Dilation and breakage dissipation of granular soils subjected to monotonic loading

Yifei Sun·Yang Xiao·Hua Ji

Dilation and breakage dissipation of granular soils subjected to monotonic loading

Yifei Sun1,2·Yang Xiao3·Hua Ji2

?The Chinese Society of Theoretical and Applied Mechanics;Institute of Mechanics,Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg 2016

Dilation and breakage energy dissipation of four different granular soils are investigated by using an energy balance equation.Due to particle breakage,the dilation curve does not necessarily pass through the origin of coordinates. Breakage energy dissipation is found to increase significantly at the initial loading stage and then gradually become stabilised.The incremental dissipation ratio between breakage energy and plastic work exhibits almost independence of the confining pressure.Accordingly,a plastic flow rule considering the effect of particle breakage is suggested.The critical state friction angle is found to be a combination of the basic friction between particles and the friction contributed by particle breakage.

Dilation·Particle breakage·Granular soil· Energy dissipation

DOI 10.1007/s10409-016-0569-z

1 Introduction

There are two important fundamentals in traditional critical state constitutive modelling:one is the yielding surface, which describes the plastic yielding of soil,and the other is the relationship between stress and dilation,which characterises the plastic flow of the corresponding plastic strains[1,2].The yielding criterion has been widely investigated by many researchers[3–7]up until now.It is commonly accepted that the yielding surface should be convex to satisfy the Drucker’s postulate[8].Stress dilation of granular soils was found to be significantly influenced by the confining pressure and particle breakage[9,10].The yielding surface can be derived from the corresponding stress dilation equation if an associated flow rule is used.For example, McDowell[3]proposed a family of yielding surfaces for granular soils by incorporating the breakage energy dissipation into the Cam-clay model.The total plastic work was assumed to be only dissipated by particle friction and breakage.However,according to Ueng et al.[11,12],dilation of granular soils was not only influenced by the frictional sliding between particles but also by the particle rotation and breakage.Therefore,the energy consumed during loading should be decomposed into three parts:frictional dissipation,particle rearrangement dissipation,and breakage dissipation.To study the breakage and energy dissipation mechanism during loading,many experimental numerical researches using discrete element method have been conducted[13–17].It was found that the extent of breakage had a great influence on the energy dissipation of granular aggregates.The higher the extent of breakage was,the higher the plastic dissipation would be.Moreover,most of the particle breakage and the associated energy dissipation was found to be at the initial loading stage[16].The discrete element approach provides a comprehensive micromechanical perspective on the energy dissipation of granular soils.However,limited experimental studies on the energy dissipation during loading have been reported.

The aim of this paper is to investigate the dilation and energy dissipation of granular soils subjected to monotonic triaxial loading.A substantial amount of experimental resultsfrom the available literature are collected and analysed by using the energy conservation equation proposed by Ueng and Chen[11].A plastic flow rule considering particle breakage is proposed.

2 Stress dilation

For a granular soil subjected to triaxial loading,the following equation for stress dilation was suggested by Rowe[18]

where d Svis the increment of the specific surface area per unit volume,and k is a material constant depending on the strength of the particle.It is noted that Eq.(2)contains unknown items,which makes it hard to be used in practical constitutive modelling without further modification.The basic friction angle should also be determined by empirically fitting the experimental data.In this study,the breakage energy consumption is assumed to be proportional to the increase of the particle surface areas.For a given particle, the specific surface area per unit volume(Sv)is

where D is the diameter of the given particle.ω denotes the shape factor of the particle.ω approaches unity when the particle becomes increasingly spherical.Accordingly,the total specific area of an entire sample can be stochastically obtained by[19]

where DMand Dmare the maximum and minimum diameters in the sample.ρ(D)is the current probability of a particle to exist in the fraction,d D,and can be correlated to the initial (ρ0(D))and ultimate(ρu(D))distribution probabilities of the particle size by the following formula

