In rock mechanics, stability and rock blasting are typically analyzed using stress as the primary controlling parameter, along with stress-based failure criteria. First-principle physics-based stress approaches require field stress measurements, which are difficult to obtain and often limited to a few monitoring points representing vast rock domains (Ljunggren et al., 2003). Stress is usually inferred from strain or deformation using rock constitutive relations, which are notoriously difficult to evaluate accurately due to numerous parameters, even in simple cases. This complexity increases when trying to define nonlinear relations for rock stability or blast modeling, further limiting the effectiveness of stress-based approaches.

Geomechanics and blasting models that rely on first-principle physics face inherent limitations due to the difficulty of measuring stress and rock constitutive properties in the field. As a result, these models often involve numerous parameters that are hard to quantify, requiring expert judgment to adjust the models to match measured results (Bawden, 2015). Furthermore, these blast models typically simulate only a few blastholes (Rong et al., 2024; Yang et al., 1996) and cannot account for the full range of blast design parameters in large-scale production blasts, which may involve hundreds or even thousands of blastholes (Yang, 2022a, 2022c). While empirical models simplify the problem, they often oversimplify key interactions between blasting parameters and rock stability, reducing their effectiveness in practical applications within mining economics (Cunningham, 2005).

Advancements in explosives, electronic initiation systems, and field measurement techniques have made it essential to simulate full-blast parameters to optimize designs for improved productivity and economic outcomes in mining. Modeling the interaction among blast design parameters in large-scale blasts with hundreds or thousands of blastholes is far more impactful than single-hole simulations.

To further understand the limitations of first-principle physics-based stress approaches, the complexity and challenges involved in in-situ stress and rock mass property measurement need to be highlighted. Several methods for measuring in-situ stresses and strains are reported in the literature, including the flat-jack, hydraulic fracturing, the USBM borehole deformation gauge, and the CSIRO over-coring gauge (Hudson & Harrison, 1997). All these methods have drawbacks, such as difficulties in implementation, particularly with over-coring or borehole drilling (Fischer, 1982). To obtain three-dimensional stress/strain data, separate tests at different nearby locations and at different time instances must be conducted (Cai & Peng, 2011; Ghosh, 2008; Lin et al., 2006). Consequently, discrepancies arising from different measurement setups, locations, and times could result in significant measurement errors (Kim & Franklin, 1987; Sazid et al., 2023; Widarsono et al., 1998).

As illustrated in Figure 1, slope faces with complex arrays of rock types and discontinuities pose significant challenges for measuring key properties, such as stress, Young's modulus, and Poisson's ratio. As shown in Figure 1, in-situ rock masses are commonly discontinuous and inhomogeneous with local joints and cracks, a conventional stress or strain gauge placed at a single spot on a rock surface cannot provide representative measurements for the strain in the rock mass on an engineering scale (e.g., a few meters). The influence of the rock mass structural elements on rock constitutive relations is equally difficult to determine. Consequently, stress-based models often rely on arbitrary adjustments to match observed rock responses, resulting in limited reliability. Given these complexities, a simplified approach that avoids the use of unmeasurable parameters may offer a more practical solution for engineering applications.

In the context of rock blasting, the dominant strain state is extensional. There is significant potential to optimize blasting to reduce energy consumption and emissions, but the blast mechanism involves a complex interaction of rock properties, explosive detonation, and design parameters. In the field, multiple charge elements create dynamic, additive strains at specific points in the rock, complicating the modeling of rock breakage. Due to the transient and violent nature of the process, dynamic stress is difficult, if not impossible, to measure during blasting in the field, and data on this subject is scarce or nonexistent in the literature.

Additionally, rock failure is traditionally evaluated using stress-based criteria derived from laboratory tests and adjusted for field conditions. However, decades of rock testing have revealed that many rocks exhibit permanent, nonisotropic strains before reaching failure, particularly under specific loading paths. These anelastic strains indicate rock damage and can lead to premature failure. A strain-based failure criterion could provide a more accurate framework for predicting rock failure across different loading paths, focusing on strain rather than stress.

