A constitutive model coupling damage and material anisotropy for wide stress triaxiality

A constitutive model that can describe the damage evolution of anisotropic metal sheets during the complex forming processes which experience wide stress triaxiality history is essential to accurately predict the deformation and rupture behaviors of the processes.In this study,a modified Lemaitre damage criterion which couples with the anisotropic Barlat 89 yield function is established.The effects of stress triaxiality,Lode parameter and shear stress on damage accumulation are considered in the constitutive model.The model is numerically implemented and applied to fracture prediction in tensile tests with different stress triaxialities and a complex deformation process with wide stress triaxiality history.The good consistency of predictions and experiments indicates that the modified Lemaitre damage model has excellent fracture prediction ability.Finally,the accuracy of the model is analyzed and discussed.

Mechanical Behavior and Microstructure Evolution of a Rolled Magnesium Alloy AZ31B Under Low Stress Triaxiality

Plastic deformation up to final rupture failure of a rolled magnesium（Mg） alloy Mg-3.0Al-1.0Zn-0.34Mn（AZ31B） under low stress triaxiality was investigated.Local strain evolution was quantified by the digital image correlation（DIC） technique analysis with tensile,combined tensile-shear,and shear specimens,corresponding to the stress triaxiality of 1/3,1/6 and 0,respectively.Stress-strain curves show that the yield stress reduces with the decrease in the stress triaxiality,and obviously exhibits different strain hardening response.Electron backscatter diffraction（EBSD） observations reveal that the twinning behavior depends on stress triaxiality.Before fracture,double twinning is the dominant mechanism at the stress triaxiality of 1/3,while extension twinning is prevalent at the stress triaxiality of 0.Moreover,scanning electron microscopy（SEM） shows that the fracture mechanism is transformed from microvoid growth and coalescence to internal void shearing as the stress triaxiality decreases from 1/3 to 0.

Mechanical responses of anchoring structure under triaxial cyclic loading

Dynamic load on anchoring structures(AS)within deep roadways can result in cumulative damage and failure.This study develops an experimental device designed to test AS under triaxial loads.The device enables the investigation of the mechanical response,failure mode,instability assessment criteria,and anchorage effect of AS subjected to combined cyclic dynamic-static triaxial stress paths.The results show that the peak bearing strength is positively correlated with the anchoring matrix strength,anchorage length,and edgewise compressive strength.The bearing capacity decreases significantly when the anchorage direction is severely inclined.The free face failure modes are typically transverse cracking,concave fracturing,V-shaped slipping and detachment,and spallation detachment.Besides,when the anchoring matrix strength and the anchorage length decrease while the edgewise compressive strength,loading rate,and anchorage inclination angle increase,the failure intensity rises.Instability is determined by a negative tangent modulus of the displacement-strength curve or the continued deformation increase against the general downward trend.Under cyclic loads,the driving force that breaks the rock mass along the normal vector and the rigidity of the AS are the two factors that determine roadway stability.Finally,a control measure for surrounding rock stability is proposed to reduce the internal driving force via a pressure relief method and improve the rigidity of the AS by full-length anchorage and grouting modification.

THE ANALYSES OF THE THREE-DIMENSIONAL STRESS STRUCTURE NEAR THE CRACK TIP OF MODE Ⅰ CT SPECIMENS IN ELASTICPLASTIC STATE(Ⅱ)--THE ANALYSES OF THE STRESS STRUCTURE

Based on[1],the stress structures of the smooth region and shear lip of the specimens have been investigated in the paper.The characteristics of the stress structure in the smooth region have been found that the variable z can separated out;the stresses in the midsection can be obtained by the plane strain FEM results or HRR structure modified by the stress triaxiality.The effects of load level and thickness on the stress structure can be reflected by the distribution of CTOD along the thickness direction.The obtained expressions of the stresses are very simple and visualized.The analyses of the stress structure in the shear lip show that the stresses can be obtained by different methods of interpolation to a certain precise degree.A new degree parameter of the plane strain state has been put forward and studied.The parameter can reflect relatively well the variation of the kind and thickness of the specimen as well as the load level.The fracture parameter has also been investigated to be sure that it can be obtained by modified CTOD with the stress triaxiality.

