Assessment of liquefaction potential based on shear wave velocity:Strain energy approach

Liquefaction assessment based on strain energy is significantly superior to conventional stress-based methods.The main purpose of the present study is to investigate the correlation between shear wave velocity and strain energy capacity of silty sands.The dissipated energy until liquefaction occurs was calculated by analyzing the results of three series of comprehensive cyclic direct simple shear and triaxial tests on Ottawa F65,Nevada,and Firoozkuh sands with varying silt content by weight and relative densities.Additionally,the shear wave velocity of each series was obtained using bender element or resonant column tests.Consequently,for the first time,a liquefaction triggering criterion,relating to effective overburden normalized liquefaction capacity energy(WL=s’c)to effective overburden stresscorrected shear wave velocity(eVs1)has been introduced.The accuracy of the proposed criteria was evaluated using in situ data.The results confirm the ability of shear wave velocity as a distinguishing parameter for separating liquefied and non-liquefied soils when it is calculated against liquefaction capacity energy(WL=s’c).However,the proposed WL=s’c-Vs1 curve,similar to previously proposed cyclic resistance ratio(CRR)-Vs1 relationships,should be used conservatively for fields vulnerable to liquefaction-induced lateral spreading.

Numerical modeling on strain energy evolution in rock system interaction with energy-absorbing prop and rock bolt

The interaction mechanism between coal and rock masses with supporting materials is significant in roadway control, especially in deep underground mining situations where dynamic hazards frequently happened due to high geo-stress and strong disturbed effects. This paper is to investigate the strain energy evolution in the interaction between coal and rock masses with self-designed energy-absorbing props and rock bolts by numerical modeling with the finite difference method. The interaction between rock and rock bolt/prop is accomplished by the cables element and the interface between the inner and outer props. Roadway excavation and coal extraction conditions in deep mining are numerically employed to investigate deformation, plastic zone ranges, strain energy input, accumulation, dissipation,and release. The effect on strain energy input, accumulation, dissipation, and release with rock deformation, and the plastic zone is addressed. A ratio of strain energy accumulation, dissipation, and release with energy input a, β, γ is to assess the dynamic hazards. The effects on roadway excavation and coal extraction steps of a, β, γ are discussed. The results show that:(1) In deep high geo-stress roadways, the energyabsorbing support system plays a dual role in resisting deformation and reducing the scope of plastic zones in surrounding rock, as well as absorbing energy release in the surrounding rock, especially in the coal extraction state to mitigate disturbed effects.(2) The strain energy input, accumulation is dependent on roadway deformation, the strain energy dissipation is relied on plastic zone area and disturbed effects, and strain energy release density is the difference among the three. The function of energyabsorbing rock bolts and props play a key role to mitigate strain energy release density and amount, especially in coal extraction condition, with a peak density value from 4×10^(4) to 1×10^(4)J/m^(3), and amount value from 3.57×10^(8) to 1.90×10^(6)J.(3) When mining is advanced in small steps, the strain energy accumulation is dominated. While in a large step, the released energy is dominant, thus a more dynamic hazards proneness. The energy-absorbing rock bolt and prop can reduce three times strain energy release amount, thus reducing the dynamic hazards. The results suggest that energy-absorbing props and rock bolts can effectively reduce the strain energy in the coal and rock masses, and prevent rock bursts and other hazards.The numerical model developed in this study can also be used to optimize the design of energyabsorbing props and rock bolts for specific mining conditions.

Mechanical properties and energy evolution of Beishan shallow-layer granite under different unloading paths

Rock has mechanical characteristics and a fracture damage mechanism that are closely related to its loading history and loading path. The mechanical properties, fracture damage features, acoustic emission(AE) characteristics, and strain energy evolution of the Beishan shallow-layer granite used in triaxial unloading tests were investigated in this study. Three groups of triaxial tests, namely, conventional triaxial compression test(Group Ⅰ), maintaining deviatoric stress synchronously unloading confining pressure test(Group Ⅱ), and loading axial pressure synchronously unloading confining pressure test(Group Ⅲ), were carried out for the cylindrical granite specimens. AE monitoring device was utilized in these tests to determine the degree to which the AE waves and AE events reflected the degree of rock damage. In addition, the crack stress thresholds of the specimens were determined by volumetric strain method and AE parameter method, and strain energy evolution of the rock was explored in different damage stages. The results show that the shallow-layer granite experiences brittle failure during the triaxial loading test and unloading test, and the rock has a greater damage degree during the unloading test. The crack stress thresholds of these samples vary greatly between tests, but the threshold ratios of all samples are similar in the same crack damage stage. The Mogi-Coulomb strength criterion can better describe the unloading failure strength of the rock. The evolution of the AE parameter characteristics and strain energy differs between the specimens used in different stress path tests. The dissipative strain energy is the largest in Group Ⅱ and the smallest in Group Ⅰ.