For example, triaxial compression tests can generate tensile strains if one or two principal stresses are insufficient to counteract the Poisson effect of the maximum compressive stress (Figure 2). Even when all principal stresses are compressive, tensile failure can still occur, underscoring the importance of strain-based analysis in understanding failure mechanisms. A recently pioneered strain-based failure criterion is briefly introduced later in this paper.

Recent advancements in high-precision global positioning systems (GPS), laser scanning, and drone technologies have made it easier to measure deformation quantities such as displacement. These technological improvements have made comprehensive strain measurements more accessible (Yang, 2020), as discussed later in this paper. As measurement accuracy improves with technological advancements, strain data is becoming more precise and directly reflective of rock behavior. As mentioned earlier, dynamic stress is difficult, if not impossible, to measure during blasting. However, it is straightforward to monitor blast vibrations using multiple sensors placed near the blastholes. From these blast vibration measurements, dynamic strain around the blastholes can be calculated (Yang, 2012), as demonstrated in the following section of this paper.

The strain-based approach offers a practical and reliable solution to many of the challenges in rock mechanics. By eliminating the need for complex constitutive relations and relying on more easily measurable field data, such as particle velocity, strain, and displacement, this method simplifies the modeling process. Strain-based models can accurately capture blasting trends, real blast geometries, and comprehensive blast design parameters with minimal calibration.

This paper introduces a novel strain-based modeling framework for blasting and geomechanical applications. The framework includes field strain measurements, model construction based on measurable variables, and laboratory-derived strain-failure criteria, all of which hold significant potential for further development. The approach is demonstrated through the multiple blasthole fragmentation (MBF) model, which is capable of simulating production blasts with hundreds of blastholes and modeling rock fragmentation and strain during multiple blasthole detonations (Yang, 2015a). This model illustrates the practicality and effectiveness of strain-based modeling for handling real-world production blasts. Furthermore, the strain-based approach has versatile applications beyond blasting, including highwall stability monitoring and other geomechanical challenges, offering a simpler yet effective solution for modeling complex rock systems.

2 NEW METHOD TO MEASURE STATIC STRAIN CHANGE FOR ROCK MASS

As discussed earlier, measuring changes in the strain state around an open pit or underground excavation is crucial for strain-based analysis and modeling. Building on displacement measurements, a strain assessment method has been developed by the author in recent years (Yang, 2020). Emerging measurement techniques such as GPS, radar systems, three-dimensional photogrammetry, and multi-sensor displacement monitoring (Wilkins & Shwydiuk, 2012) can now provide detailed three-dimensional strain distributions in rock slopes and underground structures. Although the accuracy of current technology may not yet be sufficient for high-resolution measurements, ongoing developments in these techniques show great promise. For the sake of completeness, the strain measurement technique is briefly introduced below.

2.1 Triaxial strain change from relative displacement of points

In most rock masses, cumulative displacement or strain over a long period, or after several nearby blasts, can be significant. However, over shorter periods, the deformation is generally small, meaning strain changes are minimal. Additionally, strain variation within a small zone is expected to be relatively smooth. Therefore, analyzing small deformations is useful for assessing strain changes over short periods or after a few nearby blasts. Lastly, it is assumed that the reference point for the survey is located far enough from the area of concern, ensuring that its coordinates are unaffected by excavation activities.

Figure 3 shows two points, A and B, in a rock mass before the deformation. After the deformation, the points move to A′ and B′. Under the reference coordinate system, the four points are shown in Figure 3.

The displacement of point A:

The displacement of point B:

The displacements between points A and B are:

(1)

where, , , , and , are measured before and after the deformation. Under small strain condition, the following equations can be established:

(2)

where, are the knowns from the measurement as shown in Equation ( 1) and the nine displacement gradients are the unknowns. To solve the unknowns, nine independent equations can be obtained from four measuring points that are not coplanar.

To improve the measurement accuracy, measurement can be made at more than four no-coplanar points to establish equations to obtain the least-square solution. After the displacement gradients are obtained, the tri-axial engineering strain can be calculated as:

(3)

The principal strain change, the directional cosines of the principal strain change, and other strain quantities and invariants can be calculated. If an initial strain state is assumed, the strain state can be calculated.

