Anisotropic time-dependent behaviors of shale under direct shearing and associated empirical creep models

Understanding the anisotropic creep behaviors of shale under direct shearing is a challenging issue.In this context,we conducted shear-creep and steady-creep tests on shale with five bedding orientations (i.e.0°,30°,45°,60°,and 90°),under multiple levels of direct shearing for the first time.The results show that the anisotropic creep of shale exhibits a significant stress-dependent behavior.Under a low shear stress,the creep compliance of shale increases linearly with the logarithm of time at all bedding orientations,and the increase depends on the bedding orientation and creep time.Under high shear stress conditions,the creep compliance of shale is minimal when the bedding orientation is 0°,and the steady-creep rate of shale increases significantly with increasing bedding orientations of 30°,45°,60°,and 90°.The stress-strain values corresponding to the inception of the accelerated creep stage show an increasing and then decreasing trend with the bedding orientation.A semilogarithmic model that could reflect the stress dependence of the steady-creep rate while considering the hardening and damage process is proposed.The model minimizes the deviation of the calculated steady-state creep rate from the observed value and reveals the behavior of the bedding orientation's influence on the steady-creep rate.The applicability of the five classical empirical creep models is quantitatively evaluated.It shows that the logarithmic model can well explain the experimental creep strain and creep rate,and it can accurately predict long-term shear creep deformation.Based on an improved logarithmic model,the variations in creep parameters with shear stress and bedding orientations are discussed.With abovementioned findings,a mathematical method for constructing an anisotropic shear creep model of shale is proposed,which can characterize the nonlinear dependence of the anisotropic shear creep behavior of shale on the bedding orientation.

A creep model for ultra-deep salt rock considering thermal-mechanical damage under triaxial stress conditions

To investigate the specific creep behavior of ultra-deep buried salt during oil and gas exploitation,a set of triaxial creep experiments was conducted at elevated temperatures with constant axial pressure and unloading confining pressure conditions.Experimental results show that the salt sample deforms more significantly with the increase of applied temperature and deviatoric loading.The accelerated creep phase is not occurring until the applied temperature reaches 130℃,and higher temperature is beneficial to the occurrence of accelerated creep.To describe the specific creep behavior,a novel three-dimensional(3D)creep constitutive model is developed that incorporates the thermal and mechanical variables into mechanical elements.Subsequently,the standard particle swarm optimization(SPSO)method is adopted to fit the experimental data,and the sensibility of key model parameters is analyzed to further illustrate the model function.As a result,the model can accurately predict the creep behavior of salt under the coupled thermo-mechanical effect in deep-buried condition.Based on the research results,the creep mechanical behavior of wellbore shrinkage is predicted in deep drilling projects crossing salt layer,which has practical implications for deep rock mechanics problems.

Experiment and constitutive modelling of creep deformation in the frozen silt-concrete interface

To ensure the long-term safety and stability of bridge pile foundations in permafrost regions,it is necessary to investigate the rheological effects on the pile tip and pile side bearing capacities.The creep characteristics of the pile-frozen soil interface are critical for determining the long-term stability of permafrost pile foundations.This study utilized a self-developed large stress-controlled shear apparatus to investigate the shear creep characteristics of the frozen silt-concrete interface,and examined the influence of freezing temperatures(−1,−2,and−5°C),contact surface roughness(0,0.60,0.75,and 1.15 mm),normal stress(50,100,and 150 kPa),and shear stress on the creep characteristics of the contact surface.By incorporating the contact surface’s creep behavior and development trends,we established a creep constitutive model for the frozen silt-concrete interface based on the Nishihara model,introducing nonlinear elements and a damage factor.The results revealed significant creep effects on the frozen silt-concrete interface under constant load,with creep displacement at approximately 2-15 times the instantaneous displacement and a failure creep displacement ranging from 6 to 8 mm.Under different experimental conditions,the creep characteristics of the frozen silt-concrete interface varied.A larger roughness,lower freezing temperatures,and higher normal stresses resulted in a longer sample attenuation creep time,a lower steady-state creep rate,higher long-term creep strength,and stronger creep stability.Building upon the Nishihara model,we considered the influence of shear stress and time on the viscoelastic viscosity coefficient and introduced a damage factor to the viscoplasticity.The improved model effectively described the entire creep process of the frozen silt-concrete interface.The results provide theoretical support for the interaction between pile and soil in permafrost regions.

