Modelling long-term deformation of granular soils incorporating the concept of fractional calculus

Transcription

1 University of Wollongong Research Online Faculty of Engineering and Information Sciences - Paers: Part A Faculty of Engineering and Information Sciences Modelling long-term deformation of granular soils incororating the concet of fractional calculus Yifei Sun University of Wollongong, ys9@uowmail.edu.au Yang Xiao Chongqing University, hhuxyanson@3.com Changjie Zheng University of Newcastle Khairul Fikry Hanif University of Western Australia Publication Details Sun, Y., Xiao, Y., Zheng, C. & Hanif, K. Fikry. (. Modelling long-term deformation of granular soils incororating the concet of fractional calculus. Acta Mechanica Sinica, 3 (, -. Research Online is the oen access institutional reository for the University of Wollongong. For further information contact the UOW Library: research-ubs@uow.edu.au

2 Modelling long-term deformation of granular soils incororating the concet of fractional calculus Abstract Many constitutive models exist to characterise the cyclic behaviour of granular soils but can only simulate deformations for very limited cycles. Fractional derivatives have been regarded as one otential instrument for modelling memory-deendent henomena. In this aer, the hysical connection between the fractional derivative order and the fractal dimension of granular soils is investigated in detail. Then a modified elastolastic constitutive model is roosed for evaluating the long-term deformation of granular soils under cyclic loading by incororating the concet of factional calculus. To describe the flow direction of granular soils under cyclic loading, a cyclic flow otential considering article breakage is used. Test results of several tyes of granular soils are used to validate the model erformance. Discilines Engineering Science and Technology Studies Publication Details Sun, Y., Xiao, Y., Zheng, C. & Hanif, K. Fikry. (. Modelling long-term deformation of granular soils incororating the concet of fractional calculus. Acta Mechanica Sinica, 3 (, -. This journal article is available at Research Online: htt://ro.uow.edu.au/eisaers/5

3 Modelling long-term deformation of granular soils incororating the concet of fractional calculus Yifei Sun Yang Xiao Changjie Zheng 3 Khairul Fikry Hanif Received: June 5 / Revised: 8 June 5 / Acceted: July 5 / Published online: 9 July 5 The Chinese Society of Theoretical and Alied Mechanics; Institute of Mechanics, Chinese Academy of Sciences and Sringer-Verlag Berlin Heidelberg 5 Abstract Lots of constitutive models exist to characterise the cyclic behaviour of granular soils but can only simulate deformations for very limited cycles. Fractional derivatives have been regarded as one otential instrument for modelling memory-deendent henomena. In this aer, the hysical connection between the fractional derivative order and the fractal dimension of granular soils is investigated in detail. Then a modified elasto-lastic constitutive model is roosed for evaluating the long-term deformation of granular soils under cyclic loading by incororating the concet of factional calculus. To describe the flow direction of granular soils under cyclic loading, a cyclic flow otential considering article breakage is used. Test results of several tyes of granular soils are used to validate the model erformance. Keywords Constitutive model Fractional order Fractional calculus Long-term deformation Introduction Static and cyclic stress strain resonses of granular soils have long been a critical issue that attract many researchers attention. Lots of exerimental and theoretical studies have been erformed [-] to investigate the mechanical behaviour of granular soils. For examle, Xiao et al. [5] conducted a series of true triaxial tests on the rockfill material under drained loading condition. Khalili et al. [7] and Liu et al. [] studied the cyclic behaviour of gravelly soil under cyclic loading with low frequency. Suiker et al. [] and Indraratna et al. [7, 8] investigated the deformation and degradation behaviours of both ballast and subballast under static and cyclic loading, from which the influences of loading history and confining ressure as well as loading frequency were observed. These notable contributions rovide fundamental tools for further constitutive modelling of the cyclic behaviour of granular soils. Corresonding author: Yang Xiao hhuxyanson@3.com Faculty of Engineering and Information Sciences; Univ. of Wollongong, Wollongong, NSW 5, Australia College of Civil Engineering; Chongqing Univ., Chongqing 5, China 3 ARC Centre of Excellence for Geotechnical Science and Engineering; Univ. of Newcastle, Newcastle, NSW 38, Australia Faculty of Engineering, Comuting and Mathematics; Univ. of Western Australia, Perth 97, Australia U until now, constitutive models have been develoed based on various concet, including the incremental theory [9], fractal theory [, ], shear strain and kinematic hardening theories [8,, ], and the bounding surface lasticity [7,, 3]. Some of the models [8,, 3] can simulate the real stress strain behaviour but are comlex, whereas others [, ] are relatively simle but cannot take into account the deformation and degradation of granular soils under comlicated loading conditions. Most imortantly, these models [7, ] can only simulate the cyclic behaviour of granular soils for very limited loading cycles, say less than cycles. For the long-term strain under high loading cycles (N > 3, these models usually failed due to the unintentional accumulation of numerical errors and the huge calculation effort, esecially in the finite element analysis. There is little ossibility for these theoretical models to be used in ractical engineering where the loads usually have at least tens of thousands of cycles. To overcome this limitation, lots of emirical and semiemirical models were roosed. For examle, Indraratna et al. [] roosed a sohisticated elasto-lastic model by introducing emirical arameters to consider the effect of article breakage, stress ratio, and number of loading cycles. In fact, the ermanent deformation of granular soils under cyclic loading is not only influenced by the current loading stress but also affected by revious loading cycles. It is indeed a memory-intensive and ath-deendent henomenon which might be mathematically reresented by using the concet of fractional calculus [-]. By using the fractional calculus theory [7-9], Yin et al. [5, ] successfully roosed a framework for modelling strain hardening and softening of geomaterials. The model could be easily incororated in engineering-oriented finite element method due to its exlicit exression. However, their model was just henomenological, and the hysical origin of the fractional order still remains unknown and oen for discussion. Most imortantly, the model only dealt with the mechanical behaviour under static loading. For soils tested under cyclic loading, lots of efforts still need to be done. The aim of this aer is to investigate the hysical origin of the fractional derivative order, and then develo a more rigorous constitutive model for granular soils subjected to cyclic loading by incororating the theory of fractional calculus. The ability of the roosed model in redicting long-term deformation with a large number of loading cycles is demonstrated by simulating a series of long-term cyclic tests.

