In the process of earthquake formation, there are some deformation waves that propagate slowly in the crustal medium. The deformation wave is not an elastic wave, and its velocity is obviously lower than that of the elastic wave represented by shear wave and longitudinal wave in earthquake. Richter proposed the concept of epicenter migration wave (Richter 1985), and calculated the velocity of epicenter migration wave in earthquake zones, which is about tens of kilometers per year. As for the generation of deformation waves in the Earth’s crust, Feng (Feng 1986; Kasahara 1979) believes that it is the result of the combination of external trigger and the change of medium properties in the source area. Geng et al. (Sherman et al. 2008; Geng et al. 1990) believe that there is an interlayer composed of easily rheological media in the interior of the Earth’s rock mass, which may stimulate the propagation of slow deformation waves in the interior under certain conditions, which plays a non-negligible role in the breeding and occurrence of earthquakes. Studies on the plastic flow network and plastic flow wave in continental lithosphere show that (Wang et al. 1994; Scholz 1977; Psakhie 2001), the driving force of continental plate margin mainly realizes its remote transmission in the form of plastic flow wave through the plastic flow network in the lower lithosphere (including the lower crust and lithosphere mantle), and controls the tectonic deformation and seismic activity in the plate. For example, under the influence of the Indian plate, there are different quasi-periodic plastic deformation flow waves in the central and eastern Asian continent.

The research on the deformation and failure of solid under the action of external causes is one of the basic topics in the field of materials and engineering (Fang et al. 2022; Lo et al. 2017; Ma et al. 2023; Song et al. 2023; Kromer et al. 2018; Ding et al. 2019; 2023). In recent years, the scope of space and time scale for the study of solid deformation and failure has gradually expanded (Zhang et al. 2016), from geological scale to atomic scale, from fast loading deformation and failure to material deformation and failure at creep speed (Feng 2019; Zheng et al. 2021; Ben-Haim 1993). From seismic and volcanic eruptions on the Earth scale, to tunnel deformation and failure on the engineering scale, and to strain localization and cracking on the small specimen scale, all are actually material deformation and failure problems. If the mechanism of material deformation and failure is deeply studied, it will provide important value for the study of problems on various scales. In terms of theoretical research, literature (Qi et al. 2017) established a theoretical model of slow deformation wave transmission in deep tunnel surrounding rock by using continuous phase transition theory and Lagrange mechanics. In large-scale field observation, the characteristics of slow deformation wave are determined by measuring the significant spatiotemporal variation of geophysical field. The observation shows that the directional migration of earthquake source is a common phenomenon, such as in the Hindu Kush region (Malamud et al. 1985; 1989), Baikal region of Siberia (Sherman et al. 2008; Levina et al. 2015), and China (Wang 1987; Barannikova et al. 2010). On a small scale in the laboratory, the characteristics of slow deformation waves in specimens of potassium salt, marble and alkali metal halide crystals were studied by compression and tension tests (Zuev et al. 1997), and it was found that the properties of slow deformation waves in specimens of rock and alkali metal halide crystals were similar to those previously observed in metal materials (Zuev et al. 2012). It is of great significance to study the migration of slow deformation wave and earthquake.

Throughout the above research on slow deformation waves of earthquakes, the experimental research focuses more on homogeneous test materials such as metal or metal halide, although rock materials are mentioned, the description is not comprehensive. However, the slow deformation wave mainly propagates in the crust, and the rock is the main medium, so it is more important to study the propagation and evolution law of the slow deformation wave in the rock mass. In addition, research and analysis on Luders band and nucleation phenomenon are mostly conducted in metal materials (Mazière et al. 2015; Suna et al. 2003), but the above phenomena also exist in the compression process of rock materials. However, there are very few scholars conducting analysis and research, so relevant analysis was conducted in this study. Red sandstone has a relatively low stiffness, and compared to materials with greater stiffness such as granite rocks, it is less likely to undergo brittle failure. During the loading process, the presence of slow deformation is more readily observed at the laboratory scale. In this study, red sandstone samples were used to carry out uniaxial compression laboratory tests at three loading speeds of 0.1 mm/min, 0.5 mm/min and 1 mm/min respectively. Based on the micro-characterization of speckle photography and DIC processing method, the slow deformation wave in the small size test was quantitatively analyzed. The propagation law of strain component \(\varepsilon_{xx}\) nder three loading speeds was studied, and the propagation velocity of deformation under three loading speeds was obtained, and the relationship between it and the loading speed was analyzed; The possible formation mechanism and development process of strain localization were studied by analyzing the flow channel, nucleation phenomenon and Luders band evolution in cloud images. The above research has important guiding significance for earthquake occurrence and migration.

