Christophe Dano

Carlos Ovalle

Zhen-Yu Yin

Ali Daouadji

Pierre-Yves Hicher

Behavior of Granular Materials

Affected by Grain Breakage 1

Introduction

In geotechnical and particularly in dam engineering, observations in the field and, in some cases, dramatic failures of structures have often incited research and increased knowledge. For instance, the first rockfill dams were elevated without any prior knowledge of the mechanical capacities of the rockfill mass. As natural and local resources are usually preferred for economic and environmental issues, their non-optimal properties can lead to potential severe damage to the hydraulic structure. Similarly, a basic principle of construction of rockfill dams or embankments consists in sluicing rockfill after deposition and before compaction: this recommendation has been adopted for a long time [CLE 81, HUN 02]. However, the beneficial effects of such a practice have been understood only in light of recent studies on unsaturated soil and multi-scale analyses.

Extensive research on the mechanical behavior of rockfills was initiated in North America, in the 1960s and 1970s, with the simultaneous development of high Concrete Face Rockfill Dams (CFRD) for which a better estimation of both stiffness and shear strength properties and optimized design was required. Researchers faced technical challenges related to the size of the testing apparatus, in proportion to the maximum size of rockfill particles close to 200 mm [MAR 72, MAR 73]. Their experimental databases still constitute a reference in geotechnical literature, since the main features of rockfill behavior were clearly highlighted. Particle crushing has thus been identified as a key mechanism of their behavior, inducing a decrease of both the secant stiffness and the shear strength, associated with an increase of the compressibility. Subsequent research [CHA 80, COO 04, DAO 01, DAO 10, HAR 85, LAD 96, LEP 70, MCD 02, NAK 01a, NAK 01b, VES 68] showed that grain breakage is favored by higher stress levels, weak, large, elongated and angular particles, uniform and loose grain packings, as well as by the presence of water. The amount of grain breakage also depends on the stress path and degree of humidity [CLE 81,OLD 07]. In this way, shearing paths (triaxial more than oedometric paths) and cyclic drained loadings induce a greater amount of particle crushing.

It becomes quite obvious that a better prediction of the behavior of crushable soils and rockfills requires a deep understanding of the local mechanisms at the grain scale: interactions between grains through the distribution of local forces, mode of degradation of the grains and interactions between solid and liquid phases. It is also necessary to take into account the size effect, currently associated with failure of brittle elements.

Experimental and numerical advances relative to the behavior of crushable soils will be addressed here. The chapter has been divided into three sections: the first deals with the prediction of scale effects through a scale transition approach, the second presents numerical developments, particularly to better describe the evolution of the grain size distribution upon loading and the final one discusses the influence of water on the behavior of crushable soils.

Size effects in rockfill materials

Micro and macro-scale effects

Typical sizes of particles deposited in rockfill dams range from a few centimeters up to about 100 centimeters. Some difficulties then arise. The estimation of the mechanical properties of a rockfill mass would require testing set-ups involving huge experimental samples, of at least 10 times the maximum grain size. In the absence of such huge devices (the largest triaxial or oedometer cells nowadays allow gradations to be tested with a maximum grain size less than 200mm), alternative strategies must be developed to estimate useful engineering parameters, for instance friction angle and stiffness. Whatever this strategy may be, it implies working on a reduced grain size distribution compatible with available devices using an appropriate reduction method from among the different methods as summarized by WU et al. [WU 14]. Among these, the most commonly adopted, the similar grading method, consists in testing a grain size distribution (referred to as the model) as closely parallel as possible to the original one (referred to as the prototype); that is, to divide all the sizes in a homothetic way. Some assumptions must be strictly respected: the two homothetic particle masses must have, as minimal conditions, the same mineralogy, angularity and (relative) density, so that the tested reduced sample is the exact replica of the original material at a smaller scale.

Nevertheless, once test results are obtained, the model stress-strain relationship cannot be extrapolated directly to the original material, since the model and prototype materials may behave differently because of size effects on grain breakage. Size effect is generally described as the dependence of a material property (strength) on a characteristic geometrical dimension (particle diameter), as a result of fracture mechanics. In the present case, size effects occur at both the micro-scale (grain level) as well as at the macro-scale (sample). On the one hand, experimental results through Franklin point load tests or Protodyakonov tests on individual particles [MAR 72, MIU 79, NAK 99] outline how, statistically, the resistance to failure of an individual particle decreases as the grain size increases. Marsal [MAR 73] states that the average force f g required to break a particle of mean size d g follows:

g g f k d ζ = × [ 4 . 1 ]
Where k and ζ are experimentally identified. In accordance with [JAE 67], the crushing stress σ g inducing grain breakage is calculated as follows:

2 2 g g g g f k d d ζ σ - = = × [ 4 . 2 ]
Typical values for ζ are close to 1.5 for a large variety of rocks and, at the grain scale, brittle heterogeneous materials. Assuming that crack propagation in a brittle grain is a volumetric process, the survival probability P s of a particle of volume V under a stress σ is assumed to obey a Weibull distribution:

( )

0 0 0 0 , exp exp d m n m s V d P V V d σ σ σ σ σ                 = - × = - ×                         [ 4 . 3 ]
Where m is the Weibull modulus, n d is the geometric similarity coefficient (usually assumed to be equal to 3 for volumetric similarities) and σ 0 , V 0 and d 0 are the characteristic strength, volume and size respectively. Equations [4.2] and [4.3] can be connected in order to relate the representative failure parameters ζ and m. For a given survival probability, m is then related to ζ by:

2 d n m ζ = - [ 4 . 4 ]
With a central value of ζ close to 1.5, a typical value for the Weibull modulus m is about 6 if n d = 3.

On the other hand, the literature provides many examples in which the friction angle of a soil mass decreases when the maximum grain size is increased. The latter can be seen as the consequence of the former process: particle breakage induces more contractancy due to the rearrangement of the fragments and, consequently, less shear strength resistance of the grain assembly. The challenge has then been to establish the extrapolation rules and their conditions of applicability. Based on an indepth review of past experimental results, Frossard et al. [FRO 12] proposed such a predictive method and applied it to parabolic failure envelopes representative of crushable materials [DEM 77].

Theoretical developments

In the following, consider two homothetic gradations, the coarser P (for prototype) and the finer M (for model). Macroscopic stresses σ ij applied at the boundaries of a given volume of soil induce a network of local forces ( )

n p f 
exerted by particle n on particle p at their common contact point. Following the Love-Weber approach of stress averaging, the macroscopic stress results from the integration of contact forces and branch vectors ( )

( ) ( ) / / 1 n p n p n p N t f l V σ < ≤ = ⊗    [ 4 . 5 ]
Where N is the total number of particles in the volume V t .

Under the same stress condition, the amount of particle breakage is higher in the coarser material. Inversely, the breakage ratio is similar in both fractions, assuming that Marsal's equation is suitable for a large range of grain sizes (similar k and ζ values), only if the force magnitude on the prototype is lower than in the finer fraction.

In order to obtain the same probability of particle rupture in both fine and coarse fractions, the crushing forces f M and f P of individual particles must be related as:

P P M M d f f d ζ   = ×     [ 4 . 6 ]
The homologous contact forces in the granular assemblies should be in proportion to the crushing resistances of homologous particles, so that:

( ) ( ) 2 / / d n m P P n p M n p M d f f d -   = ×     [ 4 . 7 ]
As the scaling ratio is the same for all contact forces, this relation, valid locally for each contact force, also holds globally for the complete sets of contact forces. Introducing this scaling law in equation [4.5], also considering the geometrical scaling laws on distances and volumes, it gives:

( ) ( ) M m n M P p n M N p n p n M M m n M P P d d d d l f V d d σ ⋅         = ⊗         = σ - ≤ < -  / / 1 [4.8]
Standard shear tests may allow the identification of the shear strength envelope of the model material, expressed as a parabolic function [DEM 77]:

( ) , ' M b M M nM A τ σ = ⋅ [ 4 . 9 ]
where τ is the peak shear stress, σ' n the normal effective stress and A and b two material constants. Then the application of the previous scaling laws leads to the estimation of the shear strength envelope for the prototype coarse material, such as:

( ) ( ) 1 , ' d d M M n m n b m b P P P M M n P M M d d A d d τ τ σ - - -     = ⋅ = ⋅ ⋅         [4.10]

Experimental validation

The methodology requires performing crushing tests on individual particles of different sizes to identify m and n d parameters and shear tests on a model grain size distribution to identify A M and b M , and then calibrating the shear strength envelope of the prototype material. Two materials were selected, namely a calcareous limestone (subsequently referred to as CP) exhibiting porosity of about 10% and sub-angular grains, from Préfontaines (100 km south of Paris, France) and a quartzite shale (STV), with angular and flattened particles, from Sainte Honorine La Chardonne (250 km west of Paris, France).

