Statistical Damage Model Research Articles

Summary of Rock Damage Mechanics Classification

As an important means to study the mechanical properties and failure mechanism of rock materials and engineering mechanics, damage mechanics is often used to assist in the analysis of the failure process and mechanism of rock, metal, and other materials in the current research. In the use of damage mechanics to study the properties of rock materials and explore the deformation and failure law of rock, important research results have been achieved. The research methods of damage mechanics can be roughly divided into three types: statistical damage model, mesoscopic damage model, and continuous damage model. Based on the existing literature in hand, according to the division of damage mechanics research methods in the academic community, this paper collates and considers that the existing damage model construction can be summarized from the mechanical point of view. The pure theory is derived or fitted from the mathematical point of view. One of them is derived from the mechanism point of view, and the other is derived from the phenomenon point of view. The stress, strain, acoustic emission, and other data are analyzed and reconstructed, and both have advantages and disadvantages.

Comparative study of multiple statistical damage mechanics models for rock behaviors under high temperature

Under the influence of external factors such as high temperature, chemical corrosion, and loading-unloading cycles, micro-cracks and pores may develop within rock structures. Continuum damage mechanics (CDM) characterizes these micro-defects arising within rocks, investigating the physico-mechanical behaviors of rocks in damaged states. This study, grounded in the theoretical framework of CDM and utilizing the Mohr-Coulomb failure criterion, integrates different probability distributions and damage coupling methods to develop three statistical damage models. The investigation primarily explores the effects of strength criteria, probability functions, and damage coupling mechanisms on the statistical damage models. In addressing the complex interplay between rock micro-defects and external stressors such as high temperatures and chemical corrosion, this study leverages the principles of CDM within the Mohr-Coulomb criterion framework. It advances the field by developing three innovative statistical damage models, enriched by diverse probability distributions and damage coupling methods. The research delineates the substantial impact of strength criteria, probability functions, and damage coupling mechanisms on the behavior of these models. Notably, it achieves a synthesis of theoretical constructs and experimental validation, demonstrating the models' capacity to reflect the nuanced behaviors of damaged rock accurately. Furthermore, this investigation contributes a methodological blueprint for statistical damage model construction, underscoring the critical selection of strength criteria and probability distributions. These efforts fortify the theoretical underpinnings necessary for the enhanced engineering application of statistics-driven damage analysis.

Study on the energy and damage characteristics of gas-containing coal under confining pressure unloading process

Research on energy and damage evolution patterns of gas-containing coal under constricting pressure unloading conditions is urgently needed in the process of deep underground mining and is crucial for understanding the mechanisms underlying coal and gas composite rockburst occurrences. Prior reaching peak stress, cyclic loading and unloading experiments were carried out on gas-containing coal specimens under varied confining pressures and unloading circumstances. According to the experimental findings, gas pressure significantly degrades the mechanical characteristics of coal specimens, with a higher gas pressure causing worsening of the mechanical properties. The degree of mechanical property deterioration in coal specimens caused by gas pressure steadily reduces as confining pressure rises, indicating that confining pressure has an inhibitory influence on the deterioration of mechanical characteristics brought on by gas. Additionally, the degradation of coal mechanical characteristics is a result of the unloading of confining pressures. The peak stress and elastic modulus of coal decrease as confining pressure unloading increases in magnitude. Prior to the peak stress, the capacity of coal to convert external work into elastic energy is mostly indicated by the compression storage energy coefficient, whereas the peak elastic energy predominantly denotes the upper bound of compressed storage elastic energy for coal. Based on the Weibull statistical damage theory and employing elastic energy as the distribution variable, a statistical damage model is proposed. Comparisons of coal damage evolution curves indicate that both the confining pressures unloading process and gas pressure exert a promoting effect on the damage evolution of coal.

