The occurrence of geological hazards and the instability of geotechnical engineering structures are closely related to the time-dependent behavior of rock.However,the idealization boundary condition for constant stres...The occurrence of geological hazards and the instability of geotechnical engineering structures are closely related to the time-dependent behavior of rock.However,the idealization boundary condition for constant stress in creep or constant strain in relaxation is not usually attained in natural geological systems.Therefore,generalized relaxation tests that explore the simultaneous changes of stress and strain with time under different stress levels with constant pore-water pressure are conducted in this study.The results show that in area Ⅰ,area Ⅱ,and area Ⅲ,the stress and strain both change synchronously with time and show similar evolutionary laws as the strain-time curve for creep or the stress-time curve for relaxation.When the applied stress level surpasses the δ_(ci) or δ_(cd) threshold,the variations in stress and strain and their respective rates of change exhibit a significant increase.The radial deformation and its rate of change exhibit greater sensitivity in response to stress levels.The apparent strain deforms homogeneously at the primary stage,and subsequently,gradually localizes due to the microcrack development at the secondary stage.Ultimately,interconnection of the microcracks causes the formation of a shear-localization zone at the tertiary stage.The strain-time responses inside and outside the localization zone are characterized by local strain accumulation and inelastic unloading during the secondary and tertiary stages,respectively.The width of the shear-localization zone is found to range from 4.43 mm to 7.08 mm and increased with a longer time-to-failure.Scanning electron microscopy(SEM)reveals a dominant coalescence of intergranular cracks on the fracture surface,and the degree of physiochemical deterioration caused by water-rock interaction is more severe under a longer lifetime.The brittle sandstone’s time-dependent deformation is essentially controlled by microcrack development during generalized relaxation,and its expectancy-life is determined by its initial microstructural 展开更多
This paper studies the backward-forward linear-quadratic-Gaussian(LQG)games with major and minor agents(players).The state of major agent follows a linear backward stochastic differential equation(BSDE)and the states ...This paper studies the backward-forward linear-quadratic-Gaussian(LQG)games with major and minor agents(players).The state of major agent follows a linear backward stochastic differential equation(BSDE)and the states of minor agents are governed by linear forward stochastic differential equations(SDEs).The major agent is dominating as its state enters those of minor agents.On the other hand,all minor agents are individually negligible but their state-average affects the cost functional of major agent.The mean-field game in such backward-major and forward-minor setup is formulated to analyze the decentralized strategies.We first derive the consistency condition via an auxiliary mean-field SDEs and a 3×2 mixed backward-forward stochastic differential equation(BFSDE)system.Next,we discuss the wellposedness of such BFSDE system by virtue of the monotonicity method.Consequently,we obtain the decentralized strategies for major and minor agents which are proved to satisfy the-Nash equilibrium property.展开更多
基金financially supported by the National Natural Science Foundation of China(Grant Nos.52304099,52172625)Shenzhen Science and Technology Program(Grant No.RCYX20221008092903013).
文摘The occurrence of geological hazards and the instability of geotechnical engineering structures are closely related to the time-dependent behavior of rock.However,the idealization boundary condition for constant stress in creep or constant strain in relaxation is not usually attained in natural geological systems.Therefore,generalized relaxation tests that explore the simultaneous changes of stress and strain with time under different stress levels with constant pore-water pressure are conducted in this study.The results show that in area Ⅰ,area Ⅱ,and area Ⅲ,the stress and strain both change synchronously with time and show similar evolutionary laws as the strain-time curve for creep or the stress-time curve for relaxation.When the applied stress level surpasses the δ_(ci) or δ_(cd) threshold,the variations in stress and strain and their respective rates of change exhibit a significant increase.The radial deformation and its rate of change exhibit greater sensitivity in response to stress levels.The apparent strain deforms homogeneously at the primary stage,and subsequently,gradually localizes due to the microcrack development at the secondary stage.Ultimately,interconnection of the microcracks causes the formation of a shear-localization zone at the tertiary stage.The strain-time responses inside and outside the localization zone are characterized by local strain accumulation and inelastic unloading during the secondary and tertiary stages,respectively.The width of the shear-localization zone is found to range from 4.43 mm to 7.08 mm and increased with a longer time-to-failure.Scanning electron microscopy(SEM)reveals a dominant coalescence of intergranular cracks on the fracture surface,and the degree of physiochemical deterioration caused by water-rock interaction is more severe under a longer lifetime.The brittle sandstone’s time-dependent deformation is essentially controlled by microcrack development during generalized relaxation,and its expectancy-life is determined by its initial microstructural
基金support partly by RGC Grant 502412,15300514,G-YL04.ZWu acknowledges the Natural Science Foundation of China(61573217),111 project(B12023)the National High-level personnel of special support program and the Chang Jiang Scholar Program of Chinese Education Ministry.
文摘This paper studies the backward-forward linear-quadratic-Gaussian(LQG)games with major and minor agents(players).The state of major agent follows a linear backward stochastic differential equation(BSDE)and the states of minor agents are governed by linear forward stochastic differential equations(SDEs).The major agent is dominating as its state enters those of minor agents.On the other hand,all minor agents are individually negligible but their state-average affects the cost functional of major agent.The mean-field game in such backward-major and forward-minor setup is formulated to analyze the decentralized strategies.We first derive the consistency condition via an auxiliary mean-field SDEs and a 3×2 mixed backward-forward stochastic differential equation(BFSDE)system.Next,we discuss the wellposedness of such BFSDE system by virtue of the monotonicity method.Consequently,we obtain the decentralized strategies for major and minor agents which are proved to satisfy the-Nash equilibrium property.
