle crushing. However, if included, particle damage would be an additional component of dissipated plastic energy, i.e. bond energy [57]. Particle damage has been shown to generate high levels of AE in experiments ([33, 37]). The authors’ hypothesis was that \({dW}_{\mathrm{dissipation}}\) would be strongly correlated with AE behaviour, given that energy converted to sound must be dissipated (lost) from the system, and hence irrecoverable (plastic). The sum of friction and damping energy, \({dW}_{\mathrm{dissipation}}\), is referred to as ‘AE related energy’ in the following sections.

The increment of input energy transmitted to the system across its boundaries by the motion of rigid walls is calculated as the product of the force acting on each wall, \({F}_{\mathrm{wall}}\), and the incremental displacement of each wall in a timestep \(\Delta t\), \(\Delta {x}_{\mathrm{wall}}^{\Delta t}\), which is given by:

$${\textit d}{\mathrm{W}}_{\mathrm{boundary}}=\sum {F}_{\mathrm{wall}}\Delta {x}_{\mathrm{wall}}^{\Delta t}$$
(4)

The incremental body work done by gravity in a timestep \(\Delta t\) is calculated by:

$${\textit d}{\mathrm{W}}_{\mathrm{body}}=\sum {m}_{\mathrm{particle}}g\Delta {x}_{\mathrm{particle}}^{\Delta t}$$
(5)

where \({m}_{\mathrm{particle}}\) is the mass of a particle; \(\Delta {x}_{\mathrm{particle}}^{\Delta t}\) is the displacement of a particle in a timestep. According to [22], for the adopted linear model, the absolute form of elastic strain energy stored at contacts is updated as:

$${E}_{\mathrm{strain}}=\frac{1}{2}\left(\frac{{(F}_{n}^{l}{)}^{2}}{{k}_{n}}+\frac{{\Vert {F}_{s}^{l}\Vert }^{2}}{{k}_{s}}\right)$$
(6)

where \({F}_{n}^{l}\) is the linear normal force, \({F}_{s}^{l}\) is the linear shear force, and \({k}_{n}\) and \({k}_{s}\) are the normal and shear stiffness at the contacts, respectively. Due to the quasi-static nature of the simulation, kinetic energy, \({dE}_{\mathrm{kinetic}}\), can be negligible in comparison with other energy partitions (e.g. Figure 4). The frictional slip energy is updated as:

Fig. 4
figure 4

Error in energy balance as a percentage of external work: a plotted against time for 1D compression; b plotted against shear strain for direct shear with constant shearing velocity; c plotted against shear strain for direct shear with increasing velocity; and d plotted against shear strain for cyclic shearing

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$${E}_{\mathrm{friction}}:={E}_{\mathrm{friction}}-\frac{1}{2}(\left({F}_{s}^{l}{)}_{0}+{F}_{s}^{l}\right)\cdot \Delta {\delta }_{s}^{\mu }$$
(7)

where \({({F}_{s}^{l})}_{0}\) is the linear shear force at the beginning of the timestep and \(\Delta {\delta }_{s}^{\mu }\) is the relative slip shear displacement, ‘: = ’ is the assignment operator. The kinetic energy at any timestep is calculated as:

$${\mathrm{E}}_{\mathrm{kinetic}}=\frac{1}{2}{{m}_{i}v}_{i}^{2}$$
(8)

The local damping energy used in PFC3D is similar to that described in [10], which removes kinetic energy twice per cycle, and at any timestep is calculated as:

$${E}_{\mathrm{damping}}=2\left[\frac{1}{2}(\frac{{m}_{i}}{1-\beta }-\frac{{m}_{i}}{1+\beta }){v}_{i}^{2}\right]$$
(9)

where \({m}_{i}\) and \({v}_{i}\) are generalised mass and velocity of particles, respectively, and \(\beta\) is a damping constant, which is taken as 0.7 as it is a commonly employed value to achieve quasi-static conditions [9, 42].