THE ANALYSES OF THE THREE-DIMENSIONAL STRESS STRUCTURE NEAR THE CRACK TIP OF MODE I CT SPECIMENS IN ELASTICPLASTIC STATE(Ⅰ)--THE ANALYSES OF CONSTRAINT PARAMETERS AND FRACTURE PARAMETERS

In the present paper,three dimensional analyses of some general constraint parameters and fracture parameters near the crack tip of Mode I CT specimens in two different thicknesses are carried out by employing ADINA program.The results reveal that the constraints along the thickness direction are obviously separated into two parts:the keeping similar high constraint field(Z_(1))and rapid reducing constraints one(Z_(2)).The two fields are experimentally confiremed to correspond to the smooth region and the shear lip on the fracture face respectively.So the three dimensional stress structure of Mode I specimens can be derived through discussing the two fields respectively.The distribution of the Crack Tip Opening Displacement(CTOD)along the thickness direction and the three dimensional distribution of the void growth ratio(V_(g))near the crack tip are also obtained.The two fracture parameters are in similar trends along the thickness direction,and both of them can reflect the effect of thickness and that of the loading level to a certain degree.

Experimental Evaluation of Ductile Fracture of Sheet Metals under Plane Stress States

With the application of lightweight materials such as advanced high-strength steel and aluminum alloy in the automotive industry, it is necessary to quantitatively evaluate the ultimate deformation capacity of materials under various plane stress states for the digital simulation of these materials. Conventional Nakajima test can only provide three regular plane stress states, such as tension, plane strain tension and bulging, and FLC curve is affected by deformation path, mold lubrication and other variables. More importantly, Nakajima test cannot provide shear, tension shear, which are extremely important loading conditions in automobile collisions. Therefore, the research work of this paper focuses on the evaluation of the ultimate ductile fracture behavior of sheet metals under various conditions of plane stress states. The four variables Mohr-Coulomb model was established to study the ductile fracture of metal sheets under plane stress states. Beginning with the recorded minor and major strain distributing on the deformation area of uniaxial tension samples, Moving Regression Algorithm was deployed to reveal the inherent relationship among the key parameters involved in the M-C model, which also provided an experimental technique for monitoring the instantaneous changing of triaxiality over the whole loading period. Three or four typical types of uniaxial-loading specimens were well designed to determine the M-C curve. As a result, M-C curve and the transformed major stain vs. minor strain curve provide further information about the material arrest to the ductile fracture in the area of shear loading, in comparison with the conventional FLD test.

Investigation of the influence of intermediate principal stress on the dynamic responses of rocks subjected to true triaxial stress state

Precisely understanding the dynamic mechanical properties and failure modes of rocks subjected to true triaxial stress state(σ1>σ2>σ3,whereσ1,σ2,andσ3 are the major principal stress,intermediate principal stress,and minor principal stress,respectively)is essential to the safety of underground engineering.However,in the laboratory,it is difficult to maintain the constant true triaxial stress state of rocks during the dynamic testing process.Herein,a numerical servo triaxial Hopkinson bar(NSTHB)was developed to study the dynamic responses of rocks confronted with a true triaxial stress state,in which lateral stresses can maintain constant.The results indicate that the dynamic strength and elastic modulus of rocks increase with the rise of intermediate principal stressσ2,while the dynamic elastic modulus is independent of the dynamic strain rate.Simulated acoustic emission distributions indicate that the intermediate principal stressσ2 dramatically affects dynamic failure modes of triaxial confined rocks.Asσ2 increases,the failure pattern switches from a single diagonal shear zone into two parallel shear zones with a small slant.Moreover,a recent triaxial Hopkinson bar experimental system using three bar pairs is also numerically established,and the measuring discrepancies are identified between the two numerical bar systems.The proposed NSTHB system provides a controllable tool for studying the dynamic triaxial behavior of rocks.