2.1 Sample preparation and test equipment

The DRTS-500 triaxial testing machine was used to carry out room temperature mechanical uniaxial compression test. The test equipment consisted of two parts: loading system and data acquisition system. The test samples and equipment were shown in Fig. 1. The maximum axial load of the testing machine is 500kN, and the axial pressure is generated by an external actuator. The loading mode of the triaxial testing machine has three different control modes, namely, stress control, strain control and displacement control. The purpose of this test is to analyze the deformation evolution and propagation of rock mass samples under different loading speeds, so the displacement controlled loading method is adopted.

Red sandstone sample and related test equipment

The experiment used red sandstone samples, which were processed and made from the third type of red sandstone blocks. The sample preparation follows the recommended method of ISRM (Billingsley 2001), with a length, width, and height dimension of 50 mm × 50 mm × 100 mm, each opposite face of the sample is parallel and smooth, and is made from stones extracted from the same batch in the mining site. Samples with defects and joints need to be removed.

In order to measure and record the deformation field of the surface of the specimen during loading, the test adopts the speckle microvision shooting technology, so it is necessary to pretreat the surface of the test block. The specific steps are as follows: (1) Spray a layer of white ordinary particle paint uniformly on the surface of the sample to form a coating, and then dry naturally to form a white bottom layer; (2) A layer of black granular paint is appropriately sprayed on the white coating to form random black specks (Fairhurst et al. 1999). The specific process is shown in Fig. 2.

The making process of speckle. a Protolith; b Spray with white latex paint; c Spray speckle

2.2 Test scheme

2.2.1 Uniaxial compression test

In order to determine the basic physical and mechanical parameters of red sandstone samples, uniaxial compression test was conducted on a group of rock samples before the deformation test, and the stress–strain curve of the samples was shown in Fig. 3 (The phenomenon of strain occurring at 0.3 in Fig. 3 is likely caused by a sudden internal fracture during the loading process of the rock-like material). Basic material parameters are as follows: Density ρ = 2.6 × 10³ kg/m³, Young’s modulus E = 1.3 × 10⁴ MPa, Poisson’s ratio ν = 0.25, Average compressive strength of sample σ_s = 80 MPa, Elastic compressive strain limit ε_s≈1 × 10⁻².

Stress–strain curve under uniaxial loading

2.2.2 Deformation transfer test

During the test, the displacement controlled loading method was adopted, and the samples were loaded at three loading speeds of 0.1 mm/min, 0.5 mm/min and 1.0 mm/min respectively. When the load was increased to the uniaxial compressive strength of 90%, the loading was stopped and the load was kept unchanged (for 400 s). A high-definition industrial camera (resolution 2560 × 1922) was used for data acquisition, with a sampling frequency of 0.5 Hz and a frame count of 14.2 fps.

During the test, microvision characterization technology was used to record the change process of the specimen surface deformation field. Later, DIC processing technology was used to track and record the evolution of the displacement vector field of the specimen surface points, providing data support for exploring the transfer law of the specimen strain field.

2.3 DIC technology principles

DIC is a non-contact optical measurement method, which can be used to measure the deformation state of the specimen surface, and has been widely used in the deformation experimental study of rocks and rock materials (Mao et al. 2021; Ju et al. 2019; 2020). DIC processing technology can calculate the surface deformation of the sample by comparing the relative displacement changes of the same speck in the reference image and the current image, as shown in Fig. 4.

DIC technical schematic (Mao et al. 2021)

Each component of the strain is calculated by Eq. (1):

\(\varepsilon_{xx} = \frac{\partial u}{{\partial x}}\) is the component of the strain vector in the compression direction x, \(\varepsilon_{yy} = \frac{\partial v}{{\partial y}}\) is the transverse strain component,\(\varepsilon_{xy} = \varepsilon_{yx} = \frac{1}{2}\left( {\frac{\partial u}{{\partial y}} + \frac{\partial v}{{\partial x}}} \right)\) is the shear strain component, \(\omega_{z} = \frac{1}{2}\left( {\frac{\partial u}{{\partial y}} - \frac{\partial v}{{\partial x}}} \right)\) is the rotating component, u and v are the components of the displacement vector r along the x and y axes, respectively.

