Mechanical behavior of rock under uniaxial tension:Insights from energy storage and dissipation

Many rock engineering projects show that the growth of tensile cracks is often an important cause of engineering disasters,and the mechanical behavior of rocks is essentially the transmission,storage,dissipation and release of energy.To investigate the tensile behavior of rock from the perspective of energy,uniaxial tension tests(UTTs)and uniaxial compression tests(UCTs)were carried out on three typical rocks(granite,sandstone and marble).Different unloading points were set before the peak stress to separate elastic energy and dissipated energy.The input energy density ut,elastic energy density ue,and dissipated energy density ud at each unloading point were calculated by integrating stress-strain curves.The results show that there is a strong linear relationship between the three energy parameters and the square of the unloading stress in UCT,but this linear relationship is weaker in UTT.The ue and ud increase linearly with the increase in ut in UCT and UTT.Based on the phenomenon that ue and ud increase linearly with ut,the applicability of W_(et)^(p) index in UTT was proved and the relative energy storage capacity and absolute energy distribution characteristics of three rocks in UCT and UTT were evaluated.The tensile behavior of marble and sandstone in UTT can be divided into two stages vaguely according to the energy distribution,but granite is not the case.In addition,based on dissipated energy,the damage evolution of three types of rocks in UCT and UTT was discussed.This study provides some new insights for understanding the tensile behavior of rock.

Mechanical response and dilatancy characteristics of deep marble under different stress paths:A sight from energy dissipation

Dilatancy is a fundamental volumetric growth behavior observed during loading and serves as a key index to comprehending the intricate nonlinear behavior and constitutive equation structure of rock.This study focuses on Jinping marble obtained from the Jinping Underground Laboratory in China at a depth of 2400 m.Various uniaxial and triaxial tests at different strain rates,along with constant confining pressure tests and reduced confining pressure tests under different confining pressures were conducted to analyze the mechanical response and dilatancy characteristics of the marble under four stress paths.Subsequently,a new empirical dilatancy coefficient is proposed based on the energy dissipation method.The results show that brittle failure characteristics of marble under uniaxial compression are more obvious with the strain rate increasing,and plastic failure characteristics of marble under triaxial compression are gradually strengthened.Furthermore,compared to the constant confining pressure,the volume expansion is relatively lower under unloading condition.The energy dissipation is closely linked to the process of dilatancy,with a rapid increase of dissipated energy coinciding with the beginning of dilatancy.A new empirical dilatancy coefficient is defined according to the change trend of energy dissipation rate curve,of which change trend is consistent with the actual dilatancy response in marble under different stress paths.The existing empirical and theoretical dilatancy models are analyzed,which shows that the empirical dilatancy coefficient based on the energy background is more universal.

A model for the mean velocity of debris flow movement based on the minimum energy dissipation principle

The geomorphic minimum energy dissipation principle is important in the development of gully evolutionary theory.The impact of debris flows on channels during movement also adheres to this theory.A minimum energy dissipation model for debris flows has been obtained from previous studies,which is derived from the flow rules of runoff along a channel under rainfall or ice-snow meltwater conditions.However,the lack of consideration for erosion characteristics has hindered a comprehensive understanding of the movement characteristics of debris flow.In this paper,the phenomenon of volume increase resulting from the entrainment along debris flow movement is considered in order to derive a model for the mean velocity,reflecting the minimum energy dissipation principle.The entire expression of the mean velocity model is determined through 38 typical glacial and rainstorm debris flow cases.To evaluate the reliability of the proposed model,we employed 164 monitoring data from 1995 to 2000 in the Jiangjia gully,Yunnan,China.The results show that the velocity calculated by the proposed model are highly correlated with those obtained from the monitoring data.Additionally,a comparison is made between the mean velocities calculated by the proposed model and those obtained from previous studies,highlighting the exceptional applicability of the proposed model.This study will contribute to reveal the movement laws of debris flow along the channel.