Numerical Simulation Study on the Constitutive Model of Fully-Graded Concrete Based on Statistical Damage Theory

A statistical damage model (SDM) of fully-graded concrete was created using statistical damage theory, based on the mechanical properties of axial tension and axial compression of the material. The SDM considers two damage modes, fracture and yield, and explains the intrinsic connection between the mesoscopic damage evolution mechanism and the macroscopic nonlinear mechanical behavior of fully-graded concrete. The artificial bee colony (ABC) algorithm was used to obtain the optimal parameter combination through an intelligent search of parameters εa, εh, εb and H in the constitutive model by taking the test data as the target value, and the sum of the squares of the differences between the target value and the predicted value as the objective function. The SDM numerical simulation model of fully-graded concrete is proposed by compiling subroutines in FORTRAN by constructing two modules of data model and damage analysis. The numerical results under uniaxial and biaxial forces are in agreement with the experimental results, which verifies the accuracy of the program. The model also analyzes the characteristics of mesoscopic damage evolution and predicts the mechanical properties under triaxial forces. The results show that the proposed numerical simulation model can reflect the salient features for fully-graded concrete under uniaxial, biaxial and triaxial loading conditions, and the evolution law of mesoscopic parameters. Therefore, the proposed model serves as a basis for the refined finite element analysis of hydraulic fully-graded concrete structures and reveals the mesoscopic damage mechanism of concrete under different load environments.

Seepage-stress combined experiment and damage model of rock in different loading and unloading paths

In the excavation of water-related underground projects such as hydropower and energy reserves, the surrounding rock surfers complex stress path and stress state redistribution, resulting in damage and failure under the hydro-mechanical coupling condition. However, the rock hydro-mechanical coupling characteristics under complex stress paths are unclear and corresponding theoretical models are scarce. In this study, a series of tests such as triaxial compression, unloading confining pressure and cyclic loading and unloading were carried out to study the effects of different stress paths, stress levels and seepage pressure on rock deformation, strength, failure and permeability. Based on test results, the damage evolutions under three different testing paths were analyzed, a new seepage-stress coupling statistical damage model which can better simulate the compaction stage is proposed. The prediction results of the proposed model under different stress paths are in good agreement with the experimental results. Under different stress paths, the fitting relationship between parameters R0 and n and σeff is similar and has good correlation.

Fractional derivative statistical damage model of unsaturated expansive soil based on unified hydraulic effect

Fractional derivative statistical damage model of unsaturated expansive soil based on unified hydraulic effect

Constitutive Damage Model of Foamed Lightweight Concrete Using Statistical Damage Theory.

Foamed lightweight concrete has been applied in different fields of civil engineering because of its superior properties, but the related research considering internal pore damage is limited. Based on statistical damage theory and considering the uneven distribution of fracture damage and strength between the pores of light concrete, a damage constitutive model of foamed lightweight concrete was established based on the Weibull function. The parameters of the damage model were determined through a triaxial compression test, and the rationality was verified by combining the existing test data. Comparative tests show that the theoretical calculation results of the proposed statistical damage model of foamed light soil are consistent with the general trend of the experimental results, reflecting the value of the peak stress and strain and describing the overall development law of the stress and strain. The best fit was obtained when the confining pressure was 0.3 MPa and the density was 700 kg·m-3. The suggested damage constitutive method is highly applicable, which is of great significance to the microscopic mechanical analysis of foamed light concrete and the structural design in civil engineering.

Triaxial direct shear properties, cohesive damage behavior and shear constitutive model of sandstone under high confining pressure

The shear mechanical properties of rock under high confining stresses are required in deep underground construction projects for stability evaluation. This paper tested the sandstone shear properties under high confining pressures by a novel triaxial direct shear device with a self-balancing structure for both low and high confining pressure. The triaxial direct shear experiments on Zigong sandstone from a deep tunnel were carried out to evaluate its strength and deformability. Our experimental results suggested that the shear stress-displacement curves exhibited two distinct characteristics under different confining pressures. The damaging stage occurs initially and turns wider in the shear stress-displacement curve as the confining pressure exceeds the threshold stress. The confining pressure threshold of Zigong sandstone is observed to be 10–20 MPa. Meanwhile, a statistical damage model based on Beta distribution was proposed to describe the shear stress (cohesion contribution) during the whole shearing process, which demonstrates better prediction performance than previously existing models, especially under high confining pressures. This study provides important insights into the high-pressure induced rock shear properties and sheds light on the engineering stability assessment under high stress of tunnels.