Study on creep deformation and energy development of underground surrounding rock under four‐dimensional support

There is an urgent need to develop optimal solutions for deformation control of deep high‐stress roadways,one of the critical problems in underground engineering.The previously proposed four‐dimensional support(hereinafter 4D support),as a new support technology,can set the roadway surrounding rock under three‐dimensional pressure in the new balanced structure,and prevent instability of surrounding rock in underground engineering.However,the influence of roadway depth and creep deformation on the surrounding rock supported by 4D support is still unknown.This study investigated the influence of roadway depth and creep deformation time on the instability of surrounding rock by analyzing the energy development.The elastic strain energy was analyzed using the program redeveloped in FLAC3D.The numerical simulation results indicate that the combined support mode of 4D roof supports and conventional side supports is highly applicable to the stability control of surrounding rock with a roadway depth exceeding 520 m.With the increase of roadway depth,4D support can effectively restrain the area and depth of plastic deformation in the surrounding rock.Further,4D support limits the accumulation range and rate of elastic strain energy as the creep deformation time increases.4D support can effectively reduce the plastic deformation of roadway surrounding rock and maintain the stability for a long deformation period of 6 months.As confirmed by in situ monitoring results,4D support is more effective for the long‐term stability control of surrounding rock than conventional support.

Seismic Liquefaction Resistance Based on Strain Energy Concept Considering Fine Content Value Effect and Performance Parametric Sensitivity Analysis

Liquefaction is one of the most destructive phenomena caused by earthquakes,which has been studied in the issues of potential,triggering and hazard analysis.The strain energy approach is a common method to investigate liquefaction potential.In this study,two Artificial Neural Network(ANN)models were developed to estimate the liquefaction resistance of sandy soil based on the capacity strain energy concept(W)by using laboratory test data.A large database was collected from the literature.One group of the dataset was utilized for validating the process in order to prevent overtraining the presented model.To investigate the complex influence of fine content(FC)on liquefaction resistance,according to previous studies,the second database was arranged by samples with FC of less than 28%and was used to train the second ANN model.Then,two presented ANN models in this study,in addition to four extra available models,were applied to an additional 20 new samples for comparing their results to show the capability and accuracy of the presented models herein.Furthermore,a parametric sensitivity analysis was performed through Monte Carlo Simulation(MCS)to evaluate the effects of parameters and their uncertainties on the liquefaction resistance of soils.According to the results,the developed models provide a higher accuracy prediction performance than the previously publishedmodels.The sensitivity analysis illustrated that the uncertainties of grading parameters significantly affect the liquefaction resistance of soils.

Using strain energy-based prediction of effective elastic properties in topology optimization of material microstructures

An alternative strain energy method is proposed for the prediction of effective elastic properties of orthotropic materials in this paper. The method is implemented in the topology optimization procedure to design cellular solids. A comparative study is made between the strain energy method and the well-known homogenization method. Numerical results show that both methods agree well in the numerical prediction and sensitivity analysis of effective elastic tensor when homogeneous boundary conditions are properly specified. Two dimensional and three dimensional microstructures are optimized for maximum stiffness designs by combining the proposed method with the dual optimization algorithm of convex programming. Satisfactory results are obtained for a variety of design cases.

Theoretical verification of the rationality of strain energy storage index as rockburst criterion based on linear energy storage law

The rationality of using strain energy storage index(Wet)for evaluating rockburst proneness was theoretically verified based on linear energy storage(LES)law in this study.The LES law is defined as the linear relationship between the elastic strain energy stored inside the solid material and the input strain energy during loading.It is used to determine the elastic strain energy and dissipated strain energy of rock specimens at various loading/unloading stress levels.The results showed that the Wetvalue obtained from experiments was close to the corresponding theoretical one from the LES law.Furthermore,with an increase in the loading/unloading stress level,the ratio of elastic strain energy to dissipated strain energy converged to the peak-strength strain energy storage index(Wp et).This index is stable and can better reflect the relative magnitudes of the stored energy and the dissipated energy of rocks at the whole pre-peak stage than the strain energy storage index.The peak-strength strain energy storage index can replace the conventional strain energy storage index as a new index for evaluating rockburst proneness.