Novel energy dissipative method on the adaptive spatial discretization for the Allen–Cahn equation

We propose a novel energy dissipative method for the Allen–Cahn equation on nonuniform grids.For spatial discretization,the classical central difference method is utilized,while the average vector field method is applied for time discretization.Compared with the average vector field method on the uniform mesh,the proposed method can involve fewer grid points and achieve better numerical performance over long time simulation.This is due to the moving mesh method,which can concentrate the grid points more densely where the solution changes drastically.Numerical experiments are provided to illustrate the advantages of the proposed concrete adaptive energy dissipative scheme under large time and space steps over a long time.

Many-body dissipative particle dynamics with energy conservation:temperature-dependent long-term attractive interaction

Heat and mass transfer during the process of liquid droplet dynamic behaviors has attracted much attention in decades.At mesoscopic scale,numerical simulations of liquid droplets motion,such as impacting,sliding,and coalescence,have been widely studied by using the particle-based method named many-body dissipative particle dynamics(MDPD).However,the detailed information on heat transfer needs further description.This paper develops a modified MDPD with energy conservation(MDPDE)by introducing a temperature-dependent long-term attractive interaction.By fitting or deriving the expressions of the strength of the attractive force,the exponent of the weight function in the dissipative force,and the mesoscopic heat friction coefficient about temperature,we calculate the viscosity,self-diffusivity,thermal conductivity,and surface tension,and obtain the Schmidt number Sc,the Prandtl number P r,and the Ohnesorge number Oh for 273 K to 373 K.The simulation data of MDPDE coincide well with the experimental data of water,indicating that our model can be used to simulate the dynamic behaviors of liquid water.Furthermore,we compare the equilibrium contact angle of droplets wetting on solid surfaces with that calculated from three interfacial tensions by MDPDE simulations.The coincident results not only stand for the validation of Young’s equation at mesoscale,but manifest the reliability of our MDPDE model and applicability to the cases with free surfaces.Our model can be extended to study the multiphase flow withcomplex heat and mass transfer.

Effect of thermal treatment on energy dissipation of granite under cyclic impact loading

High temperature treatment causes thermal damage to rocks in deep mining.To study the thermal effect on the energy dissipation of rocks during the dynamic cyclic loading,cyclic impact loading experiments of heat-treated rocks were carried out using the splitting Hopkinson pressure bar(SHPB)experimental system.The correlations among the energy dissipation,energy dissipation rate,impact times,accumulated absorbed energy per volume,failure mode and temperature were analyzed.The results show that the reflected energy under the first impact increases and finally exceeds the absorbed energy when the temperature increases;however,the total reflected energy decreases above 200℃.The absorbed energy under the first impact and the total absorbed energy all decrease as the temperature increases,the rates of which decrease accordingly.And the same phenomenon appears for the transmitted energy and the rate of the transmitted energy.On the contrary,the rate of the reflected energy increases with the rising temperature.When the temperature increases,the fewer impact times are needed to destroy the sample.In addition,the failure modes are different when the rock is treated at different temperatures;that is,when the temperature is high,even though the absorbed energy is low,the sample breaks into powder after several impacts.

Quantitative calculation for the dissipated energy of fault rock burst based on gradient-dependent plasticity

The capacity of energy absorption by fault bands after rock burst wascalculated quantitatively according to shear stress-shear deformation curves considering theinteractions and interplaying among microstructures due to the heterogeneity of strain softeningrock materials. The post-peak stiffness of rock specimens subjected to direct shear was derivedstrictly based on gradient-dependent plasticity, which can not be obtained from the classicalelastoplastic theory. Analytical solutions for the dissipated energy of rock burst were proposedwhether the slope of the post-peak shear stress-shear deformation curve is positive or not. Theanalytical solutions show that shear stress level, confining pressure, shear strength, brittleness,strain rate and heterogeneity of rock materials have important influence on the dissipated energy.The larger value of the dissipated energy means that the capacity of energy dissipation in the formof shear bands is superior and a lower magnitude of rock burst is expected under the condition ofthe same work done by external shear force. The possibility of rock burst is reduced for a lowersoftening modulus or a larger thickness of shear bands.