3 Interpretation of results

3.1 Energy balance

It is important to confirm that all energy partitions are balanced in DEM simulations to avoid problems with numerical instability. A generally accepted criterion for energy balance in DEM simulations is that the difference between the total external energy and total internal energy is within a tolerance. Here, the percentage error in energy balance (\(D\)) was calculated using a similar equation to that employed by [20]:

$$\begin{aligned} D(\%)=100 \times \frac{{dW}_{\mathrm{boundary}}+{dW}_{\mathrm{body}}-(d{E}_{\mathrm{strain}}+{dE}_{\mathrm{kinetic}}+{dE}_{\mathrm{friction}}+{dE}_{\mathrm{damping}})}{{dW}_{\mathrm{boundary}}+{dW}_{\mathrm{body}}}\end{aligned}$$
(10)

The incremental form is used here to eliminate the effects of initial energy stored in contacts due to normal or isotropic compression. Figure 4 shows the percentage error in energy balance against simulation time or shear strain. At the beginning of 1D compression, relatively high errors with a maximum value of approximately 6% can be observed. The samples were prepared by isotropic compression with zero interparticle friction coefficient, which was then increased to 0.42 for 1D compression and subsequently shearing. According to Bernhardt et al. [3], when this sample preparation method is employed, the interparticle contacts which were sliding, or were about to slide, are suddenly artificially stabilised as the frictional sliding limit at these contacts was enhanced. Excluding this artefact of the method used for sample preparation, the average percentage error was negligible in each case: 0.25% in 1D compression, 0.02% in direct shear with a normal stress 300 kPa, 2.7 × 10–4% for accelerating shearing velocity, and 2.4 × 10–7% for shear displacement cycles.

3.2 Compression

The 1D compression simulations terminated when a target normal stress (75, 150, 225 or 300 kPa) was reached. Figure 5 shows the results from the simulation of 1D compression with a target normal stress of 225 kPa. It can be seen from Fig. 5a that, following the onset of top wall displacement, the normal stress initially increased slowly, which was followed by a greater rate of normal stress increase with time. This is because the soil contracts and the void ratio reduces (coordination number increases) with increments of top wall displacement. It is notable that 1D compression in the DEM simulations was displacement rate-controlled (i.e. constant downward velocity of the top wall), whereas 1D compression in the experiments by Smith et al. [51] was stress controlled.

Fig. 5
figure 5

Simulation results from 1D compression tests to 225 kPa normal effective stress. a vertical displacement and normal stress plotted against time; b cumulative energy partitions plotted against time; c cumulative damping energy, body energy and kinetic energy partitions plotted against time; and d cumulative AE RDC (experimental data) and dissipated energies (simulation data) plotted against normal stress

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Figure 5b shows the cumulative energy partitions plotted against time during 1D compression to 225 kPa. The majority of the external boundary energy resulted in elastic strain energy and frictional slip energy. The external body energy was negligible in comparison with the external boundary energy. Figure 5c focuses on the damping, body and kinetic energies, which are small in Fig. 5b. Figure 5d compares the DEM simulation results from the 1D compression test (dissipated plastic energy, which is the sum of friction and damping energies) with the experimental results in Smith et al. [51] (cumulative AE RDC), both of which were to a target normal stress of 225 kPa. The cumulative dissipated plastic energy versus normal stress relationship from the simulation is comparable to the cumulative AE versus normal stress relationship measured in the experiment.

Space precludes inclusion of time-series measurements from all compression simulations; however, the general trends in behaviour were the same, and results from all compression simulations were compared in relation to the change in normal effective stress in Fig. 6. Figure 6 shows that the total dissipated plastic energy (simulation) and total generated AE (experiment) both follow a comparable relationship with the change in normal effective stress during compression. The power relationships shown in Fig. 6 best describe the data because both normal effective stress and density (i.e. number of particle contacts per particle) increase during compression, having a combined effect on energy dissipation and AE generation (as described in [47, 51]).