More than 100 crushing tests on individual particles of each material were carried out following Protodyakonov conditions, that is to say splitting between two stiff steel parallel plates, at a constant velocity of 0.1 mm/min. The loaddisplacement curves show a quasi-brittle behavior, whatever the particle size may be. The crushing strength σ g is therefore calculated from the load peak. The experimental data were statistically analyzed by size fractions. More details about the parameters identification can be found in Ovalle et al. [OVA 14]. The identified values were, respectively, m = 2.18 and n d = 0.63 for CP and m = 1.54 and n d = 0.81 for STV. The geometric similarity coefficient n d is far from the usual assumed value of 3, and the physical notion in Weibull's theory is no longer valid for the current heterogeneous and randomly shaped rock particles.

Afterwards, drained triaxial compression tests were performed on dry specimens of CP and STV with two homothetic grain size distributions of d max = 40 mm for the smallest samples CP1 and STV1 of 250 mm in diameter and 375 mm in height (model specimen) and d max = 160 mm for the largest samples CP2 and STV2 of 1000 mm in diameter and 1500 mm in height (prototype specimen) (Figure 4.1) [HUW 11a]. The initial uniformity coefficients were, respectively, 2 for CP and 5 for STV. Grain size distributions before and after shearing up to an axial strain of 20% are presented in Figure 4.2. For a dry sample preparation, granular soils were compacted by vibration in several layers. The average initial void ratios were 0.85 ± 0.03 for CP and 0.53 ± 0.04 for STV. Three confining pressures were used: 100 kPa, 200 kPa and 400 kPa. Friction angles (Figure 4.3) were then calculated from stressstrain curves, not shown here but presented in Ovalle et al. [OVA 14]. Size effects affect both rockfills with (a) a higher Marsal's breakage ratio [MAR 67] and (b) a slightly lower maximum friction angle for the larger specimens, for example the coarser specimens.

As previously suggested, the shear envelopes were fitted to the De Mello's parabolic criterion (equation [4.9]), characterized by coefficients A and b. The fitting for the model grain size distributions give (A M = 0.80; b M = 0.82) for CP1 and (A M = 0.87; b M = 0.77) for STV1. By introducing these parameters in equation [4.10], it is possible to compare the prediction (equation [4.10]) assuming the aforementioned values of m and n d with the shear strength envelopes of the coarser (prototype) specimens, characterized by (A P = 0.68; b P = 0.72) for CP2 and (A P = 0.81; b P = 0.78) for STV2. The comparison is shown in Figure 4.4, indicating a reasonable good prediction of size effects for both rockfills, even if it appears to fall on the safe side for STV. This could be explained because the method takes into account neither the magnitude of particle strength, nor the effect of the initial grain size distribution.

Challenges in modeling rockfill behavior

The methodology exposed in the previous section can be coupled with the stability analysis of dams for which the friction angle is the key factor, as developed by Frossard et al. [FRO 12]. A second way to describe dam behavior is to model their existence, from the construction stages to the serviceability conditions in the long term. In that case, constitutive models able to simulate the specific features of rockfills are required in numerical computations. One of these features is the dependence of the mechanical properties with the change in the grain size distribution. For about twenty years, many valuable models have appeared in the geotechnical literature.

Figure 4.2. Grain size distributions before and after tests for CP (top) and STV (bottom). For a color version of the

Figure 4.3. Evolution of the breakage ratio (top) and maximum friction angle (bottom). For a color version of the figure, see www.iste.co.uk/millet/advances.zip

0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 CP1 CP2 STV1 STV2
Breakage ratio (%)

Confining pressure (kPa)

0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 CP1 CP2 STV1 STV2
Maximum friction angle (degrees)

Confining pressure (kPa) The first models dedicated to crushable soils were developed in the framework of plasticity theories (in a non-exhaustive list: [BAU 09, DAO 01, DAO 10, LIL 16, MUI 08]). Critical state-based elastoplastic models will be more extensively discussed later. Among the plasticity models, it is worth noting the capabilities offered by the hypoplastic model by Bauer and co-workers (see for instance, [LIL 16]) whose central concept is the solid hardness, a measurement of the degree of weathering of rock particles. By relating the solid hardness to density, the degree of humidity and time, Bauer's model is also able to capture time-dependent mechanisms occurring in weathered and moisture-sensitive coarse-grained materials (creep or relaxation due to progressive grain crushing).

A second family of models has gained in popularity. It follows a thermodynamics approach (also called hyperplasticity by some authors) to derive constitutive equations from energy considerations (free energy and dissipation functions). One can cite the pioneering Continuum Breakage Mechanics by Einav and co-workers [EIN 07a, EIN 07b], and new developments by Wang and Arson [WAN 16] or Salami [SAL 16]. These models also introduced the partitioning between the consumed energy due to the creation of new surfaces related to the propagation of a crack in a grain and the consumed energy by friction due to the local re-organization of the granular skeleton when breakage occurs. Also note that many recent contributions to the behavior of crushable soils use the discrete element method (DEM). However, because of huge computational time demands, this method is not operative for geotechnical projects such as dams. Moreover, the description of the breakage of a grain is usually biased in the sense that the size and shape of fragments are pre-determined [OVA 16]. However, DEM can provide valuable information to validate some assumptions on the behavior of granular materials.

In particular, it has been postulated that grain crushing inducing a change in the grain size distribution also affects the critical state. This key point has surprisingly not been investigated so much in terms of modeling. However, the standard definition of the critical state (constant stress ratio, null volumetric strain rate) must be completed by an additional condition: no evolution of the grain size distribution. That condition is achieved at very large strains (more than 100%), as shown during ring shear tests [COO 04]. So the identification of the critical state parameters for crushable soils, at relatively small strain levels (usually 20%), constitutes a great, even impossible, challenge. Liu

Experimental investigation on critical state characteristics

Glass beads Hostun sand

Specimens (100 mm in diameter and about 200 mm in height) of angular Hostun sand mixtures and spherical glass bead mixtures were subjected to isotropically consolidated drained triaxial compression tests, with confining pressures of 100, 200 and 400 kPa, so that no particle breakage can occur. The grain size distributions, similar for glass beads and Hostun sand specimens, are shown in Figure 4.5. First, the tested specimens exhibit a unique critical stress ratio M = q/p' upon compression regardless of C u (M=0.81 for glass beads and M = 1.13 for Hostun sand). The uniqueness of the critical stress ratio confirms that the shear resistance is not influenced by the change in GSD.

The critical state lines in the ep' plot are represented in Figure 4.6. It is confirmed by experiments that the location of the CSL varies as C u changes, the higher the C u value, the lower the CSL location in the ep' plot. In the range of confining stresses tested, it seems that all the critical state lines are almost parallel. In other words, a well-graded material has a smaller void ratio at critical state than does a more uniform material under the same stress level. Then, it is assumed that the void ratio at critical state e c is related to the mean effective stress, as follows: 4.1 provides the constant values of these parameters for glass beads and Hostun sand specimens, respectively.

Elasto-plastic models with an implicit description of GSD evolution

A constitutive model for rockfills must describe the main features of the behavior of granular materials, namely nonlinear elasticity, shear strength due to intergranular friction, strain hardening and dilatancy, as well as peculiar features of rockfills (size effects, cap yield, evolution of the critical state line due to the change in the grain size distribution). Most of the models presented in the literature are structured in a similar way and differ only in the assumed equations for the yield surfaces, flow rule and evolution of the critical state. In this section, we present one of these models, in which grain breakage affects both the elastic stiffness and shear strength of the granular material.

The nonlinear elasticity involves both the bulk modulus K and shear modulus G, dependent on the mean effective stress p', void ratio e, as well as grain breakage through the breakage index B r by Yin et

-   = ⋅ ⋅ -   +   [4.15]
Where K 0 , G 0 and n p are material constants. The parameter θ controls the degradation rate and also the final degradation amplitude of the stiffness when B r = 1. It also leads to the degradation of the plastic stiffness, in agreement with experimental observations. The model is also composed of two yield surfaces: the first -associated for simplicity -yield surface f comp deals with the compression behavior. For η ≤ M p :

( ) 3 1 ' ' exp 2 comp m r p f p p p b B M η   = + - ⋅ -⋅       [4.16a] 0 0 1 exp p m m v p e e p p ε λ κ   + = ⋅     -   [4.16b] ( ) 0 1 ' e e p K κ + ⋅ = [4.16c]
where p m is the hardening variable controlling the size of the yield surface (initially at p m0 ), depending on the volumetric plastic strain ε p v and M p is the mobilized peak strength (stress ratio η = q/p' at peak, related to the Mohr-Coulomb peak friction angle ϕ p ). λ p and κ e are the plastic and elastic compressibility factors respectively. A degradation effect is introduced through the breakage amount B r [YIN 16].

