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Railway Engineering Science
Erschienen in: Railway Engineering Science 3/2022

Open Access 05.12.2021

A geomechanics classification for the rating of railroad subgrade performance

verfasst von: António Gomes Correia, Ana Ramos

Erschienen in: Railway Engineering Science | Ausgabe 3/2022

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Abstract

The type of subgrade of a railroad foundation is vital to the overall performance of the track structure. With the train speed and tonnage increase, as well as environmental changes, the evaluation and influence of subgrade are even more paramount in the railroad track structure performance. A geomechanics classification for subgrade is proposed coupling the stiffness (resilient modulus) and permanent deformation behaviour evaluated by means of repeated triaxial loading tests. This classification covers from fine- to coarse-grained soils, grouped by UIC and ASTM. For this achievement, we first summarize the main models for estimating resilient modulus and permanent deformation, including the evaluation of their robustness and their sensitivity to mechanical and environmental parameters. This is followed by the procedure required to arrive at the geomechanical classification and rating, as well as a discussion of the influence of environmental factors. This work is the first attempt to obtain a new geomechanical classification that can be a useful tool in the evaluation and modelling of the foundation of railway structures.
Abkürzungen
PI
Plasticity index
\(\gamma\)
Dry unit weight
S
Degree of saturation
a
Initial tangent modulus
q u
Unconfined compressive strength
% finer #200
Percent passing #200 sieve
LL
Liquid limit
% clay
Percent finer than 0.002 mm
W L
Liquid limit
W P
Plastic limit
IP
Index plasticity
W
Moisture content
W
Optimum moisture content
C u
Coefficient of uniformity
C c
Coefficient of gradation
D n
Diameter corresponding to n% of passed material in a grain size distribution curve
CBR
California bearing ratio

1 Introduction

The understanding and knowledge about the deformation and failure mechanisms of the geomaterials under repeated loadings are extremely important for the proper design and maintenance planning in the railway structures [1‐4]. Typically, the subgrade layer of railway track is composed of geomaterials (mainly coarse-grained soils due to its better performance when compared to the fine-grained soils) and presents two types of deformation when submitted to cyclic loads: the recoverable (quasi-elastic, resilient or reversible) and permanent (or plastic) deformations, which have significant importance in the long-term performance of the substructure [5‐7]. Regarding the resilient deformation, the concepts were introduced by AASHTO in 1986 [8]. This property is often used to characterize the materials in several railway layers [9‐11]. About the permanent deformation, an accurate estimation of the amount of cumulative settlement will benefit railway structures to avoid a mediocre performance and consequently higher annual maintenance costs [12‐14]. Overall, resilient and permanent deformations are both important parameters for the design of railway structures [15‐18]. However, the complexity and time-consuming of laboratory tests, mainly for permanent deformation evaluation, conduct to the development of predictive models based on the available data from the existing laboratory and/or field results.
In this line, one of the main objectives of this work is to review the available resilient and permanent deformation models in the scope of the geomaterials published in the specialized bibliography considering the information about the conditions for its development (properties of the materials tested, compaction tests, degree of compaction, moisture content, etc.) and the main variables/factors that can influence the response of the material. This means that this analysis also includes external factors as the physical state of the soil, which is often difficult to control because it depends on other environmental aspects. A first attempt was already performed for permanent deformation models in the work developed by [19]. Consequently, this work intends to complement it with the resilient deformation models, including comparisons among some of the selected models that are based on different materials with different classifications (UIC and ASTM), properties, granulometry and physical states. By coupling both resilient- and permanent-based deformations, we rate the geomaterial classifications according to the predicted resilient and permanent deformation performances, which is believed to be a helpful tool in the design of the railway structures. It is also noted that this rating should be interpreted under normalized conditions because it depends on several properties and soil state conditions. Based on these results, a novel geomechanical classification for track subgrade/ formation that couples the resilient modulus and the permanent deformation is purposed using a similar approach developed by [20].

2 Influence of subgrade in design and performance of railroad track structures

The quality and the support given by the subgrade are important aspects in the performance of the railroad track structure. The subgrade is an integrant component of the track structures (ballasted and ballastless tracks), and its properties are important in the performance of track and track quality. Indeed, this is why subgrade performance indicators are important and should be implemented into the track designs and assessments. Thus, there are numerous means of measuring and predicting subgrade performance, which include the laboratory and in situ tests. The in situ tests can be advantageous since the results are representative of the real conditions and some can be obtained quickly and more cost-effective. However, laboratory tests could be also very interesting and necessary to carry out parametric studies considering variations occurring during the service life of the structure. Indeed, recently, new methods have been developed to determine the resilient modulus, such as the performance of the cyclic lightweight deflectometer test [21].
Thus, despite the importance of the subgrade and its significant impact on the track quality and required maintenance, the evaluation of the subgrade is not straightforward since there is an extensive list of factors that can affect its short- and long-term performance: the moisture content and/ or, suction, and/ or degree of saturation, temperature (freezing and thawing), shear strength, stiffness parameters, consolidation, etc. Indeed, recent studies show the influence of soil suction on the deformation characteristics of railway formation materials [22]. Furthermore, other variables can affect the performance of the subgrade related to the train type (passenger or freight), axle load, train speed, train configuration, drainage, rail condition, tie spacing, and wheel condition [23‐25]. Thus, these factors added to the environmental factors can affect the performance of the subgrade and also the global performance of the track structure. Indeed, the poor performance of the subgrade can lead to some issues in the railroad track structure: ballast fouling, ballast pockets, pumping of soil fines through the ballast, and slope stability failure. Thus, there are two main types of subgrade failure (mostly related to the fine subgrade soils): subgrade progressive shear failure and excessive subgrade plastic deformation which are well documented [15, 17, 26]. More recent works in Refs. [27] and [28] emphasize that low- and medium-plasticity soils can also be subjected to different types of failure such as fluidization and mud pumping due to excessive increase in axle loads of freight trains in recent years. Nevertheless, these phenomena and many others like liquefaction (large displacement caused by repeated loading, saturated silt, and fine sand), massive shear failure (slope stability—this failure is associated with the weight of the train, inadequate soil strain and increase of water content), consolidation settlement (the static soil stress increased due to the embankment weight and saturated fine-grained soils), frost action (the track become rough caused by periodic freezing), swelling/shrinkage, slope erosion and soil collapse caused by ground settlement are out of scope of this paper.
The design of the subgrade (design period corresponds to 50–100 years) is defined according to the type of traffic, bearing capacity of the subgrade, configuration of the track, climatic and hydro-geological conditions [15, 29, 30]. Therefore, the subgrade should be designed to show a good short- and long-term performance, resisting to failure and excessive deformation, respectively, induced by repeated loads. The short performance is influenced directly by the resilient modulus (that has influence on the stress levels on the subgrade) and by the strength parameters, among other factors. The stress paths and the strength parameters (such as cohesion and friction angle) have a significant effect on the development of the permanent deformation. (This topic will be analysed in more detail further in this work.) Usually, the design of the ballasted track involves the determination of the deformation and stresses at critical locations in the track (sleeper-ballast or ballast-sub-ballast contacts). Posteriorly, the magnitude of the settlements and stresses (induced by the passage of the trains) is compared with allowable values in an iterative process where the dimensions of the sleepers and/or thickness of the granular layers are adjusted. These methodologies are applied by several authors and standards:
  • Reference [15] developed a methodology that prevents progressive shear failure and also excessive plastic deformation where the plastic strain (εp) and the cumulative deformation should be limited to 2% and 25 mm, respectively.
  • Reference [31] presented a set of references for the design and maintenance of track substructure. According to [31], there are four soil quality classes: QS0 (unsuitable soils), QS1 (poor soils), QS2 (average soils) and QS3 (good soils). This concept will be explored further in this work. From the soils’ classification and thickness of the granular layers, the track is evaluated in three categories: P1, P2 and P3 (mediocre, average and good, respectively).
  • British Rail Method and Network Rail Code defined recommendations related to the thickness of the trackbed layers [17].
  • West Japan Railways defined a limit value for bearing capacity (288 kPa). Below this value, the improvement of the foundation is required. Thus, the bearing capacity of 288 kPa equates to a compressive strength σs of approximately 112 kPa assuming a cohesion model plastic solution to a simple strip footing where σb = 2.57σs.
These methods allow preventing the development of an overstressed subgrade that can present cumulative permanent deformation (formation of ballast pocket) or progressive shear failure (subgrade squeezing) under repeated loads. Indeed, these are common failure modes of typical subgrades composed of fine-grained cohesive soils [15, 26, 30, 32, 33]. These types of problems can lead to an increase in dynamic loads and accelerate the deterioration of the track.
Regarding the ballastless track, the design theories of the ballastless track vary across the world according to the experience and background of each country [34].
Thus, two important parameters should be included in the design of railway tracks related to the short- and long-term performance: the resilient modulus (Mr) and the permanent deformation (εp). These two parameters should be analysed together in an integrated approach since, from a simplistic point of view, the stress levels at the rail track subgrade vary with the value of its Mr, which have impact on the permanent deformation. These parameters will be analysed together with the UIC classification in this work to purpose a novel geomechanical classification.

3 Resilient modulus

The resilient modulus is used to characterize the recoverable, reversible or quasi-elastic behaviour. In the traditional theory of elasticity, the elastic properties (which are material parameters) are defined, mostly, by the Poisson’s ratio (ν) and Young’s modulus (E). In the analysis of the response of the material when submitted to cyclic loads, the modulus of the elasticity is replaced by the resilient modulus to consider the nonlinearity (i.e. dependence on stress level) on the performance of the material, as well as the inelastic properties of the materials, i.e. the loading and unloading of the stress–strain curve are not totally overlapped due to the dissipation of the energy, as depicted in Fig. 1.
Indeed, in the case of repeated load tests with constant confining pressure, the resilient modulus and the Poisson’s ratio are defined, respectively, by the following expressions:
$${M}_{\mathrm{r}}=\frac{{\sigma }_{\mathrm{d}}}{{\varepsilon }_{1;\mathrm{r}}}, \upsilon =-\frac{{\varepsilon }_{3,\mathrm{r}}}{{\varepsilon }_{1;\mathrm{r}}},$$
(1)
where σd is the deviatoric stress, ε1;r is the recoverable axial deformation and ε3;r is the recoverable horizontal deformation.
The test method used to determine the experimental value of the resilient modulus is described in [8] and is based on the application of deviatoric stress under constant confining stress. The response of the materials varies according to its own nature. The cohesive materials are more sensitive to the deviator stress than the confining pressure since the experimental results show that there is a decrease of the resilient modulus with the increase in the deviatoric stress. On the other hand, the granular materials show a different tendency since the resilient modulus increases with confining stress. The more recent models are dependent on important factors regarding environmental aspects such as moisture content, suction and degree of saturation [9, 35‐40]. Additionally, recent work by [41] also proposes a methodology to obtain the damping from these type of tests using the standard AASHTOT-307–93 [42].
The computational modelling of resilient behaviour is a very difficult task, due to the necessity of decoding complex behaviours into simple mathematical expressions and procedures for routine analysis.