Fig. 6
figure 6

Results from 1D compression: total energy dissipation (friction and damping) plotted against the change in normal effective stress (simulation data) and total generated AE (RDC) plotted against the change in normal effective stress (experimental data)

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The Kaiser effect [27] has been observed in soil bodies and soil/structure systems during load–unload–reload (LUR) compression experiments [47, 51], whereby AE activity is negligible until the current stress conditions exceed the maximum that has been experienced in the past. In this study, LUR compression was performed in DEM by moving the top wall downwards (loading), upwards (unloading) and then downwards (reloading) at a constant velocity to target stress levels. Figure 7 shows 1D LUR compression simulation results in comparison with the experimental measurements. Figure 7a shows vertical strain (ratio of the vertical displacement of initial sample height) plotted against normal stress. Figure 7b shows the cumulative energy partitions plotted against normal stress (simulation) as well as the cumulative AE measured in the experiment by Smith et al. [51]. Figure 7c focuses on the damping, body and kinetic energies, which are negligible in Fig. 7b. Figure 7d compares the cumulative dissipated plastic energy in the simulation with the measured cumulative AE in the experiment. Figure 7e shows selected energy partitions and cumulative AE, normalised by their maximum value, plotted against normal stress and on logarithmic scales. These plots highlight that specific energy partitions exhibit behaviour comparable to the AE Kaiser effect, whereby energy is negligible until the previous maximum stress has been exceeded. Specifically, the cumulative dissipated plastic energy in Fig. 7d exhibits comparable behaviour to the cumulative AE (i.e. negligible energy in unloading–reloading cycles until the previous maximum stress is exceeded).

Fig. 7
figure 7

Results from load–unload–reload (LUR) 1D compression: a vertical strain plotted against normal stress; b cumulative energy partitions (simulation data) and cumulative AE RDC (experimental data) plotted against normal stress; c damping, kinetic and body energy plotted against normal stress; d cumulative friction and damping energy (dissipated energy) and cumulative AE RDC (experimental data) plotted against normal stress; e normalised cumulative energy partitions (simulation data) and cumulative AE RDC (experimental data) plotted against normal stress on logarithmic scales; and f conceptual normal compression line and unloading–reloading lines for energy partitions with and without the Kaiser effect

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Normalised energy partitions \(({\text{current}} \,{\text{cumulative}} \,{\text{energy}}/{\text{maximum}} \,{\text{cumulative}} \,{\text{energy}})\) were plotted against normal stress on logarithmic scales in Fig. 7d because equations can be established to describe these relationships in a bilinear form (as shown in Fig. 7f):

$${\mathrm{Loading}} \, {\mathrm{on}} \, {\mathrm{normally}} \, {\mathrm{consolidated}} \, {\mathrm{soil}}: E={e}^{a\mathrm{ln}{\sigma }_{n}+b}$$
(11)
$${\mathrm{Loading}} \, {\mathrm{and}} \, {\mathrm{unloading}} \, {\mathrm{on}} \, {\mathrm{overconsolidated}} \, {\mathrm{soil}}: E={e}^{c\mathrm{ln}{\sigma }_{n}+d}$$
(12)

where E represents normalised energy partition, \(a\) and \(c\) are the gradients of these lines, and \(b\) and \(d\) are the intercepts at \({\sigma }_{n}=1\) kPa, all of which can be calibrated from experimental results or numerical simulation. It is notable that elastic strain energy is recoverable while plastic energy partitions are irrecoverable. As a result, \(a\) is always greater than \(c\), and \(c=0\) for the Kaiser effect. At the onset of plastic deformation (yielding), the slope of the unloading–reloading line, \(c\), changes to \(a\), which is consistent with the experimental observations of Smith et al. [51].

3.3 Shearing

3.3.1 Stress level

Figure 8 compares the simulation and experimental results from soil/steel interface shearing under different normal stress levels. Figure 8a shows shear stress plotted against shear strain (ratio of shear displacement over sample length in shear direction) for normal stresses of 50, 150, 200 and 300 kPa. The simulation and experimental results exhibit comparable behaviour, whereby the shearing resistance rapidly mobilised and then remained relatively constant thereafter. This shear stress against shear strain behaviour is characteristic of interface shear between steel and granular soil (e.g.) [21]. Generally, the initial stiffness and shearing resistance at small and large shear strains in the simulations were comparable to the experimental results. Figure 8e shows the large strain interface friction coefficients obtained from the simulations and experiments, which have good agreement and demonstrate that the simulations were able to replicate the mechanical behaviour. The volumetric strain versus shear strain relationships in Fig. 8b capture the transition in behaviour following the initial mobilisation of shearing resistance, and the total volumetric strain in the simulations and experiments has the same order of magnitude. The experimental results all show contractive behaviour, whereas the simulations at the lower normal stresses (50 and 150 kPa) exhibit modest dilative behaviour.