The second yield function deals with the shear sliding behavior:

( ) ( ) 0 ' p p d shear p p p R d M f M p G G ε η ε ⋅ = - = + [4.17]
The hardening rule depends on the deviatoric plastic strain ε p d and the hardening rate variable G pR . The plastic potential function needed to compute the plastic strains, causing the model to be non-associated, is defined by:

pt q g L n M η η   = +     [4.18]
With M pt as the phase transformation stress ratio. An alternative to these two yield surfaces may consist of adopting an egg-shaped yield surface, as done by Alonso et al. [ALO 16], which presents the advantage of possibly more easily integrating anisotropic features.

It is well-known that the volumetric behavior of a granular mass upon shearing depends on the stress state and the relative distance between the void ratio e and the critical void ratio e c . In the present model, the assumption is: As a last ingredient of the model, the evolution of the GSD or the amount of grain crushing that depends on stresses and strains must be related to the mechanical loading. This relationship is usually implicit, since the precise measurement of the GSD is optional. Daouadji

( ) * ' p p p v d W p d q d ε ε = + ⋅  [4.22] * * p r p W B a W = + [4.23]
where a is a fitting parameter. The MacCauley brackets in equation [4.22] imply that the shear dilation (dε d p <0) is not accounted for in the expression of the plastic work. An equivalent function has been proposed by Muir-Wood and Maeda [MUI 08], introducing the grading index I G . In case the GSD needs to be predicted, an additional relation between the breakage ratio B r and the GSD must be established, as has been done for instance by Einav [EIN 07a, EIN 07b] in assuming the existence of an ultimate fractal GSD, enabling the calculation of the equation F(d) of the GSD as:

( ) ( ) ( ) ( ) , 0 , 1 r Einav r Einav ult F d B F d B F d = - ⋅ + ⋅ [4.24]
Where F 0 (d) is the initial GSD, F ult (d) the ultimate one defined as F ult (d) = (d/d max ) 0.3 and B r,Einav the breakage ratio defined by Einav [EIN 07a]. Some comparisons between experimental and numerical stress-strain curves and GSDs are detailed in Yin [YIN 16].

Explicit description of GSD evolution

The implicit description of the GSD suffers from some shortcomings. It requires the choice of a specific form F(d) for the GSD. It also assumes an ultimate GSD which is experimentally difficult to identify since it can be obtained only at very large strains. Moreover, it doesn't contain any information about the hardness of the grains themselves. A strategy, called confined comminution (CC), has therefore been developed to circumvent the aforementioned assumptions [OVA 13a]. Based on stronger physical considerations, it inherently integrates size effects and relates micromechanics at the grain level to macro properties of the grain assemblies. The CC model found inspiration from grinding theories commonly encountered in mining engineering problems for which the optimization of particle fragmentation processes represents a key concern.

The basic principle of this CC model is a population balance, where the rate of mass accumulation in the i th size fraction is equal to the rate of the i th size production from all coarser size fractions j, reduced by the rate of breakage in the i th fraction [REI 65] (Figure 4.8). The change in each size fraction can be described by the following discrete population balance:

i ij i i dm A m dξ = i n i i n f f ξ = [4.25]
where f n i is the intergranular normal contact force, <f n i > the mean value for the i th size fraction and the matrix A ij is defined as:

0 ij j ij i b S i j A S if i j i j <   = - =   >  [4.26]
S i is the breakage probability of grains in the i th size fraction and b ij the fragmentation function that describes the mass contribution to the i th fraction of fragments generated by the crushing of a coarser particle in the j th size fraction. The fragmentation function b ij should be related to the mineralogical and petrographic properties of the particles, but also to any factor influencing the state of tensile or shear stress inside a particle (degree of weathering, grain angularity, coordination number or orientation of contact forces). Unfortunately, no simple mathematical expression can capture the complexity of the fragmentation process.

A self-similar distribution for the retained mass between two successive size fractions (i and i+1) is therefore assumed:

3 3 1 b b D D i i ij j j d d b d d - - -     = -             [4.27]
and is characterized by the fractal dimension D b .

From a physical point of view, particle crushing develops because the intergranular contact forces induce a tensile (mode I) or a shear stress (mode II) that exceeds particle strength. Therefore, the probability of breakage S i must combine, on the one hand, statistics on particle strength (function of their size in particular) and, on the other hand, the distribution of contact forces in better and better graded granular packings, as proposed by Marketos and Bolton [MAR 07]. In terms of probability, these two events are assumed to be independent.

The particle strength statistics have already been introduced in Section 4.2.1. Considering that the particles behave according to the linear elastic brittle mechanics theory, assuming that the breakage is mainly due to tensile stresses within particles induced by normal intergranular contact forces (fracture in mode I prevailing on mode II, which justifies the use of Brazilian-type tests to characterize the load at failure), then the survival probability of a grain of size d subjected to a tensile stress σ follows a Weibull distribution (equation [4.3]) where n d = 3.

Due to the lack of experimental evidence on the contact force distributions, the use of DEM computations becomes almost indispensable. Then, to obtain a genuine description of the force network, 3D DEM simulations of isotropic compression of dense polydisperse frictionless uncrushable sphere packings were performed with the Contact Dynamics Method [MOR 04]. Silbert et al. [SIL 02] showed that the friction coefficient has a small influence on the statistical distribution of contact forces. Furthermore, to avoid boundary effects, periodic boundary conditions without gravity were used [VOI 09]. The size of the smallest particle is constant (d min = 1 mm) and the maximum grain size d max increases with the size span R = d max /d min , which varies from 1.5 for almost monodisperse samples to 50 for highly polydisperse samples. The grain size distribution of numerical specimens is specified assuming a power law distribution:

( )

1 1 min 1 1 max min DEM d d F d d d δ δ δ δ - - - - - = - [4.28]
Where δ is in the range 2-3. Details on the numerical protocol, in particular the method to obtain a dense initial state, can be found in In order to take into account the size effects arising from particle strength and cushioning effects (to be discussed subsequently), the distribution of normal forces is analyzed for each size fraction i. A size fraction of characteristic size d i is thus characterized by a span factor R i = d i /d max in a given sample of span R. Numerical results show that the force distribution becomes narrower as R i increases [OVA 16]. This can be explained by the fact that the cushioning effect on coarse grains becomes more and more important when polydispersity increases. In other words, during confined comminution, small particles are more likely to break than would the larger ones due to the so-called 'cushioning" effect [TSO 99], since large particles tend to have a higher coordination number and to be submitted to a nearly isotropic compressive stress preventative of crushing due to tensile stresses. This cushioning effect can also be highlighted through the anisotropy of the normal force directions: numerical simulations indeed reveal that the anisotropy for coarse grains decreases with R i , whereas it is relatively high for small particles. This finding allows the contact force network to be described regardless of i and R and the use of the probability density function p N (ξ) as proposed by Radjaï et (  )

1 1 exp 1 r N r r k p i f k α ξ ξ ξ ξ β ξ -  ⋅ <  =  > ⋅ -   [4.29]
where α and β r are empirical parameters, ξ = f n /<f n > and k r a normalizing constant (1/k r = 1/(1-α) + 1/β r ). The continuity of the function imposes that β r ² = (1-α)(2-α). Moreover, computations give that, for ξ i > 1:

( ) 0.365 1.1149 r i R β = ⋅ [4.30]
The anisotropy of normal forces is injected into the comminution model by using the maximal mean normal force in the i th size fraction defined for the principal direction at the particle scale, such as:

( ) max 1 2 i i i n n n f f a = ⋅ + ⋅ [4.31]
where the anisotropy of the normal contact forces in the i th size fraction is given by: ( )

1 3 5 / 2 i i i i n n a f χ χ = - [4.32]
where χ 1 and χ 3 refer to the eigenvalues of the fabric tensor, following the definition by Bathurst and Rothenburg [BAT 88].

Finally, the cumulated probability for a contact force to be higher than the statistical value at breakage f b i is:

( ) ( ) , , i b N i ib N i i P p d ξ ξ ξ ξ ξ +∞ > = ⋅  [4.33] with , max i i i b b n f f ξ =
. Accordingly, the higher the normal force anisotropy is, the higher the maximal mean normal force will be; therefore, a higher breakage probability will be computed with equation [4.33].