where B is the particle breakage ratio defined by Einav[20]. B was assumed to be size independent,i.e.,be identical for all of the particle fractions that satisfied B(D)=B[20].However,it can not be denied that B would not be perfectly size independent.This assumption did work well in analysing the particle breakage phenomenon of granular aggregates [21,22].It is noted that the ultimate probability obeys the power law in relation to the particle size which can be expressed by.α is the fractal dimension of the ultimate granular soil.Substituting Eq.(5)into Eq.(4)and performing differentiation on both sides,one has

where Suand S0are the ultimate and initial stochastic values of the total specific area of an entire sample and can be expressed as? and S0=respectively.Substituting Eq.(6)into Eq.(2),one has

where EB=k(Su?S0),is the total energy that can be released by particle breakage from the system.The modified stress dilation equation does not necessarily pass through the coordinate origin but rather intersects the vertical axis at [EBd B

Fig.1 a Stress dilatancy behavior of Yixing rockfill.b Stress dilatancy behavior of Sandstone.c Stress dilatancy behavior of Kish Island sand. d Stress dilatancy behavior of Cambria sand

Figure 1 shows the stress dilation relationship predicted by Eq.(7).Four different granular soils,i.e.,Yixing rockfill[23],Sandstone[10],Kish Island sand[9],and Cambria sand[24,25],are used.Yixing rockfill mainly consisted of weathered quartz sandstone.Kish Island mainly consisted of calcareous sand with angular/subangular shape while Cambria sand mainly consisted of quartz particles with round shape.Detailed physical properties along with the gradients of each dilation curve can be found in Table 1.To better study the stress dilation relationship during subsequent loading,the test data for the initial loading stage were not presented to eliminate the effect of elastic strain.It should be noted that the incremental strains in the subsequent loading stage include both the elastic and plastic parts.This is considered to be acceptable as the plastic strain is quite larger than the elastic strain for granular soils subjected to subsequent triaxial loading.

As can be seen,the test data are aligned in a narrow band with limited scatters at low stress ratios.However,due to the different extent of particle breakage,the gradients are observed to slightly vary with varying confining pressure,indicating a slight variation of the basic friction angle as shown in Fig.2.Similar observations can be found in Ref.[26].Nevertheless,it should be noted that the variation trends are different for the four different materials.This may be attributed to the different particle rearrangements associated with particle angularity.However,to further evaluate this phenomenon,detailed experimental studies are needed. With the increase of confining pressure,the value of the stress ratio at intersection decreases slightly,as indicated by Eq.(7).Moreover,Yixing rockfill and Sandstone appear to have generally higher basic friction angles than those of the Kish Island and Cambria sands.This is probably because the Yixing rockfill and Sandstone have a more angular particle shape when compared with the other two corresponding sands.In order to evaluate the breakage energy dissipation under monotonic triaxial loading,mean values of the basic friction angles as shown in Table 1 are used.

3 Breakage energy dissipation

Under the triaxial loading condition,the mean principal effective(p′)and deviator(q)stresses can be formulated as

In addition,the incremental general shear straincan be formulated as

Substituting Eqs.(8),(9),and(10)into Eq.(7),yields

Table 1 Physical properties of different granular soils

Fig.2 Basic friction angles of a Yixing rockfill.b Sandstone.c Kish Island sand.d Cambria sand

where d Eb=EBd B,is the increment of the breakage energy dissipation per unit volume;and critical state friction parameter M=6 sinφf/(3?sinφf).The rate of breakage energy dissipation,can be obtained by using Eq.(11).It should be noted that Eq.(11)is only used to measure the energy dissipation during shearing.The consolidation-related dissipation is not studied here.However,this is acceptable because the percentage of the consolidation-related dissipation was very small[27]and did not impact our study of the dissipation variation during subsequent shearing.