Fig. 8
figure 8

Comparisons between simulation results from direct shear under constant normal effective stresses of 50, 150, 200 and 300 kPa and experimental data from Smith et al. [51]: a shear stress plotted against shear strain; b vertical strain plotted against shear strain; c simulation cumulative energy partitions under a normal stress of 200 kPa; d AE rate (experimental data) and dissipated energy rates (simulation data) plotted against shear strain; e average shear stress (taken after 0.05 of shear strain) plotted against normal effective stress; and f average AE rate (experimental data) and dissipated energy rates (simulation data) (taken after 0.05 of shear strain) plotted against normal effective stress

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The differences in volumetric strains measured in the experiments and output from DEM are consistent with other DEM studies in the literature (e.g. [12, 16,17,18, 53,50,51,52,53,54,, 60, 61]). Zhang et al. [60] attribute the discrepancy in volumetric strains to differences in particle size distribution and grain shape between the simulation and experiment. In this study, the real particle size distribution was captured in the simulation and so the differences in volumetric strains can be principally attributed to differences in particle shape. The total vertical displacements measured in the direct shear experiments, by LVDTs connected to the top plate above the sample, were typically less than a millimetre and the calculated volumetric strains are then significantly influenced by the initial sample height measurements, which have an accuracy to within a few millimetres. Moreover, soil extrusion in the direct shear experiments causes overestimations of contractive volumetric strains; soil extrusion was significant in the cyclic shearing tests (successive forward and reverse shearing displacements), which was not captured in simulations. The range of volumetric strains observed in this study (DEM and the experiments) were small, typically between 0.1% and 1.0% in shearing, because the mechanism was sand deforming against a smooth steel plate. The volumetric strains generated in DEM and measured in the experiment are within the same order of magnitude; the experimental volumetric strains were on average larger than the DEM volumetric strains by a factor of 2 (excluding the cyclic shearing test, which experienced significant extrusion).

Figure 8c shows a representative plot of energy partitions against shear strain, in which the body and kinetic energies are excluded because they are negligible. The boundary and strain energies do not start at zero because of the accumulated energy from the previous normal compression stage. The strain energy initially increases during mobilisation of shearing resistance and then remains relatively constant thereafter. In contrast, the cumulative friction energy continues to increase throughout shearing at small and large strain. Figure 8d compares the dissipated plastic energy rate (rate of AE related energies, which is the sum of friction and damping energies) in the simulation with the measured AE rates in the experiment. The dissipated plastic energy rate and measured AE rate trends were comparable to the mobilised shearing resistance trends. An increase in effective normal stress caused a proportional increase in dissipated plastic energy rates and measured AE rates, which is highlighted in Fig. 8f where the average large strain (calculated following full mobilisation of shearing resistance) dissipated plastic energy rates (simulation) and AE rates (experiment) are plotted against the applied normal stress, which results in a linear relationship with strong correlation (R-squared 0.995 for dissipated plastic energy rate). Dissipated plastic energy rates and AE rates increase with normal effective stress because greater interparticle contact stresses develop, and hence, more work is required to displace particles relative to each other. These results highlight that dissipated plastic energy (friction and damping energy) is correlated with AE generation in soil/steel interface shearing.

In Fig. 8d, the AE trends for 50, 150, and 200 kPa show a marginal peak in AE activity at full mobilisation of shear strength (most noticeable at 200 kPa) and then oscillate around a constant mean value thereafter, whereas the incremental plastic dissipated energy does not exhibit this marginal peak. This difference in behaviour is likely due to artefacts with AE measurement in the experiment (data acquisition details reported in [51]) that are not present in the simulation, for example, AE is attenuated as it propagates from different parts of the sample to the sensor location; AE amplitude and frequency characteristics evolve with shear strain [47]; the sensor has a dominant frequency response, which introduces signal bias; and a filtering system was used to remove signals outside of 10–100 kHz, which is important for excluding extraneous noise but could result in loss of soil/structure-generated AE outside of this frequency range.

