Novel damage constitutive models and new quantitative identification method for stress thresholds of rocks under uniaxial compression

Four key stress thresholds exist in the compression process of rocks,i.e.,crack closure stress(σ_(cc)),crack initiation stress(σ_(ci)),crack damage stress(σ_(cd))and compressive strength(σ_(c)).The quantitative identifications of the first three stress thresholds are of great significance for characterizing the microcrack growth and damage evolution of rocks under compression.In this paper,a new method based on damage constitutive model is proposed to quantitatively measure the stress thresholds of rocks.Firstly,two different damage constitutive models were constructed based on acoustic emission(AE)counts and Weibull distribution function considering the compaction stages of the rock and the bearing capacity of the damage element.Then,the accumulative AE counts method(ACLM),AE count rate method(CRM)and constitutive model method(CMM)were introduced to determine the stress thresholds of rocks.Finally,the stress thresholds of 9 different rocks were identified by ACLM,CRM,and CMM.The results show that the theoretical stress−strain curves obtained from the two damage constitutive models are in good agreement with that of the experimental data,and the differences between the two damage constitutive models mainly come from the evolutionary differences of the damage variables.The results of the stress thresholds identified by the CMM are in good agreement with those identified by the AE methods,i.e.,ACLM and CRM.Therefore,the proposed CMM can be used to determine the stress thresholds of rocks.

A whole process damage constitutive model for layered sandstone under uniaxial compression based on Logistic function

Bedding structural planes significantly influence the mechanical properties and stability of engineering rock masses.This study conducts uniaxial compression tests on layered sandstone with various bedding angles(0°,15°,30°,45°,60°,75°and 90°)to explore the impact of bedding angle on the deformational mechanical response,failure mode,and damage evolution processes of rocks.It develops a damage model based on the Logistic equation derived from the modulus’s degradation considering the combined effect of the sandstone bedding dip angle and load.This model is employed to study the damage accumulation state and its evolution within the layered rock mass.This research also introduces a piecewise constitutive model that considers the initial compaction characteristics to simulate the whole deformation process of layered sandstone under uniaxial compression.The results revealed that as the bedding angle increases from 0°to 90°,the uniaxial compressive strength and elastic modulus of layered sandstone significantly decrease,slightly increase,and then decline again.The corresponding failure modes transition from splitting tensile failure to slipping shear failure and back to splitting tensile failure.As indicated by the modulus’s degradation,the damage characteristics can be categorized into four stages:initial no damage,damage initiation,damage acceleration,and damage deceleration termination.The theoretical damage model based on the Logistic equation effectively simulates and predicts the entire damage evolution process.Moreover,the theoretical constitutive model curves closely align with the actual stress−strain curves of layered sandstone under uniaxial compression.The introduced constitutive model is concise,with fewer parameters,a straightforward parameter determination process,and a clear physical interpretation.This study offers valuable insights into the theory of layered rock mechanics and holds implications for ensuring the safety of rock engineering.

Parameter calibration of the tensile-shear interactive damage constitutive model for sandstone failure

The tensile-shear interactive damage(TSID)model is a novel and powerful constitutive model for rock-like materials.This study proposes a methodology to calibrate the TSID model parameters to simulate sandstone.The basic parameters of sandstone are determined through a series of static and dynamic tests,including uniaxial compression,Brazilian disc,triaxial compression under varying confining pressures,hydrostatic compression,and dynamic compression and tensile tests with a split Hopkinson pressure bar.Based on the sandstone test results from this study and previous research,a step-by-step procedure for parameter calibration is outlined,which accounts for the categories of the strength surface,equation of state(EOS),strain rate effect,and damage.The calibrated parameters are verified through numerical tests that correspond to the experimental loading conditions.Consistency between numerical results and experimental data indicates the precision and reliability of the calibrated parameters.The methodology presented in this study is scientifically sound,straightforward,and essential for improving the TSID model.Furthermore,it has the potential to contribute to other rock constitutive models,particularly new user-defined models.

Constitutive model of viscoelastic dynamic damage for the material of gas obturator in modular-charge howitzer

In order to investigate the mechanical response behavior of the gas obturator of the breech mechanism,made of polychloroprene rubber(PCR), uniaxial compression experiments were carried out by using a universal testing machine and a split Hopkinson pressure bar(SHPB), obtaining stress-strain responses at different temperatures and strain rates. The results revealed that, in comparison to other polymers, the gas obturator material exhibited inconspicuous strain softening and hardening effects;meanwhile, the mechanical response was more affected by the strain rate than by temperature. Subsequently, a succinct viscoelastic damage constitutive model was developed based on the ZWT model, including ten undetermined parameters, formulated with incorporating three parallel components to capture the viscoelastic response at high strain rate and further enhanced by integrating a three-parameter Weibull function to describe the damage. Compared to the ZWT model, the modified model could effectively describe the mechanical response behavior of the gas obturator material at high strain rates. This research laid a theoretical foundation for further investigation into the influence of chamber sealing issues on artillery firing.