We finally assume that the force network during comminution (as well as the grading) evolves in the same way as it does in different uncrushable samples when polydispersity increases. Therefore, the breakage probability S i for the i th size fraction results from the combination of the two independent events: breakage probability of a particle, described by the Weibull distribution, and the probability to have a normalized normal contact force higher than the failure force f b i corresponding to the threshold stress σ I,b :

( ) ( ) , , 1 , i S i i b N i i b i S P d P σ ξ ξ   = - ⋅ >   [4.34]
This predictive model for the evolution of the GSD has been confronted with well-documented experimental data: oedometer tests by Nakata et al. [NAK 01b] and isotropic compression tests by Ezaoui et al. [EZA 10]. The latter will be detailed here. The crushable granular materials are made of spherical grains fabricated by mineral powder agglomeration, with a diameter from 1.6 to 4.0 mm. Ezaoui et al. [EZA 10] carried out 124 crushing tests on individual particles and isotropic quasi-static compression tests on granular assemblies, with an isotropic effective stress p' varying between 0.2 and 3.0 MPa. They compared the fine content (% finer than 1.6 mm) before and after tests. The initial GSD and the evolution of fine contents are presented in Figure 4.9. Analysis of crushing tests on grains gives the following parameters for the Weibull distribution: d 0 = 3mm, m = 6.5 and σ 0 = 0.91 MPa (n d is fixed at a value of 3). The parameter D b of the fragmentation function is assumed to be equal to 2. Combining equations [4.26], [4.27] and [4.34], and adopting the results of DEM computations for the distribution of normal forces [OVA 16], the fine content can be predicted. Figure 4.10 shows the comparison between experimental data by Ezaoui et al.

[EZA 10] and the comminution model predictions. A good agreement is obtained. Note that the full grain size distribution could also be predicted, although a sensitivity analysis on the effect of D b should be made. However, the physical values for D b seem to fall within a narrow range of values, between 2.0 and 2.6. The implementation of this explicit model for the GSD evolution into a mechanical model has not yet been realized. As the comminution model involves a scalar stress in the Weibull distribution, some further analysis is needed to choose which stress definition from the mechanical stress tensor is the most appropriate. In the previous example, the mean effective stress was considered, but the level of deviatoric stress should be more significant for most engineering applications. The comminution model presents certain advantages: a physical basis of the characteristic equations, no assumption on an ultimate or fractal grain size distribution, and flexibility, since new ingredients such as the effect of water could be introduced.

Detrimental effect of humid conditions on grain breakage

Much evidence of the influence of humid conditions on the behavior of crushable soils can be found in the field or in the literature. Research by Alonso et al. [ALO 05, ALO 16], Oldecop and Alonso [OLD 03], Oldecop and Alonso [OLD 13] and Ovalle et al. [OVA 15] on rockfills tested in different relative humidity conditions illustrate this detrimental effect of the interactions between the granular skeleton and the pore fluid. Detrimental means here that humidity favors grain breakage, as it will be shown later, and can induce unexpected settlements during the serviceability life of rockfill dams or embankments. Let's cite for instance:

-The case of Cogswell 85m high dam in California (1934, USA): the rockfill mass, of good quality, was dumped and compacted without any prior sluicing, which resulted in a greater susceptibility to water flooding. Indeed, after a severe rainy episode, an immediate settlement of 1.8 m at the dam crest observable in just one night reached 4.1 m 6 months later.

-Case histories of Aguamilpa dam (187m, Mexico, 1995), Campos Novos dam (202m, Brazil, 2006) or Mohale dam (145m, Lesotho, 2006): all exhibited concrete face damage due to excessive rockfill deformation during reservoir filling (due not only to the change in total stress) or after rains, in spite of sluicing during construction.

-The long-term monitoring of Shuibuya dam (233m, China, 2008) [ZHO 11] allows the reporting of the observed settlements and water levels during the construction and first filling stages, as well as during serviceability conditions (Figure 4.11). In Figure 4.11, the red frame indicates the first filling of the reservoir, accompanied by a faster rate of settlement, whereas the blue frame marks the operational conditions with an almost constant water level in the reservoir. At that stage, time-delayed deformations are still measurable, at a lower rate.

Briefly, the contact of water with a granular skeleton made of crushable particles generates a sudden collapse of the packing and also involves long-term deformations. From a modeling point of view, few models are able to take the water/skeleton interactions into account, except the hypoplastic model by Li et al. [LI 16] through the evolution of the solid hardness. Some attempts in the elastoplastic framework [ANT 99, SAL 16] should however be mentioned.

Experimental tests in the laboratory, on smaller size fractions of a granular soil, have attempted to replicate those field observations. 1D compression tests by Nobari and Duncan [NOB 72] in dry or saturated conditions or 1D compression in controlled relative humidity conditions [OLD 01] and triaxial tests [ALO 16, LAD 10] gave a consistent framework of the behavior of crushable soils in humid environments. Figure 4.12 shows typical experimental results of compression on crushable soils, similar to previous findings by Nobari and Duncan [NOB 72]. The upper blue curves, represented in an axial (vertical) strain vs. axial (vertical) stress plot, correspond to the material compression in dry conditions. The lower red curves are the compression curve of the same material that was initially imbibed by water, which reveals a larger compressibility. If an initially dry material is flooded (arrow in Figure 4.12) under a constant vertical stress, as illustrated by black lines in Figure 4.12, the specimen showed a significant settlement until its stress-strain behavior almost reached the compressibility curve of initially wet samples. Oldecop and Alonso [OLD 07] added the fact that there are intermediate compressibility curves related to intermediate relative humidity or suction levels.

Are these mechanisms related to grain crushing? Similar compression tests on uncrushable soils (glass beads, 1 to 2 mm in diameter to avoid capillary effects, in a certain range of stress, typically below 2 MPa) do not show any collapse of the granular structure when water is added. The most convincing evidence comes from comparing the grain size distributions of the tested materials, before loading and after complete unloading.

Water level

Settlement

The uniaxial compression tests shown in Figure 4.12 have been performed on cylindrical specimens (19 mm high and 70 mm in diameter) with a uniform distribution of angular grains of STV quartzite shale already presented in this chapter. Particle size ranges from 2.0 to 2.5 mm. In order to avoid any breakage during specimen preparation, loose samples of the dry material were prepared at an initial void ratio e 0 of 1.019 ± 0.046. 23 stress controlled compression tests were carried out at similar stress levels (0.15 MPa, 0.4 MPa, 0.8 MPa, 1.3 MPa and 2.1 MPa), each of them maintained for 24 hours. As shown in Table 4.2, three test conditions were implemented: (1) dry, (2) saturated after the preparation and (3) initially dry then flooded with demineralized water at a certain vertical stress. Note that the saturation degree was not measured but the added volume of water was enough to fill the voids. The flooding process took about one minute. No chemical reactions of dissolution occurred between water and grains (constant dry mass of grains before and after test). After unloading, samples were slowly dried in order to avoid any additional breakage due to heating, then sieved. Grain size distributions are shown in Figure 4.13. Grain breakage obviously occurred, due to the mechanical loading and due to the interactions between water and grains. An outstanding result is that initially saturated and flooded samples present similar final GSD, which can be correlated to the increase in specimen compressibility. To better highlight the differences in the final gradings, the increase of the surface area of grains due to crushing was estimated through Blaine tests according to the standard ASTM C204-11 [AST 11]. This test is suitable for particles smaller than 0.125 mm, but it is worth noting that the major contribution to the change in surface area Δ(SSA) comes from fine particles [MIU 79]. Δ(SSA) is plotted against the plastic work W p , where the plastic work is the sum of all the increments ΔW p = Δσ' v *(ε-ε e ) in which ε e is the elastic strain recovered after unloading. As shown in Figure 4.14, saturated and flooded samples follow a unique curve, separately from dry samples which have experienced less crushing. Consequently, the fact that saturated and flooded samples present a unique compression curve can be explained by the same final GSD. required to propagate a crack is weakened in humid conditions (subcritical crack growth lower than the critical threshold in dry environment). Hu [HU 07a, HU 07b] gave an appropriate modeling approach to this mechanism, coupling mechanics and chemical interactions (local dissolution of the solid that induces crack propagation) between the corrosive fluid and the solid material at the crack tip.

The previous experimental data have been analyzed more thoroughly to identify a micro-characteristic of the grains. Let's assume that the increment of plastic work δW p is the sum of the dissipations by friction δΦ friction and by breakage δΦ B . The latter dissipation can be distributed into the energy required to create new surfaces, by an increment δS, by grain breakage in a given volume as well as into kinetic energy of the fragments and surrounding grains due to breakage. McDowell et al.

[MCD 96] proposed:

( ) ( ) 1 1 c B s S R V e δ δ Γ ⋅ Φ = ⋅ + ⋅ + [4.35]
Where V s is the volume of solids, e is the void ratio and Γ c the specific surface energy from Griffith's theory [GRI 21]. The variable R represents the fraction of energy consumed in kinetic energy and contact redistribution triggered by crushing [NGU 09, NIE 11, RUS 11]. Russell [RUS 11] predicted values of R around 13 to 15. Note that plastic dissipation due to inelastic deformation around the crack or dissipation by heat production, even if, in some cases, significant [SAL 16], are therefore neglected.

