The occurrence of earthquakes is closely related to the crustal geotectonic movement and the migration of mass,which consequently cause changes in gravity.The Gravity Recovery And Climate Experiment(GRACE)satellite da...The occurrence of earthquakes is closely related to the crustal geotectonic movement and the migration of mass,which consequently cause changes in gravity.The Gravity Recovery And Climate Experiment(GRACE)satellite data can be used to detect gravity changes associated with large earthquakes.However,previous GRACE satellite-based seismic gravity-change studies have focused more on coseismic gravity changes than on preseismic gravity changes.Moreover,the noise of the north–south stripe in GRACE data is difficult to eliminate,thereby resulting in the loss of some gravity information related to tectonic activities.To explore the preseismic gravity anomalies in a more refined way,we first propose a method of characterizing gravity variation based on the maximum shear strain of gravity,inspired by the concept of crustal strain.The offset index method is then adopted to describe the gravity anomalies,and the spatial and temporal characteristics of gravity anomalies before earthquakes are analyzed at the scales of the fault zone and plate,respectively.In this work,experiments are carried out on the Tibetan Plateau and its surrounding areas,and the following findings are obtained:First,from the observation scale of the fault zone,we detect the occurrence of large-area gravity anomalies near the epicenter,oftentimes about half a year before an earthquake,and these anomalies were distributed along the fault zone.Second,from the observation scale of the plate,we find that when an earthquake occurred on the Tibetan Plateau,a large number of gravity anomalies also occurred at the boundary of the Tibetan Plateau and the Indian Plate.Moreover,the aforementioned experiments confirm that the proposed method can successfully capture the preseismic gravity anomalies of large earthquakes with a magnitude of less than 8,which suggests a new idea for the application of gravity satellite data to earthquake research.展开更多
The coupling effects of the metastable austenitic phase and the amorphous matrix in a transformation-induced plasticity(TRIP)-reinforced bulk metallic glass(BMG)composite under compressive loading were investigated by...The coupling effects of the metastable austenitic phase and the amorphous matrix in a transformation-induced plasticity(TRIP)-reinforced bulk metallic glass(BMG)composite under compressive loading were investigated by employing the digital image correlation(DIC)technique.The evolution of local strain field in the crystalline phase and the amorphous matrix was directly monitored,and the contribution from the phase transformation of the metastable austenitic phase was revealed.Local shear strain was found to be effectively consumed by the displacive phase transformation of the metastable austenitic phase,which relaxed the local strain/stress concentration at the interface and thus greatly enhanced the plasticity of the TRIP-reinforced BMG composites.Our current study sheds light on in-depth understanding of the underlying deformation mechanism and the interplay between the amorphous matrix and the metastable crystalline phase during deformation,which is helpful for design of advanced BMG composites with further improved properties.展开更多
By using GDS dynamic hollow cylinder torsional apparatus, a series of cyclic torsional triaxial tests under complex initial consolidation condition are performed on Nanjing saturated fine sand. The effects of the init...By using GDS dynamic hollow cylinder torsional apparatus, a series of cyclic torsional triaxial tests under complex initial consolidation condition are performed on Nanjing saturated fine sand. The effects of the initial principal stress direction αo, the initial ratio of deviatoric stress η0, the initial average effective principal stress Po and the initial intermediate principal stress parameter b0 on the threshold shear strain γt of Nanjing saturated fine sand are then systematically investigated. The results show that γt increases as η0,p0 and b0 increase respectively, while the other three parameters remain constant. ao has a great influence on γt, which is reduced when ao increases from 0° to 45°and increased when α0 increases from 45° to 90°. The effect of α0 on γt, plays a leading role and the effect of η0 will weaken when ao is approximately 45°.展开更多
An approach of the incompatible elements with additional internal shear strain is,in the presem paper,suggested and applied to geometrically nonlinear analysis of Mi-ndlin plate bending problem.It provides a quite cov...An approach of the incompatible elements with additional internal shear strain is,in the presem paper,suggested and applied to geometrically nonlinear analysis of Mi-ndlin plate bending problem.It provides a quite covenient way to avoid the whear loc-king troubles.An energy consistency condition for this kind of C°elements is offered.The nonlinear element formulations and some numerical results are presented.展开更多