Fractional Calculus-Based Statistical Damage Model of Unsaturated Soil under the Coupling Effect of Moistening and Stress Fields

Unsaturated soil exhibits extremely complex engineering mechanical properties under the coupling effect of moistening and stress fields. Firstly, the effective stress principle and limit equilibrium conditions of unsaturated soil under the coupling effect of moistening and stress fields were discussed based on the basic principles of unsaturated soil. Secondly, a fractional-order model considering the viscoelasticity and strain hardening of unsaturated soil was established based on the fractional calculus theory. Then, based on the principle of damage mechanics, the damage variable evolution equation under the coupling effect of moistening and stress fields was established, and the fractional calculus-based statistical constitutive damage model of unsaturated soil under the coupling effect of moistening and stress fields was developed. In turn, parameters of the developed model were solved using a triaxial test of unsaturated loess, and the calculated data using the developed model were compared with the experimental data, which demonstrated that the developed model in this paper performed well in describing the whole strain hardening process of unsaturated soil under the coupling effect of moistening and stress fields. Finally, the sensitivity of the main parameters of the developed model was discussed under the coupling effect of moistening and stress fields, which showed that the proposed model performed well in reflecting the main mechanical properties of unsaturated loess.

Evaluation method of rock damage under uniaxial compression based on unit series-parallel electrical conductive model

The evaluation of rock damage behaviour is an important requirement for ensuring stability control and safety prediction in rock engineering. However, they have not been able to obtain sufficiently accurate and dynamic results due to the insufficient evaluation method. In this study, by means of fractals and unit series division, a unit series-parallel conductive model of damaged rock is derived, and a new evaluation method of rock damage under uniaxial compression was proposed. Rock was damaged by uniaxial compression, while electrical measurements and X-ray microscopy tests were performed to obtain the damaged rock resistivity, porosity, and fractal dimension variation. By establishing the relationship between defined meso-damage factor and resistivity, rock damage evolution law under axial compression was obtained. The results indicate that the growth trend was agree with the classical statistical damage model, which verified the accuracy of the results obtained by the proposed method. Moreover, as the strain increased, the damage factor determined by resistivity gradually decreased to −0.06 firstly and then increased rapidly to 0.79. Different from previous damage evolution law, brittle failure was observed and the cracks development in each stage was considered, including the closure (negative damage) and expansion (positive damage) of cracks.

Mechanical properties and constitutive model of fractured red sandstone under acid corrosion

Red sandstone was selected for the study and specimens were prepared at 0°, 30°, 45° and 60° fracture inclinations and maintained in an acid solution at pH = 2. The specimens were subjected to uniaxial compression tests before and after maintenance to obtain the mechanical parameters and to observe the damage properties of the specimens. Based on the Lemaitre strain equivalence principle, a macroscopic fracture model for single-fissure rock masses and a statistical damage model for the Weibull distribution of micro-fissures were combined to derive macro-and micro-coupled damage variables for the specimens. The damage variables after acid solution corrosion are considered to derive the constitutive model for fractured red sandstone containing fractures after acid corrosion. The theoretical damage constitutive curves were compared with the experimental curves to analyse the applicability of the constitutive equation. The results show that the peak stress, peak strain and elastic modulus of the fractured specimens all vary regularly with the inclination angle, first decreasing and then increasing with the increase in the inclination angle; the peak compressive strength and elastic modulus of the fractured sandstone all tend to decrease and the peak strain tends to increase after corrosion by acid solution; the damage characteristics of the fractured sandstone are more influenced by the inclination angle and less influenced by acid corrosion; by comparing the theoretical curve with the experimental curve, the established damage constitutive model is in good agreement at the elastic deformation and plastic yielding stages. The mechanical properties of rock masses with different inclinations of fracture are quantified on the basis of the test results. The damage characteristics of rock masses with different fracture inclinations under normal and acid corrosion conditions are studied. The macro- and micro-coupled damage variables of fractured rock masses under acid corrosion are derived and validated.