4 Notations and definitions. Notations In the model resented, granular soils are assumed to be homogeneous. Both the elastic and elastic lastic resonses are isotroic. Comression is considered to be ositive and tension is negative. Each stress notation here reresents a corresonding effective stress unless otherwise secified. For simlicity, the following triaxial stress notations are used: = ( σ + σ 3 ( 3 q = σ σ 3 ( [, q] T σ = (3 where and q denote the mean rincial stress and deviator stress, resectively. σ and σ 3 denote the first and third rincial stresses, resectively. σ is the effective stress tensor. The corresonding volumetric strain ε v, generalised shear strain ε s, and generalised total strain ε t can be given as ε v = ε + ε 3 (a ε s = ( ε ε3 (b 3 t = ε ε 3 (5 ε + [ ε, ] T ε = v ε s ( where ε and ε 3 are the first and third rincial strains, resectively. ε is the strain tensor. The increments of the total strain under each loading ste can be decomosed into elastic and lastic arts according to e = ε ε (7 ε + where a suerimosed dot indicates an increment; the suerscrits e and denote the elastic and lastic comonents, resectively.. Definitions The fractional calculus theory deals with both fractional derivatives and integrals. In this work, two common definitions of the fractional derivative and fractional integral, known as the Riemann-Liouville fractional derivative and integral, will be used. The Riemann-Liouville fractional derivative can be formulated as d z( x d x z( τ dτ Dx z( x = =, x > dx Γ( dx ( x τ where D means derivation; is the fractional order, ranging from to. x denotes the indeendent variable and can be regarded as the loading time in static test or the loading cycles in cyclic test. Due to its integral definition of the derivative, the fractional order derivative has a strong memory of the variable, x. Accordingly, there is one articular difference between the integer order derivative and the fractional order derivative. The integer order derivative of a constant is, whereas the Riemann-Liouville fractional order derivative of a constant C is not equal to, but (8 Cx D x C = (9 Γ( where the gamma function Γ ( is defined as τ x Γ( x = e τ dτ ( Γ ( x + = xγ( x ( Another imortant formula, the Riemann-Liouville fractional integral is defined as x z( τ dτ I x z( x =, x > Γ( ( x τ ( where I means integral. Note that there are two imortant roerties for the Riemann-Liouville definition, one is ς ( I z( ς Dx z( x = Dx x x (3 which means that the Riemann-Liouville derivative oerator is a left inverse to the Riemann-Liouville integral oerator in current situation. However, the fractional derivative and integral do not commute as shown in the following Eq. ( ς ς d z( x d z( x x ( Dx z( x = ς Γ( ς ς I x ( dx dx x= 3 Connections between the fractional order and fractal dimension 3. Static loading condition To quantify the soil roerty, laboratory tests, including the oedometric, biaxial, and triaxial tests with either constant stress rate or constant strain rate, are usually conducted. Various constitutive models concerning the stress strain behaviours under these conditions were roosed. By regarding the soil as an intermediate material, lying between

