Study on Damage Statistical Constitutive Model of Triaxial Compression of Acid-Etched Rock under Coupling Effect of Temperature and Confining Pressure

Based on Lemaitre’s strain equivalence hypothesis theory, it is assumed that the strength of acid-etching rock microelements under the coupling effect of temperature and confining pressure follows the Weibull distribution. Under the hypothesis that micro-element damage meets the D-P criterion and based on continuum damage mechanics and statistical theory, chemical damage variables, thermal damage variables and mechanical damage variables were introduced in the construction of damage evolution equations and constitutive models for acid-etching rocks considering the coupled effects of temperature and confining pressure. The required model parameters were obtained by theoretical derivation, and the model was verified based on the triaxial compression test data of granite. Comparing the experimental stress-strain curve with the theoretical stress-strain curve, the results show that they were in good agreement. By selecting reasonable model parameters, the damage statistical constitutive model can accurately reflect the stress-strain curve characteristics of rock in the process of triaxial compression. The comparison between the experimental and theoretical results also verifies the reasonableness and reliability of the model. This model provides a new rock damage statistical constitutive equation for the study of rock mechanics and its application in engineering, and has certain reference significance for rock underground engineering.     

Deterioration of Mechanical Properties and the Damage Constitutive Model of Corroded Steel in an Industrial Environment

To investigate the degradation law of the mechanical properties of corroded steel, the standard specimens from machining steel members in service for 9 years in an industrial environment were subjected to tensile tests. The influences of different degrees and types of corrosion on the fracture path, stress-strain curve, and mechanical properties of specimens were discussed. Finally, the damage constitutive model of corroded steel was established based on the damage mechanics theory. The results showed that the failure modes of corroded specimens were related to the degrees and types of corrosion. The fracture morphology of specimens with general corrosion were step-like and the fractures of steel were uneven. However, those with local corrosion were mainly flat-like, and the fracture path was along the cross section where the larger corrosion pits were located. The fracture path of the specimen was related to the interaction of the corroded surface and internal material defects (holes). Meanwhile, with the increase of corrosion degree, the yield platform of stress-strain curve gradually became shorter, or even disappeared, and the ultimate strain and elongation at break decreased, implying that the ductility of steel became worse. Ultimately, the good agreement between the curves of the model and test indicated that the damage model could reflect the damage development process of corroded steel in the tensile process better. Corrosion damage resulted in the decrease in the damage threshold, and the damage variable D decreased by the time fracture occurred and the maximum reduction rate was up to 62.5%.     

Constitutive Model and Fracture Failure of Sandstone Damage under High Temperature–Cyclic Stress

Deformation and damage characteristics of high-temperature rocks during underground coal gasification are the fundamental mechanical problems encountered in coal-bed gasification production. In order to study the characteristics of deformation and damage processes of rocks under the joint action of high temperature and high stress, a damage intrinsic model under the joint action of high temperature–cyclic loading is needed. In this paper, we used the damage mechanics theory to establish the damage instantiation model of rocks under the joint action of high temperature loads based on the Mohr–Coulomb strength criterion. It was found that the higher the temperature, the lower the strength of sandstone, the lower the peak stress, and the higher the peak strain, the peak stress decreased from 95.6 MPa at room temperature to 74.8 MPa at 400 °C to 49.5 MPa at 800 °C, and the peak strain increased from room temperature to 400 °C to 800 °C by 27.9% and 33.4%, respectively. With the increase in temperature, the internal microcracks of sandstone increased and expanded, which caused the degree of damage to intensify, and the macroscopic expression was the reduction in strength and stiffness of the sandstone. The rock went through four stages from loading to damage, including damage weakening, plastic deformation, strain softening, and residual deformation. The four types of fractures that led to the overall rupture of the rock were open fracture, secondary coplanar fracture, secondary inclined fracture, and oblique fracture. The damage intrinsic model constructed in this paper could better reflect the damage process of thermally damaged sandstone under the action of periodic loading, and had certain rationality. The damage intrinsic evolution curve, as well as the damage mechanism of sandstone under the action of macroscopic high temperature-loading, were discussed from the perspective of microscopic damage evolution, and the fracture extension pattern and penetration mechanism of the rock under different temperatures were analyzed. The research results provide an important reference for the design and engineering application of gasifiers in coal-bed underground gasification projects.     