Experimental study on the mechanical and failure behaviors of deep rock subjected to true triaxial stress:A review

It has become an inevitable trend of human development to seek resources from the deep underground.However,rock encountered in deep underground engineering is usually in an anisotropic stress state(σ_(1)>σ>σ_(3))due to the influences of geological structures and engineering disturbances.It is therefore essential to study the mechanical,seepage,and dynamic disaster behaviors of deep rock under true triaxial stress to ensure the safe operation of deep rock engineering and the efficient exploitation of deep resources.In recent years,experimental techniques and research on true triaxial rock mechanics have achieved fruitful results that have promoted the rapid development of deep rock mechanics;thus,it is necessary to systematically review and summarize these developments.This work first introduced several typical true triaxial testing apparatus and then reviewed the corresponding research progress on rock deformation,strength,failure mode,brittleness,and energy as well as the 3D volumetric fracturing(dynamic disaster)properties of deep rocks under true triaxial stress.Then,several commonly used true triaxial rock strength criteria and their applicability,the permeability characteristics and mathematical models of deep reservoir rocks,and the disaster-causing processes and mechanisms of disturbed volumetric fracturing(rockburst,compound dynamic disasters)in deep rock engineering were described.This work may provide an essential reference for addressing the true triaxial rock mechanics issues involved in deep rock engineering,especially regarding the stability of surrounding rock at depth,disaster prevention and control,and oil and gas exploitation.

3D morphology and formation mechanism of fractures developed by true triaxial stress

As main part of underground rock mass,the three-dimensional(3D)morphology of natural fractures plays an important role in rock mass stability.Based on previous studies on 3D morphology,this study probes into the law and mechanism regarding the influence of the confining pressure constraints on 3D morphological features of natural fractures.First,fracture surfaces were obtained by true triaxial compression test and 3D laser scanning.Then 3D morphological parameters of fractures were calculated by using Grasselli’s model.The results show that the failure mode of granites developed by true triaxial stress can be categorized into tension failure and shear failure.Based on the spatial position of fractures,they can be divided into tension fracture surface,S-1 shear fracture surface,and S-2 shear fracture surface.Micro-failure of the tension fracture surface is dominated by mainly intergranular fracture;the maximum height of asperities on the fracture surface and the 3D roughness of fracture surfaces are influenced by σ_(3) only and they are greater than those of shear fracture surfaces,a lower overall uniformity than tension fracture surface.S-1 shear fracture surface and S-2 shear fracture surface are dominated by intragranular and intergranular coupling fracture.The maximum height of asperities on the fracture surface and 3D roughness of fracture surface are affected by σ_(1),σ_(2),and σ_(3).With the increase of σ_(2) or σ_(3),the cutting off of asperities on the fracture surface becomes more common,the maximum height of asperities and 3D roughness of fracture surface further decrease,and the overall uniformity gets further improved.The experimental results are favorable for selecting technical parameters of enhanced geothermal development and the safety of underground mine engineering.

Acoustoelastic effects on mode waves in a fluid-filled pressurized borehole in triaxially stressed formations

Based on the nonlinear theory of acoustoelasticity, considering the triaxial terrestrial stress, the fluid static pressure in the borehole and the fluid nonlinear effect jointly, the dispersion curves of the monopole Stoneley wave and dipole flexural wave prop- agating along the borehole axis in a homogeneous isotropic formation are investigated by using the perturbation method. The relation of the sensitivity coefficient and the velocity-stress coefficient to frequency are also analyzed. The results show that variations of the phase velocity dispersion curve are mainly affected by three sensitivity coefficients related to third-order elastic constant. The borehole stress concentration causes a split of the flexural waves and an intersection of the dispersion curves of the flexural waves polarized in directions parallel and normal to the uniaxial horizontal stress direction. The stress-induced formation anisotropy is only dependent on the horizontal deviatoric terrestrial stress and independent of the horizontal mean terrestrial stress, the superimposed stress and the fluid static pressure. The horizontal terrestrial stress ratio ranging from 0 to 1 reduces the stress-induced formation anisotropy. This makes the intersection of flexural wave dispersion curves not distinguishable. The effect of the fluid nonlinearity on the dispersion curve of the mode wave is small and can be ignored.