Here is proposed the principle of interaction between plastic volumetric and shear strains, revealing the main origin of generating the complexity and variety of deformations for geotechnical materials. Here are also ...Here is proposed the principle of interaction between plastic volumetric and shear strains, revealing the main origin of generating the complexity and variety of deformations for geotechnical materials. Here are also explained the manners of the interaction between plastic volumetric and shear strains and the conditions of generating shear dilatancy. It is demonstrated that dependency of the stress path exists and is a combination of effects of this interaction. According to this principle, it is theoretically proved that the space critical state line exists, and is unique and independent of the stress history. Based on this principle, the constitutive models that are able completely and accurately to characterize the basic behavior features for geotechnical materials have been constructed within the framework of thermodynamics. What is determined is a general expression of the constitutive relation as well as the inequality of the dissipative potential increment for obeying the second law of thermodynamics.展开更多
The principle of interaction between plastic volumetric and shear strains for rock and soil has been extended to the field of unsaturated soils.Two new interactions of suction-plastic volumetric strain and pore air pr...The principle of interaction between plastic volumetric and shear strains for rock and soil has been extended to the field of unsaturated soils.Two new interactions of suction-plastic volumetric strain and pore air pressure-plastic volumetric strain appear in the unsaturated state of a soil except the interaction between plastic volumetric and shear strains.It is very important to find that the suction possesses a dual property,which is the origin of generating its special functions.Thereby the effect of the suction on volumetric strain includes two opposite aspects.By means of this property of suction,the physical significance of effective stress parameter,effects of suction on volume change and preconsolidation pressure,and the mechanism of collapse upon wetting all can be explained.In addition,it is theoretically proved by application of this principle of interaction that the critical state line for unsaturated soils exists,and is unique and independent of the stress history.展开更多
We study thermoelectric transport under shear strain in two spatial dimensional quantum matter using the holographic duality.General analytic formulae for the DC thermoelectric conductivities subjected to finite shear...We study thermoelectric transport under shear strain in two spatial dimensional quantum matter using the holographic duality.General analytic formulae for the DC thermoelectric conductivities subjected to finite shear strain are obtained in terms of black hole horizon data.Off-diagonal terms in the conductivity matrix also appear at zero magnetic field,resembling an emergent electronic nematicity,which cannot nevertheless be identified with the presence of an anomalous Hall effect.For an explicit model study,we numerically construct a family of strained black holes and obtain the corresponding nonlinear stress-strain curves.We then compute all electric,thermoelectric,and thermal conductivities and discuss the effects of strain.While the shear elastic deformation does not affect the temperature dependence of thermoelectric and thermal conductivities quantitatively,it can strongly change the behavior of the electric conductivity.For both shear hardening and softening cases,we find a clear metal-insulator transition driven by the shear deformation.Moreover,the violation of the previously conjectured thermal conductivity bound is observed for large shear deformation.展开更多
This paper presents an improved strain-softening constitutive model considering the effect of crack deformation based on the triaxial cyclic loading and unloading test results.The improved model assumes that total str...This paper presents an improved strain-softening constitutive model considering the effect of crack deformation based on the triaxial cyclic loading and unloading test results.The improved model assumes that total strain is a combination of plastic,elastic,and crack strains.The constitutive relationship between the crack strain and the stress was further derived.The evolutions of mechanical parameters,i.e.strength parameters,dilation angle,unloading elastic modulus,and deformation parameters of crack,with the plastic strain and confining pressure were studied.With the increase in plastic strain,the cohesion,friction angle,dilation angle,and crack Poisson's ratio initially increase and subsequently decrease,and the unloading elastic modulus and the crack elastic modulus nonlinearly decrease.The increasing confining pressure enhances the strength and unloading elastic modulus,and decreases the dilation angle and Poisson's ratio of the crack.The theoretical triaxial compressive stress-strain curves were compared with the experimental results,and they present a good agreement with each other.The improved constitutive model can well reflect the nonlinear mechanical behavior of granite.展开更多
The mechanical behavior of graphene under in-plane shear is studied using molecular dynamics simulations.We show that the shear behavior of chiral graphene is dependent on the loading direction due to its structural a...The mechanical behavior of graphene under in-plane shear is studied using molecular dynamics simulations.We show that the shear behavior of chiral graphene is dependent on the loading direction due to its structural asymmetry.The maximum shear failure strain of graphene in one direction may be 1.7 times higher than that in the opposite direction.We discuss also the influence of the cut-off parameters on the calculations.Our findings are useful for the understanding of mechanical behavior of graphene and the potential applications of graphene in nanodevices.展开更多