By measuring the difference between the grain size distributions after a dry test and after a flooded one at the same maximal vertical effective stress, the surface area increment simultaneous to the collapse δS can be computed. Using equation [4.35], the quantity Γ c . (1+R) can be identified, as shown in Figure 4.15. When the plastic work increment tends to 0 (at low stress), the R value is assumed to become negligible: at low stresses, inter-particle contact forces on a broken grain can be redirected to nearby grains with small amount of redistribution. Therefore, the surface fracture energy Γ c can be determined. It is close to a value of about 2.5 N/m, in the same order of magnitude of surface fracture energies found in the literature for the similar rock properties [ASH 86]. On the other hand, at high stress levels, surrounding grains are not able to sustain the redistributed forces and an overall rearrangement of the granular skeleton should take place with significant collapse, which induces high values of R, of about 10, in accordance with Russell's estimation [OVA 13b, RUS 11]. During compression experiments, it was also observed that settlements do not completely stabilize, presenting then a sort of creep as observed in real dams. It has been decided to conduct preliminary long-term (more than 2 years) compression tests, one in dry conditions (e 0 = 0.895), one in saturated conditions (e 0 = 0.924 before saturation), at the same vertical stress (1.145 MPa). The soil was made of a calcareous crushable uniform sand (particle size between 2 and 2.5 mm) placed in a cylindrical mold 70 mm in diameter and about 20 mm high.

Conclusion

Crushable soils behave as granular soils usually do; that is to say, friction is the main dissipative process when subjected to compression or shearing. However, they also dissipate energy in other forms, in particular through the energy consumed to create new surfaces when fragments are generated. Grain breakage also results in more compressibility, a shift of the critical state line in the void ratio-mean effective stress plane with a relatively constant maximum friction angle provided that particle shape remains relatively unchanged (angular) and consequently in a decrease of the shear strength. For engineering purposes, the decrease of the material strength is of importance in assessing the stability of hydraulic structures (dams, embankments).

The behavior of crushable soils also highlights the role of micro-mechanics in the macroscopic behavior. Hence, size effects observed at the macro-scale on relatively initially uniform granular material, which are materialized by a decrease of both stiffness and shear strength for coarse samples compared to specimens with a finer homothetic grain size distribution and similar density, find their origin in the probability of failure more important for large grains than small ones. Nevertheless, once crushing is fully activated, for example when grain size distribution becomes better and better graded, the DEM simulations show that the cushioning effect protects the larger grains from breakage and a combination of the probability of grain breakage and the distribution of contact forces must then be considered to account for this effect.

Humid environmental conditions (relative humidity) also affect the behavior of crushable soils: they emphasize the mechanisms observed in dry conditions, since water favors the propagation of cracks. Integration of the chemo-mechanical interactions at the crack tip, following a scale transition approach, represents a challenge for modeling but, in the long run, it could help time-delayed deformation of granular materials to be better predicted.

Figure 4 . 1 .

 41 Figure 4.1. Large triaxial cells Φ1000 mm (left) and Φ250 mm (right). For a color version of the figure, see www.iste.co.uk/millet/advances.zip

Figure 4 . 4 .

 44 Figure 4.4. Shear failure envelopes for CP (top) and STV (bottom). For a color version of the figure, see www.iste.co.uk/millet/advances.zip

  Biarez and Hicher [BIA 97], Daouadji et al. [DAO 01] and Hu et al. [HUW 11b] suggested that, as the amount of grain breakage increases, the critical state line (CSL) shifts downwards in the ep' plane, where e is the void ratio and p' the mean effective stress. Based on 2D DEM simulations, Muir-Wood and Maeda [MUI 08] confirmed that the critical state line of a well-graded material lies below that of a poorly graded material. Yan and Dong [YAN 11], through 3D DEM simulations, suggested a linear relationship between the critical state parameters and the coefficient of uniformity C u . However, their study focused on small values of C u limited to 2.2. Liu et al. [LIU 14] and Li et al. [LI 15] extended the study to larger C u (up to 20) and mainly investigated the effect of the soil uniformity on the CSL position from an experimental point of view.

Figure 4 . 5 .

 45 Figure 4.5. Grain size distributions for glass beads and Hostun sand specimens

  where e ref is a reference void ratio that allocates the position of the CSL, λ controls the slope of the CSL, ξ p is a material constant fixed here at 0.9, as proposed by Yan and Dong[YAN 11] and p at the normalizing atmospheric pressure (101.3 kPa). The evolution of the two parameters e ref and λ of equation [4.11] is plotted in Figure 4.7 and the following fitting curves are calibrated: where a e,ref ,, b e,ref ,, c e,ref ,, a λ ,, b λ and c λ are fitting parameters. Table

Figure 4 . 6 .

 46 Figure 4.6. Evolution of the CSL with the coefficient of uniformity C u

  ϕ μ is the friction angle at critical state and β a material constant. The critical void ratio e c depends on the amount of grain breakage through equation [4.11]. A variant version can be found in Yin [YIN 16].

Figure 4 . 8 .

 48 Figure 4.8. Principle of confined comminution

Figure 4 . 9 .

 49 Figure 4.9. Initial GSD and fine content after testing [EZA 10]. For a color version of the figure, see www.iste.co.uk/millet/advances.zip

Figure 4 .

 4 Figure 4.11. Core and crest settlements of Shuibuya dam (redrawn from data by Zhou et al. [ZHO 11]). For a color version of the figure, see www.iste.co.uk/millet/advances.zip

Figure

  Figure 4.12. Compressibility curves for quartzite shale specimens. For a color version of the figure, see www.iste.co.uk/millet/advances.zip

Figure 4 . 14 .

 414 Figure 4.14. Increase in surface area vs. plastic work. For a color version of the figure, see www.iste.co.uk/millet/advances.zip

Figure 4 .

 4 Figure 4.15. Identification of ΓC.(1+R) after collapse due to flooding

  Figure 4.16 represents the time-delayed settlement after the initial consolidation stage from a vertical stress of some kPa to 1.145 MPa. Although less important in the dry case, creep does occur in both cases. Mechanisms at the origin of this time-dependent behavior have not been clearly identified until now, but recent experiments on grain-to-grain creep tests by Michalowski and Nadukuru [MIC 15] suggest plasticization and creep of stressed surface asperities.

Figure 4 .

 4 Figure 4.16. Long-term settlement of a crushable calcareous soil. For a color version of the figure, see www.iste.co.uk/millet/advances.zip

  figure, see www.iste.co.uk/millet/advances.zip

		CP1 before test CP1 after test (400 kPa) CP2 before test CP2 after test (400 kPa)
	% passing
	1	1 0	1 0 0
		Particle size (mm)
		STV1 before test STV1 after test (400 kPa) STV2 before test STV2 after test (400 kPa)
	% passing
	1	1 0	1 0 0
		Particle size (mm)

  et al. [LIU 14] and Li et al. [LI 15] chose to depict qualitatively the evolution of the CSL parameters through tests on uncrushable sands of different coefficients of uniformity, also supported by 3D DEM computations, and to extrapolate the findings to crushable soils.

Table 4

 4

	HOSTUN
	SAND

.1. Fitted parameters

  al. [YIN 16] according to experimental observations by Iwasaki and Tatsuoka [IWA 77]:

	e ref	0.6 0.8 1 G G p 0 at	(	2.97 1	fitting-curve fitting-curve Hostun sand Glass beads ) 2 ' at e p e p	n	p		λ ( exp	0.006 0.009 0.012 0.015 r B θ	)	Glass beads fitting-curve fitting-curve Hostun sand
		0.4																	0.003
	(a)	0.2	1	6			11	C u	16		21				(b)	0	1	6	11 C u	16	21
		0 K K p at = ⋅	(	2.97 1 + -e	e	)	2	  	' at p p	  	n	p	⋅	exp	(	θ -	r B	)	[4.14]

  et al. [DAO 01] and Hu et al. [HUW 11b] for instance have considered the amount of grain breakage as a function of the plastic work W p *, such as:

  Radjai and Voivret [RAD 11], Voivret [VOI 13] and Ovalle et al. [OVA 16].

  al. [RAD 96]:

	( )	( )

Table 4 .