Figure 3 illustrates the variation of the rate of breakage energy dissipation of different granular soils with increasing shear strain.Figure 3a shows the energy dissipation rate of Yixing rockfill tested under drained conditions with four different confining pressures.It is observed that the dissipation rate increases drastically at the initial loading stage,implying the occurrence of a significant increase of the breakage extent and the associated energy dissipation.This is in accord with the discrete element analysis performed in Refs.[14,16]. A generally higher energy dissipation rate is observed in samples loaded under higher confining pressure.However, with the development of loading strain,the energy dissipation rates of all samples become gradually stabilised.

Figure 3b shows the breakage energy dissipation rate of Sandstone tested under drained conditions with three different confining pressures.A significant increase followed by a decrease of the energy dissipation rate at the initial loading stage is also observed.The energy dissipation rate of the tests conducted at higher confining pressures(2–3 MPa)becomes stable afterwards.However,there is a slight increase of the energy dissipation rate for the sample testedMPa even though the shear strain continuously increases to some extent.This is probably because the critical state of Sandstone has not been reached under the current confining pressure[10].

Figure 3c shows the energy dissipation rate of Kish Island sand.Unlike those two larger aggregates,i.e.,Yixing rockfill and Sandstone,the energy dissipation rate continuously increases and then stabilises with increasing shear strain.This may be because the Kish Island sand mainly consisted of calcareous particles and was uniformly graded,which made the sample vulnerable to breakage during the entire loading period.With the increase of shear strain,significant breakage continuously occurred.A higher energy dissipation rate is also observed in samples loaded under higher confining pressure.

Fig.3 a Variation of the rate of breakage dissipation with shear strain for Yixing rockfill.b Variation of the rate of breakage dissipation with shear strain for Sandstone.c Variation of the rate of breakage dissipation with shear strain for Kish Island sand.d Variation of the rate of breakage dissipation with shear strain for Cambria sand

Figure 3d illustrates the rate of breakage energy dissipation of Cambria sand tested under both drained and undrained conditions with various confining pressures.A higher dissipation rate is observed in samples tested under drained conditions.This is because both the volumetric and shear strains contribute to the development of particle breakage in drained triaxial tests.However,only shear strain contributes to the particle breakage of samples tested under undrained conditions.The breakage dissipation rate of all the samples becomes stabilised when the shear strain increases to some extent,irrespective of the testing condition.Further comparison between the Fig.3c,d indicates a slight different pattern of evolution trends between the Kish Island sand and Cambria sand.This can be attributed to the different physical properties and confining pressures of the two different granular aggregates.The Kish Island sand mainly consisted of calcareous particles and was uniformly graded,which made the sample vulnerable to breakage during the entire loading period.With the increase of shear strain,significant breakage continuously occurred.However,the Cambria sand mainly consisted of sub-round quartz sand.During the initial loading stage where samples were closely held together due to high-pressure isotropic compression,any further shearing should be achieved by aggregate breakage.However,after sheared to certain extent,aggregates were also free to slide or rotate. Breakage and the associated breakage energy thus became stabilised.

Apart from the rate of breakage energy dissipation,the accumulated breakage dissipation,Eb,should be also studied.It can be derived by rearranging Eq.(11)as

Figure 4 shows the variation of the accumulated energy dissipation of four different granular soils.It can be observed from Fig.4a that the relationship between the breakage energy dissipation,Eb,and the total plastic work of Yixing rockfill is weakly dependent on the confining pressure. The curve of breakage energy dissipation slightly bends downwards with increasing plastic work.By calculating the average gradient of the data points in Fig.4a,the energy dissipated by breakage is found to be 32.5%–38.7% of the total plastic dissipation.Similar evolution trends can be found in Fig.4b where the Sandstone was tested.However,the average breakage energy dissipation for Sandstone is found to be 26.7%–29.4%of the total plastic work.