Simulating analogue film damage to analyse and improve artefact restoration on high‐resolution scans

AbstractDigital scans of analogue photographic film typically contain artefacts such as dust and scratches. Automated removal of these is an important part of preservation and dissemination of photographs of historical and cultural importance. While state‐of‐the‐art deep learning models have shown impressive results in general image inpainting and denoising, film artefact removal is an understudied problem. It has particularly challenging requirements, due to the complex nature of analogue damage, the high resolution of film scans, and potential ambiguities in the restoration. There are no publicly available high‐quality datasets of real‐world analogue film damage for training and evaluation, making quantitative studies impossible. We address the lack of ground‐truth data for evaluation by collecting a dataset of 4K damaged analogue film scans paired with manually‐restored versions produced by a human expert, allowing quantitative evaluation of restoration performance. We have made the dataset available at https://doi.org/10.6084/m9.figshare.21803304. We construct a larger synthetic dataset of damaged images with paired clean versions using a statistical model of artefact shape and occurrence learnt from real, heavily‐damaged images. We carefully validate the realism of the simulated damage via a human perceptual study, showing that even expert users find our synthetic damage indistinguishable from real. In addition, we demonstrate that training with our synthetically damaged dataset leads to improved artefact segmentation performance when compared to previously proposed synthetic analogue damage overlays. The synthetically damaged dataset can be found at https://doi.org/10.6084/m9.figshare.21815844, and the annotated authentic artefacts along with the resulting statistical damage model at https://github.com/daniela997/FilmDamageSimulator. Finally, we use these datasets to train and analyse the performance of eight state‐of‐the‐art image restoration methods on high‐resolution scans. We compare both methods which directly perform the restoration task on scans with artefacts, and methods which require a damage mask to be provided for the inpainting of artefacts. We modify the methods to process the inputs in a patch‐wise fashion to operate on original high resolution film scans.

Mechanical Properties and Constitutive Relationship of Cretaceous Frozen Sandstone under Low Temperature

During the construction of coal mine shafts through Cretaceous water-rich stratum using the freezing method, the frozen shaft lining can break and lose stability. Hence, it is necessary to examine the mechanical properties and constitutive relationship of Cretaceous water-rich sandstone under the effect of surrounding rocks. To this end, in this work, the mechanical properties of red sandstone at different confining pressures and freezing temperatures were examined by using a ZTCR-2000 low-temperature triaxial testing system, wherein the 415–418 m deep red sandstone in the Lijiagou air-return shaft of Wenjiapo Mine was taken as the research object. The test results indicated that the stress–strain curves of rock under triaxial compression and uniaxial compression presented four stages: pore compaction, elastic compression, plastic yield, and post-peak deformation. The difference between the two cases was that the post-peak curve of the former was abrupt, while the latter exhibited a post-peak strain softening section. As the freezing temperature was constant, with the raise in the confining pressure, the elastic modulus and peak strength of the rock rose linearly, while the Poisson’s ratio decreased quadratically. As the control confining pressure was constant, the elastic modulus and rock’s peak strength increased with the decrease in the temperature, and under the condition of negative temperature, the two parameters were linearly correlated with the temperature, while the Poisson’s ratio showed the opposite trend. The two-part Hooke’s model and the statistical damage model based on Drucker–Prager (D-P) yield criterion were used to establish the stress–strain relationship models before and after the rock yield point, optimize the model parameters, and optimize the junction of the two models. The results revealed that the optimized model curve was in good agreement with the experimental curve, which suggests that the proposed model can accurately describe the stress–strain characteristics of rock under three-dimensional stress. This verified the feasibility and rationality of the proposed model for examining the constitutive relationship of rocks.

Modeling for strain-softening rocks with lateral damage based on statistical physics.

Statistical physics is widely used to study the nonlinear mechanical behaviors of rock. For the limitations of existing statistical damage models and Weibull distribution, a new statistical damage with lateral damage is established. In addition, by introducing the maximum entropy distribution function and the strict constraint on damage variable, a expression of the damage variable matching the proposed model is obtained. Through comparing with the experimental results and the other two statistical damage models, the rationality of the maximum entropy statistical damage model is confirmed. The proposed model can better reflect the strain-softening behavior for rocks and respond to the residual strength, which provides a theoretical reference for practical engineering construction and design.

