Statistical Damage Constitutive Model Research Articles

An Improved Statistical Damage Constitutive Model for Granite under Impact Loading

To study the impact properties of granite, the parameters (including the stress‐strain curve, elasticity modulus, peak strength, and peak strain) of the test pieces in each group were determined via standard split‐Hopkinson pressure bar tests. The results revealed that the prepeak stress‐strain curves are approximately linear; the postpeak stress‐strain curve declined sharply and exhibited the characteristics of brittle material failure after the stress exceeded the peak strength. In terms of the specimen form following failure, for increasing strain rate, the granite specimen became increasingly fragmented after failure. In addition, the single‐parameter statistical damage constitutive model was improved, and a double‐parameter statistical damage constitutive model for describing the total stress‐strain curve of granite under the action of impact loading was proposed. The parameters of the statistical damage model, m and a, were obtained via fitting. The results revealed that the parameter m decreases with increasing elasticity modulus, whereas the parameter a increases. Similarly, the peak strength and the peak strain increased (in general) with increasing strain rate.

Theoretical and Experimental Study on Damage Properties of Surrounding Rock under High‐Frequency Constant Impact Load

This paper aims to investigate the mechanics and damage properties of granite by drilling a center hole in Φ50 × 100 mm standard granite specimens under high‐frequency constant impact load and then applying them to the uniaxial compression experiment through the INSTRON 1346 universal material testing machine. According to the experimental results, under constant impact load, as the center hole diameter increases, the peak stress of the rock specimen increases accordingly and the effective elastic modulus of the rock specimen decreases first and then increases gradually. In this study, by theoretical analysis of the surrounding rock damage caused under high‐frequency impact load, a statistical damage constitutive model that has taken surrounding rock damage into consideration is established on the basis of the Weibull distribution. Meanwhile, the experimental curve is obtained to analyze the damage and damage radius of surrounding rock under high‐frequency impact load. The results show that the surrounding rock damage and damage ratio (the ratio of damage radius to center hole radius) of rock decrease with the increase of the center hole diameter. This model that can effectively describe the damage laws of surrounding rock under impact load serves as a guide for the design and development of composite rock‐drilling tools.

Statistical Damage Constitutive Model for Cemented Sand considering the Residual Strength and Initial Compaction Phase

Based on continuum damage mechanics and the assumption of volume invariance, a damage constitutive model of cemented sand under triaxial stress was established while considering residual strength. Statistical theory was then introduced into this model. Assuming that the microunit strength of cemented sand obeys a Weibull random distribution, an expression of microunit strength based on the Mohr–Coulomb criterion was derived. Additionally, a damage evolution equation and a statistical damage constitutive model of cemented sand under triaxial stress were established. In order to consider the nonlinear deformation and volume change in the initial pore compaction stage, the critical point reflecting the completion of the initial compaction stage was determined. This was done by applying the volume invariance assumption to the linear portion of the stress and strain curve and performing a coordinate transformation. The nonlinearity of the initial compaction stage was fitted by a quadratic function. A triaxial compression test of cemented sand was then carried out to verify this proposed method. The results show that the calculated values by the damage constitutive model fit well with the actual experimental values and that the calculated results can reflect the stress softening, residual strength, and initial compaction characteristics of cemented sand, which shows the rationality and feasibility of the model.

A study of test and statistical damage constitutive model of multi-size polypropylene fiber concrete under impact load

To investigate the effects of mixing polypropylene fiber of different sizes and the effect of fiber size on the impact characteristics of concrete, two sizes of polypropylene fine fiber and one size of polypropylene coarse fiber were selected to design and fabricate nine groups of polypropylene fiber-reinforced concrete test specimens by controlling the fiber mixing ratio and conducting a split Hopkinson pressure bar test to obtain the stress–strain curves of the test specimens in various groups, and their parameters such as the elasticity modulus, peak strength, and peak strain. The incorporation of fiber improved the pre-peak impact properties of concrete to different extents. Strain hardening did not occur in the post-peak curves, and different types of fibers exhibited different characteristics. Thus, the fine fiber could significantly improve the peak strain, while the coarse fiber could more significantly improve the elasticity modulus and peak strength. The improving effects exerted by incorporating three types of fiber were better than those exerted by incorporating two types of fiber. Moreover, the statistical damage model was used to obtain the parameters by fitting and analyzing their variation rules based on the statistical damage constitutive model and the particle swarm optimization algorithm.