Modal Strain Energy Based Structural Damage Localization for Offshore Platform using Simulated and Measured Data

Modal strain energy based methods for damage detection have received much attention. However, most of published articles use numerical methods and some studies conduct modal tests with simple 1D or 2D structures to verify the damage detection algorithms. Only a few studies utilize modal testing data from 3D frame structures. Few studies conduct performance comparisons between two different modal strain energy based methods. The objective of this paper is to investigate and compare the effectiveness of a traditional modal strain energy method(Stubbs index) and a recently developed modal strain energy decomposition(MSED) method for damage localization, for such a purpose both simulated and measured data from an offshore platform model being used. Particularly, the mode shapes used in the damage localization are identified and synthesized from only two measurements of one damage scenario because of the limited number of sensors. The two methods were first briefly reviewed. Next, using a 3D offshore platform model, the damage detection algorithms were implemented with different levels of damage severities for both single damage and multiple damage cases. Finally, a physical model of an offshore steel platform was constructed for modal testing and for validating the applicability. Results indicate that the MSED method outperforms the Stubbs index method for structural damage detection.

An Improved Modal Strain Energy Method for Damage Detection in Offshore Platform Structures

The development of robust damage detection methods for offshore structures is crucial to prevent catastrophes caused by structural failures. In this research, we developed an Improved Modal Strain Energy （IMSE） method for detecting damage in offshore platform structures based on a traditional modal strain energy method （the Stubbs index method）. The most significant difference from the Stubbs index method was the application of modal frequencies. The goal was to improve the robustness of the traditional method. To demonstrate the effectiveness and practicality of the proposed IMSE method, both numerical and experimental studies were conducted for different damage scenarios using a jacket platform structure. The results demonstrated the effectiveness of the IMSE method in damage location when only limited, spatially incomplete, and noise-polluted modal data is available. Comparative studies showed that the IMSE index outperformed the Stubbs index and exhibited stronger robustness, confirming the superiority of the proposed approach.

Vibro-acoustic Radiation Characteristics Analysis of Railway Vehicle Wheel with Damping Ridges Based on Modal Strain Energy

The existing researches on the damping wheel mainly focus on investigating the influence of damping structure change on the vibro-acoustic control.The changes include the geometric size of the damping structure,the damping material parameters,and the placement,and so on.In order to further understand the mechanism in reducing the acoustic radiation of railway wheel with layer damping treatment,in this paper,the wheel is simply modified by a full-sized circular plate.The circle plate side has stuck circumference constrained damping ridges and radial constrained damping ridges on it.Based on a hybrid finite element method-boundary element method（FEM-BEM）,the paper develops a vibro-acoustic radiation model for such a distributed constrained damping structure.The vibration and acoustic radiation of the circular plate is analyzed.In the analysis,the dynamic response of the system is obtained by using the 3D finite model superposition method.The obtained vibration response is used as the initial boundary condition in solving Helmholtz boundary integral equation for the sound radiation analysis.In the procedure,firstly,the modal analysis of the circular plate is performed to get the distribution of the system modal strain energy.Secondly,the vibro-acoustic radiation characteristics of the plate with different kinds of circumference damping ridges and radial damping ridges are compared in order to try to find the best effective damping ridge structure.Thirdly,using the distribution of the plate modal strain energy investigates the effect of the ridge distribution locations on the circular plate on its vibro-acoustic radiation.The calculation and analysis research results show that,the sticking circumference and radial damping ridges on the plate can control the vibro-acoustic radiation of the plate effectively in different frequency range.The distribution of the constrained damping ridge has an effect on reduction in vibro-acoustic radiation of the circular plate.The present research is very useful in the design of railway wheel with low noise level.