Effect of CO_(2)on coal P-wave velocity under triaxial stress

As P-wave velocity is sensitive to the variations in coal reservoir parameters,it is possible to monitor the injected CO_(2)through P-wave velocity during CO_(2)sequestration in coal.However,the effects of CO_(2)on the coal P-wave velocity under triaxial stress are not clearly discerned.In the present study,different boundary conditions and gases were utilised to investigate the factors affecting the P-wave velocity after the interaction of coal with CO_(2).Experiments with helium indicated that the pore pressure primarily affected the P-wave velocity by altering the effective stress.Experiments with CH4 and CO_(2)indicated that matrix swelling induced-cleats porosity decline significantly promoted P-wave velocity.Moreover,CO_(2)caused a wider scale and severe weakening of coal matrix than CH4,thereby significantly decreasing the P-wave velocity,and the decline in P-wave velocity increases with vitrinite content.Furthermore,experiments under different boundary conditions showed that with the boundary condition having more constraints,the decrement of pore pressure on P-wave velocity is more weaken,whereas the improvement of matrix swelling on P-wave velocity is more evident.This study contributes to understanding the mechanism of effect of CO_(2)on P-wave velocity under triaxial stress condition and provides guidance for monitoring CO_(2)sequestration in coal.

New artificial neural networks for true triaxial stress state analysis and demonstration of intermediate principal stress effects on intact rock strength

Simulations are conducted using five new artificial neural networks developed herein to demonstrate and investigate the behavior of rock material under polyaxial loading. The effects of the intermediate principal stress on the intact rock strength are investigated and compared with laboratory results from the literature. To normalize differences in laboratory testing conditions, the stress state is used as the objective parameter in the artificial neural network model predictions. The variations of major principal stress of rock material with intermediate principal stress, minor principal stress and stress state are investigated. The artificial neural network simulations show that for the rock types examined, none were independent of intermediate principal stress effects. In addition, the results of the artificial neural network models, in general agreement with observations made by others, show （a） a general trend of strength increasing and reaching a peak at some intermediate stress state factor, followed by a decline in strength for most rock types; （b） a post-peak strength behavior dependent on the minor principal stress, with respect to rock type; （c） sensitivity to the stress state, and to the interaction between the stress state and uniaxial compressive strength of the test data by the artificial neural networks models （two-way analysis of variance; 95% confidence interval）. Artificial neural network modeling, a self-learning approach to polyaxial stress simulation, can thus complement the commonly observed difficult task of conducting true triaxial laboratory tests, and/or other methods that attempt to improve two-dimensional （2D） failure criteria by incorporating intermediate principal stress effects.

NEW NON QUADRATIC ORTHOTROPIC YIELD FUNCTION FOR TRIAXIAL STRESS STATE

A new Don-quadratic orthotropic yield function is developed in the present paper.It does not have those limitatioins which existing non-quadratic anisotropic yield functions have,such as being usable only for the plane stress problems and in-plane isotropic sheet metals,and that the directions of principal stress or the ex ponent in yield function can not be arbitrary,etc.Furthermore all of the material constants involved in this yield function can be determined by performing only uniaxial tension lest.This yield function contains three new parameters,of which each one is present for one principal plane of anisotropy.Their values can be.generally,selected to equal 3.Other methods to determine the value of these parmeters are discussed and given in this paper.From the regression estimate for the yield stress in five directions of several kinds of titanium metal sheet.it is obtained that the suitable value of exponent in yield function for titanium sheets is 6 or 8.This is confirmed from the use for several plastic deformation problems of titanium sheets.