A Statistical Constitutive Model considering Deterioration for Brittle Rocks under a Coupled Thermal-Mechanical Condition

Due to active actions of groundwater and geothermal, the stability of underground engineering is important during geological structure active area. The damage mechanical theory and statistical mesoscopic strength theory based on Weibull distribution are widely used to discuss constitutive behaviors of rocks. In these theories, a statistical method is used to capture mesoscopic properties of rocks in order to generate a realistic behavior at a macroscopic scale. Based on the above theories, this paper aims at establishing a constitutive relation of brittle rocks under thermal-mechanical coupling conditions. First, a statistical damage constitutive model was established by considering the thermal effects and crack initiation strength. Subsequently, the parameters of the model were determined and expressed according to the characteristics of stress-strain curve. Third, the model was verified by conventional triaxial experiments under thermal-mechanical actions, and the experimental data and theoretical results were compared and analyzed in the case study. Finally, the physical meaning of the parameters and their effects on the model performance were discussed.

Mechanical Behavior and Damage Constitutive Model of Granite Under Coupling of Temperature and Dynamic Loading

Dynamic compression tests of Huashan granite were conducted using an improved split–Hopkinson pressure bar. The effects of the treatment temperature and strain rate on the mechanical behaviors (for example, the stress–strain curve, dynamic strength, elastic modulus, energy absorption, and failure mode) of the granite samples were explored. In addition, a statistical damage constitutive model for the rock was developed based on a Weibull distribution, and the influencing factors of the model parameters were analyzed. The results show that the enhancement effect of the strain rate on dynamic compressive strength under high temperatures still exists. However, the strain rate has no significant effect on the elastic modulus. The influences of the treatment temperature on the dynamic strength and elastic modulus are complex. There is a positive linear correlation between the energy absorbed by the sample and the incident energy. As the strain rate or incident energy increases, the failure modes of heat-treated samples change from axial splitting to pulverization. Under the same dynamic loading, an increase in the temperature can exacerbate the fragmentation degree of the sample. The proposed statistical damage constitutive model can accurately describe the effects of the treatment temperature and strain rate on the stress–strain responses of rock, and its parameters have definite physical meanings. Thus, the model is a very good tool for the analysis of thermo-mechanical coupling problems involved in deep rock mass engineering.

Compressive behavior of concrete under high strain rates after freeze-thaw cycles

The dynamic compressive behavior of concrete after freezing and thawing tests are investigated by using the split Hopkinson pressure bar (SHPB) technique. The stress-strain curves of concrete under dynamic loading are measured and analyzed. The setting numbers of freeze-thaw cycles are 0, 25, 50, and 75 cycles. Test results show that the dynamic strength decreases and peak strain increases with the increasing of freeze-thaw cycles. Based on theWeibull distribution model, statistical damage constitutive model for dynamic stress-strain response of concrete after freeze-thaw cycles was proposed. At last, the fragmentation test of concrete subjected to dynamic loading and freeze-thaw cycles is carried out using sieving statistics. The distributions of the fragment sizes are analyzed based on fractal theory. The fractal dimensions of concrete increase with the increasing of both freeze-thaw cycle and strain rate. The relations among the fractal dimension, strain rates and freeze-thawing cycles are developed.

Investigation on the Nonlinear Strength Properties and Damage Statistical Constitutive Model for Frozen Sandy Soils

There are many flaws, such as fissures, cavities, and inclusions, in geomaterials, which make their mechanical properties with great randomness and uncertainty. Upon loading, the soil structure gradually losses the bearing capacity due to the transformation from microdefects to macroscopic breakage bands. Based upon the experimental data of frozen sandy soils, a new nonlinear strength equation between the first and third principal stresses was proposed, and then the nonlinear strength properties for frozen sandy soils in σ‐τ plane were analyzed. In addition, by assuming that the microstrength of frozen sandy soil obeys the Weibull distribution function, a statistical damage constitutive model was established based upon the framework of continuum damage mechanics (CDM), with few parameters and a high accuracy. Compared with experimental data, the new model can well grasp the nonlinear strength properties and simulate the stress‐strain relationships under different confining pressures for frozen sandy soils.