Energy dissipation of coal and rock during damage and failure process based on EMR

The physical and mechanical change processes of coal and rock are closely related to energy transformation,and the destruction and failure of coal and rock is an instability phenomena driven by energy change.However,the energy change of large-scale coal rock in the mine site is hardly calculated accurately,making it difficult to monitor coal-rock systematic failure and collapse from the perspective of energy.By the energy dissipation EMR monitoring system,we studied the damage and failure of coal and rock with bursting liability from the energy dissipation point using the geophysical method-EMR,and explored the energy dissipation characteristics during uniaxial compression and their main influencing factors.The results show that under displacement-control loading mode,there are 2 types of energy dissipation trends for both coal and rock with bursting liability.The type Ⅰ trend is a steady increase one during the whole process,therein,the energy dissipation of rock samples is accelerated at the peak load.The type Ⅱ trend energy is a W-shaped fluctuating one containing 6 stages.Under load-control loading mode,there is one energy dissipation trend of shock downward-steady rise.Besides that,rock samples also present a trend of 4 stages.The energy dissipation characteristics of coal and rockduring loading failure process can be used as effective criteria to assess whether they are in a stable or destructive stage.The factors influencing energy dissipation in the loading failure process of coal and rock mainly include strength,homogeneity,and energy input efficiency.

Mechanical and energy dissipation characteristics of granite under cyclic impact loading

This study investigated the effect of repeated blasting on the stability of surrounding rock during the construction of a tunnel or city underground engineering.The split Hopkinson pressure bar(SHPB)was used to carry out cyclic impact tests on granite samples,each having a circular hole,under different axial pressures,and the cumulative specific energy was proposed to characterize the damage characteristics of the rock during the cyclic impact.The mechanical properties and the energy absorbed by the granite samples under cyclic impact loads were analyzed.The results showed that under different axial pressures,the reflected waveform from the samples was characterized by“double-peak”phenomenon,which gradually changed to“single-peak”wi th the increase in damage value.The dynamic peak stress of the sample first increased and then decreased with an increase in impact times.The damage value criterion established based on the energy dissipation could well characterize the relationship between the damage and the number of impacts,which showed a slow increase,steady increase,and high-speed increase,and the damage value depended mainly on the last impact.Under the action of different axial pressures,all the failure modes of the samples were axial splitting failures.As the strain rate increased,with an increase in the dimension of the block,the sizes of the rock fragments decreased,and the fragmentation became more severe.

Energy dissipation of a ring-like metal rubber isolator

Metal rubber （MR） is a kind of homogeneous poroelastic damping material made of metal wire. In this paper, by ana- lyzing the forces on the MR isolator and the MR element, the hysteresis loops of the force and deformation are studied and verified by experiments. The results show that the force and displacement hysteresis loop of the MR isolator is described by the force and deformation hysteresis loops of the MR elements. In addition, the relationship between the energy dissipation coefficient of the MR element and that of the MR isolator is derived. The energy dissipation coefficient is programmed and calculated by MATLAB using experimental data, and the results are compared with the theoretical value. It is the basis for the design and applied research of the MR isolator in a future study.

Experimental and analytical study on seismic behavior of steel-concrete multienergy dissipation composite shear walls

In this paper, a steel-concrete multi-energy dissipation composite shear wall, comprised of steel-reinforced concrete （SRC） columns, steel plate （SP） deep beams, a concrete wall and energy dissipation strips, is proposed. In order to study the multi-energy dissipation behavior and restorability after an earthquake, two stages of low cyclic loading tests were carded out on ten test specimens. In the first stage, test on five specimens with different number of SP deep beams was carried out, and the test lasted until the displacement drift reached 2%. In the second stage, thin SPs were welded to both sides of the five specimens tested in the first stage, and the same test was carried out on the repaired specimens （designated as new specimens）. The load-bearing capacity, stiffness, ductility, hysteretic behavior and failure characteristics were analyzed for both stages and the results are discussed herein. Extrapolating from these results, strength calculation models and formulas are proposed herein and simulations using ABAQUS carried out, they show good agreement with the test results. The study demonstrates that SRC columns, SP deep beams, concrete wall and energy dissipation strips cooperate well and play an important role in energy dissipation. In addition, this study shows that the shear wall has good recoverability after an earthquake, and that the welding of thin SP＇s to repair a deformed wall is a practicable technique.