A Constitutive Material Model Applied to Microforming Processes

The plastic treatment of products reduced to sizes corresponding to the microscale poses difficulties, due to the occurrence of the so-called size effect, which is responsible for the different behavior of the material during the realization of microforming. In this study, a constitutive equation was elaborated taking into account two types of size effects, with the use of a surface model as well as a composite material model. The influence of the size effect referring to both the material grain size and the geometric scaling of the sample size on the material’s flow stresses was considered. The surface model took into account the different grain shapes present in actual polycrystalline materials. After the application of the presented model for titanium Grade 2, a good agreement of the experimental results with the FEM simulation results was obtained. Thus, the proper FEM modeling of microforming processes should be conducted with the use of a material model, taking into account the occurring size effects.     

Deformation Behavior and Constitutive Model of 34CrNi3Mo during Thermo-Mechanical Deformation Process

With higher creep strength and heat resistance, 34CrNi3Mo has been widely used in the production of engine rotors, steam turbine impellers, and turbine blades. To investigate the hot deformation behaviors of 34CrNi3Mo steel, hot compressive tests were conducted on a Gleeble-3500 thermomechanical simulator, under the temperature range of 1073 K–1373 K and strain rate ranges of 0.1 s⁻¹–20 s⁻¹. The results show that the flow stress of 34CrNi3Mo steel under high temperatures is greatly influenced by the deformation temperature and strain rate, and it is the result of the interaction between strain hardening, dynamic recovery, and recrystallization. Under the same deformation rate, as the deformation temperature increases, the softening effect of dynamic recrystallization and dynamic recovery gradually increases, and the flow stress gradually decreases. Under the same deformation temperature, with the increase of strain rate, the influence of strain hardening on 34CrNi3Mo steel is gradually in power, and the flow stress gradually increases. To predict the flow stress of 34CrNi3Mo steel accurately, a modified Arrhenius-type constitutive model considering the effects of strain, temperature, and strain rate at the same time was made based on the experiment data. On this basis, the evolution law of deformation activation and instability characteristics of 34CrNi3Mo steel were investigated, and the processing map of 34CrNi3Mo steel was established. The formability of 34CrNi3Mo steel under high temperature deformation was revealed, which provided a theoretical foundation of the equation of reasonable hot working process.     

Constitutive Modeling of the Densification Behavior in Open-Porous Cellular Solids

The macroscopic mechanical behavior of open-porous cellular materials is dictated by the geometric and material properties of their microscopic cell walls. The overall compressive response of such materials is divided into three regimes, namely, the linear elastic, plateau and densification. In this paper, a constitutive model is presented, which captures not only the linear elastic regime and the subsequent pore-collapse, but is also shown to be capable of capturing the hardening upon the densification of the network. Here, the network is considered to be made up of idealized square-shaped cells, whose cell walls undergo bending and buckling under compression. Depending on the choice of damage criterion, viz. elastic buckling or irreversible bending, the cell walls collapse. These collapsed cells are then assumed to behave as nonlinear springs, acting as a foundation to the elastic network of active open cells. To this end, the network is decomposed into an active network and a collapsed one. The compressive strain at the onset of densification is then shown to be quantified by the point of intersection of the two network stress-strain curves. A parameter sensitivity analysis is presented to demonstrate the range of different material characteristics that the model is capable of capturing. The proposed constitutive model is further validated against two different types of nanoporous materials and shows good agreement.     

Dynamic Characteristics and Damage Constitutive Model of Mudstone under Impact Loading

The mechanical response characteristics of mudstone from the ingate roadway of the west ventilation shaft in Yuandian No. 2 coal mine, Huaibei City, Anhui Province, China to dynamic loads were quantified in single- and cyclic-impact compression tests, using the split-Hopkinson pressure bar test device. The dynamic stress–strain relationships and the failure characteristics of mudstone samples under different impact loads were analyzed systematically. Considering the “rate effect” of the mudstone dynamic strength, the dynamic strength criterion of mudstone was proposed, and the dynamic damage constitutive model of mudstone was established, based on the statistical damage theory. In response to single-impact loads, with increasing impact pressure, the mudstone peak stress and strain gradually increased, and the peak stress and average strain rate increased nonlinearly. In response to cyclic-impact loads, with an increasing number of impacts, the mudstone peak stress first increased and then decreased, and the peak strain increased gradually. With increasing impact pressure, the number of impacts to the samples’ failure decreased gradually. By parameter identification and comparative analysis of the test results, the proposed dynamic damage constitutive model of mudstone was validated. The model can be used for stability analysis of roadway-surrounding rock under dynamic loads.     