Mechanism of Permeability Evolution for Reservoir Sandstone with Different Physical Properties

Permeability of sandstones with different properties taken from Chongqing reservoirs has been measured and deeply discussed under increasing deviatoric stress. Corresponding to the distinct features in the stress-strain behaviors, the permeability of sandstones is found to evolve with a clear permeability decrease in the initial closure region, a constant permeability value in the elastic region, a permeability increase in the crack initiation and propagation region, a sharp permeability increase in the crack growth region, and a decrease permeability in the residual stage. The results also show that the variation patterns of permeability are similar for two reservoir sandstones under combination of confining pressure and water pressure; however, the strength and permeability are smaller for the sandstone with mud than that without mud, deeply indicating that mud-like materials have a relatively great impact on the mechanical properties and permeability, so mud components cannot be ignored for prediction of reservoir permeability. Furthermore, a statistical damage constitutive model considering hydraulic-mechanical coupling process is presented to calculate the damage variable D, illustrating that larger water pressure will result in a relatively smaller damage variables D and corresponding maximum, which explains that the permeability increases more rapidly and is larger for the sandstone without mud than that with mud, and sandstone damage related to corresponding circumferential crack strain and permeability has been investigated, also implying the evolution mechanism of permeability for two sandstones with different physical properties. Therefore, it is worth pointing out that rock physical properties have a great influence on the reservoir permeability under complex extraction conditions and cannot be ignored, which is necessary to improve the recovery ratio and productivity.

Researches on Damage Evolution and Acoustic Emission Characteristics of Rocks

Mechanical parameters of the rock are important for the design of geotechnical, mining engineering, and petroleum reservoir projects. Many researches have suggested that the mechanical variables of rock specimens, such as compressive strength and elastic modulus, do not have a single fixed value. Uncertainty in the basic mechanical variables of the rock material can significantly affect the structural performance and safety. In this study, a series of compression experiments with acoustic emission have been performed on rock specimens. The damage evolution characteristics of the rock in the process of loading were studied, and the macromechanical behaviors were obtained at the same time. Distribution characteristics of the strength and elastic modulus as random variables are illustrated, and the statistical damage model is presented by the authors to formulate analytical constitutive relations for deformation behavior. The comparisons between predicted results and experimental data show that the statistical damage constitutive model could well reproduce the deformation process of rock materials.

Constitutive Model of Rock Uniaxial Damage Based on Rock Strength Statistics

Based on the assumption that rock strength follows the log normal distribution statistically, this paper establishes a damage constitutive model of rock under uniaxial stress conditions in combination with the Mohr–Coulomb strength criterion and damage mechanical theory. Experiments were carried out to investigate the damage evolution process of rock material, which can be categorized into nondamaging, accelerated growth, constant‐speed, similar growth, and speed‐reducing growth stages. The evolution process had a good corresponding relationship with the rock stress‐strain curves. Based on the statistical damage constitutive model proposed in this paper, a numerical fitting analysis was conducted on the uniaxial compression testing data of laboratory sand rock and on experimental data from other literature, in order to validate the rationality of the constitutive equation and the determination of its parameters and to analyze the effect of internal friction variables on damage variables and compression strength. The research outcomes presented in this paper can provide useful reference for the theory of rock mechanics and for rock engineering.

A New Equivalent Statistical Damage Constitutive Model on Rock Block Mixed Up with Fluid Inclusions

So far, there are few studies concerning the effect of closed “fluid inclusions” on the macroscopic constitutive relation of deep rock. Fluid-matrix element (FME) is defined based on rock element in statistical damage model. The properties of FME are related to the size of inclusions, fluid properties, and pore pressure. Using FME, the equivalent elastic modulus of rock block containing fluid inclusions is obtained with Eshelby inclusion theory and the double M-T homogenization method. The new statistical damage model of rock is established on the equivalent elastic modulus. Besides, the porosity and confining pressure are important influencing factors of the model. The model reflects the initial damage (void and fluid inclusion) and the macroscopic deformation law of rock, which is an improvement of the traditional statistical damage model. Additionally, the model can not only be consistent with the rock damage experiment date and three-axis compression experiment date of rock containing pore water but also describe the locked-in stress experiment in rock-like material. It is a new fundamental study of the constitutive relation of locked-in stress in deep rock mass.