Figure 9 shows the typical evolution of the change in energy partitions as a proportion of the change in input boundary energy with shear strain (the body energies were negligible as discussed earlier and hence are not considered here). At the onset of shearing, the input boundary energy is primarily stored as elastic strain energy. During shear strength mobilisation, the proportion of elastic strain energy reduces while the plastic energy dissipated in frictional sliding increases. As shearing continues following full mobilisation of shearing resistance, friction energy dominates during plastic strain (average percentage of 96.95% after shear strain reached 0.05) while elastic strain energy reduces to a very small value (average percentage of 2.99% after shear strain reached 0.05). Figure 9 shows that friction energy and elastic strain energy are inversely proportional, and the friction energy versus shear strain relationship is analogous to the AE rate versus shear strain relationships shown in Fig. 8d.

Fig. 9
figure 9

Percentage change in energy partitions over the change in input boundary energy plotted against shear strain in direct shear under a normal stress of 200 kPa

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Table 4 provides a summary of the change in energy partitions as a proportion of the change in input boundary energy at large strain under different normal stress levels. Following full mobilisation of shearing resistance, an average of approximately 96% of the input boundary energy was dissipated by frictional sliding. This, combined with Fig. 9, confirms that the relatively constant AE rates observed following shear strength mobilisation in soil element experiments conducted by [46], and [31] occurs principally due to frictional sliding. It is notable that although the magnitude of damping energy is small compared to friction energy, it also exhibits comparable behaviour to, and correlates with, AE generation (Fig. 7c). The change in energy partitions as a proportion of the change in input boundary energy was not significantly influenced by stress level for the range of normal effective stresses investigated in this study.

Table 4 Assessment of average energy proportions (%) at large shear strain
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3.3.2 Shearing velocity

Figure 10 shows the results from a simulation and experiment where stepped increases in shear velocity were imposed during the tests to investigate the dissipated plastic energy (simulation) and AE activity (experiment) response to accelerating deformation behaviour, which is typical of progressive failure processes. Figure 10a shows the stepped shear velocity stages imposed in simulation and in the experiment: the same trend was imposed in the simulation, but the velocities were increased by a factor of 10 due to computational constraints. Figure 10b shows the shear stress and vertical strain versus shear strain relationships from the simulation and experiment. Figure 10c compares dissipated plastic energy rates (AE related energy rate) from the simulation with measured AE rates (RDC/min) from the experiment against shear strain, both of which exhibit a relationship comparable to the imposed velocity versus shear strain relationship.

Fig. 10
figure 10

Results from simulation of increasing shear velocity under a normal stress of 150 kPa, together with experimental data from Smith et al. [51]: a shear velocities plotted against shear strain; b shear stress and vertical strain plotted against shear strain; c AE rate (experimental data) and dissipated energy rate (simulation data) plotted against shear strain; and d average AE rate (experimental data) and dissipated energy rate (simulation data) plotted against shear velocity

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Average dissipated plastic energy rate (simulation) and measured AE rate (experiment), calculated during each constant shear velocity stage, are plotted against shear velocity in Fig. 10d. A polynomial function best described the measured AE rate versus shear velocity relationship reported in Smith et al.[51]. However, there is no physical reason for the rate of increase in AE rates to reduce with increasing velocity: this relationship obtained from the experiments can be attributed to artefacts within the AE measurement system (e.g. frequency limits due to the band-pass filter, sampling rate limits, comparator voltage threshold used to compute RDC). A linear function best describes the dissipated plastic energy rate versus shear velocity relationship obtained from the simulations, which does not suffer from artefacts in an AE measurement system. Dissipated plastic energy rates and AE rates should increase linearly with imposed velocity because the number of particle interactions occurring per unit time should increase linearly proportionally with the imposed velocity, and hence, the linear relationship obtained in this study best represents the real behaviour.

