Study On Constructing Constitutive Relation,Influencing Factors Analysis,and Application Of The Non-Linear Viscoelasticity Of Rocks

It is usually assumed that the stress-strain relationship of rocks follows Hooke’s law which is perfectly elastic and there is a linear relationship between stress and strain.In the current study,it is known that rocks are strongly non-linearly elastic.The constitutive relation has been established to describe the non-linear elasticity of rocks,and it has also been found that the non-linear elastic behavior of rocks is accompanied by viscoelasticity.In recent years,a variety of theoretical and phenomenological models of non-linear elastic constitutive relations have been developed around the non-linear elastic behavior of rocks.However,there is no non-linear viscoelastic constitutive relation which is relatively simple in mathematical form and at the same time has both physical significance and a good match with experimental data.Therefore,it is important to construct a new non-linear viscoelastic constitutive relation that satisfies the above conditions.A new constitutive relation can not only help us to further understand the non-linear viscoelastic mechanism of rocks,but also help us to construct a new nonlinear viscoelastic wave equation,and then further extract the high-precision geophysical field information in the actual seismic exploration.It has been found that environmental factors such as temperature,humidity and atmospheric pressure affect the non-linear elastic behavior of rocks.However,the influence mechanism of nonlinear viscoelasticity of rocks has not been clarified.Research on the relationship between non-linear viscoelasticity of rock and temperature,humidity,atmospheric pressure is helpful to clarify the mechanism of non-linear viscoelasticity of rock.The research of rock non-linear elasticity has been applied to field exploration,but most current studies focus on large scale or around a single spatial and temporal scale(time or space).Therefore,for the engineering monitoring and detection of urban spatial structure,it provides a new idea and challenge that studying the change of non-linear elastic behavior caused by structural change at spatial and temporal scale within small area.Based on the nonlinear viscoelasticity of rocks,we studied its constitutive relation,influencing factors and application in field exploration.(1)In this paper,we firstly combine the classical non-linear elastic and classical viscoelastic constitutive relation models to construct a new non-linear viscoelastic constitutive relation theoretical model.First,based on the classical non-linear elastic model,we introduced Kelvin model.Then we concretize the classical non-linear elastic model according to Kelvin model and constructed the classical non-linear elastic component.Finally,taking the classical non-linear elastic constitutive relation as the framework and the Kelvin model as the core component,then we constructed the nonlinear viscoelastic component,while further introducing the error balance component,we constructed a new rock non-linear viscoelastic constitutive relation theoretical model.This new constitutive model contains both non-linear elasticity and viscoelasticity,which is closer to the stress-strain relationship of rocks in real situations.(2)We constructed a phenomenological model of non-linear viscoelastic constitutive relation of rocks,and compared and analyzed the model with the theoretical model.We set up a high-precision rock non-linear viscoelastic measurement experiment platform,and obtained high-precision non-linear viscoelastic modulus variation and strain data through performing the copropagating longitudinal acoustic wave experiment(CLAWE).Based on the obtained experimental data,we analyzed the variation characteristics of the non-linear viscoelastic behavior of rocks under different strain states and established a phenomenological model of rock non-linear viscoelastic constitutive relation.By comparing the theoretical model with the phenomenological model,the rationality and correctness of the established theoretical model are verified,and the non-linear viscoelastic parameters of the theoretical model are obtained.By further studying these parameters,we obtained the Q value during the propagation of high-frequency ultrasonic waves in the Crab Orchard sandstone.The magnitude of this value is consistent with the propagation characteristics of high-frequency ultrasonic waves in sandstone.Therefore,we believe that the new theoretical model is consistent with the physical mechanism to a certain extent.The new theoretical model of nonlinear viscoelastic constitutive relation is of great significance for further study of nonlinear viscoelastic wave equation and extraction of high-precision geophysical information from field data.(3)We revealed the nonlinear viscoelastic characteristics of rocks with temperature,humidity and atmospheric pressure.Based on the original CLAWE experimental platform,we added temperature,humidity and atmospheric pressure control equipment and monitoring sensors.In the process of rock non-linear viscoelastic measurement,the changes of temperature,humidity and atmospheric pressure are introduced.We firstly analyzed the infulence of temperature,humidity and atmospheric pressure changes on the variation of the non-linear viscoelastic modulus of rocks.We found that the variation of non-linear viscoelastic modulus of rocks is positively correlated with the changes of temperature and atmospheric pressure.And with the increasing of the temperature and atmospheric pressure,the variation of nonlinear viscoelastic modulus increases,and the absolute value decreases.The non-linear viscoelastic behavior weakens.The variation of non-linear viscoelastic modulus of rocks is negatively correlated with the change of humidity.And with the decreasing of humidity,the variation of non-linear viscoelastic modulus increases,and the absolute value decreases.The non-linear viscoelastic behavior weakens.Meanwhile,we also inverted the non-linear viscoelastic parameters of rocks by Grey Wolf Optimizer(GW O)at different temperatures and humidity,and investigated the relationship between the non-linear viscoelastic parameters and the changes of temperature and humidity.The disturbance term C is not sensitive to the variation of humidity.However,the secondorder non-linear viscoelastic parameter β is insensitive to the changes of humidity,it is extremely sensitive to the changes of temperature and can respond to temperature change less than 1℃.The viscosity parameter η/E,the third-order non-linear viscoelastic parameter δ,and the balance component parameter γ are all extremely sensitive to the changes of humidity.Among them,the η/E and δ are negatively correlated with humidity change,while γ is positively correlated with humidity change.This is important for further investigation of the rock non-linear viscoelastic mechanism.(4)We developed a new method to extract non-linear viscoelastic characteristics of shallow rocks based on a high-density seismic array.We conducted continuous ambient noise monitoring in Singapore using a dense seismograph survey line during four days.During the monitoring period,a rock mass was damaged on the survey line.We extracted the rock non-linear viscoelastic modulus variation from the coda wave of the time domain empirical Green’s function between the station pairs by using the moving window cross-spectrum analysis(MWCSA)method,and analyzed the nonlinear viscoelastic characteristics of rock strata during partial damage at high spatial and temporal resolution.The results show that local damage can affect the non-linear viscoelastic behavior observed throughout the entire array,however,there are obvious differences in the non-linear viscoelastic behavior between the undamaged and damaged areas and at different damage stages.The observation and finding play an essential role for monitoring the changes in urban underground structure and early warning of natural hazards.In summary,we constructed the theoretical model and the phenomenological model of the non-linear elastic constitutive relationship of rocks,and verified the rationality and correctness of the theoretical model through the comparison between the two models.The variation of rock non-linear viscoelastic modulus with temperature,humidity,and atmospheric pressure is revealed.Based on the non-linear viscoelastic constitutive relation theoretical model,we inverted the non-linear viscoelastic parameters by GWO,and study the changes of the non-linear viscoelastic parameters with temperature and humidity.We also extended the study of the non-linear viscoelasticity of rocks in field exploration and developed a new method to extract nonlinear viscoelastic characteristics of shallow rocks based on a high-density seismic array.