An improved creep model for unsaturated reticulated red clay

Establishment of a creep model is an important method to analyze the relationship between soil creep deformation and time,and the element model is widely used for studying soil creep.However,the element creep model is employed for fitting saturated soil,and the mechanical element model is generally linear,which cannot well fit the nonlinear deformation of the soil with time in practice.The creep process of the soil is not only time-dependent,but also related to the deviatoric stress level.Therefore,the fractional calculus theory and a parameter n reflecting the effect of deviatoric stress level on the creep properties of the soil were introduced into the element model,and the fractional qBurgers creep model was established by using the fractional Koeller dashpot and Caputo fractional calculus.The proposed model was used to fit the triaxial test data of reticulated red clay under different net confining pressures and matric suctions by unsaturated triaxial apparatus.The proposed model can well describe the nonlinearity of unsaturated reticulated red clay,has memory and global correlation to the creep development process of unsaturated reticulated red clay,and has clear physical meaning.The functional relationships of the model parameters with the matric suction,net confining pressure and deviatoric stress level were deduced,so that the creep curves of unsaturated reticulated red clay can be obtained for any conditions,which is of great value for the study of unsaturated soils.

Creep experiment and rheological model of deep saturated rock

By the methods of uniaxial single-stage loading and graded incremental cyclic loading, the creep experiments were performed on the deep saturated rock from Dongguashan Mine, and the creep curves of saturated rock under different loading stresses were obtained. By comparing with the creep rule of dry rock in the same location, the creep rule of deep saturated rock was analyzed. Based on the united rheological mechanical model, the rheological model of deep saturated rock was recognized, and the parameters of the model were determined. The results show that the creep curves are very smooth under low stress, but the phenomena of wave and catastrophe turn up under high stress, and the bearing capacity of rock is weakening over time. The rheological properties of saturated and dry rocks are very different under tlie condition of deep high stress, especially when unloading, degradation and damage of rock quality is more serious, and the effect of water cannot be neglected. The H--HIN--NJS model （Schofield-Scott-Blair model） was selected to represent the rheology rule of deep saturated rock, and the fitting curves of model agree well with the experiment data, so the selected model is reasonable.

Effect of pre-deformation on aging creep of Al-Li-S4 alloy and its constitutive modeling

A set of uniaxial tensile creep tests at different pre-deformations, aging temperatures and stress levels were carried out for Al-Li-S4 alloy, and the creep behavior and the effects of pre-deformation on mechanical properties and microstructures were determined under basic thermodynamics conditions of aging forming. The results show that pre-deformation shortens the time of primary creep and raises the second steady-state creep rate. Then, the total creep strain is greater, but in the range of test parameters it is still smaller than that without pre-deformation. In addition, transmission electron microscopy（TEM） observation shows that pre-deformation promotes the formation of T1 phase and θ′ phase and makes them distribute more dispersively, while inhibits the generation of δ′ phase, which leads to the improvement of mechanical properties of the alloy. A unified constitutive model reflecting the effects of aging mechanism, stress levels and different pre-deformations was established. The fitting results agree with the experimental data well.

Constitutive modeling and springback simulation for 2524 aluminum alloy in creep age forming

The constitutive modeling and springback simulation for AA2524 sheet in creep age forming（CAF） process were presented.A series of creep aging tests were performed on AA2524 at the temperature of 180-200 °C and under the stress of 140-210 MPa for 16 h.Based on these experimental data,material constitutive equations which can well characterize creep aging behaviors of the tested alloy were developed.The effect of interior stress distributed along the sheet thickness on springback was simulated using FE software MSC.MARC by compiling the established constitutive models into the user subroutine.The simulation results showed that the amount of sheet springback was 61.12% when merely considering tensile stress existing along the sheet thickness;while sheet springback was up to 65.93% when taking both tensile and compressive stresses into account.In addition,an AA2524 rectangular sheet was subjected to CAF experiment in resistance furnace.The springback value of the formed rectangular sheet was 68.2%,which was much closer to 65.93%.This confirms that both tensile and compressive stresses across the sheet thickness should be considered in accurately predicting springback of the sheet after forming,which can be more consistent with experimental results.