Bearing capacity of foundation on slope determined by energy dissipation method and model experiments

To determine the ultimate bearing capacity of foundations on sloping ground surface in practice, energy dissipation method was used to formulate the beating capacity as programming problem, and full-scale model experiments were investigated to analyze the performance of the soil slopes loaded by a strip footing in laboratory. The soil failure is governed by a linear Mohr-Coulomb yield criterion, and soil deformation follows an associated flow rule. Based on the energy dissipation method of plastic mechanics, a multi-wedge translational failure mechanism was employed to obtain the three bearing capacity factors related to cohesion, equivalent surcharge load and the unit gravity for various slope inclination angles. Numerical results were compared with those of the published solutions using finite element method and those of model experiments. The bearing capacity factors were presented in the form of design charts for practical use in engineering. The results show that limit analysis solutions approximate to those of model tests, and that the energy dissipation method is effective to estimate bearing capacity of soil slope.

ENERGY DISSIPATION FOR WEAK SOLUTIONS OF INCOMPRESSIBLE MHD EQUATIONS

In this article, we mainly study the local equation of energy for weak solutions of 3D MHD equations. We define a dissipation term D（u, B） that steins from an eventual lack of smoothness in the solution, and then obtain a local equation of energy for weak solutions of 3D MHD equations. Finally, we consider the 2D case at the end of this article.

Energy dissipation rate: An indicator of coal deformation and failure under static and dynamic compressive loads

Dynamic disasters in Chinese coal mines pose a significant threat to coal productivity. Thus, a thorough understanding of the deformation and failure processes of coal is necessary. In this study, the energy dissipation rate is proposed as a novel indicator of coal deformation and failure under static and dynamic compressive loads. The relationship between stress-strain, uniaxial compressive strength, displacement rate, loading rate, fractal dimension, and energy dissipation rate was investigated through experiments conducted using the MTS C60 tests(static loads) and split Hopkinson pressure bar system(dynamic loads). The results show that the energy dissipation rate peaks are associated with stress drop during coal deformation, and also positively related to the uniaxial compressive strength. A higher displacement rate of quasi-static loads leads to an initial increase and then a decrease in energy dissipation rate, whereas a higher loading rate of dynamic loads results in larger energy dissipation rate. Theoretical analysis indicates that a sudden increase in energy dissipation rate suggests partial fracture occurring within coal under both quasi-static and dynamic loads. Hence, the energy dissipation rate is an essential indicator of partial fracture and final failure within coal, as well as a prospective precursor for catastrophic failure in coal mine.

Magneto-thermo-elastic waves in an infinite perfectly conducting elastic solid with energy dissipation

The generalized thermo-elasticity theory, i.e., Green and Naghdi （G-N） Ⅲ theory, with energy dissipation （TEWED） is employed in the study of time-harmonic plane wave propagation in an unbounded, perfectly electrically conducting elastic medium subject to primary uniform magnetic field. A more general dispersion equation with com- plex coefficients is obtained for coupled magneto-thermo-elastic wave solved in complex domain by using the Leguerre＇s method. It reveals that the coupled magneto-thermoelastic wave corresponds to modified dilatational and thermal wave propagation with finite speeds modified by finite thermal wave speeds, thermo-elastic coupling, thermal diffusivity, and the external magnetic field. Numerical results for a copper-like material are presented.