3.1 Mechanistic-empirical modelling approach

The empirical approaches can be used in the design procedures of railway structures. This means that a relationship is established between the design inputs (materials, loads, environment, geometry) and, for example, rail track failures through experience, experimentation or a combination of both. The more complex empirical approaches are based on empirical equations derived from experimentations. The relationship between rail track failure and the physical phenomena is described by empirical equations.
The mechanistic-empirical modelling approaches used to determine the resilient modulus are divided according to the type of material tested (granular or cohesive). Furthermore, the models more complex depending on the physical state of the material are also presented. Moreover, in appendix, the state of the art of resilient models developed based on laboratory tests (mostly cyclic triaxial tests) considering all types of materials (such as clays, silts, sands and gravels) is summarized.

3.1.1 Cohesive materials

In the cohesive materials, the fines content and its mineralogy are important factors in the analysis, which means that the resilient modulus should be dependent on the soil physical state. Indeed, in cohesive materials, certain characteristics may have more importance in the value of the resilient modulus, besides the stress state, such as the moisture content, suction and saturation degree. In the work developed by [43], the authors state that the new generation models not only provide a better fit than the older models, but they also provide a reasonable fit to the data that can capture the effects of stress state, soil type, soil structure and the soil physical state quite effectively. Furthermore, according to [44], there are prediction models that combine the effect of the stress state and matric suction on the resilient modulus. These models not only reflect the relationship between the resilient modulus of subgrade soils and the stresses but also the effects of seasonal variation of moisture content on the resilient modulus. The authors also conclude that the degrees of stress and moisture content have a significant impact on the resilient modulus of compacted cohesive soils. Moreover, the results show that the resilient modulus of the tested cohesive soils increases with the increase in effective confining pressure, matric suction, and degree of compaction and decreases with the increase in deviator stress and moisture content. Therefore, lower moisture content concomitantly with a higher degree of compaction is beneficial to the stiffness of the subgrade, mainly in those not sensitive to water content changes.
Considering the extensive bibliography about this topic, several models were developed to describe the resilient modulus and are summarized in Table 1.
Table 1
Resilient moduli for cohesive materials (adapted from [45])
Models
Expressions
Parameters
Authors
Bilinear
\({M}_{\mathrm{r}}={K}_{1}+{K}_{2}{\sigma }_{\mathrm{d}}\) when σd < σdi
\({M}_{\mathrm{r}}={K}_{3}+{K}_{4}{\sigma }_{\mathrm{d}}\) when σd > σdi
K1, K2, K3 and K4
Thompson and Robnett [46]
Power
\({M}_{\mathrm{r}}=k{\sigma }_{\mathrm{d}}^{n}\)
k and n are parameters dependent on the type of soil and physical state; the parameter n usually has a negative value
Moossazadeh and Witczak [47]
Power (with the influence of the confining stress for overconsolidated saturated soils)
\({M}_{\mathrm{r}}=k{\left(\frac{{\sigma }_{\mathrm{d}}}{{\sigma }_{3}^{{\prime}}}\right)}^{n}\)
\({\sigma }_{3}^{\prime}\) is the confining stress
Brown, Lashine [48]
Semi-log
\({M}_{\mathrm{r}}={10}^{(k-n{\sigma }_{\mathrm{d}})}\)
k and n are empirical parameters
Fredlund, Bergan [49]
Hyperbolic
\({M}_{\mathrm{r}}=\frac{k+n{\sigma }_{\mathrm{d}}}{{\sigma }_{\mathrm{d}}}\)
Mr in ksi and σd in psi
Drumm and Pierce [50]
Octahedral
\({M}_{\mathrm{r}}=k\frac{{\sigma }_{\mathrm{oct}}^{n}}{{\tau }_{\mathrm{oct}}^{m}}\)
σoct and τoct are the shear and normal octahedral stresses
Shackel [51]
Furthermore, it is also important to take into account the type of approach in terms of total and/or effective stresses. In the recent work developed by [52], the developed model can describe the phenomena of modulus development during the cyclic undrained condition and takes into account the actual values of effective stress (p’), excess pore water pressure, the loading characteristic and the position of the effective stress path to the failure line.

3.1.2 Granular materials

The modelling of the resilient behaviour in the case of granular materials can be performed through two different approaches: simulation of the resilient modulus considering a constant value of the Poisson’s ratio and by separation of the deformation response of the material into the volumetric and shear components, which is more complex [53].
The models expressed by the resilient modulus and constant Poisson’s ratio are very easy to understand, very simple and easy to implement numerically. In fact, initially, in terms of variables, the models were only dependent on the confining stress and material parameters and/or the sum of the principal stresses [54, 55]. Due to the necessity to include the influence of the deviator stress, [56] modified the well-known k–θ model, introducing the influence of the mean and deviatoric stress (p and q, respectively):
$$E={k}_{1}{\theta }^{{k}_{2}}{q}^{{k}_{3}},$$
(2)
where θ is the sum of principal stresses, and k1, k2 and k3 are parameters dependent on the state of the soil and its characteristics.
Posteriorly, some models were improved considering, for example, the porosity of the material and variable confining stress [57, 58].
However, some of the previous models show drawbacks concerning the assumption of a constant value of the Poisson’s ratio: despite the good results regarding the axial deformations, it was difficult to simulate correctly the volumetric and radial deformations. Indeed, some authors also suggested that the use of these models could lead to Poisson’s ratios superiors to 0.5. However, some of these models are still used and were also improved because of their simplicity [59].
Nevertheless, in order to solve Poisson’s ratio problem, other models were developed, expressing the resilient behaviour through the shear modulus (G) and bulk modulus (K). This formulation (in terms of G and K) is more comprehensive than the models only characterized by the resilient modulus and Poisson’s ratio since it is dependent on more complex laboratory tests and with a more generic interpretation than a cyclic axial loading.
Indeed, Ref. [48] cited by [60] identifies three important conditions in the application and formulation of these models:
  • In each increment of calculus, a linear elastic behaviour is adopted.
  • The shear and volumetric components of stress and strain are analysed independently.
  • Better adjust to the experimental results, mostly when the solicitation presents a 3D character.
Reference [61] developed a well-known model dependent on bulk modulus and shear modulus:
$$K=\frac{p}{{\varepsilon }_{\mathrm{v}}},$$
(3)
$$G=3\frac{q}{{\varepsilon }_{\mathrm{q}}},$$
(4)
where εv and εq are respectively the volumetric and shear resilient strains:
$${\varepsilon }_{\mathrm{v}}=\frac{1}{{K}_{1}}{p}^{n}\left[1-\beta {\left(\frac{q}{p}\right)}^{2}\right],$$
(5)
$${\varepsilon }_{\mathrm{q}}=\frac{1}{{3G}_{1}}{p}^{n}\left(\frac{q}{p}\right),$$
(6)
K1, β and G1 are parameters of the model, with β = (1 − n)K1/6G1.
In this model, there is a clear division between the volumetric and shear deformation and the values of K and G are stress-dependent (p and q, respectively). Posteriorly, this model was updated by other authors [53, 62‐64] in order to consider the anisotropy of the material, which is an important characteristic regarding the deposition and compaction conditions. Indeed, [63] and [53] purpose and apply an orthotropic version of Boyce’s model introducing an anisotropic coefficient (γ).

3.2 Models dependent on the physical state—suction and degree of saturation

Recently, some studies were developed in order to include the physical state into the resilient models [9, 35, 36, 65].
In this section, a brief reference is made regarding the models dependent on the suction and degree of saturation. These models were developed based on experimental results and include suction as one of the main parameters. In the case of non-saturated soils, three important variables influence recoverable/reversible behaviour [66]:
  • net confining stress: σ3−ua;
  • deviatoric stress: σ1−σ3;
  • suction matrix: ψm = ua−uw,
where σ3 is the confining stress, σ1 is the axial stress, ua is the air pressure and uw is the water pressure.
Based on Uzan’s model, Ref. [67] developed a model that includes the suction in the determination of the resilient modulus:
$${M}_{\mathrm{r}}={k}_{1}{p}_{\mathrm{a}}{\left(\frac{{\theta }_{\mathrm{net}}-3\Delta {u}_{\mathrm{w}-\mathrm{sat}}}{{p}_{\mathrm{a}}}\right)}^{{k}_{2}}{\left(\frac{{\tau }_{\mathrm{oct}}}{{p}_{\mathrm{a}}}+1\right)}^{{k}_{3}}{\left(\frac{{\psi }_{\mathrm{m}}-\Delta {\psi }_{\mathrm{m}}}{{p}_{\mathrm{a}}}+1\right)}^{{k}_{4}},$$
(7)
where \({p}_{\mathrm{a}}\) is atmospheric pressure; \({\tau }_{\mathrm{oct}}\) is octahedral shear stress; θnet is net bulk stress, θnet = θ−3ua, and θnet is bulk stress; Δψm is the variation of the suction matrix regarding the initial suction (Δuw−sat = 0); and Δuw−sat represents the increase of the pore pressure in saturated conditions (ψm = 0).
Despite the simplicity of the model formulation (which is based on the work developed by [56]), the characterization of the suction increases the complexity and application of these types of models since this is a parameter that is difficult to characterize.
Recent studies show the influence of the water content (with special attention on the role of the suction in the interpretation of the results) and consider the possibility to apply an effective stress approach to characterize the changes in normalized resilient modulus as a function of a single parameter that includes the total stresses and suction [37]. Moreover, some studies are focused on the importance of the degree of saturation (and the controlling in the soil compaction) and its relationship with the soil structure design. In fact, the degree of saturation and also the CBR (California bearing ratio) are parameters that are easy to obtain and easy to use when compared to the suction. The authors conclude that CBR (soaked and unsoaked) and also the elastic shear modulus (representative of the stress–strain behaviour at small strains of the subgrade), unconfined compression strength, cyclic undrained shear strength, among other parameters, of unsaturated soil are controlled by ρd and Sr at the end of compaction [39]. More specific studies about compaction conditions and percentage of fines can be found in [68] and [69].