To obtain a 3D distribution of measurement points, boreholes can be drilled into the rock mass, where a straight rod is fixed at the borehole's bottom. One end of the rod is anchored at the bottom, while the other end remains exposed outside the borehole collar. The directional cosines and length of the rod are known. By surveying the exposed end of the rod, the coordinates of the bottom end can be determined, allowing for the calculation of 3D strain. As illustrated in Figure 4, the coordinates of the rod's bottom end can be derived from the surveyed top end and the rod's directional cosines.

The spatial resolution of the strain calculation depends on the accuracy of the coordinate measurements and the strain variation in the zone. Future advancements in measurement precision will allow this method to capture strain changes with significantly higher spatial resolution and accuracy compared to current techniques.

Strain change measurement at an open-pit mine in Chile since late 2015 has been reported in previous publications (Yang, 2020). Target points on a slope surface were selected, and their 3D coordinates were surveyed periodically. Strain changes in the rock mass surrounded by these points were calculated from each two consecutive measurements. The maximum shear strain and extensional strain history were obtained, with extensional strain defined as the sum of all tensile principal strains (Yang et al., 1996). Large strain increases over time served as an alarm for highwall stability.

Controlling the blast vibration peak particle velocity (PPV) is important for highwall stability. The history of the PPV at the target points from nearby blasts can be compared with the history of the strain changes, although the PPV may not be the sole driving force for strain change in the highwall. In addition to relating blast vibration PPV to the strain changes in the highwalls, the correlation between strain change and other potential causes of the highwall displacement may be examined. With such multiple variable analysis, dominant factors affecting the highwall stability may be identified for a particular operation or a mining site. This example demonstrates the rock deformation/strain analysis applied to rock stability problems.

3 DYNAMIC STRAIN CALCULATED FROM BLAST VIBRATION

The propagation of strain or stress waves through rock or structures near a blast characterizes the nature of blast vibration. Traditionally, blast vibration has been assessed using PPV or peak particle acceleration (PPA), without explicit reference to dynamic strain or stress. However, a method for converting blast vibration into three-dimensional dynamic strain waves was developed by Yang (2012). For completeness, this method is briefly presented below in the context of the present paper.

4 STRAIN-BASED FAILURE CRITERIA

Kwaśniewski and Takahashi (2010) categorized strain-based failure criteria for rocks based on several strain types: maximum principal strain (extension or compression), maximum shear strain, and mean or octahedral strain. Staat and Trần (2022) proposed strain-based brittle failure criteria for rocks. Unfortunately, there is limited available data on failure strains in the literature, particularly under triaxial loading conditions.

Yang and Green (Forthcoming) introduced a mathematical formulation of strain-based failure criteria using the well-known Hoek–Brown (HB) criterion (Hoek & Brown, 2019) alongside Hooke's law of elasticity. This formulation relies on three parameters, which can be determined through regression if triaxial failure strain data is available, allowing the rock's nonlinear behavior to be captured. As more strain data become available, these criteria can be refined further.

4.2 Converting the HB criterion to a strain-based form

The HB criterion can be converted into a strain-based counterpart by applying Hooke's linear elasticity law. Assuming average Young's modulus E and Poisson's ratio μ (Małkowskia et al., 2018), the relations for stress and strain in triaxial tests are:

(8)

(9)

where . Solve these Equations ( 7)–( 9) for :

(10)

where:

(11)

(12)

(13)

This equation provides a possible strain-based failure criterion. By recording triaxial failure strains in laboratory tests, the parameters a, b, and c can be determined through regression, making the model more realistic and less dependent on linear elastic assumptions.

When the strain-based HB failure criterion is established with the parameters a, b, c in Equation (10) determined by rock properties , the failure criterion depends on HB stress-based criterion and Hooke's linear elasticity. For hard and brittle rock, such converted strain-failure criterion may be applicable. Jaeger and Cook pointed out that linear elasticity up to failure is a reasonable assumption in many cases (Jaeger & Cook, 1976). It may be particularly true for modeling blasting rock crack initiations or strain localizations.