The influences of die parameters on shear strain were investigated by using two-dimensional finite element simulation.New formulas of shear strain were proposed.According to the results of formulas,the shear strain sh...The influences of die parameters on shear strain were investigated by using two-dimensional finite element simulation.New formulas of shear strain were proposed.According to the results of formulas,the shear strain showed a linear dependence on the difference between internal and external fillet radius and the slope was determined by the intersection angle.The simulation results indicated that the velocities of the points from different zones were different in the specimen and the motion trajectories of different points did not follow geometrical laws.The influences of the average velocity and the motion trajectory on shear strain were incorporated in the formula to calculate the shear strain produced during equalchannel angular pressing process.The reliability of simulation results has been partially validated by experiments.展开更多
The correlation between crystal rotation and redundant shear strain in rolled single crystals was investigated by using the crystal plasticity finite element(CPFE) model in this paper. The deformation in aluminium sin...The correlation between crystal rotation and redundant shear strain in rolled single crystals was investigated by using the crystal plasticity finite element(CPFE) model in this paper. The deformation in aluminium single crystals of four representative orientations(rotated-Cube, Goss, Copper, and Brass) after rolling and plain strain compression was simulated, and the predictions have been validated by the experimental observations. In the rotated-Cube and Goss, the redundant shear strain and crystal rotation were in the same pattern, alternating along the thickness, while the relation between them was not obvious for the Copper and Brass due to their asymmetrical distributions of activated slip systems. The relations between slip system activation, crystal rotation, and shear strain were investigated based on the CPFE model, and the correlation between shear strain and crystal rotation has been built.展开更多
The seismic loading on saturated soil deposits induces a decrease in effective stress and a rearrangement of the soil-particle structure, which may both lead to a degradation in undrained stiffness and strength of soi...The seismic loading on saturated soil deposits induces a decrease in effective stress and a rearrangement of the soil-particle structure, which may both lead to a degradation in undrained stiffness and strength of soils. Only the effective stress influence on small strain shear modulus Gmax is considered in seismic response analysis nowadays, and the cyclic shearing induced fabric changes of the soil-particle structure are neglected. In this paper, undrained cyclic triaxial tests were conducted on saturated sands with the shear wave velocity measured by bender element, to study the influences of seismic loading on Gmax. And Gmax of samples without cyclic loading effects was also investigated for comparison. The test results indicated that Gmax under cyclic loading effects is lower than that without such effects at the same effective stress, and also well correlated with the effective stress variation. Hence it is necessary to reinvestigate the determination of Gmax in seismic response analysis carefully to predict the ground responses during earthquake more reasonably.展开更多
A shear-lag theory was developed to investigate the strain transfer from the metal substrate to the surface acoustic wave (SAW) resonator through a bonding layer. A three-layer model of host structure-adhesive layer...A shear-lag theory was developed to investigate the strain transfer from the metal substrate to the surface acoustic wave (SAW) resonator through a bonding layer. A three-layer model of host structure-adhesive layer-resonator layer was established. The strain transfer was theoretically analyzed, and the main factors impacting the SAW sensor measurement were studied. The relationship between the sensor response and the individual effect of all these factors under static loads was discussed. Results showed that better accuracy could be achieved with increase in the adhesive stiffness or resonator length, or decrease in the adhesive thickness. The values of the strain transfer rate calculated from the analytical model agreed well with that from the available experiment data.展开更多
The realization way of snake rolling was introduced. Flow velocity, strain and stress distribution of 7075 aluminum alloy plate during snake rolling and symmetrical rolling were analyzed in Deform 3D. Effects of veloc...The realization way of snake rolling was introduced. Flow velocity, strain and stress distribution of 7075 aluminum alloy plate during snake rolling and symmetrical rolling were analyzed in Deform 3D. Effects of velocity ratio, offset distance between two rolls and pass reduction on the distribution of equivalent strain and shear strain were analyzed. The results show that flow velocity and equivalent strain on the lower layer of the plate are larger than those of the upper layer because of the larger velocity of the lower roll and the gap is increased with the increase of velocity ratio and pass reduction. The shear strain of roiling direction in the center point is almost zero during symmetrical rolling, while it is much larger during snake rolling because of the existence of rub zone. The shear strain is increased with the increase of velocity ratio, offset distance and pass reduction. This additional shear strain is beneficial to improve the in_homogeneous strain distribution.展开更多
基金supported by the National Key Research and Development Program of China(Grant No.2019YFC1509202)the National Natural Science Foundation of China(Grant Nos.41772350,61371189,and 41701513).