A numerical study of rock burst development and strain energy release

We consider rock burst to be a dynamic disaster similar to earthquakes,rapid land sliding,or coal mine gas dynamic disasters.Multi-scale mechanical principles imply the same mechanism of damage evolution proceeds the catastrophe.Damage may occur at various scales from a meso-scopic scale to a macroscopic,or engineering scale.Rock burst is a catastrophe at the scale of the engineering structure,such as a tunnel cross section or the work face of a long wall mine.It results from dynamic fracture of the structure where microscopic damage nucleates,expands,and finally propagates into a macroscopic sized fracture band.Rock burst must,therefore,undergo a relatively long development,or gestation,time before its final appearance.In this paper,a study of rock burst within a deeply buried tunnel by numerical methods is described.The results show that during rock burst gestation the distributed microscopic damage in the rock surrounding the tunnel localizes,intersects,and then evolves into a set of concentrated ''V'' shaped damage bands.These concentrated damage bands propagate in the direction of maximum shear as shearing slide bands take shape.Rock burst happens within the wedge separated by the shear bands from the native tunnel rock.An analysis of the wedge fracture shows that the unloading effects result in rock burst and rapid release of the strain energy.The implications for rock burst prediction in tunnels are that:(1) rock burst develops in the upper arch corners of in the tunnel cross section prior to developing in other zones,so good attention must be paid there;(2) all monitoring,prevention,and treatment of rock burst should be done during the gestation phase;(3) the shear bands contain abundant information concerning the physics and mechanics of the process and they are the foundation of physical and mechanical monitoring of acoustic emission,micro seismic events,stress,and the like.Thus a special study of the shearing mechanism is required.

Accurate theoretical evaluation of strain energy of all-carboatomic ring(cyclo[2n]carbon),boron nitride ring,and cyclic polyacetylene

Cyclocarbon fully consists of sp-hybridized carbon atoms,which shows quite unusual electronic and geometric structures compared to common molecules.In this work,we systematically studied strain energy(SE)of cyclocarbons of different sizes using regression analysis method based on electronic energies evaluated at the very accurate DLPNO-CCSD(T)/ccp VTZ theoretical level.In addition,ring strain of two systems closely related to cyclocarbon,boron nitride(BN)ring,and cyclic polyacetylene(c-PA),is also explored.Very ideal relationships between SE and number of repeat units(n)are built for cyclo[2n]carbon,B_(n)N_(n),and[2n]c-PA as SE=555.0·n^(-1),145.1·n^(-1),and 629.8·n^(-1)kcal·mol^(-1),respectively,and the underlying reasons of the difference and similarity in their SEs are discussed from electronic structure perspective.In addition,force constant of harmonic potential of C-C-C angles in cyclocarbon is derived based on SE values,the result is found to be 56.23 kcal·mol^(-1)·rad^(-2).The possibility of constructing homodesmotic reactions to calculate SEs of cyclocarbons is also explored in this work,although this method is far less rigorous than the regression analysis method,its result is qualitatively correct and has the advantage of much lower computational cost.In addition,comparisons show thatωB97XD/def2-TZVP is a good inexpensive alternative to the DLPNO-CCSD(T)/cc-p VTZ for evaluating energies used in deriving SE,while the popular and very cheap B3LYP/6-31G(d)level should be used with caution for systems with global electron conjugation such as c-PA.

Fault on-off versus strain rate and earthquakes energy

We propose that the brittle-ductile transition （BDT） controls the seismic cycle. In particular, the movements detected by space geodesy record the steady state deformation in the ductile lower crust, whereas the stick-slip behavior of the brittle upper crust is constrained by its larger friction. GPS data allow analyzing the strain rate along active plate boundaries. In all tectonic settings, we propose that earthquakes primarily occur along active fault segments characterized by relative minima of strain rate, segments which are locked or slowly creeping. We discuss regional examples where large earthquakes happened in areas of relative low strain rate. Regardless the tectonic style, the interseismic stress and strain pattern inverts during the coseismic stage. Where a dilated band formed during the interseismic stage, this will be shortened at the coseismic stage, and vice-versa what was previously shortened, it will be dilated. The interseismic energy accumulation and the coseismic expenditure rather depend on the tectonic setting （extensional, contractional, or strike-slip）. The gravitational potential energy dominates along normal faults, whereas the elastic energy prevails for thrust earthquakes and performs work against the gravity force. The energy budget in strike-slip tectonic setting is also primarily due elastic energy. Therefore, precursors may be different as a function of the tectonic setting. In this model, with a given displacement, the magnitude of an earthquake results from the coseismic slip of the deformed volume above the BDT rather than only on the fault length, and it also depends on the fault kinematics.