3.3.3 Shear displacement cycles

Figure 11 shows results from the simulation of shear displacement cycles, together with experimental data from Smith et al. [51]. Figure 11a shows the amplitudes of displacement cycles in the simulation and experiment. The simulation time was 1/30th of the experiment because the shearing length in the simulation was 1/3rd of the experiment and the shearing velocity in the simulation was 10 times greater than the experiment. Figure 11b and c presents the shear stress and vertical strain plotted against shear strain. The evolution of shear stress with time from the simulation shows behaviour comparable to the experimental results. Although dilative behaviour took place during the shear strength mobilisation phases of each displacement cycle, the specimens in the simulation and experiment both experienced net volume reduction. The total vertical strain was greater in the experiment, which can be attributed to differences in particle shape between the simulation and experiment, and some extrusion of sand during shearing in the experiment.

Fig. 11
figure 11

Comparison between simulation results from shear displacement cycles under a constant normal effective stress of 150 kPa and experimental data from Smith et al.[51]: a shear strain plotted against time; b shear stress plotted against time; c vertical strain plotted against time; d shear stress plotted against shear strain; e AE rate (experimental data) and dissipated energy rate (simulated data) plotted against shear strain; and f AE rate (experimental data) and dissipated energy rate (simulation data) plotted against shear stress. A, B, C, D and E relate to shear strain cycle amplitudes of 0.0033, 0.013, 0.03, 0.063 and 0.09, respectively

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Shear stress is plotted against shear strain in Fig. 11d, which shows that within each cycle the differences between forward and reverse mobilised shearing resistances were minimal. Figure 11e shows comparable hysteretic behaviour occurred in the dissipated plastic energy rate (simulation) and AE rate (experiment) measurements, which were defined as negative during the reverse phases of each displacement cycle to allow graphical visualisation. These dissipated plastic energy rate and AE rate measurements are plotted against shear stress in Fig. 11f. These results demonstrate that dissipated plastic energy and AE activity exhibit comparable behaviour in cyclic shearing, and AE monitoring could be used to interpret mobilised shearing resistance.

Figure 12 focuses on specific shear displacement cycles, with the annotated stages 1 to 5 highlighted in Fig. 11a. The dissipated plastic energy rate (AE related energy rate; simulation results) reduces to zero from a maximum absolute value at the transition in shearing direction (i.e. when the shearing velocity goes to zero), which is then followed by an increasing trend to the maximum absolute value during constant shearing velocity. Due to limited data points in the experiment at the transition in shearing direction (i.e. when shearing velocity goes to zero), the drop in AE rates to zero is less clear. Dissipated plastic energy rate and shear stress are only zero together at the onset of shearing in the first displacement cycle. The input energy is zero only when shearing velocity is zero, and energy is dissipated when shearing velocity not zero.

Fig. 12
figure 12

Evolution of AE rate (experimental data) from Smith et al. [51] and dissipated energy rate with shear stress during shearing displacement cycles

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4 Discussion

A programme of DEM simulations has been performed on granular soil/steel structure interfaces to establish links between energy dissipation processes at the particulate scale with AE activity at the macro-scale. Results from experiments performed by Smith et al. [51] were used to calibrate the mechanical behaviour of the DEM model and compare observed AE behaviour in physical tests with energy partitions in DEM simulations. A soil/steel large strain interface friction angle of 30.3° was obtained from the DEM results, which is consistent with mechanical sand-steel interface behaviour reported in the literature ([21, 51]).