Mechanical properties and damage constitutive model of sandstone after acid corrosion and high temperature treatments

Aiming at the problem of temperature-mechanics-chemical(T-M-C)action encountered by rocks in underground engineering,sandstone was selected as the sample for acid corrosion treatment at pH 1,3,5 and 7,the acid corrosion treated samples were then subjected to high-temperature experiments at 25,300,600,and 900℃,and triaxial compression experiments were conducted in the laboratory.The experimental results show that the superposition of chemical damage and thermal damage has a significant impact on the quality,wave velocity,porosity and compression failure characteristics of the rock.Based on the Lemaitre strain equivalent hypothesis theory,the damage degree of rock material was described by introducing damage variables,and the spatial mobilized plane(SMP)criterion was adopted.The damage constitutive model can well reflect the stress-strain characteristics of the rock triaxial compression process,which verified the rationality and reliability of the model parameters.The experiment and constitutive model analyzed the change law of mechanical properties of rock after chemical corrosion and high temperature thermal damage,which had certain practical significance for rock engineering construction.

Predicting the electromechanical properties of small caliber projectile impact igniter using PZT dynamic damage constitutive model considering crack propagation

Block piezoelectric ceramics(PZTs)are often used in impact igniters to provide activation energy for electric initiators.Under the action of strong impact stress,PZTs release electric energy accompanied by crack initiation,propagation and crushing.At present,the electrical output performance of PZTs in projectile is usually calculated by quasi-static piezoelectric equation without considering the dynamic effect caused by strong impact and the influence of crack propagation on material properties.So the ignition parameters are always not accurately predicted.To tackle this,a PZT dynamic damage constitutive model considering crack propagation is established based on the dynamic impact test and the crack propagation theory of brittle materials.The model is then embedded into the ABAQUS subroutine and used to simulate the electromechanical response of the impact igniter during the impact of a small caliber projectile on the target.Meanwhile,the experiments of projectile with impact igniter impact on the target are carried out.The comparison between experimental and numerical simulation results show that the established dynamic damage model can effectively predict the dynamic electromechanical response of PZTs in the missile service environment.

An extended micromechanical-based plastic damage model for understanding water effects on quasi-brittle rocks

Water effects on the mechanical properties of rocks have been extensively investigated through experiments and numerical models.However,few studies have established a comprehensive link between the microscopic mechanisms of water-related micro-crack and the constitutive behaviors of rocks.In this work,we shall propose an extended micromechanical-based plastic damage model for understanding weakening effect induced by the presence of water between micro-crack’s surfaces on quasi-brittle rocks,based on the Mori-Tanaka homogenization and irreversible thermodynamics framework.Regarding the physical mechanism,water strengthens micro-crack propagation,which induces damage evolution during the pre-and post-stage,and weakens the elastic effective properties of rock matrix.After proposing a special calibration procedure for the determination of model parameters based on the laboratory compression tests,the proposed micromechanical-based model is verified by comparing the model predictions to the experimental results.The model effectively captures the mechanical behaviors of quasibrittle rocks subjected to the weakening effects of water.

Phase-field modeling for anisotropic ductile damage of magnesium alloys at finite deformations

The damage anisotropy of an extruded ZK60 Mg alloy is characterized using tensile tests and scanning electronic microscopy.The accumulation of anisotropic deformations leads to the great differences of the dimple evolution and strains at fracture along different loading directions.To introduce the anisotropic deformation information into the damage constitutive relationship,a thermodynamically consistent phase-field model of ductile damage fully coupled with elastoplastic finite deformations is developed in this study.Using the user-defined constitutive relationship and displacement-temperature coupling element,the finite element simulations are conducted.The results show that:(1)ZK60 Mg alloys presents clear R-value difference in 0°,45°,and 90°tests of intact specimens.The 45°test possesses the greatest R-value(1.50)and the greatest strain at fracture,however,the R-value for 0°is less than 1,indicating the thinning is preferential.(2)The higher ultimate stress leads to a larger average dimension of the dimples,whereas the higher density correlates with a larger elongation ratio at the fracture.The disappearance of the stress-bearing area indicates that the phase-field assumption on stress degradation is completely compatible with the dimple analysis on fractography.(3)The simulation results of the stress-strain relationships and damage paths correlate well with the experimental ductile damage of magnesium alloys at 200◦C.Slight errors are basically attributed to the modeling parameters and finite element iteration algorithm.The proposed model presents fine applicability and reliability for the predictions of plastic deformations,ductile damage,and fracture of anisotropic Mg alloys.