3.3 Main differences of the resilient modulus formulation—analysis and comparison

As previously described, the resilient behaviour is dependent on several factors such as the stress level, fines contents, particle shape, grading, density, maximum grain size, aggregate type, moisture content, stress history and number of load cycles. From all of them, two important factors stand out: the applied stress level and moisture content. The stress (in terms of confining stress, sum of principal stresses and deviator stress) is the most important factor in the analysis of the resilient behaviour since its impact on the response of the material is very significant. Indeed, most of the geomaterials (especially the granular materials) show a stress dependency under repeated loads. To replicate this nonlinearity and time dependency, the resilient response of the materials is not solved by the traditional elasticity theory. However, as mentioned previously, there are other formulations to characterize the response of the material such as the replacement of the resilient modulus and Poisson’s ratio by the shear and bulk modulus.
This formulation is simple and based on curve-fitting. Indeed, this approach deals better with the nonlinear response of the geomaterials (mostly in the case of granular materials) from a theoretical point of view, and it is more realistic regarding the physical meaning in a 3D stress regime. However, as mentioned previously, these models have a more complex formulation and the parameterization is also difficult to obtain through the available test data [70].
Another relevant aspect is related to the fact that some of the resilient models only fit the laboratory data used for their own development, which means that the mathematical formulation can only be used in a particular situation.
The models that include the influence of the moisture content and/or degree of saturation describe better the material conditions. In the case of the granular materials, the granulometry and the fines content (in a compacted well-graded material) have a significant influence on the response of the material since the water cannot drain freely since it is kept in the pores of the materials [71]. Indeed, when the saturation is close to the maximum, the resilient behaviour is affected [72]. Nevertheless, these models are more complex since it is difficult to characterize these parameters related to the suction and degree of saturation.
More recently, some resilient models have been developed using machine learning and statistical techniques and optimization tools as described in the work developed by [73] and [74].
Despite the several advantages of the empirical models, the main limitation is related to the confidence in the extrapolation of the analysis beyond the conditions in which they were defined. The tables presented in appendix (Tables 17–21) contain information regarding the type of soils (UIC and ASTM classification), source (authors), the mathematical models and respective variables and empirical constants. This information is quite significant in the modelling of the subgrade of the railway tracks since supports information about the materials used to define the model, as well as its physical conditions such as moisture content and dry density.

4 Permanent deformation

The permanent deformation is induced by repeated traffic loading and may occur in railway structures. The development of permanent deformation is the result of the accumulation of deformation throughout millions of cycles, which is a complex process and depends on several factors such as the number of load cycles (N), stress levels, loading history, the effect of the principal stress rotation, moisture content, density, aggregate type and particle size distribution. This development of permanent deformation during N loading cycles can stabilize or, in the worst situation, may lead to the ultimate collapse of the structure (excessive rutting). Regarding the moisture content, despite the extensive research about the influence of this parameter on the reversible deformation, a recent study was developed to study the effects of the compaction moisture content on the plastic behaviour of fine soils [75].
To predict permanent deformation, several approaches were proposed: numerical simulations using elastoplastic models, shakedown theory or mechanistic-empirical permanent deformation models based on laboratory tests such as cyclic triaxial tests or the hollow cylinder tests [19]. Throughout the analyses presented by these authors, more emphasis is given to the mechanistic-empirical permanent deformation models from the laboratory testing until the modelling. In fact, the work done in this paper for the resilient modulus is a mirror of the one for permanent deformation.