To visualize Equation ( 10), the parameters at the crack initiation of Lac du Bonnet granite reported by Chandler ( 2013) shown in Table 1 is used to generate the strain failure envelope for Lac du Bonnet granite in Figure 5. The corresponding equation of the crack initiation envelope is:

(14)

Table 1. Property of Lac du Bonnet granite (Chandler, 2013).

Failure stages	HB parameter m	UCS (MPa)	Poisson's ratio	Young's modulus (MPa)
Crack initiation	15	106	0.24	67 000

• UCS is Uniaxial compressive strength.

4.3 Preliminary example—regression from failure strain data to obtain a strain-based criterion

Conducting laboratory testing to determine failure strains (crack initiation, onset of strain localization, and peak strains) for specific rock types is the optimal method for establishing a strain-based failure criterion since the failure strain data and the established strain failure criterion capture the nonlinear response of the rock upon failure.

Figure 6 shows a suite of complete stress-strain curves, together with the corresponding axial strain-radial strain (negative) curves obtained from triaxial compression tests at different confining pressures on specimens of rock taken from a single piece of argillaceous Witwatersrand quartzite (after Hojem et al., 1975). From the radial strain corresponding to the brittle failure (red dots), it is worth noting that the substantial dilation only starts at or close to the peak axial failure stress. By digitizing these failure strains and regressing them to Equation (10), a strain-based HB criterion is established, as shown in Figure 7, for argillaceous Witwatersrand quartzite.

Based on the limited data, the regression equation of the strain-based HB failure criterion for argillaceous Witwatersrand quartzite is:

(15)

The valid regression interval is from −0.20% to −0.04% for the principal failure strain based on the available data. This example demonstrates that by recording failure strains from triaxial compressive testing, regressing the measured failure strains to Equation (10) can establish a strain-based HB failure criterion. In such a way the established failure criterion captures the nonlinear response of the rock in the failure process, and it is more realistic than using elastic and HB criterion parameters.

5 MODELING ROCK FRAGMENTATION, MAXIMUM STRAIN, AND MICROFRACTURING DURING BLASTING

In recent years, the MBF model has been developed to effectively simulate the impact of various blast design parameters on rock fragmentation, using easily measurable variables. The MBF model has been widely applied across numerous mine sites globally for predicting rock fragmentation (Yang & Assenso-Onyinah, 2018; Yang & Patterson, 2017; Yang et al., 2019, 2023). More recently, the model was expanded to simulate microfractures within the blasted volume (Yang & Green, 2024). This capability to model strain and microfracturing within rock fragments offers valuable insights for optimizing blast designs to enhance fragment microfracturing, ultimately improving downstream rock crushing and comminution, increasing production efficiency, and reducing operational costs.

The MBF model is grounded in the concept that blast-induced vibration is directly related to dynamic strain (Yang, 2012, 2016) and rock fragmentation (Seaman et al., 1976; Yang, 2015a, 2018; Yang et al., 2016, 2023). The model simulates the contribution of multiple charges in multiple blastholes to the vibration at a point in the rock using nonlinear charge weight scaling, which accounts for the delay timing and location of the explosive charges (Yang & Scovira, 2008). Figure 8 illustrates the framework of the MBF model—the interaction between explosive charge detonation and a point in the rock near a blasthole. In the model, an explosive column is treated as a stack of charge elements, with each element's length corresponding to the borehole diameter. The detonation time for each element is calculated based on the primer initiation time and the velocity of detonation (VoD) of the explosive charge. In a real-world blast, multiple blastholes contribute to the blast waves (and strain) at different times and from various distances, resulting in nonlinear additive effects that cause rock fragmentation and the formation of microfractures within the fragments (Yang & Green, 2024). The following Sections 5.1–5.6 briefly describe some key concepts that the MBF model is based on to model the nonlinear additive effects and interactions between explosives and rock. These concepts were reported in previous publications, and they are established from field measurements and validated in the applications (Yang, 2018, 2022c).