Creep model for unsaturated soils in sliding zone of Qianjiangping landslide

The mechanical behavior of sliding zone soils plays a significant role in landslide. In general, the sliding zone soils are basically in unsaturated state due to rainfall infiltration and reservoir water level fluctuation. Meanwhile, a large number of examples show that the deformation processes of landslides always take a long period of time, indicating that landslides exhibit a time-dependent property. Therefore, the deforma- tion of unsaturated soils of landslide involves creep behaviors. In this paper, the Burgers creep model for unsaturated soils under triaxial stress state is considered based on the unsaturated soil mechanics. Then, by curve fitting using the least squares method, creep parameters in different matric suction states are obtained based on the creep test data of unsaturated soils in the sliding zones of Qianjiangping landslide. Results show that the predicted results are in good agreement with the experimental data, Finally, to fur- ther explore the creep characteristics of the unsaturated soils in sliding zones, the relationships between parameters of the model and matric suction are analyzed and a revised Burgers creep model is developed correspondingly. Simulations on another group of test data are performed by using the modified Burgers creep model and reasonable results are observed,

Viscoelastoplastic constitutive model for creep deformation behavior of asphalt sand

A uniaxial viscoelastoplastic model that can describe whole creep behaviors of asphalt sand at different temperatures was presented.The model was composed of three submodels in series,which describe elastoplastic,viscoelastic and viscoplastic characteristics respectively.The constitutive equation was established for uniaxial loading condition,and the creep representation was also obtained.The constitutive parameters were determined by uniaxial compression tests under controlled-stress of 0.1 MPa with five different test temperatures of 20,40,45,50 and 60 ℃.Expressions of the model parameters in terms of temperatures were also given.The model gave prediction at various temperatures consistent with the experimental results,and can reflect the total deformation characterization of asphalt sands.

Creep behaviour and constitutive model of coal filled with gas

Coal exhibits different creep behaviours when filled with different amounts of gas. Creep tests of coal filled with 0 and 0.5 MPa gas were performed, and strain under different axial stress was compared.The three creep constitutive models which were analysed using the method fitting experimental data for determining which creep model can reflect the creep process of the test best. The results show that the deformation of coal filled with 0.5 MPa gas is more higher than that of coal filled with 0 MPa gas under the same axial stress. Gas plays a positive effect on the deformation of coal process and will accelerate creep process. And gas will reduce coal intensity and change coal creep properties.Compared with Nishihara Model and Extensional Nishihara Model, Burgers Model can reflect the three stages of creep process of coal filled with gas better. The research results can contribute to reveal coal and gas outburst mechanism.

Unsaturated creep tests and empirical models for sliding zone soils of Qianjiangping landslide in the Three Gorges

Creep of sliding zone soils may cause significant displacement in large-scale landslides in the Three Gorges reservoir area.To investigate the effects of water on the soil creep behavior of the Qianjiangping landslide,a series of unsaturated triaxial creep tests on the sliding zone soils were performed.Based on the analyses of testing results,a new stress intensity incorporating matric suction was defined and an unsaturated Singh-Mitchell creep model was developed.Predicted results are in good agreement with the experimental results,which indicates that the established unsaturated model can reasonably simulate the effects of water on the soil creep behavior of the landslide.Finally,relationships between matric suction and the parameters of the model were analyzed.This study provides a calculation model and parameters for the evaluation of long-term stability of landslides under the influence of water.