A Nonlinear Viscoelastic Constitutive Model for Solid Propellant with Rate-Dependent Cumulative Damage

Solid propellant is a composite material exhibiting classic nonlinear viscoelastic mechanical characteristic, which is due in a large part to a cumulative damage process caused by the formation and growth of microflaws inside. The standard relaxation tests and uniaxial tension tests under different velocities of hydroxyl-terminated polybutadiene (HTPB) propellant are carried out in this paper, where Digital Image Correlation (DIC) technique is applied to record deformation. The experimental results show that the material mechanical behavior is rate-dependent. It is also observed that the yield stress and failure stress are significantly rate-dependent on the tensile velocity. Based on these experimental results, it can be inferred that the stiffness degradation and damage evolution of HTPB propellant are a rate-dependent processes. Therefore, the damage accumulation of HTPB propellant is considered rate-dependent in this research. In order to describe the mechanical characteristic precisely, a nonlinear viscoelastic constitutive model with rate-dependent cumulative damage is developed. The damage model is developed based on the concept of pseudo strain, in which a Prony series representation of viscoelastic material functions is applied. Besides, a rate-dependent damage variable is introduced into the model through considering the rate-dependent characteristics of cumulative damage process. In addition, a new normalized failure criterion is derived on the basis of the proposed damage model, which is independent of strain-rate after normalization. Finally, it is implemented in commercial finite element software for stress analysis to verify the predictive capacities of the damage model. The accuracy of the constitutive model and failure criterion is validated under uniaxial tensile tests of various strain rates.     

Influence of Microstructure on Dynamic Mechanical Behavior and Damage Evolution of Frozen–Thawed Sandstone Using Computed Tomography

Frost-induced microstructure degradation of rocks is one of the main reasons for the changes in their dynamic mechanical behavior in cold environments. To this end, computed tomography (CT) was performed to quantify the changes in the microstructure of yellow sandstone after freeze–thaw (F–T) action. On this basis, the influence of the microscopic parameters on the dynamic mechanical behavior was studied. The results showed that the strain rate enhanced the dynamic mechanical properties, but the F–T-induced decrease in strength and elastic modulus increased with increasing strain rate. After 40 F–T cycles, the dynamic strength of the samples increased by 41% to 75.6 MPa when the strain rate was increased from 75 to 115 s⁻¹, which is 2.5 times the static strength. Moreover, the dynamic strength and elastic modulus of the sample were linearly and negatively correlated with the fractal dimension and porosity, with the largest decrease rate at 115 s⁻¹, indicating that the microscopic parameters have a crucial influence on dynamic mechanical behavior. When the fractal dimension was increased from 2.56 to 2.67, the dynamic peak strength of the samples under the three impact loads decreased by 43.7 MPa (75 s), 61.8 MPa (95 s), and 71.4 MPa (115 s), respectively. In addition, a damage evolution model under F–T and impact loading was developed considering porosity variation. It was found that the damage development in the sample was highly related to the strain rate and F–T damage. As the strain rate increases, the strain required for damage development gradually decreases with a lower increase rate. In contrast, the strain required for damage development in the sample increases with increasing F–T damage. The research results can be a reference for constructing and maintaining rock structures in cold regions.     