5.1 Site-specific data collection for model input and calibration: A standard testing method

The collection of relevant site-specific data is crucial for both blast and geomechanics modeling. A systematic method for obtaining this information should be established as a standard practice for these models. At every site where modeling is to be conducted, a standardized test should be implemented to collect the necessary data for model input and calibration. Currently, no such standardized testing exists for geomechanical models based on finite/distinct element methods or similar approaches. Establishing a standard test for modeling and analysis inputs would enable comparisons across different sites, allowing for the application of knowledge gained from other locations.

For blast modeling, the author has developed a standardized test method. This test combines a multiple signature hole blast with a calibration production blast. The purpose of the test blast is to acquire site characterization parameters related to near-field blast vibrations through signature hole blast monitoring. These parameters include:

• 1.

  Vibration attenuation with distance and charge weight—for modeling blast vibrations.

• 2.

  Multiple seed waveforms at various distances—for blast vibration modeling.

• 3.

  Ground frequency response—for blast vibration control and frequency shifting.

• 4.

  Ground sonic velocity—for modeling blast vibration and fragmentation as an approximation to shock wave velocity in rock. The ground sonic velocity can also be used to estimate rock mechanics properties, which are useful for blast design and other geomechanics analyses.

• 5.

  Measurement of rock fragmentation and blast vibration from the calibration production blast to validate the MBF modeling for the site.

The data from the near-field signature hole blast vibration may be the most relevant site characterization for modeling near-field blast vibration at highwalls and rock fragmentation, among other measurements for the rock. A geological domain is usually large and contains many blasts. Within a domain, the rock property may be defined with spatially high resolution by drill monitoring. In one blast, a few blocks of different rock properties may exist. The effects of rock density ( ), UCS, and p-wave velocity (c) on rock fragmentation can be modeled (Yang, 2022a, Warden et al., 2022).

Figure 9a depicts a test blast layout, which includes a series of seven signature blastholes and a production blast. Figure 9b shows the recorded blast vibration from both the signature hole and production blasts at a monitoring location. The initiation time of the third signature hole is recorded in the fourth channel of the monitor for measuring ground sonic velocity. The blasthole diameter is 216 mm, with a depth of 13.5 m and a stemming length of 4.6 m. The charge weight per hole in the signature hole blast is 365 kg. The delay between successive signature blastholes is 1.6 s, which is also the delay between the firing of the last signature blasthole and the first blasthole of the production blast. The production blast consists of 243 blastholes, each initiated with millisecond delays for controlled blast vibration and rock fragmentation outcomes.

5.2 PPV charge weight scaling law for nonlinear rock-explosive interactions

A complete charge weight scaling law of the form:

can be established with multiple-segment regression from near-field blast vibration monitoring and theoretical extrapolations, as shown in Figure 10a,b. The upper portion of the scaling law, for scaled distances ( ) less than 0.5 m/kg 0.5 (extending up to the blasthole wall), can be derived theoretically based on the properties of the explosive and the rock. Using the explosive properties (VoD and , the borehole pressure ( P b) can be calculated. Based on rock properties ( , s, c), the borehole particle velocity ( ) can be estimated. The corresponding scaled distance is calculated by assuming that the contributing charge segment has a length equal to the diameter of the charge (Figure 10a).

π (16)

where, r is the charge radius and is the density of explosive (Yang, 2018). The lower portion of the charge weight scaling law, shown in Figure 10b, is established from field signature hole blast vibration monitoring. Multiple-segment regression can be performed on the signature hole vibration data to determine the variable PPV attenuations at different charge weight scaled distances. This allows for modeling the nonlinear higher PPV attenuation around blastholes and lower attenuation further away from blastholes. Since blast vibrations exhibit random behavior, both the best-fit (average) and the 97.5% prediction upper bound scaling laws are used for Monte Carlo modeling to account for geological randomness and explosive-rock interactions. At a specific site, for a given charge weight scaled distance ( sd), the measured PPV is assumed to follow a normal distribution N ( μ, σ) around its mean ( μ), with a standard deviation ( σ) based on a prior study (Yang & Lownds, 2011).