Analysis of the mechanic characteristics of the damage propagation of rock under triaxial stress condition

The advanced computerized tomography is applied to study the damage propagation of rock. The real time CT scanning is carried out to the damage propagation of rock under triaxial stress condition. The damage propagation constitutive relation of rock under triaxial stress condition is analyzed at last.

An open-end high-power microwave-induced fracturing system for hard rock

Microwave pre-treatment is considered as a promising technique for alleviating cutter wear. This paper introduces a high-power microwave-induced fracturing system for hard rock. The test system consists of a high-power microwave subsystem (100 kW), a true triaxial testing machine, a dynamic monitoring subsystem, and an electromagnetic shielding subsystem. It can realize rapid microwave-induced fracturing, intelligent tuning of impedance, dynamic feedback under strong microwave fields, and active control of microwave parameters by addressing the following issues: the instability and insecurity of the system, the discharge breakdown between coaxial lines during high-power microwave output, and a lack of feedback of rock-microwave response. In this study, microwave-induced surface and borehole fracturing tests under true triaxial stress were carried out. Experimental comparisons imply that high-power microwave irradiation can reduce the fracturing time of hard rock and that the fracture range (160 mm) of a 915-MHz microwave source is about three times that of 2.45 GHz. After microwave-induced borehole fracturing, many tensile cracks occur on the rock surface and in the borehole: the maximum reduction of the P-wave velocity is 12.8%. The test results show that a high-power microwave source of 915 MHz is more conducive to assisting mechanical rock breaking and destressing. The system can promote the development of microwave-assisted rock breaking equipment.

Mechanical parameter evolutions and deterioration constitutive model for ductile-brittle failure of surrounding rock in high-stress underground engineering

The deep surrounding rock is usually in the true triaxial stress state,and previous constitutive models based on the understanding of uniaxial and conventional triaxial test results have difficulty characterizing the degradation and fracture process of rock ductile–brittle failure under true triaxial stress state.Therefore,this study conducted a series of true triaxial tests to obtain the understanding of the ductile–brittle behaviour of rock,and then combined the test results and the Mogi–Coulomb strength criterion,and proposed calculation methods for the elastic modulus E,cohesion c and internal friction angle u and the evolution functions of E,c and u of rock under true triaxial stresses.With the decreasing of the minimum principal stress r3 or increasing of the intermediate principal stress r2,the marble post-peak stress drop rate gradually increases,the ductility gradually weakens,and the brittleness significantly strengthens.The calculation method and evolution function of rock E,c and u under true triaxial stress were proposed.E decreased at first and then tended to remain stable with the increasing of equivalent plastic strain increment dep.c and u slowly increased at first and then rapidly decreased.With a method of parameter degradation rate to realize post-peak stress drop rate to reflect the ductile–brittle characteristics,a new three-dimensional ductile–brittle deterioration mechanical model(3DBDM)was established.The proposed model can accurately characterize the influence of r2 and r3 on mechanical parameters,the ductile–brittle behaviour of rock under true triaxial stresses,and the asymmetric failure characteristics of surrounding rock after excavation of deep underground engineering.The proposed model can be reduced to elastic–perfectly plastic,elastic–brittle,cohesion weakening friction strengthening(CWFS),Mohr–Coulomb,and Drucker–Prager models.