Time-Dependent Damage Investigation of Rock Mass in an In Situ Experimental Tunnel

In underground tunnels or caverns, time-dependent deformation or failure of rock mass, such as extending cracks, gradual rock falls, etc., are a costly irritant and a major safety concern if the time-dependent damage of surrounding rock is serious. To understand the damage evolution of rock mass in underground engineering, an in situ experimental testing was carried out in a large belowground tunnel with a scale of 28.5 m in width, 21 m in height and 352 m in length. The time-dependent damage of rock mass was detected in succession by an ultrasonic wave test after excavation. The testing results showed that the time-dependent damage of rock mass could last a long time, i.e., nearly 30 days. Regression analysis of damage factors defined by wave velocity, resulted in the time-dependent evolutional damage equation of rock mass, which corresponded with logarithmic format. A damage viscoelastic-plastic model was developed to describe the exposed time-dependent deterioration of rock mass by field test, such as convergence of time-dependent damage, deterioration of elastic modules and logarithmic format of damage factor. Furthermore, the remedial measures for damaged surrounding rock were discussed based on the measured results and the conception of damage compensation, which provides new clues for underground engineering design.     

Research on Soft Rock Damage Softening Model and Roadway Deformation and Failure Characteristics

To determine a reasonable control strategy for deep buried soft rock roadways, a study on deformation and failure characteristics was carried out. The Weibull distribution damage variable was introduced to construct a damage-softening model considering the lateral deformation of the rock mass, and the functional relationship between the model parameters F₀ and m and the confining pressure were discussed. The nonlinear fitting method was used to correct the model parameters. Using the model, the failure characteristics of deep buried soft rock roadways were analyzed. A comprehensive and step-by-step joint support control strategy was proposed based on the numerical simulation results. The research results showed that the damage-softening model curve established could genuinely reflect the whole process of mudstone failure. The apparent stress concentration phenomenon occurred in the surrounding rock. The surrounding rock deformation showed that roadway floors had larger plastic failure areas than sides and vaults. The plastic failure depth could reach 2.45 m. After a comprehensive and step-by-step joint support control strategy was adopted, the deformation rate of the roadway at the section was less than 0.1 mm/d. The optimized support scheme can effectively improve the stability of the roadway.     

Research on the Springback Behavior of 316LN Stainless Steel in Micro-Scale Bending Processes

Hydrogen fuel cells have been used worldwide due to their high energy density and zero emissions. The metallic bipolar plate is the crucial component and has a significant effect on a cell’s efficiency. However, the springback behavior of the metallic bipolar plate will greatly influence its forming accuracy in the micro-scale sheet metal forming process. Therefore, accurate calculation of the springback angle of the micro-scale metallic bipolar plate is urgent but difficult given the state of existing elastoplastic theory. In this paper, a constitutive model that simultaneously considers grain size effect and strain gradient is proposed to analyze micro-scale bending behavior and calculate springback angles. The specialized micro-scale four-point bending tool was designed to better calculate the springback angle and simplify the calculation step. A pure micro-bending experiment on a 316LN stainless steel sheet with a thickness of 0.1 mm was conducted to verify the constitutive model’s accuracy.     

Investigation of Damage Evolution in Heterogeneous Rock Based on the Grain-Based Finite-Discrete Element Model

Granite exhibits obvious meso-geometric heterogeneity. To study the influence of grain size and preferred grain orientation on the damage evolution and mechanical properties of granite, as well as to reveal the inner link between grain size‚ preferred orientation, uniaxial tensile strength (UTS) and damage evolution, a series of Brazilian splitting tests were carried out based on the combined finite-discrete element method (FDEM), grain-based model (GBM) and inverse Monte Carlo (IMC) algorithm. The main conclusions are as follows: (1) Mineral grain significantly influences the crack propagation paths, and the GBM can capture the location of fracture section more accurately than the conventional model. (2) Shear cracks occur near the loading area, while tensile and tensile-shear mixed cracks occur far from the loading area. The applied stress must overcome the tensile strength of the grain interface contacts. (3) The UTS and the ratio of the number of intergrain tensile cracks to the number of intragrain tensile cracks are negatively related to the grain size. (4) With the increase of the preferred grain orientation, the UTS presents a “V-shaped” characteristic distribution. (5) During the whole process of splitting simulation, shear microcracks play the dominant role in energy release; particularly, they occur in later stage. This novel framework, which can reveal the control mechanism of brittle rock heterogeneity on continuous-discontinuous trans-scale fracture process and microscopic rock behaviour, provides an effective technology and numerical analysis method for characterizing rock meso-structure. Accordingly, the research results can provide a useful reference for the prediction of heterogeneous rock mechanical properties and the stability control of engineering rock masses.     