5 the ideal solids which obey Hooke s law and the Newtonian fluids which satisfy Newton s law of viscosity, Yin et al. [] roosed a general framework for fractional order constitutive modelling of soils under static loading. For clarification, their work is briefly introduced here. The basic constitutive law of an intermediate material can be simly defined as σ ( t = Eθ Dt ε ( t, (5 where E and θ are material constants. Note that the Hooke s law ( = and Newton s law ( = are just the secial cases of Eq. (5. For soils tested under a constant strain rate ( c condition with ε ( t = ct, Eq. (5 can be further derived by using Riemann-Liouville definition as ce ( cθ σ = ε, Γ( ( For soils tested under a constant stress rate ( c condition with σ ( t = ct, Eq. (5 can be further derived as + σ ε =, E( cθ Γ( + (7 Note that for the same soil, the fractional orders in Eqs. ( and (7 should be correlated by ( ( + =. This is because the ower law connection between the strain and stress should be the consistent, irresective of the loading methods. As already validated by Yin et al. [], the roosed model can well cature the stress strain behaviour of different geomaterials under different static loading condition even with fewer model arameters, when comared with other existing constitutive models. However, in their model, the fractional order was just regarded as a material constant. No hysical origins of the fractional order were roosed. In the next section, an attemt will be made to demonstrate the correlation of the fractional order with the fractal dimension of granular soils under static loading. 3. Cyclic loading condition There are three common aroaches in modelling the cyclic behaviour of granular soils. The first is the traditional theoretical method, including the elasto-lastic [7, ] and elasto-visco-lastic models [3, 3], where the deformation of each loading cycle is strictly counted. This method reveals the deformation mechanism of soils subjected to cyclic loading. However, it cannot be used to redict the long-term deformation with large amount of loading cycles, due to the intrinsic accumulation of numerical errors with increasing loading stes. The second is the semi-emirical method [,, 3, 33], where the theoretical models are modified by incororating emirical equations to rovide a better simulation of the cumulative deformation for a large number of cycles. However, these models usually have a lot of model arameters and therefore are hardly to be used in ractical engineering. The third is the ure emirical method, which is usually roblem targeted and flexible for engineering alication. However, it does not reflect the essential mechanism behind the deformation henomenon of granular soils. The cyclic behaviour of granular soils is actually a history-deendent henomenon where the current soil deformation is often influenced by the revious loading history. The fractional derivative may be acted as an alternative way in modelling the cyclic behaviour of soils considering its definition in an integral form. de /dn (% de /dn (% = 3kPa = kpa = kpa = kpa (a 3 Number of cycles N = 3kPa = kpa = kpa = kpa Best-fit line 3 Number of cycles N Fig. Variation of the strain accumulation rate with number of loading cycles : (a normal scale and (b log-scale (modified after [] In the context of cyclic tests, the loading cycles N instead of the time t is used [3, 3], as the cumulative strain rate during cyclic loading means a derivative with resect to the number of N. Fig. (a shows the variation of the axial strain accumulation rate with the number of loading cycles, where the strain accumulation rate for one close cycle of loading decreases significantly at first and then aroaches stable with increasing number of loading cycles. However, (b