5 Geomechanics classification and the rating

5.1 Resilient modulus: parametric study

Considering the extended information presented in Sect. 3 and also the tables presented in appendix, a first attempt was done to compare the resilient modulus of different materials under a certain stress level. This analysis is important to comprehend how the models differ from each other in terms of formulation and the importance of certain variables.
The models were calibrated considering the same reference stress path for all materials during the calibration process. The stress path described by [76] was used. This stress path is characterized by a cyclic deviator stress of 24 kPa and a constant confining stress of 60 kPa. During the cyclic tests, the stress ratio (σd/σc) was kept constant at 0.4, which is a representative ratio in the subgrade of a rail track full-scale model test. It is noted that other confining stresses from 60 to 210 kPa were also tested.
In this analysis, four models were applied in order to compare the resilient modulus for the coarse-grained soils, as depicted in Table 2. In order to facilitate the interpretation of Table 2, C and m are material constants and \({\sigma }_{\text{v}}^{^{\prime}}\) is the effective vertical stress (more details in Tables 19 and 20)).
Table 2
Models considered in the analysis of the coarse-grained soils
Refs.
Models
[55, 77, 78]
\({M}_{\mathrm{r}}={k}_{1}{\left(\theta \right)}^{{k}_{2}}\)
[56]
\({M}_{\text{r}}={k}_{1}{p}_{0}{\left(\frac{\theta }{{p}_{0}}\right)}^{{k}_{2}}{\left(\frac{q}{{p}_{0}}\right)}^{{k}_{3}}\)
[79]
\({E}_{\mathrm{v}}=C{{\sigma }_{\text{v}}^{^{\prime}}}^{m}\)
[59]
\({M}_{\text{r}}={k}_{1}{p}_{0}{\left(\frac{\theta }{{p}_{0}}\right)}^{{k}_{2}}{\left(\frac{{\tau }_{\text{oct}}}{{p}_{0}}+1\right)}^{{k}_{3}}\)
The first part of this work is based on the experimental results presented in [80]. The authors studied six granular materials representative of the base and sub-base materials used on flexible pavements. [80] performed rapid shear tests and repeated load tests in order to determine the shear strength parameters (cohesion and friction angle), resilient modulus, rutting potential and moisture susceptibility. The properties of the materials are presented in Table 3.
Table 3
Properties of the materials (coarse-grained soils)
No.
UIC and ASTM classification
Materials
Properties
Observations
1
QS3
GW
Well-graded gravel
Cu = 27; Cc = 2
(PImin = NP;
PImax = 6)
WL;máx = 25
W = 6.3%
WOMC = 6.8%
Granular material used as based and sub-base in the flexible pavement sections;
2
QS2/QS3
SW/SM-ML
Sand
Cu = 20; Cc = 0.8–1.1
(PImin = NP;
PImax = 6)
WL;máx = 25
W = 7.7%
WOMC = 7.7%
The target moisture content and densities were selected based on AASHTO T99 test results and field-measured values [80]
The constants of each model were determined by regression analyses. The values of these constants reflect the material properties as well as the loading cycle applied in each test that may vary from author to author. For example, in the work developed by [80], the specimens were conditioned for 1000 load repetitions while in the work developed by [9], the authors follow the procedures described in AASHTO T307-99. However, for practical purposes, the conditioning adopted for resilient modulus determination following the standard procedures (AASHTO T307-99, [40]) intends to assure that the number of cycles during resilient modulus determination will not affect the results. The results of the four models and the two materials are presented in Table 4. It is important to refer that for material 2, due to the lack of data, it was not possible to present the regression results for the model developed by [79] and MEPDG model. Analysing Table 4, in general, the models present similar r-squared values. Therefore, considering the obtained constants and the stress path selected, the models were compared in terms of the resilient modulus, as depicted in Table 5. It can be observed that the models present similar results. Material 2 presents an inferior resilient modulus when compared to the well-graded gravel, as expected. Regarding the material classified as QS3-GW, it is evident the close values between the Uzan’s model, k−θ and MEPDG model.
Table 4
Summary of the values of the regression analysis based on experimental results
Models
Material classification
Parameters
Regression analysis R2
\({M}_{\mathrm{r}}={k}_{1}{\theta }^{{k}_{2}}\)
Well-graded gravel
QS3-GW
k1 = 5176
k2 = 0.64
0.997
Sand
QS2/QS3-SW/SM-ML
k1 = 36,942
k2 = 0.32
0.933
\({M}_{\mathrm{r}}={k}_{1}{p}_{0}{\left(\frac{\theta }{{p}_{0}}\right)}^{{k}_{2}}{\left(\frac{q}{{p}_{0}}\right)}^{{k}_{3}}\)
Well-graded gravel
QS3-GW
k1 = 5306
k2 = 0.59
k3 = 0.05
0.998
Sand
QS2/QS3-SW/SM-ML
k1 = 34,336
k2 = 0.45
k3 =-0.15
0.942
\({M}_{\mathrm{r}}={k}_{1}{p}_{0}{\left(\frac{\theta }{{p}_{0}}\right)}^{{k}_{2}}{\left(\frac{{\tau }_{\mathrm{oct}}}{{p}_{0}}+1\right)}^{{k}_{3}}\)
Well-graded gravel
QS3-GW
k1 = 1.02
k2 = 0.62
k3 =-0.01
0.997
\({E}_{\mathrm{v}}=C{{\sigma }_{\text{v}}^{^{\prime}}}^{m}\)
Well-graded gravel
QS3-GW
C = 6768
m = 0.65
0.988
Table 5
Determination of the resilient modulus for the coarse-grained materials
Models
Material classification
Resilient modulus (MPa)
\({M}_{\mathrm{r}}={k}_{1}{\theta }^{{k}_{2}}\)
Well-graded gravel
QS3-GW
156
Sand
QS2/QS3-SW/SM-ML
109
\({M}_{\mathrm{r}}={k}_{1}{p}_{0}{\left(\frac{\theta }{{p}_{0}}\right)}^{{k}_{2}}{\left(\frac{q}{{p}_{0}}\right)}^{{k}_{3}}\)
Well-graded gravel
QS3-GW
143
Sand
QS2/QS3-SW/SM-ML
131
\({M}_{\mathrm{r}}={k}_{1}{p}_{0}{\left(\frac{\theta }{{p}_{0}}\right)}^{{k}_{2}}{\left(\frac{{\tau }_{\mathrm{oct}}}{{p}_{0}}+1\right)}^{{k}_{3}}\)
Well-graded gravel
QS3-GW
159
\({E}_{\mathrm{v}}=C{{\sigma }_{\text{v}}^{^{\prime}}}^{m}\)
Well-graded gravel
QS3-GW
118
After the analysis of the coarse-grained soils, the fine-grained soils were evaluated. In the case of the fine-grained soils, four models (Table 6) and two materials (Table 7) were compared. The materials were tested through the triaxial test and both materials were compacted at optimum water content and 2% above the optimum. The measured resilient modulus is based on the work developed by [9].
Table 6
Models selected for the fine-grained soils
Refs.
Models
[59]
\({M}_{\mathrm{r}}={k}_{1}{p}_{0}{\left(\frac{\theta }{{p}_{0}}\right)}^{{k}_{2}}{\left(\frac{{\tau }_{\mathrm{oct}}}{{p}_{0}}+1\right)}^{{k}_{3}}\)
[35]
\({M}_{\mathrm{r}}={k}_{1}{({\sigma }_{\mathrm{d}}+\chi {\psi }_{\mathrm{m}})}^{{k}_{2}}\)
[9]
\({M}_{\mathrm{r}}={k}_{1}{p}_{\mathrm{a}}{\left(\frac{\theta +\chi {\psi }_{\mathrm{m}}}{{p}_{\mathrm{a}}}\right)}^{{k}_{2}}{\left(\frac{{\tau }_{\mathrm{oct}}}{{p}_{\mathrm{a}}}+1\right)}^{{k}_{3}}\)
[81]
\({E}_{\mathrm{r}}=A+B{p{^{\prime}}}_{0}+C{q}_{\mathrm{r}}\)
Table 7
Properties of the fine-grained materials
No.
UIC and ASTM classification
Materials
Properties
Observations
1
QS1-CL
Low plasticity
Lean clay
WL = 30.8%
WP = 18.4%;
IP = 12.3%;
The effect of the matric suction will be included in the effective stress
2
QS1-CL-ML
Low plasticity
Silt-lean clay
WL = 27.8%;
WP = 19.8%;
IP = 8%;
The triaxial tests following the AASHTO T307-99 procedure were conducted on both materials
Considering the results obtained by [9] regarding the measured resilient modulus, the constants of the parameters were determined through regression analyses (Table 8). However, it is important to refer that the MEPDG model is defined in terms of total stresses, which means that the regression analysis is performed for each material on both moisture contents (opt and opt + 2%). The remaining models are dependent on effective stresses, which means that the regression analysis is performed considering the whole set of moisture content (opt and opt + 2%). This type of information allows analysing the sensibility of the resilient modulus regarding the moisture content. The models developed by [9, 35] and [81] are dependent on the ψm and χm parameters, which means that the results are presented considering \({M}_{\mathrm{r}}^{\mathrm{opt}}\) and \({M}_{\mathrm{r}}^{\mathrm{opt}+2\mathrm{\%}}\), although the regression is performed with the whole set (opt and opt + 2%).