5.3 Calculating PPV vector increment from a charge element

5.3.1 Nonlinear charge weight scaling

Although the direction of the PPV contribution from an explosive charge element can be determined by the location of the charge element relative to the calculation point, the magnitude of this contribution (| |) must be estimated. Additionally, since the rock close to the explosive charge may behave nonlinearly, contributions from different charge elements should not be added linearly, even when the | | contributions from two charge elements arrive simultaneously. Therefore, nonlinear charge weight scaling is applied to approximate nonlinear vibration superposition, based on the established PPV scale distance relation established in Figure 10b. The details of this scaling approach have been previously reported (Yang & Scovira, 2008; Yang et al., 2023).

At a calculation point, one dominant charge element typically has the minimum scaled distance among all charge elements from all charges in all blastholes. In most cases, the dominant charge element is the one closest to the calculation point, assuming all charge elements have equal weights. The charge weight of a charge element is scaled to its effective charge weight based on the distance of the dominant charge element to the point of calculation, assuming the charge x would produce the same magnitude of PPV at distance as it does at its original location . This follows the charge weight scaling, expressed as:

(17)

The scaled effective charge weight is then given by:

(18)

where, are obtained from the complete PPV-sd law in Figure 10b established from the site characterization test. After the nonlinear scaling is additive with a weighting function in Figure 11 accounting for the arrival delay time difference ( ) between the charge element x and the dominant charge element.

5.3.2 Delay time modeling

Assuming the wave arrival time of the dominant charge element at a calculation point in rock is To account for the blast wave arrival time difference at a calculation point between a contributing charge element and the dominant charge element ( ), a weighting function on the scaled charge weight is used. When the arrival time difference ( ) is smaller than the half width of the time window (0.5Tx), a simple exponential function is used as the weight function, as shown in Figure 11. tx is the blast wave arrival time from a contributing charge element x, Tx is the time window width for the charge element x. calculated according to the blast vibration waveform duration increase with distance ( ) established from the field measurement ( at the site (Yang et al., 2023).

5.3.3 Direction and magnitude of PPV contributed from an explosive charge element

Before the i th charge element is superimposed at the dominant charge element location, the recursion formula of the superimposed effective charge weight is:

(19)

After the i th charge element is scaled and superimposed to the dominant charge weight location, the superimposed effective charge weight is:

(20)

The magnitude of the PPV contributed from the i th explosive charge element is calculated:

(21)

The accumulated effect charge weight at a calculation point is dependent on the charge weight distribution in the blast, the VoD, and the delay time among the charges of the blastholes. Figure 12 shows the direction and the magnitude of calculation.

The direction from the charge element to the calculation point is considered as the direction of the PPV contributed ( ) from the charge element, as shown in Figure 12. By accounting for the direction and magnitude of the PPV contributed by each charge element ( ), it is possible to model the 3D strain path and strain state change in the rock from the blast. This information is crucial for predicting rock fragmentation and the resulting particle size distribution, which is essential for optimizing blasting operations in mining and construction. Moreover, by considering the motion of blast fragments, it is possible to evaluate the potential for ore-grade mixing during blasting and muckpile formation and shapes from the blast (Yang & Kavetsky, 1990; Yang et al., 1989).

5.4 Spherical strain generated by detonation of a charge element

Detonation of each charge element is a point source and generates spherical strains at a calculation point as shown in Figure 13.

In Figure 13, the radial displacement is the function of the nonlinearly scaled PPV increment due to the charge element ( in Figure 12 and the principal frequency ( ) of the blast particle velocity waveform from the charge element.

(22)

Due to spherical symmetry, there is no tangential displacement. The principal frequency can be estimated from the increase in waveform duration with distance at the site and can be estimated based on the distance between the charge element and the calculation point (Yang, 2015b, Yang et al., 2023).

The spherical strains are converted to Cartesian strains and then superimposed from all contributing charge elements (after the nonlinear superposition is considered in Sections 5.1– 5.3). Since deformation close to blastholes can be large, logarithmic strain is used for strain calculation. The formula for logarithmic strain ( ε) is given by:

₀ (23)

where ε is the logarithmic strain, L is the final length of the material after deformation, and L₀ is the initial length of the material before deformation.