文摘The occurrence of earthquakes is closely related to the crustal geotectonic movement and the migration of mass,which consequently cause changes in gravity.The Gravity Recovery And Climate Experiment(GRACE)satellite data can be used to detect gravity changes associated with large earthquakes.However,previous GRACE satellite-based seismic gravity-change studies have focused more on coseismic gravity changes than on preseismic gravity changes.Moreover,the noise of the north–south stripe in GRACE data is difficult to eliminate,thereby resulting in the loss of some gravity information related to tectonic activities.To explore the preseismic gravity anomalies in a more refined way,we first propose a method of characterizing gravity variation based on the maximum shear strain of gravity,inspired by the concept of crustal strain.The offset index method is then adopted to describe the gravity anomalies,and the spatial and temporal characteristics of gravity anomalies before earthquakes are analyzed at the scales of the fault zone and plate,respectively.In this work,experiments are carried out on the Tibetan Plateau and its surrounding areas,and the following findings are obtained:First,from the observation scale of the fault zone,we detect the occurrence of large-area gravity anomalies near the epicenter,oftentimes about half a year before an earthquake,and these anomalies were distributed along the fault zone.Second,from the observation scale of the plate,we find that when an earthquake occurred on the Tibetan Plateau,a large number of gravity anomalies also occurred at the boundary of the Tibetan Plateau and the Indian Plate.Moreover,the aforementioned experiments confirm that the proposed method can successfully capture the preseismic gravity anomalies of large earthquakes with a magnitude of less than 8,which suggests a new idea for the application of gravity satellite data to earthquake research.
基金financially supported by the National Natural Science Foundation of China(Nos.52061135207,51871016,51921001,11790293,and 51971017)111 Project(No.B07003)the Projects of SKL-AMM-USTB(Nos.2019Z-01 and 2018Z-19)。
文摘The coupling effects of the metastable austenitic phase and the amorphous matrix in a transformation-induced plasticity(TRIP)-reinforced bulk metallic glass(BMG)composite under compressive loading were investigated by employing the digital image correlation(DIC)technique.The evolution of local strain field in the crystalline phase and the amorphous matrix was directly monitored,and the contribution from the phase transformation of the metastable austenitic phase was revealed.Local shear strain was found to be effectively consumed by the displacive phase transformation of the metastable austenitic phase,which relaxed the local strain/stress concentration at the interface and thus greatly enhanced the plasticity of the TRIP-reinforced BMG composites.Our current study sheds light on in-depth understanding of the underlying deformation mechanism and the interplay between the amorphous matrix and the metastable crystalline phase during deformation,which is helpful for design of advanced BMG composites with further improved properties.
基金supported by the Key Research Project of National Natural Science Foundation of China under grant No. 90715018the Special Fund for the Commonweal Industry of China under grant No. 200808022the Key Basic Research Program of Natural Science of University in Jiangsu Province under grant No. 08KJA560001
文摘By using GDS dynamic hollow cylinder torsional apparatus, a series of cyclic torsional triaxial tests under complex initial consolidation condition are performed on Nanjing saturated fine sand. The effects of the initial principal stress direction αo, the initial ratio of deviatoric stress η0, the initial average effective principal stress Po and the initial intermediate principal stress parameter b0 on the threshold shear strain γt of Nanjing saturated fine sand are then systematically investigated. The results show that γt increases as η0,p0 and b0 increase respectively, while the other three parameters remain constant. ao has a great influence on γt, which is reduced when ao increases from 0° to 45°and increased when α0 increases from 45° to 90°. The effect of α0 on γt, plays a leading role and the effect of η0 will weaken when ao is approximately 45°.
文摘An approach of the incompatible elements with additional internal shear strain is,in the presem paper,suggested and applied to geometrically nonlinear analysis of Mi-ndlin plate bending problem.It provides a quite covenient way to avoid the whear loc-king troubles.An energy consistency condition for this kind of C°elements is offered.The nonlinear element formulations and some numerical results are presented.