Table 8
Regression results for the fine-grained materials
Models
Material classification
Parameters
Regression analysis R2
\({M}_{\mathrm{r}}={k}_{1}{p}_{0}{\left(\frac{\theta }{{p}_{0}}\right)}^{{k}_{2}}{\left(\frac{{\uptau }_{\mathrm{oct}}}{{p}_{0}}+1\right)}^{{k}_{3}}\)
QS1-CL
Low plasticity
opt
k1 = 0.55
k2 = 0.15
k3 = -1.82
0.98
QS1-CL
Low plasticity
opt + 2%
k1 = 0.35
k2 = 0.14
k3 = -1.61
0.90
QS1-CL-ML
Low plasticity
Opt
k1 = 0.97
k2 = -0.17
k3 = -1.97
0.96
QS1-CL-ML
Low plasticity
opt + 2%
k1 = 0.68
k2 = -0.10
k3 = -1.13
0.87
\({M}_{\mathrm{r}}={k}_{1}{({\sigma }_{\mathrm{d}}+\chi {\psi }_{\mathrm{m}})}^{{k}_{2}}\)
QS1-CL
Low plasticity
k1 = 5.45
k2 = 0.36
Difficult adjustment
QS1-CL-ML
Low plasticity
k1 = 72.11
k2 = -0.04
Difficult adjustment
\({M}_{\mathrm{r}}={k}_{1}{p}_{\mathrm{a}}{\left(\frac{\theta +\chi {\psi }_{\mathrm{m}}}{{p}_{\mathrm{a}}}\right)}^{{k}_{2}}{\left(\frac{{\tau }_{\mathrm{oct}}}{{p}_{\mathrm{a}}}+1\right)}^{{k}_{3}}\)
QS1-CL
Low plasticity
k1 = 0.24
k2 = 0.85
k3 = -2.29
0.62
QS1-CL-ML
Low plasticity
k1 = 0.78
k2 = 0.13
k3 = -2.01
0.61
\({E}_{r}=A+B{p{^{\prime}}}_{0}-C{q}_{\mathrm{r}}\)
QS1-CL
Low plasticity
A = 3499.51
B = 261.25
C = -190.78
0.91
QS1-CL-ML
Low plasticity
A = 60,443.76
B = -167.96
C = -474.39
0.69
The model developed by [81] is simpler than the remaining models, and the influence of the moisture content is indirectly included in the suction matrix in the initial mean normal stress (\({p}_{0}^{\mathrm{^{\prime}}}\)). From multiple linear regression analyses, the empirical constants were determined, and the values are presented in Table 8.
From the constant parameters and the selected stress path, the resilient moduli of the materials were found, as depicted in Table 9.
Table 9
Resilient modulus of the fine-grained materials
Models
Material classification
Resilient modulus (MPa)
\({M}_{\mathrm{r}}={k}_{1}{p}_{0}{\left(\frac{\theta }{{p}_{0}}\right)}^{{k}_{2}}{\left(\frac{{\tau }_{\mathrm{oct}}}{{p}_{0}}+1\right)}^{{k}_{3}}\)
QS1-CL
Low plasticity
50 (opt)
33 (opt + 2%)
QS1-CL-ML
Low plasticity
70 (opt)
56 (opt + 2%)
\({M}_{\mathrm{r}}={k}_{1}{({\sigma }_{\mathrm{d}}+\chi {\psi }_{\mathrm{m}})}^{{k}_{2}}\)
QS1-CL
Low plasticity
Difficult adjustment
QS1-CL-ML
Low plasticity
Difficult adjustment
\({M}_{\mathrm{r}}={k}_{1}{p}_{\mathrm{a}}{\left(\frac{\theta +\chi {\psi }_{\mathrm{m}}}{{p}_{\mathrm{a}}}\right)}^{{k}_{2}}{\left(\frac{{\tau }_{\mathrm{oct}}}{{p}_{\mathrm{a}}}+1\right)}^{{k}_{3}}\)
QS1-CL
Low plasticity
53 (opt)
47 (opt + 2%)
QS1-CL-ML
Low plasticity
73 (opt)
72 (opt + 2%)
\({E}_{r}=A+B{p{^{\prime}}}_{0}-C{q}_{\mathrm{r}}\)
QS1-CL
Low plasticity
83 (opt)
76 (opt + 2%)
QS1-CL-ML
Low plasticity
80 (opt)
71 (opt + 2%)
Considering the obtained results for the fine- and coarse-grained materials, the MEPDG model [59] will be adopted in the following analysis, despite its complexity when compared to the remaining models. The main reason is related to its universal character since the model can be applied in the prediction of the resilient modulus of all types of geomaterials.
This analysis shows a possible way to rank materials according to the resilient modulus, as depicted in Fig. 2. In the case of the material classified as QS2/QS3-SW/SM-ML, instead of the MEPDG model, the Uzan’s model was used due to the lack of information. (It was not possible to perform a regression analysis.) However, as the previous analysis shows, the MEPDG and Uzan’s models present similar results. All the materials present in Fig. 2 are at optimum moisture content, except the material classified as QS2/QS3 SW/SM-ML. In this case, the moisture content is close to the optimum conditions.

5.2 Permanent deformation: parametric study

The comparison regarding the permanent deformation models was already presented in the work developed by [19]. As in the previous exercise, in this parametric study, the authors attempted to compare different permanent deformation models available for different types of soils considering the empirical permanent deformation model developed by [76]. Thus, the results showed that two soils can be integrated into the same classification (UIC or ASTM), but the laboratory conditions may differ greatly and therefore lead to different results in terms of permanent deformation. The calibration was performed to find the best-fit for the experimental data through the parameters επ;o, α and B (which correspond to the model constants associated with the material properties). The models were calibrated considering the same stress paths (described by [76]). As in the resilient modulus analysis, the materials selected are representative of different types of materials.
In order to understand better the ranking, the materials selected for the preliminary analysis of the permanent deformation are depicted in Table 10. This table also includes information about the ASTM and UIC classifications. To complete the information about the materials, the physical properties, state conditions and strength properties of the materials are depicted in Tables 11, 12 and 13, which would be extremely important to understand the novel geomechanical classification purposed.
Table 10
Materials selected for the preliminary analysis of the permanent deformation [19]
Refs
Soils
ASTM classification
UIC classification
Observations
[76]
Sand
SP
QS2
Compacted at field conditions and saturated
Silt
CL-ML
QS1
Compacted at field conditions and saturated
[82] and [66]
Silty sand (42.2% fines)
SM
QS1
Compacted at optimum compaction—standard proctor
Silty sand (27.4% fines)
SM
QS1
Compacted at optimum compaction —standard proctor
[83]
Well-graded Sand
SP
QS2
Compacted at optimum compaction—standard proctor
Poor-graded Sand
SW
QS3
Compacted at optimum compaction—standard Proctor
Table 11
Physical properties of the materials—granulometry [19]
Material Properties
C. Bruynweg [83]
Crusher [83]
Silty clay
(42.2% fines content) [82]
Silty clay
(27.4% fines content)
[66]
Silt [76]
Coarse sand [76]
Cu
2.100
10.50
28.0
33.0
2.51
4.68
Cc
1.050
1.250
0.54
0.75
1.32
0.62
D10
0.148
0.217
–
–
–
–
D30
0.219
0.784
–
–
–
–
D50
0.280
1.722
–
–
–
–
D60
0.310
2.280
–
–
–
–
CBR (%)
22.00
15.70
–
–
–
–
Table 12
State conditions of the materials [19]
Material
properties
C. Bruynweg [83]
Crusher [83]
Silty clay (42.2% fines content) [82]
Silty clay (27.4% fines content) [66]
Silt [76]
Coarse sand [76]
γdry;max (kg/m3)
1723
1755
1998
2070
1620
2110
Wopt (%)
12.5
10.5
10.1
7.6
–
–
γwet (kg/m3)
1942
1937
–
–
–
–
Liquid limit (%)
Non-plastic
Non-plastic
Non-plastic
Non-plastic
35
Non-plastic
Plastic limit
Non-plastic
Non-plastic
Non-plastic
Non-plastic
24
Non-plastic
Table 13
Strength properties of the materials [19]
Material
properties
C. Bruynweg [83]
Crusher
[83]
Silty clay (42.2% fines content) [82]
Silty clay (27.4% fines content) [66]
Silt [76]
Coarse sand [76]
ϕ (o)
48.2
50.20
36.18
45.66
11.7*
0.00*
c (kPa)
5.60
8.680
15.82
10.43
16.4*
33.0*
s (kPa)**
9.90
14.90
31.80
19.10
24.8
0.00
m**
1.98
2.070
1.470
1.880
0.62
1.33
* In this case, the author described the saturated samples (silt and coarse sand) in ϕ’ and c’
** s and m are the parameters used to define the failure envelope: q = s + mp
In the work developed by [19], the ranking process is described. The final ranking is depicted in Fig. 3.