Experimental study of the damage characteristics of rocks containing non-penetrating cracks under cyclic loading

The damage evolution process of non-penetrating cracks often causes some unexpected engineering disasters.Gypsum specimens containing non-penetrating crack(s)are used to study the damage evolution and characteristics under cyclic loading.The results show that under cyclic loading,the relationship between the number of non-penetrating crack(s)and the characteristic parameters(cyclic number,peak stress,peak strain,failure stress,and failure strain)of the pre-cracked specimens can be represented by a decreasing linear function.The damage evolution equation is fitted by calibrating the accumulative plastic strain for each cycle,and the damage constitutive equation is proposed by the concept of effective stress.Additionally,non-penetrating cracks are more likely to cause uneven stress distribution,damage accumulation,and local failure of specimen.The local failure can change the stress distribution and relieve the inhibition of non-penetrating crack extension and eventually cause a dramatic destruction of the specimen.Therefore,the evolution process caused by non-penetrating cracks can be regarded as one of the important reasons for inducing rockburst.These results are expected to improve the understanding of the process of spalling formation and rockburst and can be used to analyze the stability of rocks or rock structures.

A damage constitutive model of rock-like materials containing a single crack under the action of chemical corrosion and uniaxial compression

To describe the deformation and strength characteristics of the corroded rock-like specimens containing a single crack under uniaxial compression,a damage constitutive model combining hydro-chemical damage with coupling damage of micro-flaws and macro-cracks is proposed.Firstly,based on phenomenological theory,the damage variable of the rock-like specimens subjected to water environment erosion and chemical corrosion is obtained.Secondly,a coupled damage variable for cracked rock-like specimens is derived based on the Lemaitre strain equivalence hypothesis,which combines the Weibull statistical damage model for micro-flaws and the fracture mechanics model for a macro single crack.Then,considering the residual strength characteristics of the rock-like materials,the damage variable is modified by introducing the correction coefficient,and the damage constitutive model of the corroded rock-like specimens with a single crack under uniaxial compression is established.The model is verified by comparing the experimental stress−strain curves,and the results are in good agreement with those provided in the literature.Finally,the correction coefficient of the damage variable proposed in this paper is discussed.The damage constitutive model developed in this paper provides an effective method to describe the stress−strain relationship and residual strength characteristics of the corroded rock-like specimens with a single crack under uniaxial compression.

Damage constitutive model for strain-softening rock based on normal distribution and its parameter determination

Firstly, using the damage model for rock based on Lemaitre hypothesis about strain equivalence, a new technique for measuring strength of rock micro-cells by adopting the Mohr-Coulomb criterion was developed, and a statistical damage evolution equation was established based on the property that strength of micro-cells is consistent with normal distribution function, through discussing the characteristics of random distributions for strength of micro-cells, then a statistical damage constitutive model that can simulate the full process of rock strain softening under specific confining pressure was set up. Secondly, a new method to determine the model parameters which can be applied to the situations under different confining pressures was proposed, by deeply studying the relations between the model parameters and characteristic parameters of the full stress-strain curve under different confining pressures. Therefore, a unified statistical damage constitutive model for rock softening which can reflect the effect of different confining pressures was set up. This model makes the physical property of model parameters explicit, contains only conventional mechanical parameters, and leads its application more convenient. Finally, the rationality of this model and its parameters-determining method were identified via comparative analyses between theoretical and experimental curves.

Study on damages constitutive model of rocks based on lognormal distribution

The damage constitutive relation of entire rock failure process was established using the theory of representative volume element obeying the Iognormal distribution law, and the integrated damages constitutive model of rock under triaxial compression was established. Comparing with triaxial compression test result, it shows that this model correctly reflects the relationship of stress-strain. At the same time, according to the principle of the rock fatigue failure that conforms to completely the static entire process curve, a new method of establishing cyclic fatigue damage evolution equation was discussed, this method form is simple and the physics significance is clear, it may join preferably the damage relations of the rock static entire process curve.