6 by rehrasing the strain accumulation rate and number of loading cycles in a log-log form as shown in Fig. (b, a simle linear variation deicted by Eq. (8 can be observed. log ε = β log N + log b, < β < (8 where β and b are fitting arameters, which remain as constant for a given loading state. As will be demonstrated later, b is more likely deendent on the loading stress state, for examle, and q in triaxial loading condition. denotes the strain accumulation rate with resect to N, which is scale invariant and obeys the ower law in relation to the number of loading cycles. Note that soil deformation only initiated when the external load alies. Therefore, the soil strain at N = is regarded as zero. By erforming fractional differentiation on both sides of Eq. (8, one has the following simle and yet useful exression: ε The theoretical concets were therefore incororated into a comrehensive comuter subroutine to comute the ore size distribution. Fig. 3 illustrates two secific ore size distributions corresonding to a given article size distribution. As exected, the ore sizes of a loose samle are generally larger than those of a dense one. It should be noted that the calculated ore size distribution is only a qualification of the distribution of ores in the soil samle. The actual ore size distribution can only be quantified by laboratory measurement, which is not within the scoe of this study. However, bear in mind that the densest ore size distribution (with relative density equal to unit is observed to be an almost arallel shifting from an initially loose ore size distribution. 8 Loosest ore size Densest ore size Particle size distribution d ε = bγ( dn where the fractional derivative order = β. (9 Percentage finer (% Parallel shifting To reresent how the fractal dimension of soils determines the fractional order in Eq. (9 during cyclic loading, let us assume that the lastic deformation of granular soils is only attributed to the comression and exansion of the internal ores formed by the skeleton articles within the samle. - - Pore (Particle size (mm Fig. 3 Loosest and densest ore size distributions of the corresonding article size distribution The three dimensional ores are usually distributed in fractal law [37-39], which can be well characterized by the following formula: N L > l = Bl ( ( Fig. Schematic reresentation of the equivalent ore size of granular soils under comression As shown in Fig., the ore size of a loose samle is relatively large (l L while it reduces to a small value (l D once the external loads aly. It is hardly to evaluate the true ore size distribution of a three dimensional material. However, a two dimensional theoretical aroach similar to the determination of constriction (ore size distribution of granular soils [35, 3] can be used to give an aroximate analysis of the ore size distribution of a given granular soil. The diameter of an equivalent circle that has the same area as the shaded region shown in Fig., reresents the size of the ore formed by surrounding skeleton articles. l π S π = ( π where N denotes the number of ores with diameters larger than l. B is a constant of roortionality and is the fractal dimension of the internal ores. Therefore, the total surface area and volume of the internal voids can be obtained by using Eq. (. S V Bγ = l ( Bγ = l 3(3 M 3 M l l m 3 m ( (3 where γ and γ denote the surface and volume shae factors of the internal ores, resectively; l M and l m are the current maximum and minimum ore sizes, resectively. It is assumed that the total surface areas of articles and ores are equal, given that the combined contact area of touching

7 articles is very small comared to the total surface area. The following relationshi between fractal dimension of soil articles, s, and the fractal dimension of the internal ores,, can be given as [39]: ln ( lsm lsm = + ( s ( ln( l m l M where l sm and l sm are the maximum and minimum article sizes, resectively. As already roved in Fig. 3, both the maximum and minimum ore sizes should decrease with soil comression. The change of the total ore volume should be attributed to the change of the overall ore sizes. As this toic was rarely reached, a secial case of the oedometric test where ε = εv is considered here. Therefore, the increment of axial strain accumulation can be determined by dv Bγ = = e l M dl M l m dl m (5 ( + e V 3( + e V s s where V s and e are the total volume of the internal articles and the initial void ratio, resectively. Due to the arallel shifting of the ore size distribution, the maximum and minimum article sizes should vary simultaneously by decreasing k times from their initial values. This can be also validated by ensuring the equality of Eq. (, suosing the fractal dimensions of both the ores and articles are invariable. Eq. (5 can be further derived as k k ε = µ ( N where the material constant µ is exressed as in which Bγ 3 3 m = l l (7 3( + e V M m l and M s l are the initial maximum and m minimum ore sizes within the samle. The decreasing rate k changes with the number of loading cycles. By recalling the relationshi shown in Eq. (8, and comaring the exonent with that of Eq. (, an exlicit exression for k can be obtained as k = /( 3 [(3 N + ] (8 As can be observed in Eq. (8, both the fractional derivative order and the fractal dimension influence the erformance of k. Fig. reresents the evolution of the function k and the normalized total volume of the internal ores. The function k is observed to increase significantly when the number of loading cycles is small. However, the total volume of internal ores decreases raidly at the initial thousands of loading cycles and soon becomes stable, imlying that large cumulative deformation usually occurs at the initial loading stage. This is in accordance with the exerimental observations by Suiker et al. [, 3]. Function k 8 Function k Nomalized ore volume V / (mv s a =. e =.8 =.53 3 Fig. Variation of the function k and normalized total volume of internal ores The hysical connection between the fractional derivative order ( and the fractal dimension ( s can be obtained by using Eqs. ( and (8 with N equal to..5.5 Nomalized total ore volume = ( s ϑ + χ (9 where arameters ϑ = χ ln ( lsm lsm / ln( l m l M and 3 χ = k ; k corresonds to the decreasing rate of the first loading cycle. Therefore, the fractional derivative order decreases with the increasing fractal dimension of a given granular soil, indicating that soils with higher fractal dimensions are more hardly to be comacted when comared with those with lower fractal dimensions. The changing rate deends on the value of ϑ. Constitutive model incororating fractional calculus.general constitutive equations The total increments of strains can be decomosed into the incremental lastic strain and elastic strain, resectively. Following the traditional elastic theory, the incremental elastic stress can be exressed as ε e e = C σ (3 where C e is the elastic comliance matrix, which can be defined by the the bulk and shear moduli of the samle: where K and G are exressed as e K C = (3 3G

