Peridynamic Analysis Of Deformation And Failure Of Rock-Like Materials Under Static Load

The deformation and failure of rock material under static load is a process from continuous to discontinuous,from static equilibrium to local dynamic instability.When the traditional mechanical method based on continuity assumption is used to deal with the process from static stability to dynamic failure,the type of the differential equation of motion changes substantially,losing the strong elliptic type and turning into hyperbolic type,which makes the original static boundary value problem untenable,and then leads to the strong grid dependence of the numerical results of failure.The emerging peridynamic method(Peridynamic(PD)uses the motion equation of time differential and space integral to uniformly solve the continuous and discontinuous problems of materials.It allows discontinuity to appear as part of the solution and has the ability to simulate the spontaneous generation and expansion of cracks in the failure process.At the same time,the motion equation is always hyperbolic in the solution process,avoiding the unwell problem caused by the degeneration of the equation,and there is no numerical obstacle in the process from static equilibrium to dynamic instability of the simulated material.However,for(quasi)static force under the condition of material,such as rock,the static and dynamic deformation damage problem,PD theory usually adopt dynamic relaxation method(static algorithm),the introduction of artificial damping while speeding up the efficiency of solving the static deformation,but the damage in phase can severely inhibit object acceleration effect,make the material not completely destroy the dynamic effect of development,inadequate,cannot reflect the actual damage of the material process.The explicit(dynamic)algorithm explicitly calculates the material point acceleration and can accurately reflect the dynamic effect of the system,but the calculation time is high because the time step is too small in solving the static equilibrium process.Aiming at the static and dynamic failure of rock-like materials under static load,the damage switching criterion of the system is first proposed,aiming at constructing the bond-based PD model(SD-BPD)of the joint dynamic relaxation method and display algorithm.The dynamic relaxation method is adopted in the static equilibrium phase,and the dynamic phase is automatically switched to the explicit algorithm,which overcomes the distortion problem faced by the dynamic relaxation method in solving the dynamic phase,and has the ability to solve the deformation and failure of rock-like materials under static load.Then the feasibility and accuracy of the model are verified by an example.Finally,the process of deformation localization to failure of a Brazilian disk containing prefabricated cracks is numerically studied using the SD-BPD model.The main research work and results are as follows:(1)The example of Kalthoff-Winkler again verifies the effectiveness of the explicit algorithm in solving the dynamic failure problem,which lays a foundation for the joint algorithm to solve the dynamic failure problem.Then,dynamic relaxation method is adopted to conduct numerical study on the static and dynamic failure of L-shaped specimen under static load.The results show that: The dynamic relaxation method is suitable for the static deformation stage without damage.The existence of artificial damping in the simulated dynamic instability stage directly affects the change of acceleration.The acceleration of the system presents a sick phenomenon of "disappearing and disappearing" and zero points appear for many times.When the acceleration of the system is zero,it indicates that the object is in the state of equilibrium,which is contrary to the dynamic instability stage of the object under the action of continuous load.(2)The system damage switching criterion is proposed,the joint algorithm to solve the static and dynamic failure problem of rock-like materials under static load is given,and the corresponding flow chart is drawn,thus the SD-BPD model based on the joint algorithm is established.The verification example shows that: S-BPD model cannot describe the real dynamic failure process,the SD-BPD model based on the joint algorithm has the ability to describe rock materials in the dynamic failure stage,and the obtained numerical results are highly accurate and highly consistent with the test results,which can reproduce the deformation localization and crack dynamic propagation phenomena in the process of material static deformation to dynamic failure.(3)SD-BPD model was used to reproduce deformation localization and crack propagation in the deformation and failure process of Brazilian disk,and the influence of crack characteristics and elastic modulus on the deformation and failure process was analyzed.The results show that: The initial crack angle,relative length and elastic modulus all affect the failure load of the disk,the crack inclination β=0° corresponds to the lowest failure load,35°～ 45° crack inclination corresponds to the maximum failure load,and the bearing capacity of the disk increases with the increase of the relative crack length and elastic modulus.The deformation zone is in positive relation with the crack inclination β=0° or β=90°,and in other cases,the deformation zone is approximately parallel to the crack strike.In addition,numerical results of Brazilian disk with parallel double cracks show that the pattern of crack arrangement and the length of rock bridge have influence on the transfixion mode of rock bridge,which results in significant changes in the failure pattern of the disk.