Cyclic constitutive equations of rock with coupled damage induced by compaction and cracking

In this paper,the cyclic constitutive equations were proposed to describe the constitutive behavior of cyclic loading and unloading.Firstly,a coupled damage variable was derived,which contains two parts,i.e.,the compaction-induced damage and the cracking-induced damage.The compaction-induced damage variable was derived from a nonlinear stress–strain relation of the initial compaction stage,and the cracking-induced damage variable was established based on the statistical damage theory.Secondly,based on the total damage variable,a damage constitutive equation was proposed to describe the constitutive relation of rock under the monotonic uniaxial compression conditions,whereafter,the application of this model is extended to cyclic loading and unloading conditions.To validate the proposed monotonic and cyclic constitutive equations,a series of mechanical tests for marble specimens were carried out,which contained the monotonic uniaxial compression(MUC)experiment,cyclic uniaxial compression experiments under the variable amplitude(CUC-VA)and constant amplitude(CUC-CA)conditions.The results show that the proposed total damage variable comprehensively reflects the damage evolution characteristic,i.e.,the damage variable firstly decreases,then increases no matter under the conditions of MUC,CUC-VA or CUC-CA.Then a reasonable consistency is observed between the experimental and theoretical curves.The proposed cyclic constitutive equations can simulate the whole cyclic loading and unloading behaviors,such as the initial compaction,the strain hardening and the strain softening.Furthermore,the shapes of the theoretical curves are controlled by the modified coefficient,compaction sensitivity coefficient and two Weibull distributed parameters.

Statistical damage constitutive model for concrete materials under uniaxial compression

According to the damage mechanism of concrete material during the uniaxial compressive failure process,this paper further establishes the statistical damage constitutive model of concrete subjected to uniaxial compressive stress based on the statistical damage model under uniaxial tension. The damage evolution law in the direction subjected to pressure is confirmed by the tensile damage evolution process of lateral deformation due to the Poisson effect,and then the compressive stress-strain relationship is defined. The peak nominal stress state and the critical state occurring in the macro longitudinal distributed splitting cracks are distinguished. The whole loading process can be divided into the even damage phase and the local breakage phase. The concrete specimen is divided into the failure process zone and the resting unloading zone. The size effects during the local breakage phase under the uniaxial monotonic compressive process and the hysteretic phenomenon under the cyclic compressive loading process are analyzed. Finally,the comparison between theoretical results and experimental results preliminarily verifies the rationality and feasibility of understanding the failure mechanism of concrete through the statistical damage constitutional law.

Acoustic emission activity in directly tensile test on marble specimens and its tensile damage constitutive model

For understanding acoustic emission （AE） activity and accumulation of micro-damage inside rock under pure tensile state, the AE signals has been monitored on the test of directly tension on two kinds of marble specimens. A tensile constitutive model was proposed with the damage factor calculated by AE energy rate. The tensile strength of marble was discrete obviously and was sensitive to the inside microdefects and grain composition. With increasing of loading, the tensile stress-strain curve obviously showed nonlinear with the tensile tangent modulus decreasing. In repeated loading cycle, the tensile elastic modulus was less than that in the previous loading cycle because of the generation of micro damage during the prior loading. It means the linear weakening occurring in the specimens. The AE activity was corresponding with occurrence of nonlinear deformation. In the initial loading stage which only elastic deformation happened on the specimens, there were few AE events occurred; while when the nonlinear deformation happened with increasing of loading, lots of AE events were generated. The quantity and energy of AE events were proportionally related to the variation of tensile tangent modulus. The Kaiser effect of AE activity could be clearly observed in tensile cycle loading. Based on the theory of damage mechanics, the damage factor was defined by AE energy rate and the tensile damage constitutive model was proposed which only needed two property constants. The theoretical stress-strain curve was well fitted with the curve plotted with tested datum and the two property constants were easily gotten by the laboratory testing.

GENERAL EXPRESSIONS OF CONSTITUTIVE EQUATIONS FOR ISOTROPIC ELASTIC DAMAGED MATERIALS

The general expressions of constitutive equations for isotropic elastic damaged materials were derived directly from the basic law of irreversible thermodynamics. The limitations of the classical damage constitutive equation based on the well-known strain equivalence hypothesis were overcome. The relationships between the two elastic isotropic damage models (i.e. single and double scalar damage models) were revealed. When a single scalar damage variable defined according to the microscopic geometry of a damaged material is used to describle the isotropic damage state, the constitutive equations contain two 'damage effect functions', which describe the different influences of damage on the two independent elastic, constants. The classical damage constitutive equation based on the strain equivalence hypothesis is only the first-order approximation of the general expression. It may be unduly simplified and may fail to describe satisfactorily the damage phenomena of practical materials.