 4

		1.1
		1		Flooding		Dry Test 7 Sat. Test 16 Sat. Test 15 Dry Test 8
						Test 17 Test 18 Test 19
	Void ratio	0.8 0.9				Test 20 Test 22
		0.7
		0.6
		0	500	1000	1500	2000	2500
				Vertical effective stress (kPa)
	4.12. Compressibility curves for quartzite shale specimens. For a color version of the figure, see www.iste.co.uk/millet/advances.zip

2. Uniaxial compression tests on quartzite shale specimens

Figure 4.13. Grain size distributions for quartzite shale specimens after testing. For a color version of the figure, see www.iste.co.uk/millet/advances.zip

Acknowledgements

The authors would like to thank the valuable contributions of students (M. Cisternas, G. Colombo, P. Duan, W. Hu, X. Jing, G. Li, Y. Salami, E. Tinubi) involved in the research projects related to grain crushing. Most experimental developments and tests were carried out during the ECHO project, managed by E. Frossard from Tractebel Engineering (formerly Coyne-et-Bellier) and granted by the French National Research Agency: we would like to acknowledge their confidence and financial support.


Nadia El-Nemr
email: elnemrnadia@gmail.com

Beatrice Canel-Depitre
email: beatrice.canel@univ-lehavre.fr

Atour

DETERMINANTS OF HOTEL ROOM RATES Luxury Industries Conference London 2017

Keywords: dynamic pricing, pricing information, price determinants and hotel industry

The hotel industry is one of many industries that offer fixed perishable products that should be sold over a finite time horizon to heterogeneous customers. Because of its particularity, the hotel industry faces many challenges to decide the convenient room rate to offer. In order to understand price formation, we have studied pricing information assumed for the suggested dynamic pricing models for hotel rooms. In addition, we identified some factors that may influence the rate. We compared the results and suggested a list of information that might be useful to consider for price formation to optimize the revenue.

INTRODUCTION

The hotel industry is one of many industries that face the problem of selling a perishable product over a finite time horizon [START_REF] Weatherford | A taxonomy and research overview of perishable-asset revenue management: yield management, overbooking, and pricing[END_REF]. Rooms are the most important asset of the hotel. It is perishable because its revenue-generating capabilities drop to zero right when the sales period ends. For example, at the end of the night, the loss of an empty room is lost forever [START_REF] Baker | The benefits of optimizing prices to manage demand in hotel revenue management systems[END_REF], Like other industries, the aim of hotels is to achieve the highest possible revenue. To accomplish this goal, many methods during the history were applied in order to influence customer purchase decisions such as market segmentation, product differentiation, promotions and loyalty programs. However, many authors had emphasized on the importance of pricing for firm's survival and profitability (Nagle & Holden, 1995;Finch et al.1998;Potter, 2000;[START_REF] O'connor | On-line pricing: An analysis of hotel-company practices[END_REF]. Pricing is one of the 4P's forming the marketing mix in addition to product, promotion and place (Kotler & Armstrong, 2006). A firm needs to create a successful mix of the right product, sold at the right price, in the right place and using the most suitable promotion to influence consumer decision-making and lead to profitable exchange (Borden, 1984;Peter & Donnelly, 2007;Munusamy & Hoo, 2008). Despite the importance of all the factors composing this mix, authors claimed the importance of pricing to be the only element of the marketing mix to generate revenues (Nagle & Holden, 1995;Finch et al.1998;Potter, 2000;[START_REF] O'connor | On-line pricing: An analysis of hotel-company practices[END_REF]Avlonitis et al. 2006). Hoffman et al. (2002) suggests that developing an effective pricing strategy remains the most difficult variable of the traditional marketing mix. During decades, researchers were trying to develop pricing models to maximize revenue. The first tentative to develop a pricing strategy for hotel rooms was the rule of thumb or 1$ per 1000$ investment. This cost based strategy suggests that for each 1000$ investment, the hotel room price should increase by 1$. For example, if the capital of hotel construction composed of 100 rooms was 100 000$, the room rate should be 100$. Another well-known method is the Hubbart formula that was developed for the American Hotel and Motel Association (AHMA) in 1952. It is consisted of eight steps: calculating _desired profit(1), pre-tax profits(2), fixed charges and management fees(3), undistributed operating expenses(4), estimated non-room operated department profit \loss(5), required rooms department income(6), determine rooms department revenue(7), calculating average room rate(8) (Harris & Brown, 1998). Based on this approach, the room rate is determined by costs, desired profits and expected rooms sold (Zheng [START_REF] Gu | Proposing a room pricing model for optimizing profitability[END_REF]. However, these approaches don't take into consideration the market conditions and they represent static pricing. Drucker (1993) claimed that customers are not responsible of the costs and profits of the investors. By observing the downfall of one once dominant business after another like General Motors, Sears and IBM, Drucker (1993) suggested that cost based approach for pricing is one of five deadly business sins that must be avoided for the survival of any company. In addition, static pricing are not the best solution for hotel industry. Later, the "Perishable-asset revenue management" (PARM) become the used term to denote a process that deal with yield management and dynamic pricing for perishable assets [START_REF] Weatherford | A taxonomy and research overview of perishable-asset revenue management: yield management, overbooking, and pricing[END_REF]. The yield management approach deal with cases when a limited number of products (goods or services) are available for limited time and after this limit the product perishes or ages. In such cases customers are price sensitive to the time and the quantity purchased [START_REF] Finch | An option-based approach for pricing perishable services assets[END_REF]. In contrast to static pricing, dynamic pricing can achieve a finite markets seller's goal of selling its entire inventory. "Dynamic pricing" term refers to the change of prices in a marketplace based on two perspectives that are different prices for different market segments, and change prices over time by taking advantage of the fluctuation in cumulative buyer demand considering a finite time horizon (DiMicco et al. 2003). Recently, many researchers are trying to develop dynamic pricing models based on different market conditions. The technological innovations have made it possible to gather more information on customers' behavior and preferences (Grewal et al. 2011). In addition, thanks to the digital market, the cost associated with frequent price changes are greatly diminished leading the dynamic pricing to become a common competitive maneuver particularly in markets under a finite time horizon. (DiMicco et al. 2003) This paper is presenting the literature review papers of hotel pricing classification in the section 2. In section 3, we are presenting the context and methodology of the work. The results are presented in section 4. At section 5, we conclude and suggest future directions.

LITERATURE REVIEW

By studying the literature review papers of hotels related to pricing, we have noticed that the only existing literature is related to models' classifications. Hotel price determinants were not addressed as a literature review paper. Previously, two literature review papers were published examining the issue of "an examination of hotel room pricing methods: practiced and proposed" (Gu and Steed, 2004) and "dynamic pricing in hospitality: overview and opportunities" [START_REF] Anderson | Dynamic pricing in hospitality: overview and opportunities[END_REF]. The first paper presents a classification of hotel room pricing approaches. Authors examined articles that are directly related to their study. All the studied articles are pricing strategies applied for hotel rooms. They classified the pricing strategies based on the simple approach known for pricing that are: cost based pricing, market based pricing, combined cost and market based pricing and best practice pricing. In this article they described each approach by giving information about the steps to follow to obtain the desired price. Some critics of approaches were presented as well. At the end authors has recommended some ideas for the best choice and implementation of the pricing strategy. Similarly, [START_REF] Anderson | Dynamic pricing in hospitality: overview and opportunities[END_REF] presented the dynamic pricing models developed for the hospitality industry. They discussed the works presented in this context. Next, they presented some choice based models works as distinct room types offered by hotels make hospitality related dynamic pricing very amenable to the application of choice-based approaches. At the end, they suggested unique research opportunities to be studied in the hospitality industry. These two papers presents a classification of different pricing models existing in the literature by giving some critics and advantages of the works exposed. By examining these works, hotel managers can't understand which model is best to be used or how to choose the pricing strategy that suits the most their property. Therefore, in this work, we tried to understand the formation of pricing models for hotel rooms, then we studied the elements that may affect the price. By gathering this information, we have proposed a list of elements that should be considered when taking a pricing decision. In addition, we have shown the distribution of the existing pricing works into a pricing information matrix. Therefore, we believe that in this way, we will be able to identify the gaps in the literature and provide guidance for future researchers and help managers improve their pricing decisions.

CONTEXT AND METHODOLOGY

Today, hotel revenue management is shifting toward price optimization solutions to generate optimal revenue and compete in today's market with the ease of accessibility to customers ' and competitors' information. (Kimes, 2016;Noone, 2015). In 2002, Hoffman stated "today, price remains one of the least researched and mastered areas of marketing . . . Marketers have only recently begun to focus on effective pricing. . . Although the need for effective pricing is frequently voiced at conference sessions focusing on services marketing strategy, an overview of specific topics of potential interest has yet to be developed". (Hoffman et al., 2002) Few empirical studies were conducted in service industry and its branches (Hoffman et al., 2002). [START_REF] Chung | Tourism Management. Hotel room rate pricing strategy for market share in oligopolistic competition-eight-year longitudinal study of super deluxe hotels in Seoul[END_REF] studied hotel room rate pricing strategy based on real world empirical data. His study was focused on oligopolistic market of super deluxe hotels. Avlonitis and Indounas (2006) examined pricing practices of service organizations. They tried to understand the relationship between pricing policies adopted and pricing information collected. Oxenfeldt (1983) claimed that to reach the final price needed, specific pricing policies should be considered. According to him, when setting prices, firms must take into consideration pricing information that represents the organizational and environmental conditions. Tzokas et al. (2000) confirmed that some factors influence pricing decisions of which some of them can be controlled by firms more easily than others. They emphasized on the importance of collecting the appropriate type of information when setting prices to be able to assess and control those factors. Therefore, collected information might help the firm to control more effectively the conditions surrounding its environment.