Recent progress and application on seismic isolation energy dissipation and control for structures in China

China is a country where 100% of the territory is located in a seismic zone. Most of the strong earthquakes are over prediction. Most fatalities are caused by structural collapse. Earthquakes not only cause severe damage to structures, but can also damage non-structural elements on and inside of facilities. This can halt city life, and disrupt hospitals, airports, bridges, power plants, and other infrastructure. Designers need to use new techniques to protect structures and facilities inside. Isolation, energy dissipation and, control systems are more and more widely used in recent years in China. Currently, there are nearly 6,500 structures with isolation and about 3,000 structures with passive energy dissipation or hybrid control in China. The mitigation techniques are applied to structures like residential buildings, large or complex structures, bridges, underwater tunnels, historical or cultural relic sites, and industrial facilities, and are used for retrofitting of existed structures. This paper introduces design rules and some new and innovative devices for seismic isolation, energy dissipation and hybrid control for civil and industrial structures. This paper also discusses the development trends for seismic resistance, seismic isolation, passive and active control techniques for the future in China and in the world.

Profile Evolution and Energy Dissipation for Internal SolitonTransmitting over Different Submarine Ridges

Fundamental experiments were carried out in a wave flume on internal solitary wave （ISW） of depression-type propagating over a submerged ridge. The seabed ridge included either triangular or semicircular shape - regarded as topographic obstacles. Influenced by the submarine ridge, the transmitted waves were found to always consist of a leading pulse （a solitary wave） followed by a dispersive wave train. The wave profile propagating over a triangular ridge was similar to that caused by a semicircular obstacle. Apparently, the smooth face of a semicircular ridge produced time lag of wave propagation. From experimental results available, the reduction in wave energy induced by a semicircular ridge was larger than that by a triangular one. The events of wave distortion, strong breaking, internal bolus, and stratification mixing happened in case that the crest of an ISW was great enough to interact with the topographic obstacle. The reduction in wave energy was induced by strong breaking, and it depended on the ridge height rather than the geometric shape of the ridge.

An Approximation to Energy Dissipation in Time Domain Simulation of Sloshing Waves Based on Linear Potential Theory

This paper proposes a new approximation to energy dissipation in time domain simulation of sloshing waves by use of a linear potential theory. The boundary value problem is solved by the NURBS （non-uniform rational B-spline） higher-order panel method, in which a time-domain Green function is employed. The energy dissipation is modeled by changing the boundary condition on solid boundaries. Model experiments are carried out in a partially filled rectangular tank with forced horizontal motion. Sloshing-induced internal pressures and horizontal force obtained numerically and experimentally are compared with each other. It is observed that the present energy dissipation approximation can help produce a fair agreement between experimental forces and those of numerical simulations.

ASYMPTOTIC STABILITY OF TRAVELING WAVES FOR A DISSIPATIVE NONLINEAR EVOLUTION SYSTEM

This paper is concerned with the existence and the nonlinear asymptotic stabil- ity of traveling wave solutions to the Cauchy problem for a system of dissipative evolution equations {θt=vζx＋（ζθ）x＋aθxx,ζt=-θx＋βζxx;with initial data and end states （ζθ）（x,0）=（ζ0,θ0）（x）→（ζ±,θ±）as x→∞.We obtain the existence of traveling wave solutions by phase plane analysis and show the asymptotic nonlinear stability of traveling wave solutions without restrictions on the coeffi- cients a and v by the method of energy estimates.

Relationship between energy dissipation rate and channel morphology in the development of the model braided channel

A certain pattern of channel is the product of its self-adjustment under given boundary, discharge and sediment conditions. Based upon the principle of process-response model, an experimental study with 18 runs is carried out in LESRC. This paper is focused on the variation of the energy dissipation versus the channel morphology during and after the bedmaking process of braided channel. The results show that there exists a good empirical relationship between the energy dissipation rate and channel morphology. According to this relationship and the theory of minimum rate of energy dissipation, the authors explain the metamorphosis of the model channel with the development of the braided river.