Microstructure-based prediction of UHPC's tensile behavior considering the effects of interface bonding, matrix spalling and fiber distribution

Ultra-high performance concrete (UHPC) is an ideal construction material to cut down carbon emissions due to its superior mechanical performance and durability. It is important to clarify the relationship between the microstructure and the overall tensile response of UHPC. In this paper, a microstructure-based multiscale framework is proposed to predict the stress-strain curve of UHPC under tensile load. The linear elastic stage is obtained by estimating the effective properties with the multi-level homogenization scheme based on the microstructure. The strain hardening and tension softening processes in the tensile curve are predicted with a statistical micromechanical damage model, where the characteristics of steel fibers are considered, including the effects of slip-softening, residual bonding strength of the fiber/matrix interface, matrix spalling and fiber orientation distribution. Comparisons with existing test results and previous models demonstrate the feasibility of using this multiscale framework to predict the overall tensile response of UHPC, especially the tension softening branch. Finally, the effects of interface parameters, matrix spalling degree, fiber orientation distribution, hydration degree and water/binder ratio on the tensile response of UHPC are evaluated and discussed with the proposed framework.

Experimental investigation on the compressive mechanical properties of the non-aqueous reacting polymer with AE and DIC techniques

As a new type of non-excavation repair material, non-aqueous reacting polymer (NARP) is increasingly adopted in rehabilitation engineering. Properly evaluating the compressive mechanical properties of NARP with different densities is crucial for optimizing the repair design in engineering practices. However, due to a lack of experiments, the mechanism of the compressive mechanical properties of NARP influenced by the density is still not clear, and the constitutive model applicable for NARP with a wide range of density has also not been established yet. In this study, the compression characteristics of NARP with different densities under uniaxial compressive loading were comprehensively investigated, by using the acoustic emission (AE) and digital image correlation (DIC) techniques. Consequently, a statistical damage constitutive model considering the effect of density was established. The results showed that both the compressive yield strength and the elastic modulus of the NARP were strongly correlated with the density, and the correlated relationships can be characterized by two separate quadratic polynomial functions. The damage pattern of the NARP mainly changed from a ductile mode with low-density (ρ < 0.4 g/cm3) to a brittle mode with high-density (ρ > 0.4 g/cm3). The high-density NARP tends to exhibit a significant compressive brittleness. The AE signals were detected throughout the uniaxial compression experiments and the released AE activity was stronger at the same strain with a higher polymer density. Both the pre- and post- yield stages of NARP materials with different densities can be well characterized by using the proposed statistical damage model, being capable of considering the effect of density. The findings on the deformation and damage mechanism, together with the established constitutive model, have significant implications for the design of NARP in engineering practices.

Investigation of thermal induced damage of deep shale considering in-situ thermal shock effects

Thermal shock caused by injecting cold fluids into hot underground formations, like deep shale hydraulic fracturing and geothermal extraction, is believed to cause deterioration of rock mechanical properties. Consequently, investigating thermally induced damage in deep shale under in-situ conditions is necessary to understand its mechanical behaviors. In this study, the structure and permeability characteristics of shale rock before and after different cooling modes are measured with the gas permeability measurement and CT-Scanning, the compression mechanical response of thermal shock specimens under in-situ conditions and its application prospects are discussed, and a statistical damage model is established eventually to predict deep shale mechanical constitutive. Results show that cooling rate was critical in the deterioration of rock properties, since micro-cracks were induced in water-cooling specimens, the physical properties of these samples were significantly enhanced. Moreover, rock mechanical properties were also deteriorated by thermal shock immensely and the deterioration was aggravated as the contact temperature difference increased. The understanding suggested that thermal shock was beneficial to increase fracture complexity and enhance shale seepage capacity, cryogenic fracturing techniques were applicable in enhancing stimulation response. Finally, the overall experimental and modeling methods introduced in this paper could well reflect the mechanical behaviors of deep shale considering thermal shock effects.