3.1 The spatial distribution of strain tensor field

Deformation transfer tests were carried out on red sandstone samples at different loading rates (0.1 mm/min, 0.5 mm/min and 1 mm/min), and the axial pressure remained unchanged after rising from zero to 90% of the failure load (about 70 MPa). Figure 5 shows the variation curves of stress and strain of rock samples with time at the loading speed of 0.1 mm/mim.

Curve of stress and strain at loading end of specimen with time under axial loading (0.1 mm/min)

Hd industrial camera was used to record the deformation of the sample during the loading process, and then DIC non-contact optical method was used to process the deformation of the sample, and the strain distribution characteristics of the sample surface were obtained. For ease of explanation, the spatial distribution of surface strain of the specimen is selected at time t = 8 s and loading speed of 0.1 mm/min, as shown in Fig. 6.

Strain Tensor Deformation Distribution Map. a Loaded sample; b The longitudinal strain component is distributed \(\varepsilon_{xx}\); c Transverse strain components are distributed \(\varepsilon_{yy}\); d The shear strain component is distributed \(\varepsilon_{xy}\))

As shown in Fig. 6a, the coordinate origin is located at the top left corner of the specimen, with the specimen length direction as the x-axis and the width direction as the y-axis. In the initial loading stage, the strain fluctuations in all three directions of the specimen are small. Initially, the longitudinal strain \(\varepsilon_{xx}\) fluctuates from the loading end and gradually decreases along the x-axis, indicating a certain directional deformation transfer. The transverse strain \(\varepsilon_{yy}\) shows less obvious deformation at the beginning, with weak regularity and little variation across the entire region, indicating a lack of significant directional characteristics. The shear strain \(\varepsilon_{xy}\) shows similar magnitudes of fluctuations from the loading end to the fixed end. The deformation in all three directions exhibits non-uniform distribution in both time and space. After the deformation transfer stabilizes, the distribution of strain components (\(\varepsilon_{xx}\), \(\varepsilon_{yy}\), \(\varepsilon_{xy}\)) is shown in Figs. 6b, c, and d, respectively. (The initial loading stage figures are not displayed due to the length limitations of the article).

3.2 The transmission process of slow deformation

Figure 7 shows the planar distribution of strain in the initial loading stage, with the loading end on the right and the fixed end on the left. It can be observed that the overall deformation of the specimen surface is small in the initial loading stage, and the deformation is mainly concentrated on the loading end side. As the distance increases, the deformation gradually decreases, and at the fixed end, the deformation is nearly zero. This suggests that the deformation of the specimen is likely a gradual transfer process from the loading end to the fixed end, indicating a slow and progressive deformation propagation.

Plane distribution of strain \(\varepsilon_{xx}\) (a Elevation, b Plan view)

In order to analyze the propagation law of surface deformation in the red sandstone specimen, ensure the validity and accuracy of the test data, and speed up the processing of the test data while reducing repetitive work, three lines L1, L2, and L3 are selected on the specimen surface to represent the distribution of deformation in the specimen plane by averaging their deformation characteristics. Among them, L2 represents the centerline of the surface, while lines L1 and L3 are located 10 mm to the left and right of the centerline L2, respectively. Please refer to Fig. 6a for the specific positions.

Taking a loading speed of 0.5 mm/min as an example, after processing the micro-characterization data of the specimen, vertical strain \(\varepsilon_{xx}\) distribution maps of L1, L2, and L3 at different time points are selected from the processing results for analysis (refer to Fig. 8). By comparing the strain \(\varepsilon_{xx}\) distribution at different time points, the following observations can be made:

1. (1)

   During the loading stage, as the loading time increases, the strain amplitude gradually increases. The strain values near the loading end are generally higher than those near the fixed end.

2. (2)

   During the stress-holding stage, the strain peak near the loading end remains relatively unchanged, while the strain values near the fixed end significantly increase. Eventually, the strain distribution on the entire surface of the specimen becomes more uniform.

Deformation diagram at 0.5 mm/min

From these observations, it can be inferred that during the process of strain adjustment, deformation propagates from high strain regions (high energy) to low strain regions (low energy).