文摘Here is proposed the principle of interaction between plastic volumetric and shear strains, revealing the main origin of generating the complexity and variety of deformations for geotechnical materials. Here are also explained the manners of the interaction between plastic volumetric and shear strains and the conditions of generating shear dilatancy. It is demonstrated that dependency of the stress path exists and is a combination of effects of this interaction. According to this principle, it is theoretically proved that the space critical state line exists, and is unique and independent of the stress history. Based on this principle, the constitutive models that are able completely and accurately to characterize the basic behavior features for geotechnical materials have been constructed within the framework of thermodynamics. What is determined is a general expression of the constitutive relation as well as the inequality of the dissipative potential increment for obeying the second law of thermodynamics.
文摘The principle of interaction between plastic volumetric and shear strains for rock and soil has been extended to the field of unsaturated soils.Two new interactions of suction-plastic volumetric strain and pore air pressure-plastic volumetric strain appear in the unsaturated state of a soil except the interaction between plastic volumetric and shear strains.It is very important to find that the suction possesses a dual property,which is the origin of generating its special functions.Thereby the effect of the suction on volumetric strain includes two opposite aspects.By means of this property of suction,the physical significance of effective stress parameter,effects of suction on volume change and preconsolidation pressure,and the mechanism of collapse upon wetting all can be explained.In addition,it is theoretically proved by application of this principle of interaction that the critical state line for unsaturated soils exists,and is unique and independent of the stress history.
基金partially supported by the National Natural Science Foundation of China Grant Nos.12122513,12075298,11991052 and 12047503the Chinese Academy of Sciences Project for Young Scientists in Basic Research YSBR-006.
文摘We study thermoelectric transport under shear strain in two spatial dimensional quantum matter using the holographic duality.General analytic formulae for the DC thermoelectric conductivities subjected to finite shear strain are obtained in terms of black hole horizon data.Off-diagonal terms in the conductivity matrix also appear at zero magnetic field,resembling an emergent electronic nematicity,which cannot nevertheless be identified with the presence of an anomalous Hall effect.For an explicit model study,we numerically construct a family of strained black holes and obtain the corresponding nonlinear stress-strain curves.We then compute all electric,thermoelectric,and thermal conductivities and discuss the effects of strain.While the shear elastic deformation does not affect the temperature dependence of thermoelectric and thermal conductivities quantitatively,it can strongly change the behavior of the electric conductivity.For both shear hardening and softening cases,we find a clear metal-insulator transition driven by the shear deformation.Moreover,the violation of the previously conjectured thermal conductivity bound is observed for large shear deformation.
基金financially supported by the National Natural Science Foundation of China(Grant No.52074269).
文摘This paper presents an improved strain-softening constitutive model considering the effect of crack deformation based on the triaxial cyclic loading and unloading test results.The improved model assumes that total strain is a combination of plastic,elastic,and crack strains.The constitutive relationship between the crack strain and the stress was further derived.The evolutions of mechanical parameters,i.e.strength parameters,dilation angle,unloading elastic modulus,and deformation parameters of crack,with the plastic strain and confining pressure were studied.With the increase in plastic strain,the cohesion,friction angle,dilation angle,and crack Poisson's ratio initially increase and subsequently decrease,and the unloading elastic modulus and the crack elastic modulus nonlinearly decrease.The increasing confining pressure enhances the strength and unloading elastic modulus,and decreases the dilation angle and Poisson's ratio of the crack.The theoretical triaxial compressive stress-strain curves were compared with the experimental results,and they present a good agreement with each other.The improved constitutive model can well reflect the nonlinear mechanical behavior of granite.
基金supports from the Graduate Student Program of Shanghai University (Grant No.SHUCX101079)supported by the National Natural Science Foundation of China (Grant No.11172160)+2 种基金Fok Ying Tung Education Foundation (Grant No.121005)Shanghai Shuguang Program (Grant No. 08SG39),Shanghai Rising Star Program (Grant No. 09QH1401000)Shanghai Leading Academic Discipline Project (Grant No. S30106)
文摘The mechanical behavior of graphene under in-plane shear is studied using molecular dynamics simulations.We show that the shear behavior of chiral graphene is dependent on the loading direction due to its structural asymmetry.The maximum shear failure strain of graphene in one direction may be 1.7 times higher than that in the opposite direction.We discuss also the influence of the cut-off parameters on the calculations.Our findings are useful for the understanding of mechanical behavior of graphene and the potential applications of graphene in nanodevices.