Figure 4c illustrates the variation of breakage energy dissipation with plastic work of Kish Island sand.It can be observed that the relationship between the breakage energy dissipation,Eb,and the total plastic work is slightly dependent on the confining pressure.As the confining pressure increases,the proportion of the energy dissipated by breakage in total plastic work decreases.The average breakage energy dissipation for Kish Island sand is found to be 23.1%–33.8% of the total plastic work.Figure 4d represents the variation of breakage energy dissipation with plastic work of Cambria sand.Almost independence of the relationship between the breakage energy dissipation and total plastic work on the confining pressure is observed.The average breakage energy dissipation for Cambria sand is found to be 69.28%–69.81% of the total plastic work.This may be because that the confin-ing pressures were very large which suppressed the dilation of the internal aggregates.Contraction of sand was facilitated by particle breakage.Therefore,the breakage energy dissipation is higher when compared with the other three.

Fig.4 a Evolution of the breakage dissipation with plastic work for Yixing rockfill.b Evolution of the breakage dissipation with plastic work for Sandstone.c Evolution of the breakage dissipation with plastic work for Kish Island sand.d Evolution of the breakage dissipation with plastic work for Cambria sand

To comprehensively understand the energy dissipation during triaxial loading,the energy ratio between the increments of the breakage energy dissipation and the total plastic work is represented in Fig.5.As can be seen,the incremental energy ratio increases rapidly at the initial loading stage and then becomes increasingly stable with further loading.The incremental energy ratio of Yixing rockfill slightly depends on the confining pressure while the other three exhibit weak dependence.

Accordingly,the following expression can be suggested to approximately describe the evolution of the breakage energy dissipation with total plastic work

where γ is a material parameter used to characterise the proportion of breakage energy dissipation during plastic loading. Strictly speaking,γ is not invariant at the initial loading stage, as implied in Fig.5.But it finally becomes constant with further development of the shear strain.Therefore,γ is regarded as a constant under for the consideration of practical application.

4 Particle breakage ratio

The particle breakage ratio used in this study is the modified breakage index,B,which was defined by Einav[20]to represent the fractal evolution of the particle size distribution curve.Following Eq.(5),the particle breakage ratio,B,can be obtained as

Figure 6a shows the particle breakage ratio of Yixing rock-fill and Cambria sand.It is observed that the breakage extent increases with increasing confining pressure.A higher breakage extent is found to be in samples tested under drained condition.This is in accord with Fig.3,where a higher rate of breakage energy dissipation is observed in drained tests.Furthermore,Fig.6b represents the relationship between particle breakage ratio,B,and the breakage energy dissipation,Eb. It is observed that the breakage ratio increases with increasing breakage energy dissipation.An almost linear evolution can be found if we rephrase the relationship in the linear-logarithmic coordinate,as indicated in Fig.6b.

Accordingly,the total breakage energy,EB,corresponding to the breakage ratio,B,equal to unit can be obtained.

Fig.5 a Variation of the breakage dissipation with shear strain for Yixing rockfill.b Variation of the breakage dissipation with shear strain for Sandstone.c Variation of the breakage dissipation with shear strain for Kish Island sand.d Variation of the breakage dissipation with shear strain for Cambria sand

Fig.6 a Variation of particle breakage ratio with confining pressure. b Relationship between particle breakage ratio and breakage energy dissipation

The total breakage energies of Yixing rockfill and Cambria sand are calculated to be 0.385 and 11.075 MPa,respectively. With the help of the total breakage energy,a new breakage index BEis suggested as

where the energy-based breakage index,BE,can be calculated by using Eqs.(12)and(15).

Figure 7 illustrates the performance of the newly suggested breakage ratio.The energy-based prediction shows a good agreement with the laboratory measured breakage ratio.With the increase of the modified breakage ratio,B, the energy-based breakage ratio,BEincreases.A higher breakage extent in drained triaxial condition is also successfully represented by the energy-based breakage index,BE.Itshould be noted that BEcan not be easily applied in practical engineering before the determination of the total breakage energy and breakage energy dissipation during loading.However,it can be used as an alternative way in quantifying the breakage extent of granular aggregates.