In previous studies on alloy deformation, it was found that longitudinal strain \(\varepsilon_{xx}\) propagates from the loading end to the fixed end during the loading process, and the propagation speed of deformation was calculated (Barannikova et al. 2010). In the case of constant specimen loading at a fixed speed, the vertical strain component is proportional to time. By determining the coordinates of the localized strain peaks (peaks and valleys) on the x-axis during the loading process, the movement speed of the localized strain extremum points can be calculated at different time points, which represents the propagation speed of deformation.

The strains at the coordinates of 10 mm, 30 mm, and 60 mm were chosen for calculation. The corresponding slope (deformation speed) of the resulting straight lines is calculated as follows: 7.25 × 10⁻⁵ m/s, 6.03 × 10⁻⁵ m/s, 4.68 × 10⁻⁵ m/s, respectively. Similar calculations were performed for other points, and the specific results are shown in Fig. 9. During this time, the boundary propagation speed is 0.5 mm/min (8.33 × 10⁻⁵ m/s). It can be determined that the deformation speed is proportional to the position and is similar to the boundary propagation speed, as shown in Fig. 9. The propagation speed of deformation is approximately equal to or close to the propagation speed of plastic flow waves that control seismic migration in the lower lithosphere.

Distribution of deformation velocity

A comparison reveals that the deformation propagation speed and the boundary strain advancement speed are proportional to the position coordinates. Therefore, the propagation speed of deformation can be approximated as the transmission speed of particle motion, where the term “particle motion” refers to the positional changes of particles rather than their dynamic motion in the traditional sense.The local thickening and uplift caused by boundary advancement continue to propagate forward with the assistance of elastic forces and inertia forces. In this process, due to the concentration of local stress, there exists a stress gradient (imbalance force). Under the effect of this stress gradient, a new round of particle motion occurs, perpetuating the deformation propagation. This cyclic process repeats itself, allowing the deformation to be transmitted.

Regarding the triggering conditions for deformation transfer, preliminary analysis suggests that during the loading process, the specimen is generally compressed and in the compaction or elastic stage. However, the internal microstructure of the specimen has locally entered an upward segment similar to the loading curve in the plastic zone, and as the loading curve segment in the plastic zone gradually increases, the tangent modulus also gradually decreases. According to the equation \(v = \sqrt {{{E^{\prime}} \mathord{\left/ {\vphantom {{E^{\prime}} \rho }} \right. \kern-0pt} \rho }}\), the wave speed decreases accordingly; Different wave velocities occur during the deformation process, depending on the magnitude of the tangent modulus at different stages.

3.3 Deformation characteristics and strain localization process

The strain components \(\varepsilon_{xx}\),\(\varepsilon_{yy}\) and \(\varepsilon_{xy}\) during the specimen loading process are shown in Fig. 10 (with the loading end located below the specimen and the upper portion being the fixed end). In the test range, compressive strain is defined as negative, while tensile strain is defined as positive. In the figure, the blue area represents the compressed region, while the red and yellow areas represent the tensile region.

Strain nephogram \(\varepsilon_{xx}\) of loaded specimen (a, b, c, and d is the formation process of flow channels and nucleation phenomena; e, f, g, and h is the formation process of Lüders bands phenomenon)

As shown in deformation component \(\varepsilon_{xx}\) in Figs. 10a, b, c, and d, during the initial stages of loading, the surface microvolume particles of the specimen appear to be randomly dispersed in space. Additionally, during the loading process, there exist stretched deformation regions between the compressed regions. As the loading continues, these tiny particles start to gather towards the direction of the channel (The lines drawn in Fig. 10d are called flow channels. The positions of these flow channels cross each other symmetrically, forming a rhomboid distribution with symmetrical positions). These channel positions are symmetrically intercrossed, forming a mesh-like distribution, gradually connecting in the channel positions.

In the strain cloud maps shown in Figs. 10e, f, g, and h, nucleation phenomena can be observed during the specimen loading process. When they first appear, these nuclei have the shape of small elliptical sections and occur in pairs: one in the compressive strain region, and the other in the tensile strain region. The nucleation phenomenon of deformation indicates the origin of dislocations, and the surface undergoes greater deformation relative to the volume of the specimen. During the deformation process of the specimen under loading, dislocations are quickly released and formed. After the deformation nuclei of the particles are formed, they gradually develop into band-like structures (Lüders bands (Mazière et al. 2015; Suna et al. 2003)), which represent the concentrated deformation regions. At the beginning, these bands are relatively vague, but they become clearer as the loading progresses, gradually increasing in prominence. They start to develop simultaneously from both sides of the specimen, gradually connecting and spanning horizontally. This phenomenon indicates that the Lüders bands have formed before the plastic deformation stage of the specimen. The Lüders bands mainly concentrate on half of the cross-section near the loading end, while the other half of the cross-section has relatively smaller strains.