基金Item Sponsored by Fundamental Research Funds for Central Universities of China(HEUCF20151002)
文摘The influences of die parameters on shear strain were investigated by using two-dimensional finite element simulation.New formulas of shear strain were proposed.According to the results of formulas,the shear strain showed a linear dependence on the difference between internal and external fillet radius and the slope was determined by the intersection angle.The simulation results indicated that the velocities of the points from different zones were different in the specimen and the motion trajectories of different points did not follow geometrical laws.The influences of the average velocity and the motion trajectory on shear strain were incorporated in the formula to calculate the shear strain produced during equalchannel angular pressing process.The reliability of simulation results has been partially validated by experiments.
文摘The correlation between crystal rotation and redundant shear strain in rolled single crystals was investigated by using the crystal plasticity finite element(CPFE) model in this paper. The deformation in aluminium single crystals of four representative orientations(rotated-Cube, Goss, Copper, and Brass) after rolling and plain strain compression was simulated, and the predictions have been validated by the experimental observations. In the rotated-Cube and Goss, the redundant shear strain and crystal rotation were in the same pattern, alternating along the thickness, while the relation between them was not obvious for the Copper and Brass due to their asymmetrical distributions of activated slip systems. The relations between slip system activation, crystal rotation, and shear strain were investigated based on the CPFE model, and the correlation between shear strain and crystal rotation has been built.
基金Project supported by the National Natural Science Foundation ofChina (No. 10372089) and Provincial Department of EducationZhejiang Province (No. 20010572) China
文摘The seismic loading on saturated soil deposits induces a decrease in effective stress and a rearrangement of the soil-particle structure, which may both lead to a degradation in undrained stiffness and strength of soils. Only the effective stress influence on small strain shear modulus Gmax is considered in seismic response analysis nowadays, and the cyclic shearing induced fabric changes of the soil-particle structure are neglected. In this paper, undrained cyclic triaxial tests were conducted on saturated sands with the shear wave velocity measured by bender element, to study the influences of seismic loading on Gmax. And Gmax of samples without cyclic loading effects was also investigated for comparison. The test results indicated that Gmax under cyclic loading effects is lower than that without such effects at the same effective stress, and also well correlated with the effective stress variation. Hence it is necessary to reinvestigate the determination of Gmax in seismic response analysis carefully to predict the ground responses during earthquake more reasonably.
文摘A shear-lag theory was developed to investigate the strain transfer from the metal substrate to the surface acoustic wave (SAW) resonator through a bonding layer. A three-layer model of host structure-adhesive layer-resonator layer was established. The strain transfer was theoretically analyzed, and the main factors impacting the SAW sensor measurement were studied. The relationship between the sensor response and the individual effect of all these factors under static loads was discussed. Results showed that better accuracy could be achieved with increase in the adhesive stiffness or resonator length, or decrease in the adhesive thickness. The values of the strain transfer rate calculated from the analytical model agreed well with that from the available experiment data.
基金Projects(2012CB619505,2010CB731703)supported by the National Basic Research Program of ChinaProject(CX2013B065)supported by Hunan Provincial Innovation Foundation for Postgraduate,ChinaProject(117308)supported by Postdoctoral Science Foundation of Central South University,China
文摘The realization way of snake rolling was introduced. Flow velocity, strain and stress distribution of 7075 aluminum alloy plate during snake rolling and symmetrical rolling were analyzed in Deform 3D. Effects of velocity ratio, offset distance between two rolls and pass reduction on the distribution of equivalent strain and shear strain were analyzed. The results show that flow velocity and equivalent strain on the lower layer of the plate are larger than those of the upper layer because of the larger velocity of the lower roll and the gap is increased with the increase of velocity ratio and pass reduction. The shear strain of roiling direction in the center point is almost zero during symmetrical rolling, while it is much larger during snake rolling because of the existence of rub zone. The shear strain is increased with the increase of velocity ratio, offset distance and pass reduction. This additional shear strain is beneficial to improve the in_homogeneous strain distribution.