The external and internal energy partitions in a programme of DEM simulations were monitored for comparison with the AE activity observed in direct shear experiments. These simulations and experimental tests comprised one-dimensional (1D) compression, load–unload–reload (LUR) compression, direct shear at constant and accelerating shearing velocities, and shearing displacement cycles. Comparisons between energy partitions in DEM and AE activity measured by Smith et al. [51] demonstrate that the sum of friction and damping energies (i.e. dissipated energy and irrecoverable/plastic work) strongly correlates with AE generation. Dissipated plastic energy (friction and damping energy) consistently exhibited trends in behaviour comparable to the observed AE activity under all tested conditions. This study has extended knowledge to demonstrate that particulate-scale dissipated plastic energy is influenced in the same way as measured AE activity at the macro-scale by Smith et al. [51]: unload–reload behaviour and the Kaiser effect (i.e. compression-induced energy dissipation in soil − structure systems is negligible until the current stress conditions exceed the maximum that has been experienced in the past); normal effective stress; mobilised shearing resistance; and shearing velocity.

This study has shown that the fundamental particulate-scale mechanisms of AE generation in granular media are friction between particles or at particle/structure interfaces and damping in rearrangement of particle assemblages. Comparing the magnitudes of friction and damping energy partitions demonstrated that friction is the dominant mechanism of AE generation. Particle crushing and damage were excluded from the DEM simulations in this study because of the relatively low normal stresses and observations in the experiments revealed negligible particle crushing. However, if included, particle damage would be an additional component of dissipated plastic energy, i.e. bond energy [57]. Particle damage has been shown to generate high levels of AE in experiments [33, 37]; however, DEM studies [57] show energy dissipated by particle crushing (bond energy) is small compared to friction and damping energies. Further work is required to quantify the contribution of particle damage to AE generation at the particulate scale.

Previous studies that have attempted to simulate AE generation in particulate materials have used fibre bundle models [38, 39], which assume that the mechanism of AE generation is the release of strain energy during stick–slip and model particle contacts with independent elastic mechanical elements (referred to as fibres) that have an elastic modulus and rupture threshold. This approach ignores the fundamental, particulate-scale energy dissipation mechanisms that can be observed in DEM. Studies on AE generated during wear of mechanical components concluded that the fundamental mechanism was release of strain energy during elastic asperity contact of materials [5, 15].

This study has advanced knowledge to establish the link between particulate-scale energy dissipation and AE generation. These findings are significant because this new understanding enables improved interpretation of AE measurements made in the laboratory and field and underpins the development of theoretical and numerical approaches to model AE behaviour in particulate materials. The findings have implications for a range of disciplines, including: soil mechanics, landslides (e.g.) [40], faulting and earthquakes (e.g. [23]), granular physics (e.g.) [6], additive manufacturing (e.g.) [14], mining (e.g.) [29] and pharmaceuticals (e.g.) [44].

Proportions of the friction energy (\({E}_{\mathrm{friction}}\)) and damping energy (\({E}_{\mathrm{damping}}\)) dissipated during the deformation of granular material or soil–structure interaction are converted to AE waves. Measurements of AE in the laboratory and field are affected by the AE data acquisition system and attenuation of AE energy as the AE waves propagate from the source to the sensor. A general expression that links measured AE and energy dissipation may be proposed as the following incremental form:

$${\textit d}\mathrm{AE}=\upxi {(\alpha dE}_{\mathrm{friction}}+\epsilon {dE}_{\mathrm{damping}}+\upgamma {dE}_{\mathrm{damage}})$$
(13)

where \(\alpha\), \(\epsilon ,\) and \(\upgamma\) are the proportions of friction, damping and damage energies that are converted to AE energy, respectively, and \(\upxi\) is a function that quantifies the influence of attenuation and data acquisition on the measured AE signal. It is notable that the widely used local damping in DEM is numerically convenient for reaching quasi-static conditions but not necessarily realistic. The damping energy is considered here for the completeness of the energy analysis.

For stress changes in 1D compression, Fig. 13a shows the relationship between total generated AE (experimental results) and total dissipated plastic energy (DEM outputs) (R2 of 0.996). For the example element test, mechanism, and sample volume studied, this indicates that approximately 220,000 AE RDC is generated per Joule of dissipated plastic energy in compression. Figure 13b shows the relationship between AE rate and dissipated plastic energy rate (taken as average values following full mobilisation of shear strength) in shearing (R2 of 0.967). This example indicates that per time increment, approximately 76,000,000 AE RDC is generated per Joule of dissipated plastic energy in shearing. These findings show that significantly greater RDC/J is generated in shearing compared to compression, which agrees with observations by Smith et al. [46, 47] where boundary, distortional, and volumetric work were compared with AE measurements in triaxial tests on sands.