8 (.97 e G = G a (3a + e a ( + ν K = G (3b 3( ν in which ν is the Poisson ratio. As demonstrated before, the strain accumulation behaviour can be better described by using fractional order derivatives. The strain accumulation rate remains scale invariant when reresented in the scale of N. Therefore, instead of the traditional incremental definition of the accumulation rate [3, 3], the cumulative soil strain ε can be fractionally defined as d ε dn = dε m (33 where m denotes the flow direction of the cumulative strain, which can be derived from a loading function. dε is the flow intensity. Note that Eq. (33 degrades to the form of the high-cycle accumulation model [3, 33] with =. During triaxial cyclic loading, the deformation of granular soils ( ε is usually decomosed into a shearing-related comonent ε s and a comaction-related comonent ε v, which yields where arameter a ( reflects the effect of article breakage. denotes the size of the current loading surface. M denotes the critical state friction arameter and can be obtained by using M = sinφ /(3 Sinφ where φ is the critical state friction angle. Detailed derivations of Eq. (3 can be found elsewhere in McDowell []. By recalling Eq. (8, dλ g / σ should remain as constant for a given loading state. Therefore, alying the derivation of order to both sides of Eq. (3, one has ε v f = ε s = D N dλ DN dλ q (37 which can be regarded as the cyclic flow rule of granular soils. η = q / is the stress ratio. Similar concets of the cyclic flow rule can be found elsewhere in []. Accordingly, the flow direction can be defined as T T f [ m mq ] m =, =, (38 + f + f.3 Hardening rule Alying the consistency condition at the loading surface and assuming isotroic hardening of the loading surface with the lastic volumetric comression, the derivative of the loading surface can be obtained as d εs dn d εv dn = dλ (3b q = dλ (3b + q + ε v = (39 q ε By recalling Eq. (3, the lastic volumetric strain can be rewritten as v ε v where g is the loading function and d λ is the modified lastic multilier that can be related to the flow intensity as follows dε = dλ (35 σ reresents the module of a tensor. Determination of the lastic multilier and flow intensity will be given in the following section.. Loading function and flow direction As granular soils, such as the rockfill and ballast, usually undergo internal article breakage during loading, a loading surface function considering the effect of article breakage on the mechanical resonse is used []: a+ a+ q g = + ln = (3 M dλ Γ( v = N ( ε Substituting Eq. ( into (39, the lastic multilier d λ can be derived as dλ + q q = Γ( N ε v ( where the relationshi between and ε v under isotroic loading condition can be formulated as e + ( e = v λ κ ( where λ and κ are gradients of the critical state and swell lines, resectively. e denotes the initial void ratio. However,