Let S be a strain tensor with a Spherical-Polar coordinator and C be a strain tensor with a Cartesian coordinator:

(24)

Let T be the transfer matrix:

(25)

The two sets of strain components are related by:

where ( ) and ( ) are coordinates of the calculation point and the charge element location.

5.6 Example of modeling strain and fracture density

The MBF model's capability to simulate rock fragmentation for large-scale, multi-blasthole blasts has been detailed in previous publications, as referenced in earlier sections. In this example, we focus on the modeling of strain and fracture density. The MBF model calculates distributions of the maximum principal strains (corresponding to the maximum stretch) experienced during the blasting at designated grid points and compares them with the strain failure criterion for crack initiation, as illustrated by Equation (14).

Figure 14 shows the blast pattern used for the modeling example. The blastholes have a depth of 14.6 m and a diameter of 311 mm. Each blasthole is loaded with 9 m of explosives (marked in red) and 5.6 m of stemming (marked in black). The delay timing is set at 40 ms between blastholes in a row and 20 ms between rows. Figure 15 compares the fracture density results from the same blast pattern using different types of explosives.

Figure 15a shows the modeled fracture density using an explosive with a lower VoD of 3000 m/s, while Figure 15b uses another explosive with a higher VoD of 4200 m/s. Although the modeled fracture densities are uncalibrated, the modeling demonstrates that the higher VoD generates more fractures in the rock, which is expected due to the corresponding higher borehole pressure. Due to the limitation of the paper's length, only one scenario is demonstrated here. The rock strength envelope is commonly used, indicating that only the strain states ( outside the envelope of the crack initiation strain (cracking strain) in Figure 16 contribute to cracking.

In this preliminary modeling, the crack density is correlated with the nearest distance (d) from the cracking strain state to the cracking strain envelope. The color-coded crack density in Figure 15 is uncalibrated and used only as a reference scale for comparison.

6 DISCUSSION

This paper introduces, for the first time, a comprehensive strain-based modeling framework specifically tailored for large-scale blasting operations, marking a significant departure from conventional stress-based models. By leveraging easily measurable field parameters, such as displacement, strain, and PPV, the framework simplifies blast modeling and enables the modeling of real production blasts involving hundreds to thousands of blastholes. This addresses the practical limitations associated with stress-based or empirical approaches that can only simulate blasts with a single hole or a couple of blastholes.

Traditional stress-based models face inherent challenges, primarily due to the difficulty in measuring stress and quantifying rock mass properties in the field. These models often require numerous input parameters that are hard to determine, leading to frequent adjustments to match measured results. As a result, stress-based modeling becomes more interpretive, relying heavily on user judgment and experience. Additionally, the predictive power of these models is limited, as they are typically confined to simulating single or small-scale blasthole configurations (Rong et al., 2024; Yang & Turcotte, 1994, Yang et al., 1996), due to the complexity of implementing rendering them unsuitable for real-world production scenarios involving hundreds or even thousands of blastholes. The ability to model full-blast design parameters across multiple blastholes is crucial for practical applications, as it directly impacts the economics of mining operations. While empirical models offer some utility, they often oversimplify the complex interactions among blast design parameters, explosives, and rock properties, further limiting their effectiveness in large-scale operations.

In contrast, strain-based modeling provides several distinct advantages. Strain and deformation are not only easier to measure in the field compared to stress, but they also offer more direct insights into rock failure mechanisms related to blasting and rock stability. The parameters required for strain-based models are easier to obtain, as demonstrated in the standardized field test setup for site-specific data collection (Section 5.1). Advances in field measurement technologies are further enhancing accuracy, enabling more effective input, calibration, and validation of strain-based models—challenges that remain significant for stress-based models. Moreover, due to its relative simplicity to implement, strain-based models can explicitly account for full-blast design parameters, including real-world geometries defined by today's technology, for example, laser scanning and the interactions of hundreds of blastholes, each with varying loading conditions and delay timings. This level of detail is increasingly in demand in the mining industry, where optimizing blast performance and reducing mining costs are paramount. Based on this approach, a suite of models for blasting has been developed and applied to blast optimizations (Yang, 2018, 2022a, 2022b; Yang & Scovira, 2010).