Influence of screw length and diameter on tibial strain energy density distribution after anterior cruciate ligament reconstruction

Postoperative tunnel enlargement has been frequently reported after anterior cruciate ligament（ACL）reconstruction.Interference screw,as a surgical implant in ACL reconstruction,may influence natural loading transmission and contribute to tunnel enlargement.The aims of this study are（1）to quantify the alteration of strain energy density（SED）distribution after the anatomic single-bundle ACL reconstruction;and（2）to characterize the influence of screw length and diameter on the degree of the SED alteration.A validated finite element model of human knee joint was used.The screw length ranging from 20 to 30 mm with screw diameter ranging from 7 to 9 mm were investigated.In the post-operative knee,the SED increased steeply at the extra-articular tunnel aperture under compressive and complex loadings,whereas the SED decreased beneath the screw shaft and nearby the intra-articular tunnel aperture.Increasing the screw length could lower the SED deprivation in the proximal part of the bone tunnel;whereas increasing either screw length or diameter could aggravate the SED deprivation in the distal part of the bone tunnel.Decreasing the elastic modulus of the screw could lower the bone SED deprivation around the screw.In consideration of both graft stability and SED alteration,a biodegradable interference screw with a long length is recommended,which could provide a beneficial mechanical environment at the distal part of the tunnel,and meanwhile decrease the bone-graft motion and synovial fluid propagation at the proximal part of the tunnel.These findings together with the clinical and histological factors could help to improve surgical outcome,and serve as a preliminary knowledge for the following study of biodegradable interference screw.

Evaluation of rockburst proneness considering specimen shape by storable elastic strain energy

To systematically assess the rockburst proneness considering specimen shape,multiple groups of laboratory tests were performed on 5 rock materials in cylindrical and cuboid shapes.The linear energy storage(LES)law of both cylindrical and cuboid rock specimens under uniaxial compressive load was confirmed,and the energy storage coefficient was found to be unrelated to specimen shape.On the basis of LES law,two rockburst proneness indexes,namely the strain energy storage index(W_(et))and the potential energy of elastic strain(PES),were modified.Subsequently,the W_(et),PES,peak-strength strain energy storage index(W_(et))p,and peak-strength potential energy of elastic strain(PESp)were used to assess the rockburst proneness of the cylindrical and cuboid specimens.In addition,the fragment ejection course of specimens under test was recorded by a high-speed camera.Then,the rockburst proneness judgments obtained from the 4 indexes were compared with the qualitative data during rock destruction.The results show that,under similar stress conditions,specimen shape has an ignorable effect on the rockburst proneness as a whole.The judgment accuracy of the two modified indexes,especially that of the PESp,is favorably improved to evaluate the rockburst proneness of both cylindrical and cuboid specimens.However,misjudgment ofW_(et)^(p)and PESp may still occur in the assessment of rockburst proneness as these two indexes only consider the energy state before rock peak strength and the W_(et)^(p)is formulated in a ratio form.

Determination of Material Parameters of EVA Foam under Uniaxial Compressive Testing Using Hyperelastic Models

The objective of this research was to determine the mechanical parameter from EVA foam and also investigate its behavior by using Blatz-Ko,Neo-Hookean,Mooney model and experimental test.The physical characteristic of EVA foam was also evaluated by scanning electron microscopy(SEM).The results show that Blatz-Ko and Neo-Hookean model can fit the curve at 5%and 8%strain,respectively.The Mooney model can fit the curve at 50%strain.The modulus of rigidity evaluated from Mooney model is 0.0814±0.0027 MPa.The structure of EVA foam from SEM image shows that EVA structure is a closed cell with homogeneous porous structure.From the result,it is found that Mooney model can adjust the data better than other models.This model can be applied for mechanical response prediction of EVA foam and also for reference value in engineering application.