Experimental investigation of stress unloading effects on rock damage and confining pressure-dependent crack initiation stress of porous sandstone under true triaxial stress environments

This study investigates the impact of intermediate(σ_(2))and minimum(σ_(3))principal stress unloading on damage behavior and the confining pressure influence on crack initiation stress(σci)in true triaxial stress conditions,utilizing large-scale cubic samples.Two distinct true triaxial tests were executed,examining the effects of confining stress(σ_(2)andσ_(3))unloading on porous sandstone damage and the correlation between confining stress andσci.Acoustic emission(AE)parameters,signal characteristics,and wave velocity variations were utilized to elucidate cracking mechanisms and damage development in the samples.Unloading tests reveal consistent ve-locities in three orthogonal directions(V_(11),V_(22),and V_(33))during the initial two unloading stages.In the subse-quent three stages,confining stress unloading leads to a decrease in wave velocity in the corresponding direction,while velocities in the other two directions remain nearly constant.Notably,σ_(2)unloading generates higher amplitude AE signals compared toσ_(3)unloading,with over 70%of the micro-cracks categorized as tensile.In the incremental loading tests,σ_(ci) is found to be contingent on confining pressure,withσ_(2)playing a crucial role.Duringσ_(1) loading,V_(33) decreases,indicating additional crack formation;conversely,σ_(3)loading results in V33 increase,signifying the continuous closure of existing cracks.Limitations of the experiments are summarized and prospects in this domain are outlined.

Failure properties of rocks in true triaxial unloading compressive test

Slabbing failure often occurs in the surround rock near a deep underground excavation. The mechanism of slabbing failure is still unclear. In order to reveal the influence of the intermediate principal stress （σ2） on slabbing failure, true triaxial unloading compressive test was carried out based on the stress path of the underground engineering excavation, i.e., unloading the minimum principal stress （σ3）, keeping σ2, increasing the maximum principal stress （σ1）. The initiation and the propagation of slabbing fracture in rock specimens were identified by examining the acoustic emission （AE） and the infrared radiation characterization. The test results show that the failure modes of the granite and red sandstone specimens are changed from shear to slabbing with the increase of σ2. The AE characteristic of rock specimen under low σ2 is swarm type which is the main shock type under high σ2. The infrared radiation properties of rock specimen under different σ2 are also different. The temperature change area is just along the shear fracture such as the uniaxial compression. With the increase of σ2, the temperature change area is planar of rock specimen which proofs that the failure mode of rock specimen turns into slabbing.

Theoretical study of void closure in nonlinear plastic materials

Void closing from a spherical shape to a crack is investigated quantitatively in the present study. The constitutive relation of the Void-free matrix is assumed to obey the Norton power law. A representative volume element （RVE） which includes matrix and void is employed and a Rayleigh-Ritz procedure is developed to study the deformation-rates of a spherical void and a penny-shaped crack. Based on an approximate interpolation scheme, an analytical model for void closure in nonlinear plastic materials is established. It is found that the local plastic flows of the matrix material are the main mechanism of void deformation. It is also shown that the relative void volume during the deformation depends on the Norton exponent, on the far-field stress triaxiality, as well as on the far-field effective strain. The predictions of void closure using the present model are compared with the corresponding results in the literature, showing good agreement. The model for void closure provides a novel way for process design and optimization in terms of elimination of voids in billets because the model for void closure can easily be applied in the CAE analysis.

Numerical Simulation on Interfacial Creep Failure of Dissimilar Metal Welded Joint between HR3C and T91 Heat-Resistant Steel

The maximum principal stress, von Mises equivalent stress, equivalent creep strain, stress triaxiality in dissimilar metal welded joints between austenitic（HR3C） and martensitic heat-resistant steel（T91） are simulated by FEM at 873 K and under inner pressure of 42.26 MPa. The results show that the maximum principal stress and von Mises equivalent stress are quite high in the vicinity of weld/T91 interface, creep cavities are easy to form and expand in the weld/T91 interface. There are two peaks of equivalent creep strains in welded joint, and the maximum equivalent creep strain is in the place 27-32 mm away from the weld/T91 interface, and there exists creep constrain region in the vicinity of weld/T91 interface. The high stress triaxiality peak is located exactly at the weld/T91 interface. Accordingly, the weld/T91 interface is the weakest site of welded joint. Therefore, using stress triaxiality to describe creep cavity nucleation and expansion and crack development is reasonable for the dissimilar metal welded joint between austenitic and martensitic steel.