9 considering the effect of loading history (N, Eq. ( can be further modified by using Eq. (, which yields ( + ε = N ε λ κ v (3 Note that Eq. (3 reduces to Eq. ( at virgin loading condition with N =. Strictly seaking, the effect of cyclic loading history should be reresented by internal soil variables. However, to avoid excessive comlexity in constitutive equations and esecially the alication in numerical simulation, N is included in the model formulation. A similar henomenological aroach was also used by Liu and Carter [] and Chen et al. []. Fig. 5 shows the ability of the roosed model in characterising the entire stress strain resonse of granular soils under cyclic loading. The rate of cyclic densification can be well catured by varying the fractional derivative order,. The shear strain is observed to densify more quickly with a smaller value of. deviatoric stress q (kpa a =. a = Shear strain e s (% Fig. 5 Schematic reresentation of the effect of fractional order in modelling cyclic resonse The above theoretical formula is an extension of the fractional order constitutive theory roosed by Yin et al. []. It incororates the fractional calculus theory in traditional lasticity theory. Emirical models with = / 3 by Indraratna et al. [] and Liu and Carter [] are just secial cases based on our roosed theory. The distinct difference between the roosed fractional order constitutive model and the traditional constitutive ones is its simlicity and yet vigorous in modelling the long-term deformation of granular soils. + 3 q g ( dε Γ = σ ( + ε ξ where ξ = ( λ κ and ( a η = ( a + q M (5a M a+ η = ( a + M σ = = ( a + + q (5b (5c (5d In traditional lasticity model, the long-term deformation is imlicitly calculated by accumulating the lastic strain of each loading cycle. However, due to the numerical errors built u during each ste of iteration, the redictions are usually far from the exerimental observations. To avoid comlexity in redicting the long-term deformation of granular soils under drained triaxial loading condition, the lastic strain is suosed to be only induced by loading and unloading is totally elastic. The increment (e.g., of a cyclic load from its minimum value, min, to its maximum value, max, is regarded as one comlete loading ste. Moreover, according to the research by Wichtmann et al. [, 3], the cyclic lastic flow of the long-term strain of granular soils can be aroximately characterised by the modified Cambridge model consisting of the average rincial loading stresses. Therefore, a general flow otential formed by the averaged mean rincial stress av = ( min + max / and the averaged deviatroic stress av q = ( qmin + qmax /, is used to cature the cyclic flow in relation to the cumulative strain of each loading cycle. The suerscrit av means arithmetic average. The flow intensity can be rewritten as. Exlicit formula for redicting long-term deformation Under the triaxial testing condition, the increments rincial mean stress,, and deviator stress, q, can be linked to each other as q = 3. Accordingly, combining Eqs. (33, (35, (, and (3, the exlicit exressions for the flow intensity can be given as v v 3 η η M M + dε = + v η ( + M σ v Γ( ( + ε ξ ( where η av = q av av. ξ is found to deend on the loading condition, which can be exressed as av b av b + b ( / 3 ξ = b ( q / q. Substituting Eq. (

10 into Eq. (33 yields the exlicit formula for granular soils under long-term cyclic loading. N ε v = dε m Γ( v v 3 η η N = + M M v ( + ε ξ N ε s = dε mq Γ( v v 3 η η + M M q N = + v q ( + ε ξ M v η 5 Model validation 5. Model arameters (7a (7b There are nine arameters, i.e., G, ν, φ, a,, b, b, b 3, and b, in the current model. Two elastic arameters, G and ν, can be determined by erforming resonant column test or measuring the initial stress-strain gradient of the samle subjected to static triaxial loading ( ε.%.the critical state friction angle φ can be obtained by static triaxial test or from the reosed angle of the cone of the corresonding material deosited by slow centric lifting of a funnel. Parameter a defines the flow direction and can be determined by using εv ε s = or by reresenting the relationshi between εv av ε s and η. The fractional derivative order determines the rate of strain accumulation and can be determined by fitting the relationshi between lnε and lnn. Parameters, b, b, b 3, and b, can be obtained by regressing the lastic strain of the first loading cycle. Values of the model arameters along with the test conditions are listed in Table. As only the ermanent deformation is evaluated in this study, elastic arameters are omitted here for clarity. Table Model arameters and test conditions for different granular soils Test material (kpa q (kpa min max min max φ ( o a b b b 3 b B [] B [3] B3 [3] B [3] S, [] SG3, [] SW,7,8 [] SS [3] SS3 [3] SS [3] SS [3] SS5 [3]