6 Geomechanical classification

Considering the results presented in the previous sections, two models were selected to characterize the resilient modulus and permanent deformation of the geomaterials. Indeed, this is an attempt to relate the resilient modulus with the permanent deformation under a certain stress level. As in the previous analysis, the selected stress path is characterized by a cyclic deviator stress equal to 24 kPa and a confining constant stress equal to 60 kPa. The permanent deformation was determined considering a number of load cycles equal to 30,000. The selected materials are presented in Table 14. Some of the materials are the same used in Sects. 5.1 and 5.2, and others were added to increase the robustness of the analysis, the ranking and the impact of the final result of the geomechanical classification. These “auxiliary” data only have information about the resilient modulus or permanent deformation (they were used to define the “limits” of the classification). The work developed by [84] was also included but due to the significant information about both types of deformation (resilient and permanent) under different water content conditions. It is important to refer that, for these particular materials, are presented, whenever it is possible, the values of both deformations: resilient and permanent deformation. This table also includes information about compaction conditions.
Table 14
Properties of the materials
Information
UIC and ASTM classification
Materials
Properties
Observations
[50]
QS1-CL
–
WL = 28.5%; WP = 19.2%;
IP = 9.3%; Gs = 2.73;
Clay content 20.0%;
Passing #200 73.0%;
Maximum dry density
17.36 kN/m3
–
From the resilient modulus parametric study
QS1
Low plasticity
CL
Lean clay
WL = 30.8%;
WP = 18.4%;
IP = 12.3%
The effect of the matric suction will be included in the effective stress; the triaxial tests following the AASHTO T307-99 procedure were conducted on both materials
[84]
QS1
Lean clay (CL)
Silty clay
WL = 28.19%; Gs = 2.63;
IP = 12.55% (low plasticity);
Passing #200 80%;
Maximum dry density 17.10 kN/m3;
Wopt = 17.11%;
c = 60 kPa; ϕ = 18o
Compacted at optimum compaction—standard proctor
From the permanent deformation parametric study
QS1
CL-ML
Silt
WL = 35%; WP = 24%;
IP = 11% (low to medium plasticity); Gs = 2.67;
Cu = 2.51;Cc = 1.32;
Maximum dry density 15.89 kN/m3;
Hydraulic permeability 5.3 × 10–7 m/s;
c’ = 11.7 kPa; ϕ’ = 16.4°
Compacted at field conditions and saturated
From the resilient modulus parametric study
QS1
Low plasticity
CL-ML
Silt-lean clay
WL = 27.8%;
WP = 19.8%;
IP = 8%
The effect of the matric suction will be included in the effective stress;
The triaxial tests following the AASHTO T307-99 procedure were conducted on both materials
From the permanent deformation parametric study
QS1
SM
Silty sand (42.2% fines)
Cu \(\approx\) 28; Cc \(\approx\) 0.54;
Gs = 2.68; Wopt = 10.1%;
Fines content = 42.2%;
Maximum dry density
19.6 kN/m3
Compacted at optimum compaction—standard proctor
From the permanent deformation parametric study
QS1
SM
Silty sand
(27.4% fines)
Cu \(\approx\) 33; Cc \(\approx\) 0.75;
Gs = 2.67; Wopt = 7.6%
Fines content = 27.4%;
Maximum dry density 20.3 kN/m3
Compacted at optimum compaction—standard proctor
[84]
QS1
Silt (SM)
Silty clayed sand
WL = 16.70%; Gs = 2.70;
IP = 7.50% (low plasticity);
Passing #200 38%;
Maximum dry unit weight 16.9 kN/m3;
Wopt = 19.3%; c = 103 kPa;
ϕ = 35o
Compacted at optimum compaction—standard proctor
From the permanent deformation parametric study
QS2
SP
Well-graded sand
Cu = 2.10; Cc = 1.05;
Maximum dry density 16.90 kN/m3;
Wopt = 12.5%;
c = 5.60 kPa; ϕ = 48.2°
Compacted at optimum compaction—standard proctor
[58]
QS2
SP
Poorly graded Sand
Gs = 2.74; Cu≈2.0; Cc≈1.04
–
[83]
QS2
SP
Poorly graded Sand
Cu≈1.88; Cc≈1.04;
Maximum dry density
16.56 kN/m3;
Wopt = 14.0%;
c = 6.34 kPa; ϕ = 41.8°
Standard proctor
From the permanent deformation parametric study
QS2
SP
Sand
Gs = 2.66; Cu = 4.8;
Cc = 0.62;
Maximum dry density
20.69 kN/m3;
Minimum dry density
15.89 kN/m3;
Hydraulic permeability 3.2 × 10−5 m/s;
c’ = 0 kPa; ϕ’ = 33°
Compacted at field conditions and saturated
[84]
QS2
SP
Poorly graded sand
WL = 26.40%; Gs = 2.71%;
Passing #200 0.70%;
Cu = 1.79; Cc = 0.89;
Maximum dry unit weight 15.70 kN/m3;
Wopt = 13.70%;
c = 20 kPa; ϕ = 42°
Compacted at optimum compaction—standard proctor
[85]
QS2-QS3
–
LA = 22;
MED = 15;
Fines content = 10%
Modified proctor
From the resilient modulus parametric study
QS2/QS3
SW/SM-ML
Sand
Cu = 20; Cc = 0.8–1.1;
(PImin = NP; PImax = 6)
WL;máx = 25;
W = 7.7%;
WOMC = 7.7%
Granular material used as base and sub-base in the flexible pavement sections;
The target moisture content and densities were selected based on AASHTO T99 test results and field-measured values [80]
From the permanent deformation parametric study
QS3
SW
Poor-graded sand
Cu = 10.5; Cc = 1.25;
Maximum dry density
17.21 kN/m3;
Wopt = 10.5%;
c = 8.68 kPa; ϕ = 50.2°
Compacted at optimum compaction—standard proctor
From the resilient modulus parametric study
QS3
GW
Well-graded gravel
Cu = 27; Cc = 2;
(PImin = NP; PImax = 6)
WL;máx = 25;
W = 6.3%;
WOMC = 6.8%
Granular material used as base and sub-base in the flexible pavement sections;
The target moisture content and densities were selected based on AASHTO T99 test results and field-measured values [80]
[58]
QS3
GW
Well-graded gravel
Gs = 2.69;Cu≈61; Cc≈2.11
–
The resilient modulus and permanent deformation of the selected materials (Table 14) are presented in Table 15. The materials with both values of Mr and εp allow validating the mechanical classification.
Table 15
Resilient modulus and permanent deformation of the selected materials
Information
Materials classification
Resilient modulus (Mpa)
Permanent deformation εp (%)
[50]
QS1-CL
37
–
From the resilient modulus parametric study
QS1-CL
Low plasticity
50
–
[84]
QS1-CL
–
0.01864
From the permanent deformation parametric study
QS1-CL/ML
Silt
–
0.03126
From the resilient modulus parametric study
QS1-CL-ML
Low plasticity
70
–
From the permanent deformation parametric study
Silty sand (42.2% fines)
QS1-SM
258
0.01385
From the permanent deformation parametric study
Silty sand (27.4% fines)
QS1-SM
294
0.01251
[84]
QS1-SM
511
0.00658
From the permanent deformation parametric study
Well-graded sand
QS2-SP
509
0.00003
[58]
QS2-SP
673
–
[83]
QS2-SP
–
9.85 × 10–10
From the permanent deformation parametric study
Sand
QS2-SP
–
0.01170
[84]
QS2-SP
172
0.01873
[85]
QS2-QS3
–
0.0002
From the resilient modulus parametric study
Sand
QS2/QS-SW/SM-ML
131
–
From the permanent deformation parametric study
Poor-graded sand
QS3-SW
178
0.00084
From the resilient modulus parametric study
Well-graded gravel
QS3-GW
159
0.00320
[58]
QS3-GW
862
–
From the results presented in Table 15, it is possible to define guiding/limit values associated with the UIC classification: QS1, QS2 and QS3 [31]. This work only presents a preliminary geomechanical classification. The lack of information about the models’ parameters found in the bibliography (namely the models that include the suction and other complex model coefficients) and the necessity and difficulty to performed back analysis to find important parameters (as the cohesion and friction angle used to define the failure criterion) reduced the number of models and materials that could be used in this geomechanical classification. Besides, some of the models found in the bibliography tested materials classified as QS0, which implies its elimination in the geomechanical classification, since these materials require mechanical improvement.
The geomechanical classification is an attempt to define a novel helpful guide to be used as a support for the modelling and design of the substructure. Indeed, this novel geomechanical classification tries to rank soils in a very similar way developed by [20] that was inspired by [40]. This work allows understating which are the acceptable values of the permanent deformation and resilient modulus according to the type of material and its classification. In this particular case, [31] classification was used as depicted in Fig. 4.
Analysing Fig. 4, a significant range of values was obtained for each classification. This was expected due to the mineralogical nature and physical properties of each material, which can have an influence on several parameters used in the determination of the resilient modulus and permanent deformation. Indeed, in the case of the materials classified as QS1 (fine soils), the type of the material (CL, CH, ML, or MH) associated with its plasticity’s characteristics and consistency index (as well as the percentage of fines) can have a significant impact in the permanent deformation and resilient behaviour, as depicted in Fig. 5. For the materials classified as QS2 and QS3, the type of materials (sand or gravel) as well as its granulometry (well- or poor-graded) and the percentage of fines (in the well-graded materials) can also influence the response of the material in terms of resilient and permanent deformations.
With this work, it was possible to define subsets associated with the properties of the soils, despite the need for more information. For example, as Fig. 5 shows, a silty soil (classified as QS1-ML) presents a superior resilient modulus when compared to clayed soils (QS1-CL). Regarding the granular materials, the subsets can be defined by the granulometry of the materials (well- or poor-graded). However, it is important to refer that the percentage of fines can also influence the obtained results. Indeed, despite the importance of the granulometry and plasticity properties, the fines content and the moisture content have also significant importance, mostly in the fine soils. As mentioned previously, the work developed by [84] shows the influence of the moisture content in the soils. Thus, [84] selected and studied the behaviour of the clay considering three compaction moisture content and dry unit weight conditions representing dry of optimum moisture content level, optimum moisture content level (corresponds to 95% of maximum dry unit weight), and wet of optimum moisture content level. From the results obtained by [84], an estimation was performed (in terms of resilient modulus and permanent deformation), and the results are presented in Figs. 6 and 7. These results are based on the properties of the clay described in Table 16. Analysing Fig. 7, the moisture content can influence both deformations, which means that can affect severely the track performance. In a more detailed analysis, it is possible to identify a reduction of around 200% in the resilient modulus from dry to wet conditions. This difference is significant, which means that the moisture content is an important parameter since influences the performance of the subgrade and corroborates the findings of [20, 86] in the case of non-standard unbound granular materials for pavements.
Table 16
Properties of the clay [84]
Clay
Wopt (%)
ϕ (°)
c (kPa)
Liquid limit (%)
Plasticity index (%)
Maximum dry unit weight (kN/m3)
Optimum
17.11
18
60
28.19
12.55
17.10
Dry of optimum
11.10
8
152
–
–
–
Wet of optimum
21.50
16
42
–
–
–