Based on what is presented above and after analyzing the literature review papers presented by Gu and Steed (2004) and [START_REF] Anderson | Dynamic pricing in hospitality: overview and opportunities[END_REF], our work presents an analysis of pricing models developed for hotel room pricing. We are trying to examine the conditions (pricing information) under which these methods were developed. Moreover, we studied the literature addressing attributes that influence hotel room pricing to be able to give some hints for managers when making pricing decisions. Many authors consider airlines models when studying hotel pricing practices. We believe that the hotel industry is particular and the dynamic pricing models found in the literature are not appropriate for the hotel industry due to two main reasons. First, the hotel must determine and reveal the booking price for rooms on each individual day. The reason behind is that any extension or reduction in the length of stay may require an adjustment in the payment. Secondly, explicit constraints are required to ensure that customers do not make a reservation if the hotel is out of rooms for any day during their interval of stay [START_REF] Nadarajah | Dynamic pricing for hotel rooms when customers request multiple-day stays[END_REF]. Moreover, customer relationship management play an important role in the hotel's decisions.

RESULTS

We have studied dynamic pricing models for hotel rooms and have collected the assumptions based on which the models were developed. We have noticed that almost all models share similar assumptions used to develop the pricing model. One of the earlier studies on hotel pricing was presented by [START_REF] Gu | Proposing a room pricing model for optimizing profitability[END_REF] who proposed an optimal room rate model that incorporated both variable costs and two basic features of market demand. Using similar assumptions, Ladany (1995) proposed an optimal market segmentation of hotel rooms to optimize revenue. Van Dijk & Van Der Stelt-Scheele (1993) studied price formation in tourism industry branches. They showed that cost factors and demand factors play an important role in pricing decisions. Howard [START_REF] Finch | An option-based approach for pricing perishable services assets[END_REF] presented an option-based approach for pricing perishable service assets. They proposed that selling a service at a discounted price at least makes a positive contribution to fixed coverage. Authors suggest that cutting price (discounting) in the event of less than expected demand should have positive value for the firm. The problem of this model is that the best result is just obtained when it's used to special cases: when the perishable product priced is unique and expensive; the purchase process can start well in advance of the service offering and the pool of customers is sensitive to price. Inspired by [START_REF] Gu | Proposing a room pricing model for optimizing profitability[END_REF], [START_REF] Pan | Market demand variations, room capacity, and optimal hotel room rates[END_REF] developed an optimal hotel room rate model incorporating costs, market demand variations and hotel limited room capacity. This model is developed to operate in high and low seasons [START_REF] Pan | Market demand variations, room capacity, and optimal hotel room rates[END_REF]. [START_REF] Bayoumi | Dynamic pricing for hotel revenue management using price multipliers[END_REF] presented a dynamic pricing for hotel revenue management using price multipliers. They used four variables that is known to have an influencing effect on pricing decisions: time from reservation until arrival date, the hotels remaining capacity at the time of reservation, the length of stay, and the number of room to be reserved [START_REF] Bayoumi | Dynamic pricing for hotel revenue management using price multipliers[END_REF]. [START_REF] Aziz | Dynamic room pricing model for hotel revenue management systems[END_REF] addressed the problem of optimally setting the price of the hotel rooms in order to maximize revenue by incorporating price elasticity of demand. Their proposed framework overcomes the limitations associated with the research gaps in dynamic pricing literature and can contribute in increasing the revenue of a hotel. Authors didn't consider in their work group reservations which is a typical case in the hotel industry. In addition, they didn't consider overbookings and cancellations. [START_REF] Ellerbrock | Competition and lodging room rates[END_REF] 2015) considered the length of stay in hotels to develop their model. They formulated the dynamic pricing problem as a finite-horizon and discrete-time Markov decision process and approach its solution using stochastic dynamic programming. They determined the optimal booking price of rooms for each individual day, while considering the availability of room capacity throughout the multiple-day stays requested by customers.

Recently, more studies become concerned with pricing strategies online or by coordination with web parties as the internet represent an extremely efficient medium for accessing, organizing and communicating information (Yelkur et al. 2001). Guo et al. (2012) studied the optimal pricing strategy for hotels when they operate with online distribution channel by cooperating with a third party website. At first, authors assumed that all participants are integrated in a centralized system and gave the best optimum solution. Then, they gave the best solution under a decentralized scenario through a noncooperative game model composed by Stackelberg game between hotels and the website and a Nash game among hotels. Authors showed that even under the decentralized scenario, the cooperation contract provide a high operational performance. Later, Guo et al. (2013) were the first to determine the optimal dynamic pricing strategy in using of ORS (online reservation system) of the service industries. [START_REF] Ling | Optimal Pricing Strategy of a Small or Medium-Sized Hotel in Cooperation With a Web Site: 中小型宾馆与第三方网站合作的最优定价机制[END_REF][START_REF] Ling | Optimal pricing strategy of hotel for cooperative travel agency[END_REF] has investigated the optimal pricing strategies of the hotel with a second party (website and travel agency). These studies confirmed that cooperating with a third party contributes to a win-win situation. In addition, they confirmed that higher level hotels gain more revenue from cooperation. [START_REF] Ling | Optimal Pricing Strategy of a Small or Medium-Sized Hotel in Cooperation With a Web Site: 中小型宾馆与第三方网站合作的最优定价机制[END_REF]) [START_REF] Ling | Optimal pricing strategy of hotel for cooperative travel agency[END_REF]. [START_REF] Anderson | Setting prices on priceline[END_REF] showed how a firm can use Priceline to improve its inventory allocation and prices. Each of these papers are suggesting a pricing model for hotel rooms, but they are presenting different cases and different methods to develop the model. By gathering the information used, we notice that different works share some similar assumptions like demand and market structure. While some works incorporate costs, seasonality and competition, others don't. Therefore, the information assumed to develop a pricing model for hotel rooms are: demand (deterministic \ stochastic), market structure (monopolistic\ competition), product number ( one or multiple), costs (variable and fixed costs), market segments and seasonality.

In addition to demand type, market structure, product number and product shelf life, there are many other factors that affect pricing decision of which hotels must be aware to charge right price to the right segment. We have studied the works that analyze the effect of some attributes on hotel room pricing. We will present the attributes that allow an increase of room prices, the enhancement of hotel rating and customers 'perception of the fairness of room rates. 2008), showed that hotels that locate in close proximity to higher segmented hotels benefit without making similar product and service investments themselves. In contrary, colocating with a high percentage of hotels in lower product segments erodes the prices of higher segmented properties.

Yeoman& McMahon-Beattie (2006) addressed the issue of luxuryfication of society and how organizations are using premium prices effectively by offering luxury products and services. Authors concluded that consumers are willing to pay a higher price for value and brand equity. They suggested that offering high quality and a flawless experience are key factors for customer willingness to pay higher prices. [START_REF] Lien | Online hotel booking: The effects of brand image, price, trust and value on purchase intentions[END_REF] concluded that brand image and value are the most important determinant of customer purchase intentions. Abrate et al. ( 2011) confirmed the conclusion of Yeoman& McMahon-Beattie (2006) by analyzing the relationship between quality signals and price setting through the application of hedonic price functions. They proved that star category, quality and location explain price differences. Also, Chiu & Chen (2014) confirmed that customers are willing to pay for higher service quality when they chose travel destinations and accommodation.

In addition to location and quality, researches have shown that star rate, size of hotels, age of hotels, presence of hotel safe and association to hotel chain influence formation of hotel room rates (Zhang, 2011;[START_REF] Schamel | Weekend vs. midweek stays: Modelling hotel room rates in a small market[END_REF]Costa, 2013;Carvell & Herrin, 1990). Also, special events, festivals and concerts permit optimal pricing of room rates (Herrmann, 2012). Fleischer ( 2012) and Masiero et al. (2015) suggested that customers are willing to pay higher prices for rooms with a view. White & Mulligan (2002) examined geographic variation in hotel room rates during both summer and winter seasons at the broad regional scale using the hedonic approach. Authors found that recreational amenities such as spas and pools, location, urban setting, and economic features of the surrounding county are predictors of hotel prices in both seasons. Moreover, Studies addressed the different perception of prices and willingness to pay of consumers from different segments (business and leisure travelers) [START_REF] Schamel | Weekend vs. midweek stays: Modelling hotel room rates in a small market[END_REF]. Nowadays most of customers check online reviews of each property before making a reservation. Highly rated hotels are perceived of better quality. Ye.Q et al. (2014) emphasized on the importance of star rating given by customer perception. They advised managers to consider online traveler reviews when developing pricing strategies. Schamel (2012) also emphasized the importance of online reviews.