11 5. Model erformance The model is validated with laboratory exerimental results reorted by Wichtmann et al. [3], Indraratna et al. [], Lackenby [3], as well as Lekar and Dawson []. Test conditions can be found in Table. Figs. to shows the long-term deformation of different sands [3] tested under different cyclic loading conditions. All the sands used were natural sub-angular quartz aggregates taken from a sand it near Dorsten, Germany. Detailed hysical roerties can be found elsewhere in [3] and are omitted here for simlicity. Figs. to 9 resent the rediction results of the ermanent deformation of different sands (SS, SS SS3, SS under different deviator stress amlitudes. With the increase of the number of loading cycles the residual strain increases raidly and then aroach stable. Higher stress amlitude is observed to cause higher sand deformation. Good agreements between the model redictions and the corresonding exerimental results are observed. Fig. shows the ermanent deformation of sand No. SS5 with different deviator stresses. With the increase of the deviator stress, a higher ermanent deformation is observed, which can be well catured by the roosed model. (% q = kpa q = kpa q = kpa Model rediction a =.,,, 8,, Fig. Permanent deformation of sand (SS tested under different stress amlitudes (%.9..3 q = 8 kpa q = kpa q = kpa q = kpa Model rediction a =.37,,, 8,, Fig. 7 Permanent deformation of sand (SS tested under different stress amlitudes (% 9 3 q = 8 kpa q = kpa q = kpa q = kpa Model rediciton a =.,,, 8,, Fig. 8 Permanent deformation of sand (SS3 tested under different stress amlitudes (% q = kpa q = 3 kpa q = kpa q = 3 kpa Model rediction a =.7,,, 8,, Fig. 9 Permanent deformation of sand (SS tested under different stress amlitudes (% q av = 5 kpa q av = kpa q av = 5 kpa q av = 75 kpa q av = kpa q av = 5 kpa Model rediction a =.,,, 8,, Fig. Permanent deformation of sand (SS5 tested under different deviatoric stresses Figs. to resent the long-term deformation of different crushed rocks tested under different mean rincial and deviator stresses. The crushed rock is a dark-coloured volcanic (igneous rock containing lagioclase, feldsar, and augite. The hysical and durability roerties of the

12 crushed rocks can be found in [3]. Four test results under different loading frequency are resented in Fig.. The corresonding stress amlitudes were calculated in accordance with Esveld [5] for 3 ton axle loading. As illustrated, with the increase of the loading stress, the ermanent deformation increases, which can be well characterised by the roosed model. Figs. to resents the comarison between the model rediction and the exerimental results obtained under different average mean rincial stresses. The roosed model is also caable of characterising the long-term deformation of the crushed rocks under various mean rincial stresses. Cumulative shear strain e s (% (% 8 a =. q = 39 kpa q = 383 kpa q = 37 kpa q = 9 kpa Model rediction (b 8 Fig. Permanent deformation of ballast (B tested under mean rincial and deviator stresses (a Cumulative shear strain e s (% (% 8 8 av =.8 kpa av = 5.8 kpa av =.8 kpa Model rediction (b Fig. 3 Permanent deformation of ballast (B3 tested under various mean rincial stresses Cumulative shear strain e s (% 8 a =. a =. av = 5.5 kpa av = 37.5 kpa Model rediction (a (a Cumulative shear strain e s (% 5 3 a =. av = 75.8 kpa av = 5.8 kpa av = 5.8 kpa Model rediction (a (% 8 (b Fig. Permanent deformation of ballast (B tested under different mean rincial stresses (% Fig. Permanent deformation of ballast (B tested under various mean rincial stresses (b Permanent axial strains of three different avement materials are also simulated in Fig. 5. The material tyes include Leighton Buzzard sands (S, the sand and gravel (SG, and the slate waste (SW. The SG and SW were tested in a triaxial aaratus, while the S was tested in a hollow cylinder aaratus. As can be seen, the ermanent deformation of different avement materials tested under different loading conditions can also be well reresented using the current model.