Non-linear creep damage model of sandstone under freeze-thaw cycle

To study rock damage characteristics under long-term freeze-thaw cycles and loads,rock freeze-thaw and creep damage factors were defined based on nuclear magnetic resonance porosity and volume strain,respectively.The damage factor is introduced into the basic rheological element,and the non-linear creep damage constitutive model and freeze-thaw rock equation are established to describe non-linear creep characteristics under a constant load.Simultaneously,the creep test of freeze-thaw rock under step loading is performed.Based on the test data,the applicability and accuracy of the creep damage freeze-thaw rock model are analyzed and verified.The results show that freeze-thaw cycles result in continuous rock pore structure damage and deterioration,and nuclear magnetic resonance porosity enhancement.The constant load induces increasing rock plastic deformation,volume,and creep aging damage.As the loading stress increases,the instantaneous rock elastic parameters increase,and the rheological elastic and viscosity parameters decrease.Furthermore,the damage degradation of freeze-thaw cycles weakens the rock viscoplasticity,resulting in a rapid decrease in the viscosity parameter with an increase in freeze-thaw cycles.Generally,the continuous damage of the rock is degraded,and the long-term strength decreases continuously.

Coupled model of consolidation and creep for consolidation test

For most soft soil,the consolidation process and the creep process were coupled and the character of settlement time curve was obtained by the consolidation test at the same time.A simple and continuous function was presented to express the whole process of consolidation including immediate settlement stage,the primary consolidation stage and the secondary consolidation stage.And the features of the continuous function were analyzed.The results of the long-term(duration to 18 months) consolidation test on clays from the Chek Lap Kok formation proved the rightness of the method.In the end,the parameter meanings of the continuous function were discussed.

QUICK ASSESSMENT METHODOLOGY FOR RELIABILITY OF SOLDER JOINTS IN BALL GRID ARRAY (BGA) ASSEMBLY——PART Ⅰ: CREEP CONSTITUTIVE RELATION AND FATIGUE MODEL

In this study, a new unified creep constitutive relation and a mod- ified energy-based fatigue model have been established respectively to describe the creep flow and predict the fatigue life of Sn-Pb solders. It is found that the relation successfully elucidates the creep mechanism related to current constitutive relations. The model can be used to describe the temperature and frequency dependent low cycle fatigue behavior of the solder. The relation and the model are further employed in part Ⅱ to develop the numerical simulation approach for the long-term reliability assessment of the plastic ball grid array (BGA) assembly.

Creep Model of High-Strength High-Performance Concrete Under Cyclic Loading

Concrete creep under both static and cyclic loading conditions was investigated. Four groups of high-strength high-performance concrete(HSHPC) prism specimens were fabricated, and three of these specimens were loaded periodically by the MTS Landmark Fatigue Testing Machine System. Creep characteristics and creep coefficients of HSHPC under static loading and cyclic loading, respectively, were obtained and compared. The experimental results show that the creep strains under cyclic loading with a mean stress of 0.4 fcp and an amplitude of 0.2 fcp increase significantly compared with the creep strains under static loading, and the maximum value was 1.2-2.3 times at early stages. In addition, the creep coefficient increases nonlinearly with the number of cyclic loading repetitions. The influence coefficient for cyclic loading γcyc=1.088×(N/N0)0.078 was introduced based on the previous HSHPC creep model, and then the modified creep model under cyclic loading was established. Finally, the residual method, the CEB coefficient of variation method and the B3 coefficient of variation method were applied to evaluate the modified creep model. The statistical results demonstrate that the modified creep model agrees well with the experimental measurements. Hence, it has important theoretical and practical values for accurately predicting the deflection of concrete bridges under cyclic traffic loading.