A new statistical damage model for true triaxial pre-and post-peak behaviors of rock considering intermediate principal stress and initial compaction effects

Deep underground engineering excavation will induce complex change in the true three-dimensional stress state of the surrounding rock, causing rock damage and then leading to a change in mechanical properties. In this research, the true triaxial compression tests are conducted to study the influence of σ2 and σ3 on the mechanical properties of marble, such as the characteristic strength (crack initiation stress σci, damage stress σcd, peak strength σp, and residual strength σr), post-peak deformation, ductility and brittleness. The peak strength σp, crack initiation stress σci and damage stress σcd of rock increase with the increase in σ2, while the variation of residual strength σr with σ2 is different from that of peak strength. As σ2 increases or σ3 decreases, the pre-peak plasticity of the rock decreases, the post-peak curve shape of the rock changes significantly, and the overall shape of the post-peak curve transforms from Class I to Class II. The Mogi-Coulomb strength criterion with intermediate principal stress effect was used to characterize the rock microelement strength, and then the new statistical damage constitutive model considering the initial compaction effect and damage threshold was proposed, and the model validation was carried out based on the experimental results and the effects of the proposed model parameters m and F0 on the rock strength and post-peak curve morphology were further explored.

Investigating the aging damage evolution characteristics of layered hard sandstone using digital image correlation

• Associative dimension of AE ring counts had a negative correlation with peak stress. • The peak correlative dimension implied a judgmental marker for damage aggravation. • The displacement field had a spatiotemporal dissimilation characteristics. • The damage constitutive model was established by introducing the influence factor. Layered sedimentary hard rock is an important inducing factor that deforms the tunnel structure. To explore the aging deformation and mechanical characteristics of damaged layered hard sandstone, the digital image correlation technique and acoustic emission (AE) was used to obtain the displacement and strain fields and AE signals under uniaxial compression. The results showed that mechanical parameters and failure models had significant heterogeneity. The more significant the layering effect, the less obvious the bearing capacity. The AE ringing count has fractal characteristics and associative dimensions of 0-90° samples were 1.2067, 1.3497, 1.6788, 1.7400 and 1.4799, respectively. The greater the bearing strength, the greater the associative dimension. The associative dimension had a phase response to the applied loading stress, and the rupture process had dimensionality reduction features. The peak information of correlative dimension during plastic deformation was used as an effective index for internal damage aggravation. The dissimilation phenomenon of displacement field caused synergistic features and resulted in spatiotemporal response characteristics. The displacement fields had an initial disordered distribution and evolved to an ordered structure, and then localized dissimilation occurred in banded dissociation zones. The dislocation performance of isodisplacement lines containing potential cracks and formation of band-like dissimilation zone indicated penetration of cracks. The influence factor was introduced to establish a segmental statistical damage model and the performed analyses revealed the model's accuracy. The more obvious the structural effect, the higher the influence factor.

Damage Evolution Modelling for Rock Materials Based on the Principle of Least Energy Dissipation Rate within Irreversible Thermodynamics.

The nonlinear mechanical behavior of rock significantly influences the design and construction of underground structures. Due to the complexity and diversity of the damage mechanisms of rock, the damage variable directly defined by partial-damage mechanisms is insufficient in reflecting the progressive-failure process of rock comprehensively. So, in this paper, a novel damage variable is introduced into the plastic-strain rate based on the theoretical framework of irreversible thermodynamics to overcome this defect. The general expression is derived according to the least energy dissipation rate principle. The proposed damage variable can represent the irreversible energy dissipation process and has a strictly theoretical basis in mechanics. Moreover, the granite and marble stress-strain curves are simulated and compared with the Lemaitre damage model, Mazars damage model, and statistical damage model. The results show that the form of the proposed damage variable is practical and straightforward and can better reflect the entire stress-strain relationship of rock. Furthermore, the initial value of the inelastic response strain can be given directly through the proposed damage variable. The model presented here can overcome the issue that the current models need to select the damage threshold indirectly or assume it in advance and ensures that the damage evolution characteristics follow the first principle entirely.