13 Cumulative axial strain e (% Cumulative axial strain e (% SW8, av = kpa, q av = 3 kpa SW7, av = 5 kpa, q av = 3 kpa S, av = 87.5 kpa, q av = 5.5 kpa Model rediction S: a =. SW: a =. SG: a =. S, av = 83.3 kpa, q av = kpa SW, av = kpa, q av = 5 kpa SG, av = 75.8 kpa, q av = 5 kpa SG3, av = 5.8 kpa, q av = kpa Model rediction Fig. 5 Permanent deformation of avement materials tested under different loading stresses Conclusions Fractional derivatives were found to be a owerful instrument in characterising the memory-intensive henomena, such as the soil deformation and chemical diffusion, etc. Previous work [] has demonstrated the flexible ability of fractional derivatives in modelling the static deformation of geomaterials. However, the hysical origin of the fractional derivative order has not yet been exlored. In this study, an attemt has been made to investigate the connection between the fractional derivative order and the fractal dimension of granular soils. Then a modified elasto-lastic constitutive model was roosed for granular soils by incororating the theory of fractional calculus, and was further sulemented by exerimental data. In the current model, a cyclic flow rule considering the effect of article breakage was also roosed. The concet introduced for modelling long-term deformation was simlified, and it was an extension of a monotonic loading model roosed earlier by Yin et al. []. Deformation under the first loading cycle was considered as virgin loading, and the deformation under subsequent loading cycles was considered a function of the first loading cycle. The effect of the stress history has been taken into account by using the fractional derivative order. The roosed model degrades to the traditional lasticity model with the fractional derivative order equal to unit. With the decrease of the fractional derivative order, the model was observed to exhibit an increasing rate for reaching cyclic densification. (a (b Predicted results have been comared with the exerimental data to validate the model accuracy. It was observed that the roosed model can well characterise the long-term deformation of various granular soils under different cyclic loading conditions. Acknowledgements The authors would like to thank Professor W. Chen and Dr Xiaodi Zhang in the Deartment of Engineering Mechanics, Hohai University, for their kind instruction and continuous insiration on several fundamentals of the fractional calculus during the undergraduate eriod. The authors would also like to thank Mr. Rodger Paton at University of Wollongong for his technical assistance in comuter rograming. The financial suorts rovided by the Fundamental Research Funds (Grant No. 5CDJXY8 is areciated. References. Aursudkij, B., McDowell, G.R., Collo, A.C.: Cyclic loading of railway ballast under triaxial conditions and in a railway test facility. Granular Matter (, 39- (9. Wichtmann, T., Niemunis, A., Triantafyllidis, T.: On the determination of a set of material constants for a highcycle accumulation model for non-cohesive soils. Int. J. Numer. Anal. Meth. Geomech. 3(, 9- ( 3. Wichtmann, T., Niemunis, A., Triantafyllidis, T.H.: Validation and calibration of a high-cycle accumulation model based on cyclic triaxial tests on eight sands. Soils Found. 9(5, 7-78 (9. Rahman, M., Baki, M., Lo, S.: Prediction of undrained monotonic and cyclic liquefaction behavior of sand with fines based on the equivalent granular state arameter. Int. J. Geomech. (, 5- ( 5. Indraratna, B., Thakur, P.K., Vinod, J.S.: Exerimental and numerical study of railway ballast behavior under cyclic loading. Int. J. Geomech. (, 3- (. Chang, C., Whitman, R.: Drained Permanent Deformation of Sand Due to Cyclic Loading. J. Geotech. Eng. (, -8 ( Khalili, N., Habte, M., Valliaan, S.: A bounding surface lasticity model for cyclic loading of granular soils. Int. J. Numer. Meth. Eng. 3(, (5 8. Liu, H.B., Zou, D.G.: Associated Generalized Plasticity Framework for Modeling Gravelly Soils Considering Particle Breakage. J. Eng Mech. 39(5, -5 (3 9. Sevi, A., Ge, L.: Cyclic Behaviors of Railroad Ballast within the Parallel Gradation Scaling Framework. J. Mater. Civil Eng. (7, (. Suiker, A.S., Selig, E.T., Frenkel, R.: Static and cyclic triaxial testing of ballast and subballast. J. Geotech. Geoenviron. Eng. 3(, (5. Sun, Y., Xiao, Y., Hanif, K.: Comressibility deendence on grain size distribution and relative density in sands. Sci. China Tech. Sci. 58(3, 3-8 (5

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