7 Conclusions

The capability to determine the resilient modulus and also the permanent deformation of geomaterials is imperative in the modelling, design and evaluation of the performance of railway structures. This work attempts to identify and summarize the main models to estimate the resilient modulus and also permanent deformation. At the beginning of the analysis, the most simplistic models are presented. In the case of the resilient modulus, the models should include the influence of the deviatoric and confining stresses and the constant parameters should reflect the influence of other factors as the moisture content and physical state of the material. The mechanistic-empirical models summarized in Annex were divided according to the type of material (clay, silt, sand and gravel) to better understand the model’s formulation and the conditions of the material when tested in terms of its granulometry (Cu and Cc) and plasticity properties (WL, WP, and IP). Tables also include some observations regarding the laboratory tests and classification of the material (UIC and ASTM).
From the available mechanistic-empirical approach, it was important to evaluate the robustness of the models and their sensitivity. Thus, some models and materials were selected to perform a parametric study in terms of resilient modulus. A similar analysis was already performed for the permanent deformation in the work developed by [19]. The selection depended on the data available for each material and the variability of the geomaterials (type of soils) in terms of granulometry, percentage of fines, moisture content and plasticity properties. The analysis shows better results in the case of the coarse-grained soils. (The R-squared value is higher and the obtained values for resilient modulus are similar.) This fact can be explained by the complexity of the nature of fine-grained soils, namely the variation of its properties in terms of plasticity and fines content. At the end of this analysis, one model was selected to rating the geomechanical performance. The MEPDG model shows better results for both materials. Indeed, this choice is also justified by its universal character, which means that can be applied to all types of materials. As expected, the comparison of the results of the resilient modulus shows that the values increase with the UIC classification: higher resilient modulus is related to well-graded materials (QS2 and QS3).
The geomechanical classification is based on the parametric studies of the resilient modulus (applying the MEPDG model) and the permanent deformation (applying Chen’s model). Moreover, other materials were added to this analysis because of the available information about the moisture content. The classification presents a wide range of results in terms of permanent deformation and resilient modulus. This range is a reflection of the variation of the properties of the materials with the same classification (QS1, QS2 and QS3) that influence both types of deformations. Indeed, this work is the first attempt to obtain a novel geomechanical classification that can be a helpful tool in the modelling of the foundation of the railway structures since gather information about the UIC classification (which is a reference in the railway works) and permanent and resilient deformations. Nevertheless, several limitations are recognized in this proposal, which were discussed throughout this paper. These limitations should be overcome in future work updating this proposal with a wide range of geomaterials properly characterized (physical and mechanical characteristics) and tested under different state conditions (deviation to the optimum compaction conditions), which should include variations of water content /suction/degree of saturation and density.

Acknowledgements

This work was partially carried out under the framework of In2Track, a research project of Shift2Rail. This work was partly financed by FCT/MCTES through national funds (PIDDAC) under the R&D Unit Institute for Sustainability and Innovation in Structural Engineering (ISISE), under reference UIDB/04029/2020. It has been also financially supported by national funds through FCT—Foundation for Science and Technology, under grant agreement [PD/BD/ 127814/2016] attributed to Ana Ramos.
Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://​creativecommons.​org/​licenses/​by/​4.​0/​.
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Appendix

An extensive list of resilient models based on laboratory tests is presented in this appendix. Although these tables were constructed uniformly (especially regarding the nomenclature of the variables and the parameters and characteristics of the materials), the authors maintained the original symbols described in the original paper. For example, the symbol Er is used to describe the resilient modulus, which is usually represented by the symbol Mr.
Some models were developed using certain SI units, and where possible, these units are described in the column “variables and empirical constants”.
Regarding the resilient modulus tables, [87] defined wc as optimum moisture content (which is equal to Wopt) and wcr as the relation between the actual moisture content and the optimum moisture content. In this analysis, the fines are measured through the passing sieve #200 or through the percentage of fines. Furthermore, the maximum dry unit weight is also defined as maximum dry density. The constants such as A, B, C or ki are defined as fitting, regression, material and model parameters according to the original formulation defined by the author. The symbols pa and p0 are defined as atmospheric stress/pressure and reference stress and it is different from pu which means unit pressure.
Regarding the stress variables, the symbol θ is defined as bulk stress or the sum of principal stresses, and the deviator stress can be represented by the symbols q, qr and σd. The symbol q is different from qm which represents the mean value of the deviator stress (Tables 17–21).
Table 17
Models that describe resilient modulus of fine-grained soils—clays
UIC
ASTM classification
Ref.
Equation model
Variables and empirical constants
Parameters and characteristics
Observations
QS0*
CH
(no information about the particle size curve)
High plasticity
[88]
\({M}_{\mathrm{r}}={k}_{1}{p}_{\mathrm{a}}{\left(\frac{{\theta }_{\mathrm{net}}}{{p}_{\mathrm{a}}}\right)}^{{k}_{2}}{\left(\frac{{\tau }_{\mathrm{oct}}}{{p}_{\mathrm{a}}}+1\right)}^{{k}_{3}}{\left(\frac{{\psi }_{\mathrm{m}}}{{\theta }_{\mathrm{net}}}+1\right)}^{{k}_{4}}\)
θnet is the net bulk stress (θnet = θ-3ua);
τoct is the octahedral shear stress;
ψm is the matric suction;
pa is the atmospheric stress;
ki are the material parameters
WL = 61%; WP = 25%;
IP = 36%; Gs = 2.69
The model is similar to Cary and Zapata [67] model.
The last term related to the suction parameter is normalized with the net bulk stress instead of the atmospheric pressure.
The model does not consider the effect of the pore-water pressure under saturated conditions
QS0
CL
High plasticity
[36]
\({M}_{\mathrm{r}}={k}_{1}{p}_{\mathrm{a}}{\left(\frac{\theta -3{k}_{4}}{{p}_{\mathrm{a}}}\right)}^{{k}_{2}}{\left(\frac{{\tau }_{\mathrm{oct}}}{{p}_{\mathrm{a}}}+{k}_{5}\right)}^{{k}_{3}}\)
\({\mathrm{MR}}_{\mathrm{opt}}=\frac{{M}_{\mathrm{rs}}}{{M}_{\mathrm{rs};\mathrm{opt}}} =c+d\mathrm{log}{\psi }_{\mathrm{m}}\)
MRopt is the modulus ratio;
Mrs is the modulus at certain suction (or moisture);
Mrs;opt is the modulus at optimum moisture content;
c and d are model parameters;
pa is the atmospheric pressure (101 kPa);
ki are fitting parameters
WL = 85%; IP = 52%; Gs = 2.75;
Sand content = 3.1%;
Silt content = 21.2%;
Clay content = 75.2%;
Fines content = 96.4%;
Wopt = 27.5%;
Maximum dry unit weight = 14.4 kN/m3
The model is developed based on four tested fine-grained soils.
This model demands significant regression parameters and several combinations of parametric values to fit laboratory tests.
The test specimens were initially saturated and then subjected to desorption until the matric suction was 154 or 350 kPa.
All the test soils were prepared at OPT water content and at a dry density corresponding to 98% or 103% relative compaction for a standard proctor
QS0-QS1
CL
Medium plasticity
[50]
\({E}_{\mathrm{r}}=\frac{{a}^{^{\prime}}+b{^{\prime}}{\sigma }_{\mathrm{dv}}}{{\sigma }_{\mathrm{dv}}}\, \mathrm{ for} \,{\sigma }_{\mathrm{dv}}>{\mathrm{0}}\)
\({a}^{^{\prime}}\) and \({b}^{^{\prime}}\) are material parameters:
\({a}^{^{\prime}}=318.2+0.337\left({q}_{\mathrm{u}}\right)+\)
\(0.73\left(\% \mathrm{clay}\right)+2.26\left(\mathrm{PI}\right)-\)
\(0.915\left(\gamma \right)-2.19\left(S\right)-\)
\(0.304(\% \mathrm{finer} \#200)\)
\({b}^{^{\prime}}=2.10+0.00039\left(\frac{1}{a}\right)+\)
\(0.104\left({q}_{\mathrm{u}}\right)+0.09(\mathrm{LL})\)-
0.10(% finer #200)
WL = 42.1%; WP = 22.0%; IP = 20.1%; Gs = 2.76;
Clay content = 36%;
Passing sieve #200 = 93%
Maximum dry density = 15.20 kN/m3
The hyperbola predicts excessive values of Er when σd is close to zero
QS1
CL
Low plasticity
WL = 30.5%; WP = 22.1%; IP = 8.4%;
Gs = 2.66; Clay content = 17.0%;
Passing sieve #200 = 95.0%;
Maximum dry density = 16.28 kN/m3
QS1
CL
Medium plasticity
WL = 38.8%; WP = 23.3%; IP = 15.5%; Gs = 2.65;
Clay content = 18.0%;
Passing sieve #200 = 100.0%;
Maximum dry density = 15.98 kN/m3
QS1
CL
Low plasticity
WL = 29.5%; WP = 20.1%; IP = 9.4%;
Gs = 2.67; Clay content = 16.0%;
Passing sieve #200 = 78.0%;
Maximum dry density = 17.16 kN/m3
QS1
CL
Low plasticity
WL = 28.5%; WP = 19.2%; IP = 9.3%;
Gs = 2.73; Clay content = 20.0%;
Passing sieve #200 = 73.0%;
Maximum dry density = 17.36 kN/m3
QS1
CL
Low plasticity
[36]
\({M}_{\mathrm{r}}={k}_{1}{p}_{\mathrm{a}}{\left(\frac{\theta -3{k}_{4}}{{p}_{\mathrm{a}}}\right)}^{{k}_{2}}{\left(\frac{{\tau }_{\mathrm{oct}}}{{p}_{\mathrm{a}}}+{k}_{5}\right)}^{{k}_{3}}\)
\({\mathrm{MR}}_{\mathrm{opt}}=\frac{{M}_{\mathrm{rs}}}{{M}_{\mathrm{rs};\mathrm{opt}}}=c+d\mathrm{log}{\psi }_{\mathrm{m}}\)
MRopt is the modulus ratio;
Mrs is the modulus at certain suction (or moisture);
Mrs;opt is the modulus at optimum moisture content;
c and d are model parameters;
pa is the atmospheric pressure (101 kPa);
ki are fitting parameters
WL = 26.0%; IP = 9.0%; Gs = 2.66;
Sand content = 36.3%;
Silt content = 45.3%;
Clay content = 14.5%;
Per cent fines = 59.7%;
Wopt = 16.0%;
Maximum dry unit weight = 17.70 kN/m3
Model developed based on four tested fine-grained soils.
This model demands significant regression parameters and several combinations of parametric values to fit laboratory tests.
The test specimens were initially saturated and then subjected to desorption until the matric suction was 154 or 350 kPa.
All the test soils were prepared at OPT water content and at a dry density corresponding to 98% or 103% relative compaction for a standard proctor
QS1
CL
Low plasticity
[9]
\({M}_{\mathrm{r}}={k}_{1}{p}_{\mathrm{a}}{\left(\frac{\theta +\chi {\psi }_{\mathrm{m}}}{{p}_{\mathrm{a}}}\right)}^{{k}_{2}}{\left(\frac{{\tau }_{\mathrm{oct}}}{{p}_{\mathrm{a}}}+1\right)}^{{k}_{3}}\)
θ is the bulk stress;
τoct is the octahedral shear stress;
χ is the Bishop's effective stress parameter;
ψm is the matric suction;
ki are the material parameters
WL = 30.8%; WP = 18.4%; IP = 12.3%;
Passing sieve #200 = 68.8%;
Wopt = 16.5%;
Maximum dry density = 17.70 kN/m3
The model takes into account, explicitly, the effect of the confining and octahedral shear stress.
It is used for a large range of unbound materials.
The effect of the matric suction is included in the effective stress.
The model is similar to the model developed by Uzan [56]
 ≤ QS1
(based on AASHTO classification)
–
[87]
\({M}_{\mathrm{r}}={k}_{1}{P}_{\mathrm{a}}{\left(1+\frac{{\sigma }_{\mathrm{d}}}{1+{\sigma }_{\mathrm{c}}}\right)}^{{k}_{2}}\)
Mr is the resilient modulus (MPa);
σd is the deviator stress (kPa);
σc is the confining pressure (kPa);
Pa is the atmospheric pressure (kPa);
k1 and k2 are model parameters
γd = 16.92 kN/m3; γdr = 0.97;
wcr = 1.48; wc = 20.7%;
Optimum wc = 14.0%;
Maximum dry density = 
17.4 kN/m3;
Passing sieve #200 = 89%
k1 is dependent on γdr, LL and wc.
k2 is dependent on γdr, LL, wcr and #200, where γdr and wcr are the density and moisture ratio, respectively
 