Generic creep behavior and creep modeling of an aged surface support liner under tension

Polymer-based materials have been motivated to be an alternative support system element in the mining/tunneling industry due to their logistic and geotechnical benefits.Thin spray-on liner(TSL),a term to define the application of the material on the rock surface with a layer ranging from 2 mm to10 mm in thickness,shows some promising results.TSLs are mainly composed of plastic,polymer,or cement-based ingredients to a certain proportion.This study intends to reveal the time-dependent response of TSL specimens,cured throughout 500 d,under four constant stress levels for stable laboratory conditions.The results were correlated using two interrelated equations to predict the material’s service life(creep-rupture envelopes).The proposed correlations offered an insight into both the effective permanent support time and the strain amount at the liner failure.The time-dependent deformation of TSL,whose performance is highly responsive to creep behavior,was obtained so that the design engineers may use the findings to avoid the severe problems of material creep.Experimental data were also used to develop a Burgers(four-element)creep model.Since the liner has a nonlinear time-dependent behavior,creep models were built for each stress level separately.Subsequently,a generic equation was obtained using the nonlinear parametric dependencies.There is a good agreement between the proposed model and the experimental results.The proposed model can be used as a basis for future numerical studies related to the support behavior of aged surface support liners.

A triaxial creep model for coal containing gas and its stability analysis

Triaxial creep tests on CCG specimens were systematically performed using aself-made creep seepage experimental apparatus for determining the creep law of CCG.An improved triaxial creep model of CCG was established on the basis of a Nishiharamodel and another visco-elasto-plastic model,parameters of which were fitted on test data.Furthermore,the creep model is validated according to the result of triaxial creep experiments,and the outcome shows that the proposed triaxial creep model can properly characterizethe properties of various creep deformation phases of CCG,especially the acceleratingcreep phase.At the same time,the instability conditions of CCG were presentedbased on the discussion of the improved model's stability in terms of stability theories ofdifferential equation solution.

A nonlinear creep model for surrounding rocks of tunnels based on kinetic energy theorem

The initiating condition for the accelerated creep of rocks has caused difficulty in analyzing the whole creep process.Moreover,the existing Nishihara model has evident shortcomings in describing the accelerated creep characteristics of the viscoplastic stage from the perspective of internal energy to analyze the mechanism of rock creep failure and determine the threshold of accelerated creep initiation.Based on the kinetic energy theorem,Perzyna viscoplastic theory,and the Nishihara model,a unified creep constitutive model that can describe the whole process of decaying creep,stable creep,and accelerated creep is established.Results reveal that the energy consumption and creep damage in the process of creep loading mainly come from the internal energy changes of geotechnical materials.The established creep model can not only describe the viscoelasticeplastic creep characteristics of rock,but also reflect the relationship between rock energy and creep deformation change.In addition,the research results provide a new method for determining the critical point of creep deformation and a new idea for studying the creep model and creep mechanical properties.

Effect of strain rate difference between inside and outside groove in M-K model on prediction of forming limit curve of Ti6Al4V at elevated temperatures

The influence of initial groove angle on strain rate inside and outside groove of Ti6Al4V alloy was investigated.Based on the evolution of strain rate inside and outside groove,the effect of strain rate difference on the evolution of normal stress and effective stress inside and outside groove was also analyzed.The results show that when linear loading path changes from uniaxial tension to equi-biaxial tension,the initial groove angle plays a weaker role in the evolution of strain rate in the M-K model.Due to the constraint of force equilibrium between inside and outside groove,the strain rate difference makes the normal stress inside groove firstly decrease and then increase during calculation,which makes the prediction algorithm of forming limit convergent at elevated temperature.The decrease of normal stress inside groove is mainly caused by high temperature softening effect and the rotation of groove,while the increase of normal stress inside groove is mainly due to strain rate hardening effect.

DAMAGE AND CREEP MODEL FOR ROCK SALT AS WELL AS ITS VALIDATION

The constitutive model Hou/Lux,based on continuum damage mechanics and on the material model Lubby2 is presented and the damage,the healing of damage,the tertiary creep and the creep rupture of rock salt described. An application shows how phenomena observed in laboratory tests on axially perforated cylinder sample, such as softening, dilatancy, spalling and radial deformation into the axial bore as well as spalling can be numerically simulated with the model Hou/Lux. Furthermore,it is then evaluated against the measured permeability and stress profiles of a 37-year-old drift at the Sondershausen minein,Germany. Satisfactory agreement is obtained between the calculated and measured data.