Fig.7 Energy-based breakage ratio

5 Plastic flow and critical state

Many plastic potential surfaces have been proposed for modelling the plastic flow behavior of granular soils.However,the plastic flow rule considering the effect of particle breakage was seldom investigated.Limited researches can be found in Refs.[3,19,28,29].Here an attempt is made to shed additional light on the effect of particle breakage on the plastic flow rule of granular soils.Substituting Eq.(13)into Eq.(11), one has

where the stress ratio η=q/p′.Parameter χ= γ(6+ 4 M)(3?M)/[3(6+M)].As γ has been shown to slightly depend on confining pressure,χ should be varied slightly at different loading conditions.However,for practical application in constitutive modelling,χ remains as a constant.The direction of plastic flow can be thereby derived as

Figure 8 shows the comparison of the proposed plastic flow rule with the Cam-clay model and the Rowe model.Due to the effect of particle breakage,more shears train should be developed at the initial loading stage and a relatively lower dilatant trend would occur subsequently.This phenomenon can be well predicted by the current model.

Fig.8 Plastic flow rule considering particle breakage

Figure 9 shows the comparison between the experimental results and the corresponding model simulation.As can be observed,the proposed plastic flow rule can well characterise the stress dilation behavior of various granular soils.For those granular soils sheared to a critical state, there should be no incremental volumetric strain.Therefore, Eq.(16)degrades to the following expression

As discussed before,χ is influenced by the breakage energy dissipation ratio,γ.So the critical state friction parameter,Mcr=M/(1?χ),is determined by the dissipation of breakage energy at the critical state.Figure 10 shows the comparison of the predicted results with experimental measurements where a good agreement can be observed. Therefore,the critical state friction angle is actually a combination of the basic friction between particles and the friction contributed by breakage.Breakage is not suspended at the critical state but continues until the internal aggregates arrive at an ultimate fractal distribution.The critical state of granular soils is therefore a balance between volumetric reduction due to particle breakage and dilation resulted from particle rearrangement[30].It is a transient state since the ultimate particle size distribution is usually not reached,as also indicated by Luzzani and Coop[31].

6 Conclusions

Particle breakage and stress dilation significantly influence the strength deformation behavior of granular soils.This study made an attempt to study the breakage energy dis-sipation of four different granular materials.Several major conclusions are summarised as follows

Fig.9 a Model simulation of the plastic flow of Yixing rockfill. b Model simulation of the plastic flow of Sandstone.c Model simulation of the plastic flow of Kish Island sand.d Model simulation of the plastic flow of Cambria sand

Fig.10 Critical state friction angle

(1)To obtain the basic friction angle,analysis of the relationship between stress and dilation was performed,where the relationship was found to slightly depend on the confining pressure due to significant particle breakage under higher loading stress.

(2)To predict the breakage extent during loading,an energy-based breakage index was suggested.The rate of breakage energy dissipation was found to increase significantly at the initial loading stage and then gradually become stabilised.Its value was observed to depend on the confining pressure while the final value of the incremental dissipation ratio between breakage energy and total plastic work exhibited almost an independence of the confining pressure.

(3)A constant ratio of the breakage energy dissipation to the total plastic work was suggested for constitutive modelling.Accordingly,a plastic flow rule considering the effect of particle breakage was suggested.The proposed flow rule exhibited good prediction of the corresponding test results.

(4)The critical state friction angle was suggested to be a combination of the basic friction between particles and the friction contributed by breakage.

Acknowledgments The project was supported by the Fundamental Research Funds for the Central Universities(Grant 106112015CDJXY 200008),the National Natural Science Foundation of China(Grant 51509024),and the China Postdoctoral Science Foundation(Grant 2016M590864).

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? Yang Xiao hhuxyanson@163.com

1Faculty of Engineering and Information Sciences,University of Wollongong,Wollongong,NSW 2522,Australia

2School of Civil and Transportation Engineering, Hohai University,Nanjing 210098,China

3College of Civil Engineering,Chongqing University, Chongqing 400450,China

19 August 2015/Revised:18 March 2016/Accepted:11 April 2016/Published online:7 September 2016