Damage Evolution of Hot Stamped Boron Steels Subjected to Various Stress States: Macro/Micro-Scale Experiments and Simulations

Hot stamping components with tailored mechanical properties have excellent safety-related performance in the field of lightweight manufacturing. In this paper, the constitutive relation and damage evolution of bainite, martensite, and mixed bainite/martensite (B/M) phase were studied. Two-dimensional representative volume element (RVE) models were constructed according to microstructure characteristics. The constitutive relations of individual phases were defined based on the dislocation strengthening theory. Results showed that the damage initiation and evolution of martensite and bainite phases can well described by the Lou-Huh damage criterion (DF2015) determined by the hybrid experimental–numerical method. The calibrated damage parameters of each phase were applied to the numerical simulation, followed by the 2D RVE simulations of B/M phase under different stress states. To study the influence of martensite volume fraction (Vm) and distribution of damage evolution, the void nucleation and growth were evaluated by RVEs and verified by scanning electron microscope (SEM). Three types of void nucleation modes under different stress states were experimentally and numerically studied. The results showed that with the increase of Vm and varying martensite distribution, the nucleation location of voids move from bainite to martensite.     

Complete Stress–Strain Relations of Early-Aged Cementitious Grout under Compression: Experimental Study and Constitutive Model

The compressive stress–strain behaviors of early-aged cementitious grout specimens were experimentally investigated, and the differences of characteristic parameters of the stress–strain curve and the energy evolution law of each specimen under uniaxial compression were discussed in this study. The results indicate that with an increase in the specimen age, the peak stress, peak strain, ultimate strain, elastic modulus, peak secant modulus, strain ductility coefficient, and energy-dissipation coefficient of the prism specimens gradually improved. Additionally, a comparison of the test results of cementitious grout specimens and concrete specimens with the same age reveals that the peak stress, peak strain, and ultimate strain of cementitious grout specimens were greater than that of concrete specimens, the elastic modulus and peak secant modulus of the specimens were less than that of concrete specimens, and the strain ductility coefficient and energy-dissipation coefficient show no consistent conclusions with respect to the material type. Moreover, comparing the energy evolution curves of specimens with different specimen ages shows that the decrease rate of the elastic strain rate and the increase rate of the dissipated energy rate gradually decreased with the increase in specimen age. The elastic strain rate and dissipated energy rate of the CGM−270 specimen and control specimens were greater than that of other specimens, and the decrease rate of the elastic strain rate was less than that of other specimens. Based on the statistical damage theory, a statistically stochastic damage constitutive model was derived by considering the characteristics of cementitious grout according to the compression test results. A comparison of the proposed models with the experimental results indicated that the proposed stress–strain constitutive models were sufficiently accurate.     

Experimental Investigation of Moisture Sensitivity and Damage Evolution of Porous Asphalt Mixtures

Porous asphalt (PA) mixtures are designed with a high air void (AV) (i.e., 18~22%) content allowing rainwater to infiltrate into their internal structures. Therefore, PA mixtures are more sensitive to moisture damage than traditional densely graded asphalt mixtures. However, the moisture damage evolution of PA mixtures is still unclear. The objective of this study was to investigate the moisture damage evolution and durability damage evolution of PA mixtures. The indirect tensile test (ITT), ITT fatigue test, and Cantabro loss test were used to evaluate the moisture sensitivity and durability of PA mixtures, and a staged ITT fatigue test was developed to investigate the damage evolutions under dry and wet conditions. Indirect tensile strength (ITS), fatigue life, indirect tensile resilience modulus (E), and durability decreased with the increment of moisture damage and loading cycles. The fatigue life is more sensitive to the moisture damage. The largest decrements in ITS and E were found in the first 3000 loading cycles, and PA mixtures tended to fail when the decrement exceeded 60%. Damage factors based on the ITS and E are proposed to predict the loading history of PA mixtures. The durability damage evolution and damage factors could fit an exponential model under dry conditions. Moisture had a significant influence and an acceleration function on the moisture damage evolution and durability damage evolution of PA mixtures.     