A Study on the Mechanical Behavior and Statistical Damage Constitutive Model of Sandstone

Triaxial compression test results of sandstone indicate that the peak point strain, elastic modulus, peak deviatoric stress and residual deviatoric stress of the tested sandstone increase with increasing confining pressure, and the variations in them with the confining pressure can be described with a linear function, a logistic function, the generalized Hoek–Brown criterion and the linear Mohr–Coulomb criterion, respectively. Supposing that the rock material can be divided into an elastic part and a damaged part in the rock failure process, the deviatoric stress–strain relationship of the elastic part satisfies Hooke’s law, while the damaged part provides residual deviatoric stress. On this basis, it was assumed the rock meso-element strength follows a composite power function distribution. Then, the damage evolution equation was deduced using a statistical method, and a new damage model, which can reflect the rock residual deviatoric stress, was proposed. The reasonability of the new model was verified using the test results of the sandstone. A comparison of the predicted and test results shows that this damage model can well simulate the deviatoric stress–strain response in the failure process of the tested sandstone. In particular, it can reflect the residual deviatoric stress after rock failure.

A statistical damage constitutive model under freeze-thaw and loading for rock and its engineering application

The freeze-thaw damage of rock is a critical problem to study rock engineering in cold regions. When water is frozen in pores, 9% volume expansion will induce damage in rock for a huge ice pressure. Thus, rock will be deteriorated and softened after freeze-thaw. Besides, loading damage produces when the inner stress exceeds the bearing capacity of rock. So it's crucial to establish a damage constitutive model under freeze-thaw and loading in order to evaluate the stability of rock engineering in cold regions. The freeze-thaw damage of rock is expressed by static elastic modulus in this paper, which can be accurately and simply predicted by damage evolution equation using P-wave velocity or compression strength proposed by Liu et al., 2015. Loading damage can be obtained by statistical theory assuming that micro-unit strength satisfies the Weibull distribution and the maximum-tensile-strain yield criterion. Then the final statistical damage constitutive equation under freeze-thaw and loading is derived, in which unknown parameters can be determined through carrying out compression experiment on rock after different freeze-thaw cycles. This developed model is validated by a previous experiment and applied to analyze the stability of a tunnel under coupled thermo-hydro-mechanical condition in cold regions. The results show that the proposed constitutive model is very suitable for rock suffering freeze-thaw cycles and it is of high accuracy and good practicality.

Statistical damage constitutive model for rocks subjected to cyclic stress and cyclic temperature

A constitutive model of rocks subjected to cyclic stress–temperature was proposed. Based on statistical damage theory, the damage constitutive model with Weibull distribution was extended. Influence of model parameters on the stress–strain curve for rock reloading after stress–temperature cycling was then discussed. The proposed model was initially validated by rock tests for cyclic stress–temperature and only cyclic stress. Finally, the total damage evolution induced by stress–temperature cycling and reloading after cycling was explored and discussed. The proposed constitutive model is reasonable and applicable, describing well the stress–strain relationship during stress–temperature cycles and providing a good fit to the test results. Elastic modulus in the reference state and the damage induced by cycling affect the shape of reloading stress–strain curve. Total damage induced by cycling and reloading after cycling exhibits three stages: initial slow increase, mid-term accelerated increase, and final slow increase.

Shear creep tests and creep constitutive model of marble with structural plane

The paper aims at investigating the shear creep property and creep model of marble with structural plane. Marble specimens were taken from diversion tunnel engineering of Jinping II hydropower station. A series of shear creep tests were conducted for two types of marbles with weak structural plane and hard structural plane. The results showed that the creep process of marble with weak structural plane included three creep phases when shear stress was higher. However, shear creep process of marble with hard structural plane only included transition creep phase and steady creep phase and roughness of hard structural plane affected on the shear creep property. Based on the experimental results, a time-dependent statistical damage constitutive model was proposed in terms of fractional calculus theory and damage evolution law to describe the time-dependent damage characteristics. The proposed model was analysed by utilising results of marble creep testing. The results of the analysis showed that the proposed model could describe not only the transient and steady creep phase of marble with weak or hard structural plane, but also the acceleration creep phase of marble with weak structural plane when shear stress is higher.