The results of this study show that business travelers are willing to pay higher premium price for another point obtained in online ratings. He also proved that hotel star rating and distance to the center also contributes to the willingness to pay higher rates for hotel rooms. Bulchand-Gidumal et al. (2011) suggested that hotel can improve its star rating by offering free Wi-Fi service. Heo & Hyun (2015) suggested that Wi-Fi internet access in the room is considered the most important item by customers. They also showed that customers are willing to pay higher prices for luxury amenities and the presence of Gyms, pools and spas within the hotel facility. Stringam & Gerdes Jr (2010) explored over 60,000 comments and ratings made by travelers on an online distribution site in an attempt to find which factors influence consumer ratings of hotels. The results show that location, cleanliness and service are the most important hotel attributes. Kim et al. ( 2016) have found that the availability of recently-added amenities such as free Wi-Fi, pet boarding, free self-parking, multilingual staff, and airport transportation, as well as green practice initiatives have a positive impact on online review ratings and willingness to pay high prices in addition to the revisit intention.

With the rise of green practices and environmental concerns and awareness, some hotels started the application of these practices within their organizations. Sánchez-Ollero et al. ( 2014) studied the influence of environment respect on final prices in the hospitality sector. The result show that customers positively value the implementation of green practices and are willing to pay higher prices for the service provided. This fact may encourage hotels to implement environmental practice measures and gain more customers valuing this practice and willing to pay higher prices. In a study conducted by Chen et al. (2014), results suggest a mass-marketing strategy to be adopted by medium quality hotels. For high quality hotels, promotions with price discounts, customer relationship management and customized packages are more favorable to be used as marketing strategies. Also, it is important to mention that firms applying yield management should take into consideration price perceptions of customers and that pricing decisions should be made by properly communicating changes in prices and the reasons behind them (Rosa-Diaz et al. 2012). A final suggestions for hoteliers is to make attention to the importance of the price ending strategy that may send a message of value and quality as it's explained by [START_REF] Collins | Pricing strategies to maximize revenues in the lodging industry[END_REF], not to forget that 0.001$\room might make a big difference for the revenue of the company. Therefore, we can resume what was presented above by suggesting that location, services, rating and online reviews, hotel size, quality, cleanliness, brand image, room view and green practices are important attributes that may affect the price of the hotel room and the perception of customers.

As mentioned above, many authors have emphasized on the importance of many attributes that may affect the price of the room rate. Despite their importance, none of these attributes was considered for the development of a pricing model. The matrix below represent four parts representing the different information that must be used for an optimal price decision and the distribution of studies based on the consideration of this information.

We notice that none of the studies considered customers behavior and just one study considered the supplement of existing attributes that may be useful to enhance the price and therefore the revenue. The complete piece of cake represents the optimal profit that manager can generate in case of using complete information in order to decide room rates. The cake is formed of different layers representing different kind of information that may be used for pricing. The first layer represents the basic costs that each property tries to cover before thinking of generating profit. The second one is the information used by researchers to propose pricing models that were presented in the study. The third layer is the customer who is the most important player of any business. Many studies started to include different type of customers into their studies to make the most convenient decisions about the price [START_REF] Levin | Optimal dynamic pricing of perishable items by a monopolist facing strategic consumers[END_REF]Zhao et al.2012). And the profit cake is topped by the attributes that may increase slightly the price but can make a big difference on the overall profit.

The table below represents the information that must be considered for room rates decisions. The more managers consider details; the optimal is the price obtained resulting in better profitability. Table 2: List of information that must be considered when pricing for optimal profit CONCLUSION AND FUTURE DIRECTIONS This paper has presented different pricing information that affects the pricing decisions in hotel industry. In order to suggest all useful information that may help increase the profit, we have studied pricing models, room price determinants and what influence room prices in general. Our suggested matrix has shown that a limited number of existing information is used for developing pricing models. These information are limited to demand, market structure, product nature and number, costs, market segments and seasonality. Despite the use of this information, most of studies are simplifying the assumptions. Most of works are considering monopolistic market or oligopolistic competition which doesn't reflect the reality. Market segments are almost always considered as one segment and seasonality are often neglected. In addition, consumer behavior is an important factor to consider when pricing, but it was not considered for hotel pricing. Nowadays, consumers are acting more strategically with the high availability of pricing data accessible on the internet (Schutze, 2008). Technological innovations, media and firms 'availability online made the access to information easy for all customers at any time and place. Similarly, firms are able to collect information about customers that use online platforms. By distributing the different studies within the matrix, we have noticed that some factors such as costs were considered earlier but no more in new studies. Costs should not be the most important factor based on which the price should be determined, but it should be considered as well when pricing to ensure covering the basic expenditures of the hotel. Many studies have emphasized on the importance of many attributes appreciated by customers that affect pricing, and that can help managers increase room rates. We noticed that none of these attributes were mentioned in any work suggesting a pricing model for hotels. Not just that the existence of these factors may improve the price, but managers may enhance the use of these attributes and improve them as well. For example, location is one of the most important attributes that may enhance the price. But not each segment values the same location similarly. Hotel managers must understand which segment values most their hotel location to incorporate it correctly into room rates. Another example for rating that plays an important role as an indicator of customers' satisfaction from previous experiences may be enhanced by offering free Wi-Fi service and thus capturing more customers through online bookings (Bulchand-Gidumal et al. 2011). These are two examples of how to enhance the profitability of the hotel by implementing simple thing. All attributes may be profitable and can be enhanced within the property. In general, we can suggest for an optimal pricing, better performance and long-term customer retention, managers should try to offer in their hotels a higher quality service merged by high quality tangible products. Also, they should understand how to assess each of the existing attributes to take advantage of them and make accurate pricing decisions. Finally, it is important to remind managers that long-term profitability is assured by customer satisfaction. Customer satisfaction is the salient driver of hotel performance and it has a positive influence on all of the hotel performance indexes including ADR, RevPAR, TrevPAR, restaurant F&B revenue and banquet F&B revenue (Kim et al. 2013;[START_REF] Tajeddini | Managing Swiss Hospitality: How cultural antecedents of innovation and customer-oriented value systems can influence performance in the hotel industry[END_REF]O'Neill & Mattila, 2004). Future researches are directed to take the maximum number of information and attributes into consideration to develop more realistic and contemporary pricing models. Research are also directed to study the percentage of price increase related to each attribute represented in this work that may lead hotels to a remarkable increase in their profits.

Figure 1 :

 1 Figure 1: The number of times of mentioning each attribute in the studied literature

Figure 2 :

 2 Figure 2: Piece of Cake Matrix: Matrix showing the importance of information used to enhance hotel profit

Table 1 : Information used by different studies to develop pricing models for hotel rooms

 1

	Information	Costs	Demand	Market structure seasonality	Market	Product
	used	(Variable					segments	number
		and/or fixed)
				monop	competiti			one multiple
				ole	on
	Number of	8	13	14	2	2	1	15	1
	articles
	Total number								16
	of articles

  presented an empirical model of pricing hotel rooms based on traditional microeconomic price theory. Their model incorporated: fixed costs, variable costs, advertising expenditures, location, amenities, percent occupancy rate, season and amount of competition. Despite the simplicity of the presented work, it emphasized on the importance of competition factor in pricing decisions. Also,[START_REF] Masruroh | Pricing model development for hotel industries using game theory approach considering seasonal factor and uncertain demand[END_REF] developed a pricing policy for two hotels competing for the same market. Their model is applied for deterministic and uncertain demand and it considers the seasonal factor.[START_REF] Arenoe | Game theoretic pricing models in hotel revenue management: An equilibrium choice-based conjoint analysis approach[END_REF] explored a game theoretic pricing model that determines equilibrium room rates under differentiated price competition in an oligopolistic hotel market. In this model, authors considered non-price attributes to analyze consumer choice. Concentrated on the particularity of hotel industry Lim et al. (

Costs and expenses Available Data and Predictions Hotel attributes

			 Location
	 Fixe charges and man-	 Demand	 Services available
	agement fees	 Market structure	 Online reviews
	 Undistributed operat-	 Market share	 Rating
	ing expenses	 Market segments	 Hotel size (chain)
		 Seasonality	 Quality
		 Room types	 Cleanliness
		 Customer types (stra-	 Room view
		tegic)	 Green practices
			 Amenities (quality &
			diversity)
			 Events