Damage characteristics and new constitutive model of sandstone under wet–dry cycles

The mechanical properties of rock deteriorate under repeated wet-dry(WD)cycles,causing the deformation and failure of the rock mass.A reasonable damage constitutive model can truly reflect the whole process of rock deformation and failure.Therefore,it is of great significance to study the damage characteristics and constitutive behaviour of rock subjected to numerous WD cycles.First,sandstone from Tingliang tunnel was sampled for the WD cycle experiment,and uniaxial and triaxial tests were carried out on the rock samples after various numbers of WD cycles to analyze their macroscale damage characteristics.Then,the damage mechanisms of the rock samples under the action of WD cycling were identified by X-ray diffraction(XRD)and scanning electron microscopy(SEM).Finally,based on the test data,the WD cycle-induced damage variable,Weibull distribution function,damage threshold,Drucker-Prager(D-P)yield criterion and residual strength correction coefficient were introduced,a wet-dry loading(WDL)constitutive damage model that considers the cracking stress of rock masses was established,and the expressions of the corresponding parameters were given.The results show that an increasing number of WD cycles induces considerable variations in the macroscopic physical and mechanical parameters(such as the rock sample mass,saturated water content,longitudinal-wave velocity,compressive strength and elastic modulus),and the rate of change presents two stages,the inflection point of their rate of change is the 15th WD cycle.Microscopically,the rock sample structure changes from intact and dense to fragmented and unconsolidated;additionally,the surface roughness increases,and the mineral composition changes.The established constitutive damage model exhibited good agreement with the experimental data;thus,this model can reflect the deformation and failure of rocks under WDL conditions,and the physical meaning of each parameter is clear.

Viscoplastic Damage-Softening Constitutive Model for Concrete Subjected to Uniaxial Dynamic Compression

A new viscoplastic damage-softening constitutive model is presented. It is developed by integrating a Bodner-Partom viscoplastic model with a newdamage evolution equation. A set of ordinary differential equations（ ODEs） is formulated,and a Runge-Kutta integral method is used to get stress-strain curves given by the model. Also,stress-strain curves of a wide range of strain-rates for concrete were obtained by split Hopkinson pressure bar（ SHPB） tests. By fitting the integral stressstrain curves to the experimental ones with the least square optimization method,we determined the material parameters in our model. Some properties of the newmodel,such as strain-rate sensitivity,damage evolution characteristics,strain-rate jump effects and unloading feature,are explored.These results showthat our new model can describe dynamic behaviors of concrete very well,and our integrating-fitting-optimizing method to get material parameters is valid.

Study on Carbonation Damage Constitutive Curve and Microscopic Damage Mechanism of Tailing Recycled Concrete

To improve the resource utilization of recycled aggregate concrete(RAC)and make use of the unique pozzolanic activation characteristics of iron ore tailing(IOT),the constitutive curves of tailing recycled concrete(TRC)before and after carbonization were analyzed theoretically,experimentally and microscopically.Firstly,according to the experimental data,the damage constitutive and related damage parameters of TRC were theoretically established by Weibull probability distribution function.Secondly,the comprehensive damage parameter b under different working conditions was studied.Finally,the damage mechanism was formed by EDS and SEM.The results showed that the damage constitutive model based on Weibull probability distribution function was in good agreement with the experimental results.Under each carbonization period,the b first decreased and then rose with the increase of tailings content.When its content was 30%,the b values of TRC were minimized,which were 22.14%,20.99%,25.39%lower than those of NAC,and 41.09%,34.89%,35.44%lower than those of RAC,indicating that IOT had a relatively good optimization effect on the constitutive curve of RAC.The microscopic analysis results also proved that the IOT addition with a proper amount would improve the matrix structure of RAC and increased its compactness,but when the content was higher,it would also cause harmful cracks in its matrix structure and reduced its density.Therefore,the optimal tailing content was about 30%.This paper provided a new method for damage constitutive calculation and analysis of TRC before and after carbonization.

INVESTIGATION ON GRADIENT-DEPENDENT NONLOCAL CONSTITUTIVE MODELS FOR ELASTO-PLASTICITY COUPLED WITH DAMAGE

Firstly, typical) gradient-dependent nonlocal inelastic models were briefly reviewed. Secondly, based on the principle of ‘gradient-dependent energy dissipation', a gradient-dependent constitutive model for plasticity coupled with isotropic damage was presented in the framework of continuum thermodynamics. Numerical scheme for calculation of Laplacian term of damage field with the numerical results obtained by FEM calculation was proposed. Equations have been presented on the basis of Taylor series for both 2-dimensional and 3-dimensional cases, respectively. Numerical results have indicated the validity of the proposed gradient-dependent model and corresponding numerical scheme.