Studying the dynamic damage failure of concrete based on acoustic emission

Understanding the properties of concrete under load is essential for assessing the stability and serviceability of buildings. In this work, the uniaxial compression test of concrete was carried out based on acoustic emission (AE) in order to monitor the dynamic damage of concrete in real time and study the failure mechanism. The change of AE energy and ringing counts over rising load was recorded, and the 3D evolution of AE locations during the dynamic damage failure of concrete was also determined. The damage variables during the loading process were calculated based on AE activity. The results showed that the AE activity existed throughout the entire failure process and was the most prominent in the yielding phase. The damage of concrete could be predicted qualitatively by the sharp increase in both the accumulated AE energy and the ringing counts. The test on the 3D distribution of AE events illustrated the generation, growth, and connection of microcracks. The correlation between AE parameters and damage variables was discussed based on the statistical damage constitutive model and the Weibull random distribution.

Impact characterization and modelling of cement and asphalt mortar based on SHPB experiments

The dynamic behaviour of cement and asphalt mortar (CA mortar, including types CAI with the asphalt/water-to-cement ratios of 0.85/0.78 for the Shinkansen track and CAII with the asphalt/water-to-cement ratios of 0.30/0.50 for the Bögl track) was experimentally investigated under impact loading using a Split Hopkinson Pressure Bar (SHPB). The strain rate effects on the compressive strength, elastic modulus, peak strain and specific energy absorption were obtained. The results showed that the compressive strength and specific energy absorption increased with increasing strain rate, whose strain rate sensitivity for CAI was stronger than that for CAII. The strain rate sensitivity of the elastic modulus and peak strain scattered. Under similarly high strain rates, the compressive strength, elastic modulus, peak strain and specific energy absorption of CAII were larger than those of CAI. A statistical continuous damage constitutive model involving the strain rate for CA mortar was proposed based on the theory of thermodynamics and statistical damage mechanics. The theoretical results showed a consistent comparison with the experimental data.

Effect of Pre‐strain Aging on the Damage Properties of Composite Solid Propellants based on a Constitutive Equation

AbstractAccelerated aging tests under pre‐strain were conducted on HTPB‐based composite solid propellant with the goal of investigating the effect of pre‐strain aging on its damage properties. A statistical damage constitutive model based on continuum damage theory and statistical strength theory was established. The aging damage coefficient, making aging process of propellant equivalent to a form of damage, was introduced to correct the damage variable. Experimental results show that theoretical model has good agreement with experimental results and can accurately describe the mechanical behavior of propellant during pre‐strain aging. Further analysis indicated that the damage effects caused by pre‐strain can be identified from the equation of the aging damage coefficient. Aging time influences both tensile strength and shape characteristics of the stress‐strain curve of propellant in the damage stage, while pre‐strain only decreased the tensile strength. The strain damage threshold value decreased linearly over the aging period and with increasing pre‐strain level during the aging process.

Experimental and Numerical Investigation of Permeability Evolution with Damage of Sandstone Under Triaxial Compression

A series of triaxial compression tests with permeability measurements was carried out under different confining pressure and pore pressure difference coupling conditions to investigate some mechanical properties and permeability evolution with damage of sandstone. It is found that the shapes of stress–strain curves, permeability evolution curves, and failure patterns are significantly affected by the confining pressure but are only slightly affected by the pore pressure difference. In addition, the corresponding numerical simulations of the experiments were then implemented based on the two-dimensional Realistic Failure Process Analysis-Flow (RFPA2D-Flow) code. In this simulator, the heterogeneity of rock is considered by assuming the material properties of the mesoscopic elements conform to a Weibull distribution and a statistical damage constitutive model based on elastic damage mechanics and the flow–stress–damage (FSD) coupling model. The numerical simulations reproduced the failure processes and failure patterns in detail, and the numerical results about permeability–strain qualitatively agree with the experimental results by assigning different parameters in the FSD model. Finally, the experimental results about relationship between permeability evolution and volumetric strain are discussed.