Fig. 13
figure 13

a Relationship between total generated AE (experimental results) and total dissipated plastic energy (DEM outputs); b Relationship between AE rate and dissipated plastic energy rate (taken as average values following full mobilisation of shear strength) in CDR shearing; and c AE rate (shearing) and cumulative AE RDC (1D compression) against the ratio of dissipated energy and maximum dissipated energy

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Figure 13c plots the AE rate (shearing) and cumulative AE RDC (1D compression) against the ratio of dissipated energy and maximum dissipated energy. The normalised dissipated plastic energy is used to combine compression and shearing mechanisms on the same plot, which were shown previously to generate significantly different RDC/J, and combine both ADR and CDR test results, which have different shearing velocities. For shearing, where the incremental energy and AE rate are considered, if the values of \(\alpha\), \(\epsilon\) and \(\upgamma\) in Eq. (13) are taken as 1, 1, and 0, then a linear fit with \({R}^{2}\) of 0.955 may best fit the shearing data with \(\upxi\)=915,106. For 1D compression, where the cumulative form of AE and energy is considered, if the values of \(\alpha\), \(\epsilon\) and \(\upgamma\) in Eq. (13) are taken as 1, 1 and 0, then a linear fit with \({R}^{2}\) of 0.995 may best fit the compression data with \(\upxi\)=124,738. This then enables predicted AE RDC values to be determined based on dissipated plastic energy, or vice versa.

Previous experimental studies have shown that measured AE activity is dependent on the external input work (e.g. stress level, stress path, strain rate), the properties of particles and their contacts (e.g. particle size, shape and surface roughness), the initial soil packing fabric (e.g. void ratio, relative density) and the stress history. Future work in the form of a parametric study would enable the influence of each of these variables on energy dissipation and AE generation to be investigated and quantified, which would underpin the development of a universal framework to interpret AE measurements from element and physical model tests in the laboratory and full-scale geotechnical systems in the field.

The DEM approach employed in this investigation may not be directly applicable for studies of other mechanisms, element tests, or interfaces: this work employed the linear contact model, spherical particles and without particle damage, which was shown to be appropriate for this application. In future investigations of triaxial shearing, the authors will implement particle rolling behaviour, real particle shapes, and particle damage to replicate the real mechanical behaviour in that application.

5 Conclusion

This study has gone beyond the state of the art to establish links between fundamental particulate-scale energy dissipation processes and acoustic emission (AE) activity measured at the macro-scale. This new knowledge enables improved interpretation of AE measurements made in the laboratory and field and underpins the development of theoretical and numerical approaches to model and predict AE behaviour in particulate materials. A programme of 3D DEM simulations was performed on granular soil/steel structure interfaces, and the results were compared with measurements made in direct shear experiments, which has led to the following principal findings:

  • Dissipated energy (irrecoverable/plastic work) strongly correlated with AE generation (e.g. R2 values of 0.99).

  • Dissipated plastic energy was influenced in the same way as measured AE activity by unload–reload behaviour and the Kaiser effect (i.e. compression-induced energy dissipation was negligible until the current stress conditions exceeded the previous maximum); normal effective stress; mobilised shearing resistance; stress path; and shearing velocity.

  • The fundamental particulate-scale mechanisms that contribute to AE generation are friction and damping in particulate rearrangement, with friction being the dominant mechanism (i.e. > 95% of the total energy).

  • Relationships have been established between AE and dissipated plastic energy (R2 from 0.96 to 0.99), which show AE generated per Joule of dissipated plastic energy is significantly greater in shearing than compression

  • A general expression has been proposed that links measured AE and energy dissipation, which considers the contributions of different energy dissipation processes, as well as the influence of attenuation during the propagation of AE waves and data acquisition system configurations.

  • Future work in the form of a parametric study would enable the development of a universal framework to interpret AE measurements from element and physical model tests in the laboratory and full-scale geotechnical systems in the field.