Multiscale Study on Low Temperature Crack Resistance Mechanism of Steel Slag Asphalt Mixture

The objective of this paper was to study low temperature crack resistance mechanism of steel slag asphalt mixture(SAM).Thermal stress restrained specimen test(TSRST)and three-point bending test were carried out to evaluate the low-temperature crack resistance of SAM and basalt asphalt mixture(BAM).Based on the digital image correlation technique(DIC),the strain field distribution and crack propagation of SAM were analyzed from the microscopic point of view,and a new index,crack length factor(C),was proposed to evaluate the crack resistance of the asphalt mixture.The crystal phase composition and microstructure of steel slag aggregate(SA)and basalt aggregate(BA)were studied by X-ray diffraction(XRD)and scanning electron microscopy(SEM)to explore the low-temperature crack resistance mechanism of SAM.Results show that the low-temperature crack resistance of SAM is better than that of BAM;SAM has good integrity and persistent elastic deformation,and its bending failure mode is a hysteretic quasi-brittle failure;The SA surface is evenly distributed with pores and has surface roughness.SA has the composition phase of alkaline aggregate-calcite(CaCO3),so it has good adhesion to asphalt,which reveals the mechanism of excellent low-temperature crack resistance of SAM.

Rockburst proneness considering energy characteristics and sample shape effects

Accurate prediction of rockburst proneness is one of challenges for assessing the rockburst risk and selecting effective control measures.This study aims to assess rockburst proneness by considering the energy characteristics and qualitative information during rock failure.Several representative rock types in cylindrical and cuboidal sample shapes were tested under uniaxial compression conditions and the failure progress was detected by a high-speed camera.The far-field ejection mass ratio(FEMR)was determined considering the qualitative failure information of the rock samples.The peak-strength energy impact index and the residual elastic energy index were used to quantitatively evaluate the rockburst proneness of both cylindrical and cuboidal samples.Further,the performance of these two indices was analyzed by comparing their estimates with the FEMR.The results show that the accuracy of the residual elastic energy index is significantly higher than that of the peak-strength energy impact index.The residual elastic energy index and the FEMR are in good agreement for both cylindrical and cuboidal rock materials.This is because these two indices can essentially reflect the common energy release mechanism characterized by the mass,ejection velocity,and ejection distance of rock fragments.It suggests that both the FEMR and the residual elastic energy index can be used to accurately measure the rockburst proneness of cylindrical and cuboidal samples based on uniaxial compression test.

RESEARCH FOR THE STRAIN ENERGY RELEASE RATE OF COMPLEX CRACKS BY USING POINT-BY-POINT CLOSED EXTRAPOLATION APPROACH

A new extrapolation approach was proposed to calculate the strain energy release rates of complex cracks. The point_by_point closed method was used to calculate the closed energy, thus the disadvantage of self_inconsistency in some published papers can be avoided. The disadvantage is that the closed energy is repeatedly calculated: when closed nodal number along radial direction is more than two, the displacement of nodes behind the crack tip that is multiplied by nodal forces, the closed energy has been calculated and the crack surfaces have been closed, and that closed energy of middle point is calculated repeatedly. A DCB (double cantilever beam) specimen was calculated and compared with other theoretical results, it is shown that a better coincidence is obtained. In addition the same results are also obtained for compact tension specimen, three point bend specimen and single edge cracked specimen. In comparison with theoretical results,the error can be limited within 1 per cent. This method can be extended to analyze the fracture of composite laminates with various delamination cracks.

Prediction of cutting force in ultra-precision machining of nonferrous metals based on strain energy

The effects of the nonuniform cutting force and elastic recovery of processed materials in ultra-precision machining are too complex to be treated using traditional cutting theories,and it is necessary to take account of factors such as size effects,the undeformed cutting thickness,the tool blunt radius,and the tool rake angle.Therefore,this paper proposes a new theoretical calculation model for accurately predicting the cutting force in ultra-precision machining,taking account of such factors.The model is first used to analyze the material deformation of the workpiece and the cutting force distribution along the cutting edge of a diamond tool.The size of the strain zone in different cutting deformation zones is then determined by using the distribution of strain work per unit volume and considering the characteristics of the stress distribution in these different deformation zones.Finally,the cutting force during ultra-precision machining is predicted precisely by calculating the material strain energy in different zones.A finite element analysis and experimental data on ultra-precision cutting of copper and aluminum are used to verify the predictions of the theoretical model.The results show that the error in the cutting force between the calculation results and predictions of the model is less than 14%.The effects of the rake face stress distribution of the diamond tool,the close contact zone,and material elastic recovery can be fully taken into account by the theoretical model.Thus,the proposed theoretical calculation method can effectively predict the cutting force in ultra-precision machining.