An Experimental and Analytical Study on a Damage Constitutive Model of Engineered Cementitious Composites under Uniaxial Tension

Engineered cementitious composites (ECC) exhibit ultra-high ductility and post-cracking resistance, which makes it an attractive material in civil engineering. First, a monotonic uniaxial tensile test was performed, considering the effects of polyvinyl alcohol (PVA) fiber volume content and water-binder ratio. Then, the effects of the above variables on the tensile characteristics including the tensile stress–strain relationship, deformation capacity, and fracture energy were investigated based on test results; and when the water-binder ratio is 0.28 and the fiber volume content is 2%, the deformation performance of ECC is improved most significantly. Next, combined with damage mechanics theory, the damage evolution mechanism of ECC in monotonic uniaxial tension was revealed, based on which the damage factor and damage evolution equation of ECC were developed and the expressions of model parameters were proposed. Moreover, the comparison between the proposed model and test results demonstrated the accuracy of the proposed model. Finally, to further verify the feasibility of the proposed model, a finite element (FE) simulation analysis of the tensile performance of high-strength stainless steel wire rope (HSSWR) reinforced ECC by adopting the proposed model was compared with test results and the simulation analysis results by using anther existing model, the “trilinear model of ECC”. The comparison shows that the proposed model in this paper can predict more accurately.     

A Chemical Damage Creep Model of Rock Considering the Influence of Triaxial Stress

In order to accurately describe the characteristics of each stage of rock creep behavior under the combined action of acid environment and true triaxial stress, based on damage mechanics, chemical damage is connected with elastic modulus; thus, the damage relations considering creep stress damage and chemical damage are obtained. The elastic body, nonlinear Kelvin body, linear Kelvin body, and viscoelastic–plastic body (Mogi–Coulomb) are connected in series, and the actual situation under the action of true triaxial stress is considered at the same time. Therefore, a damage creep constitutive model considering the coupling of rock acid corrosion and true triaxial stress is established. The parameters of the deduced model are identified and verified with the existing experimental research results. The yield surface equation of rock under true triaxial stress is obtained by data fitting, and the influence of intermediate principal stress on the creep model is discussed. The derived constitutive model can accurately describe the characteristics of each stage of true triaxial creep behavior of rock under acid environment.     

A Constitutive Model of Time-Dependent Deformation Behavior for Sandstone

Considering sandstone’s heterogeneity in the mesoscale and homogeneity in the macroscale, it is very difficult to describe its time-dependent behavior under stress. The mesoscale heterogeneity can affect the initiation and propagation of cracks. Clusters of cracks have a strong influence on the formation of macroscale fractures. In order to investigate the influence of crack evolution on the formation of fractures during creep deformation, a time-dependent damage model is introduced in this paper. First, the instantaneous elastoplastic damage model of sandstone was built based on the elastoplastic theory of rock and the micro-heterogeneous characteristics of sandstone. A viscoelastic plastic creep damage model was established by combining the Nishihara model and the elastoplastic damage constitutive model. The proposed models have been validated by the results of corresponding analytical solutions. To help back up the model, some conventional constant strain rate tests and multi-step creep tests were carried out to analyze the time-dependent behavior of sandstone. The results show that the proposed damage model can not only reflect the time-dependent viscoelastic deformation characteristics of sandstone, but also provide a good fit to the viscoelastic plastic deformation characteristics of sandstone’s creep behavior. The damage model can also reproduce the propagation process of mesoscopic cracks in sandstone upon the damage and failure of micro-units. This research can provide an effective tool for studying the propagation of microscopic cracks in sandstone.     

Study on Shear Creep Characteristics of the Discontinuities with Different 3D Morphologies

The rheological phenomenon of rock mass affects the long-term safety of rock mass engineering. In this study, gneiss samples with different 3D morphologies are prepared by splitting tests and are tested through multi-step creep tests. The long-term strength of rock discontinuities is determined by using several methods. The test results show that as the 3D morphological parameter increases, the creep deformation, creep rate, and the duration of failure all decrease. The long-term strength of rock discontinuities is linearly related to the 3D morphological parameter. Based on the principle of damage mechanics for rock mass, a damage variable is introduced in the creep model, and an improved non-linear Burgers model is established. Research results are of great theoretical significance and practical value for the design, construction, and long-term safety of rock mass engineering.