Regarding the \(\varepsilon_{xx}\) component, the channel positions are parallel to the Lüders bands. However, the deformed positions of the Lüders bands, which are the initial concentration points or nucleation positions, are located outside the channels. Therefore, it can be considered that their origins are different. Three points are taken from the positions of the specimen at distances of 30, 50, and 70 mm along the axial line (as shown in the strain cloud map in Fig. 10f). From the strain cloud map, it can be observed that the strains are larger near the loading end and decrease as the distance from the loading end increases. These three points are close to the centers of the three channels formed during the specimen loading process. The closer the flow channels is to the loading end, the larger the strain component is. This is determined by their respective durations, with a longer duration resulting in larger strain values. Additionally, as the width of the Lüders bands increases, they gradually join the active deformation. Therefore, it can be concluded that the formation of the Lüders bands is related to the maximum values of the channels, and as the deformation increases, the Lüders bands merge and develop along with the nearby channels.

Throughout the loading stage, the values of the strain deformation components \(\varepsilon_{yy}\) and \(\varepsilon_{xy}\) are very small, as shown in Fig. 11. The strain cloud map \(\varepsilon_{yy}\) exhibits clear vertical distribution characteristics, and the extreme strain values gradually shift towards the center as time progresses. The strain cloud map \(\varepsilon_{xy}\) shows periodic regions of strong tensile deformation. These regions of strong tensile deformation correspond to the channel positions observed in the cloud map \(\varepsilon_{xx}\). From the cloud map of component \(\varepsilon_{xy}\) near the left edge of the specimen, deformations centers can be clearly seen with a periodic distribution at approximately 10 mm intervals.

Strain nephogram \(\varepsilon_{yy}\) and \(\varepsilon_{xy}\) of loaded specimen (a, b \(\varepsilon_{yy}\); c, d \(\varepsilon_{xy}\))

The above phenomena are appropriate for describing the causes and the micro-evolutionary process and mechanism of the deformation transmission process, which can lead to a clearer understanding of the reasons behind the formation of deformation transmission.

The propagation of deformation occurs in the early stages of specimen loading, and the defect damage of the specimen develops uniformly. As the loading progresses and the specimen enters the stage of strain localization, further propagation of deformation becomes unlikely. Figure 12 shows the distribution of strain component \(\varepsilon_{xx}\) along the length of the specimen when strain localization occurs. At this point, the deformation becomes concentrated at nearby weak points and gradually develops into plastic localization. These weak points then gradually connect to form a certain fracture surface, resulting in a fracture phenomenon.

Strain distribution during localization

In this study, the surface of sandstone specimens under uniaxial compression was characterized using speckle photography, and the slow deformation transfer process was observed and analyzed using DIC (Digital Image Correlation) technology. The following specific conclusions were drawn about the deformation transfer characteristics during the loading process:

1. (1)

   Slow deformation transfers from the actively loaded end to the fixed end, resulting in a gradual convergence of strain values at both ends from an initially asymmetric state (one end with larger strain, the other with smaller strain). This demonstrates the existence of deformation transfer in small-scale experiments.

2. (2)

   Clear temporal and spatial frequency characteristics of slow deformation transfer were observed in the loaded specimens. The generation of deformation propagation in the specimen is related to elastic stress concentration zones of different scales. The propagation velocity of deformation and the boundary advancement velocity are proportional to the position, indicating that the propagation velocity of deformation is equal to the transfer velocity of the particle’s motion state. It was found that the transformation relationship between elastic and plastic strain in rock materials roughly follows the previously discovered form of strain invariants.

3. (3)

   The evolution process of channels, nucleation phenomena, and Lüders bands observed in the DIC cloud maps provided insights into the possible mechanisms and development processes of strain localization in the specimen.

Based on the above research findings, a clearer understanding of the deformation transfer characteristics in sandstone specimens during the loading process has been achieved. This provides an important basis for further understanding the deformation behavior of rock materials and geological structures. The mechanism research and evolution process exploration of earthquake migration characteristics have important reference significance. Based on the deformation-related patterns observed in small-scale experiments, we can analyze the process of large-scale earthquake transfer and provide a basis for exploring the characteristic patterns of earthquake transfer.