The review process has identified areas for improvement, particularly in the precision of strain measurements and the availability of failure strain data. However, these limitations stem more from current technological constraints in surveying and measurement than from weaknesses in the modeling framework itself. Technologies such as GPS, photogrammetry, and laser scanning, though currently limited in resolution, are advancing rapidly and will soon provide the accuracy needed to fully realize the potential of strain-based modeling. For example, the current strain error in strain measurements—approximately 0.1% for a 5-m sensor spacing—is expected to become negligible as these technologies evolve. Additionally, the lack of triaxial failure strain data can be addressed through future experimental research.

The section on strain-based failure criteria, which incorporates regression analysis of triaxial failure strain data from the literature, underscores the adaptability of this approach to different rock types. However, this area remains underexplored in the literature, offering a critical opportunity for future research. A detailed comparison between strain-based and stress-based failure criteria would further highlight the advantages of this model, although such a study would require focused, in-depth investigation.

The MBF model introduced in this paper represents a significant advancement in blast modeling. Unlike traditional finite element models, such as ABAQUS or Autodyn, which are often limited to simulating a single or a few blastholes, the MBF model utilizes the novel framework that simulates the nonlinear, time-varying additive effects of explosive charge elements across multiple blastholes on rock fragmentation and damage. The MBF model excels in simulating full-scale production blasts involving hundreds of blastholes, each subjected to varying load configurations (e.g., multiple decks) and initiation delay timings (multiple initiation points), meeting the practical demands of optimizing rock blasting.

This capability sets a new benchmark for practical and scalable blast modeling in the mining industry. The MBF model has already been successfully applied in numerous open-pit mines and aggregate quarries worldwide, predicting rock fragmentation and enhancing productivity through optimized blasting techniques. Nevertheless, like the broader strain-based framework it belongs to, the MBF model continues to evolve, with future developments expected to improve its predictive accuracy and practical utility.

7 CONCLUSIONS

This paper presents a novel strain-based modeling framework for blasting applications, offering a more accessible and scalable alternative to traditional stress-based models. By leveraging measurable field parameters like strain, displacement, and PPV, the framework simplifies the modeling of large-scale, multi-blasthole blasts, effectively bypassing the complexities and limitations associated with stress-based inputs.

The strain-based approach addresses a critical gap in current blast modeling practices: the ability to model full-blast design parameters in real production environments. With this framework, complex geometries and interactions of hundreds of blastholes can be explicitly modeled using field data—an area where stress-based and empirical models often fall short.

While current limitations exist, particularly in the precision of strain measurements and the availability of failure strain data, ongoing advancements in measurement technologies and future experimental research will further refine this framework. The revised failure criterion, based on regression analysis of triaxial failure strain data, provides a robust foundation for future developments, allowing the model to more effectively capture the nonlinear behavior of rock compared to traditional methods.

The MBF model, demonstrated within this framework, successfully simulates major trends and macro-phenomena in blasting. By incorporating both fundamental and measurement-based aspects, the MBF model can simulate full-blast design parameters for large-scale, multi-blasthole blasts. Its straightforward implementation and ease of calibration make it highly valuable for the mining industry today.

Overall, the strain-based modeling framework introduced in this paper holds significant implications for the future of blast design in the mining industry. Its simplicity, scalability, and reliance on easily measurable parameters make it a powerful tool for optimizing blast performance, improving fragmentation, and reducing environmental impacts. Furthermore, the strain-based approach has versatile applications beyond blasting, including highwall stability monitoring and other geomechanical challenges, offering a simpler yet effective solution for modeling complex rock systems. As technological advancements continue to enhance the accuracy and accessibility of strain measurements, the potential of this framework will only grow, positioning it as a key innovation in geotechnical and blasting engineering.

CONFLICT OF INTEREST STATEMENT

The author declares no conflict of interests.