γd = 19.87 kN/m3; γdr = 1.10;
wcr = 0.82; wc = 11.5%;
Optimum wc = 14.0%;
Maximum dry density = 18.4 kN/m3;
Passing sieve #200 = 44%;
QS1
CL-ML
Low plasticity
[9]
\({M}_{\mathrm{r}}={k}_{1}{p}_{\mathrm{a}}{\left(\frac{\theta +\chi {\psi }_{\mathrm{m}}}{{p}_{\mathrm{a}}}\right)}^{{k}_{2}}{\left(\frac{{\uptau }_{\mathrm{oct}}}{{p}_{\mathrm{a}}}+1\right)}^{{k}_{3}}\)
θ is the bulk stress;
τoct is the octahedral shear stress;
χ is the Bishop's effective stress parameter;
ψm is the matric suction;
ki are the material parameters
WL = 27.8%; WP = 19.8%; IP = 8%;
Passing sieve #200 = 56.3%;
Wopt = 14.2%;
Maximum dry density = 17.75 kN/m3
The model takes into account, explicitly, the effect of the confining and octahedral shear stress.
It is used for a large range of unbound materials;
The effect of the matric suction is included in the effective stress.
The model is similar to the model developed by Uzan [56]
QS1**
CL
(no information about the particle size curve)
Low to medium plasticity
[88]
\({M}_{\mathrm{r}}={k}_{1}{p}_{\mathrm{a}}{\left(\frac{{\theta }_{\mathrm{net}}}{{p}_{a}}\right)}^{{k}_{2}}{\left(\frac{{\tau }_{\mathrm{oct}}}{{p}_{\mathrm{a}}}+1\right)}^{{k}_{3}}{\left(\frac{{\psi }_{\mathrm{m}}}{{\theta }_{\mathrm{net}}}+1\right)}^{{k}_{4}}\)
θnet is the net bulk stress (θnet = θ-3ua);
τoct is the octahedral shear stress;
ψm is the matric suction;
pa is the atmospheric stress;
ki are the material parameters
WL = 37%; WP = 18%;
IP = 19%; Gs = 2.69
The model is similar to Cary and Zapata [67] model.
The last term related to the suction parameter is normalized with the net bulk stress instead of the atmospheric pressure.
The model does not consider the effect of the pore-water pressure under saturated conditions
*Classification based on the plasticity index and AASHTO classification (A-7–5: > 35% passing the 0.075 mm sieve)
** Classification based on the plasticity index and AASHTO classification (A-7–6: > 35% passing the 0.075 mm sieve)
Table 18
Models that describe resilient modulus of fine-grained soils—silts
UIC
ASTM classification
Ref.
Equation model
Variables and Empirical Constants
Parameters and characteristics
Observations
QS0
MH
(no information about the particle size curve)
High plasticity
[35]
\({M}_{\mathrm{r}}={k}_{1}{p}_{\mathrm{a}}{({\sigma }_{\mathrm{d}}+\chi {\psi }_{\mathrm{m}})}^{{k}_{2}}\)
χ is the Bishop's effective stress parameters (it is function of the degree of saturation of the soil);
ψm is the matric suction;
ki are the materials’ parameters;
WL = 54.0%; IP = 20%;
Gs = 2.71; Wopt = 18%;
Maximum dry density = 17.26 kN/m3
This model does not take into account the effect of confining stress.
The model can be applied to a large range of moisture content.
The model is dependent on only two material parameters
QS0
MH
High plasticity
[50]
\({E}_{\mathrm{r}}=\frac{{a}^{^{\prime}}+b{^{\prime}}{\sigma }_{\mathrm{d}}}{{\sigma }_{\mathrm{d}}}\,\mathrm{for}\,{\sigma }_{\mathrm{d}}>0\)
\({a}^{^{\prime}}\) and \({b}^{^{\prime}}\) are material parameters:
\({a}^{^{\prime}}=318.2+0.337\left({q}_{\mathrm{u}}\right)+\)
\(0.73\left(\% \mathrm{clay}\right)+2.26\left(\mathrm{PI}\right)-\)
\(0.915\left(\gamma \right)-2.19\left(S\right)-\)
\(0.304(\% \mathrm{finer} \#200)\)
\({b}^{^{\prime}}=2.10+0.00039\left(\frac{1}{a}\right)+\)
\(0.104\left({q}_{\mathrm{u}}\right)+0.09(\mathrm{LL})\)-
0.10(% finer #200)
WL = 68.5%; WP = 39.2%;
IP = 29.3%; Gs = 2.71;
Clay content = 28.7%;
Passing sieve #200 = 80.0%;
Maximum dry density = 12.55 kN/m3
The hyperbola predicts excessive values of Er when σd is close to zero
WL = 69.5%; WP = 42.6%;
IP = 26.9%; Gs = 2.73;
Clay content = 55.0%;
Passing sieve #200 = 80.0%;
Maximum dry density = 13.83 kN/m3
QS0-QS1
MH
(no information about the particle size curve)
Medium to high plasticity
[35]
\({M}_{\mathrm{r}}={k}_{1}{p}_{\mathrm{a}}{({\sigma }_{\mathrm{d}}+\chi {\psi }_{\mathrm{m}})}^{{k}_{2}}\)
χ is the Bishop's effective stress parameters (it is function of the degree of saturation of the soil);
ψm is the matric suction;
ki are the materials’ parameters;
Gs = 2.67; WL = 50.0%;
IP = 23%; Wopt = 17%;
Maximum dry density = 17.65 kN/m3
This model does not take into account the effect of confining stress.
The model can be applied to a large range of moisture content.
The model is dependent on only two material parameters
QS1
ML
Low plasticity
[36]
\({M}_{\mathrm{r}}={k}_{1}{p}_{\mathrm{a}}{\left(\frac{\theta -3{k}_{4}}{{p}_{\mathrm{a}}}\right)}^{{k}_{2}}{\begin{array}{c} \\ \left(\frac{{\tau }_{\mathrm{oct}}}{{p}_{\mathrm{a}}}+{k}_{5}\right)\end{array}}^{{k}_{3}}\)
\({\mathrm{MR}}_{\mathrm{opt}}=\frac{{M}_{\mathrm{rs}}}{{M}_{\mathrm{rs};\mathrm{opt}}}\)
\(=c+d\mathrm{log}{\psi }_{\mathrm{m}}\)
MRopt is the modulus ratio;
Mrs is the modulus at certain suction (or moisture);
Mrs;opt is the modulus at optimum moisture content;
c and d are model parameters;
pa is the atmospheric pressure (101 kPa);
ki are fitting parameters
WL = 28.0%; IP = 11.0%;
Gs = 2.69;
Sand content = 11.9%;
Silt content = 82.4%;
Clay content = 5.7%;
Fines content = 88.1%;
Wopt = 13.5%;
Maximum dry unit weight = 17.9 kN/m3
The model is developed based on four tested fine-grained soils.
This model demands significant regression parameters and several combinations of parametric values to fit laboratory tests.
The test specimens were initially saturated and then subjected to desorption until the matric suction was 154 or 350 kPa.
All the test soils were prepared at OPT water content and at a dry density corresponding to 98% or 103% relative compaction for a standard proctor
MRopt is the modulus ratio;
Mrs is the modulus at certain suction (or moisture);
Mrs;opt is the modulus at optimum moisture content;
c and d are model parameters;
pa is the atmospheric pressure (101 kPa);
ki are fitting parameters
WL = 42.0%; IP = 24.0%;
Gs = 2.69; Wopt = 22.0%;
Sand content = 8.9%;
Silt content = 63.8%;
Clay content = 27.3%;
Fines content = 91.1%;
Maximum dry unit weight = 15.8 kN/m3
QS1
ML
Medium plasticity
[88]
\({M}_{\mathrm{r}}={k}_{1}{p}_{\mathrm{a}}{\left(\frac{{\theta }_{\mathrm{net}}}{{p}_{\mathrm{a}}}\right)}^{{k}_{2}}\times\)
\({\left(\frac{{\tau }_{\mathrm{oct}}}{{p}_{\mathrm{a}}}+1\right)}^{{k}_{3}}{\left(\frac{{\psi }_{\mathrm{m}}}{{\theta }_{\mathrm{net}}}+1\right)}^{{k}_{4}}\)
θnet is the net bulk stress (θnet = θ-3ua);
τoct is the octahedral shear stress;
ψm is the matric suction;
pa is the atmospheric stress;
ki are the material parameters
WL = 43%;WP = 29%;
IP = 14%; Gs = 2.73;
Sand content = 24%;
Silt content = 72%;
Clay content = 4%;
Cu = 4.55; Cc = 0.61;
Wopt = 16.3%;
Maximum dry density = 17.26 kN/m3
The model is similar to Cary and Zapata [67] model.
The last term related to the suction parameter is normalized with the net bulk stress instead of the atmospheric pressure.
The model does not consider the effect of the pore-water pressure under saturated conditions
QS1
ML
Low plasticity
[50]
\({E}_{\mathrm{r}}=\frac{{a}^{^{\prime}}+b{^{\prime}}{\sigma }_{\mathrm{d}}}{{\sigma }_{\mathrm{d}}}\,\mathrm{for} \,{\sigma }_{\mathrm{d}}>0 \)
\({a}^{^{\prime}}\) and \({b}^{^{\prime}}\) are material parameters:
\({a}^{^{\prime}}=318.2+0.337\left({q}_{\mathrm{u}}\right)+\)
\(0.73\left(\% \mathrm{clay}\right)+2.26\left(\mathrm{PI}\right)-\)
\(0.915\left(\gamma \right)-2.19\left(S\right)-\)
\(0.304(\% \mathrm{finer} \#200)\)
\({b}^{^{\prime}}=2.10+0.00039\left(\frac{1}{a}\right)+\)
\(0.104\left({q}_{\mathrm{u}}\right)+0.09(\mathrm{LL})\)-
0.10(% finer #200)
WL = 36.2%; WP = 34.1%;
IP = 2.1%; Gs = 2.66;
Clay content = 18.0%;
Passing sieve #200 = 80.0%;
Maximum dry density = 13.14 kN/m3
The hyperbola predicts excessive values of Er when σd is close to zero
WL = 37.1%; WP = 27.0%;
IP = 10.0%; Gs = 2.67;
Clay content = 35.0%;
Passing sieve #200 = 68.0%;
Maximum dry density = 14.91 kN/m3
QS1
ML
Medium plasticity
[89]
\({M}_{\mathrm{r}}={k}_{1}{p}_{\mathrm{a}}{\sigma }_{\mathrm{d}}^{{k}_{2}}{\psi }_{\mathrm{m}}^{{k}_{3}}\)
σd is the deviator stress;
ψm is the matric suction;
ki are the regression parameters
WL = 46.4%; WP = 28.8%;
IP = 17.6%; Gs = 2.66
Clay content = 38%
Passing sieve #200 = 52%;
Wopt = 19.5%
Maximum dry density = 16.28 kN/m3
The model does not take into account the effect of confining stress.
It is only valid for a limited range of moisture content.
When the soils get saturated and matric suction is equal to zero (high moisture content) the resilient modulus is close to zero.
Low moisture content values lead to high (unrealistic) resilient modulus.
The suction was determined through the paper filter method
Table 19
Models that describe resilient modulus of granular soils—sands
UIC
ASTM classification
Ref.
Equation model
Variables and empirical constants
Parameters and characteristics
Observations
QS1
SM-CL
Low plasticity
[50]
\({E}_{\mathrm{r}}=\frac{{a}^{^{\prime}}+b{^{\prime}}{\sigma }_{\mathrm{d}}}{{\sigma }_{\mathrm{d}}}\,\mathrm{ for}\, {\sigma }_{\mathrm{d}}>0\)
\({a}^{^{\prime}}\) and \({b}^{^{\prime}}\) are material parameters:
\({a}^{^{\prime}}=318.2+0.337\left({q}_{\mathrm{u}}\right)+\)
\(0.73\left(\% \mathrm{clay}\right)+2.26\left(\mathrm{PI}\right)-\)
\(0.915\left(\gamma \right)-2.19\left(S\right)-\)
\(0.304\left(\% \mathrm{finer} \#200\right)\)
\({b}^{^{\prime}}=2.10+0.00039\left(\frac{1}{a}\right)+\)
\(0.104\left({q}_{\mathrm{u}}\right)+0.09(\mathrm{LL})\)-
0.10(% finer #200)
WL = 21.0%; WP = 14.1%;
IP = 6.9%;
Gs = 2.65;
Clay content = 16.0%;
Passing sieve #200 = 36.0%;
Maximum dry density = 18.04 kN/m3
The hyperbola predicts excessive values of Er when σd is close to zero
QS1
SM
Low plasticity
WL = 20.7%; WP = 19.0%;
IP = 1.7%; Gs = 2.60;
Clay content = 17.0%;
Passing sieve #200 = 20.0%;
Maximum dry density = 17.55 kN/m3
QS1
SC
(the fines are classified as CL)
Low plasticity
[67]
\({M}_{\mathrm{r}}={k}_{1}{p}_{\mathrm{a}}{\left(\frac{{\theta }_{\mathrm{net}}-3\Delta {u}_{\mathrm{w}-\mathrm{sat}}}{{p}_{a}}\right)}^{{k}_{2}}{\begin{array}{c} \\ \left(\frac{{\tau }_{\mathrm{oct}}}{{p}_{\mathrm{a}}}+1\right)\end{array}}^{{k}_{3}}{ \left(\frac{{\psi }_{\mathrm{m}}-{\Delta \psi }_{\mathrm{m}}}{{p}_{\mathrm{a}}}+1\right)}^{{k}_{4}}\)
θnet is the net bulk stress (θnet = θ-3ua);
Δuw-sat is the pore-water pressure build-up in a saturated condition (ψm = 0);
Δψm is the relative change in matric suction with respect to the initial matric suction in an unsaturated condition
(Δuw-sat = 0)
WL = 22%; WP = 18%;
IP = 5%; Gs = 2.71;
Passing sieve #200 = 47%
Wopt = 12.1%
Maximum dry density = 18.73 kN/m3
The model is based on the universal model with an additional term that includes the matric suction effects into the resilient modulus
QS1
SC
(clayed sand)
Low plasticity
[51]
\({M}_{\mathrm{r}}=k\frac{{\sigma }_{\mathrm{oct}}^{n}}{{\tau }_{\mathrm{oct}}^{m}}\)
σoct and τoct are the octahedral normal and shear stresses, respectively
WL = 26%; WP = 17.6%;
IP = 8.4%;
60% sand and 40% kaolinite by weight
This model is difficult to apply [86]
QS2
SP
Poor-graded sand
[58]
\({M}_{\mathrm{r}}=A{\left(\frac{{p}_{\mathrm{m}}}{{p}_{\mathrm{u}}}\right)}^{B}{\left(\frac{{p}_{\mathrm{u}}}{p}\right)}^{C}\)
qm is the mean value of deviator stress;
pm is the mean value of mean normal stress;
p is the mean normal stress;
pu is unit pressure (1 kPa);
A, B and C are constants
Gs = 2.64; Cu \(\approx\) 2.5; Cc \(\approx\) 1.5
This model is based on the triaxial tests carried out in six aggregates.
The model is based on laboratory triaxial testing (constant and variable confining pressure) [70]
QS2
Gs = 2.64; Cu \(\approx\) 2.0; Cc \(\approx\) 1.04
QS3
Gs = 2.61; Cu \(\approx\) 28; Cc \(\approx\) 0.54
Aggregate abrasion
value = 1.9;
QS3
Gs = 2.24; Cu \(\approx\) 30; Cc \(\approx\) 0.5
Wopt = 8–12%;
Permeability index = 0.0002 m/s
QS3
SW
Well-graded sand
[57]
\({M}_{r}={A({n}_{\mathrm{max}}-n)p}_{0}{\left(\frac{\theta }{{p}_{0}}\right)}^{0.5}\)
\({M}_{r}={B({n}_{\mathrm{max}}-n)p}_{0}{\left(\frac{\theta }{{p}_{0}}\right)}^{0.7}{\left(\frac{q}{{p}_{0}}\right)}^{-0.2}\)
θ is the sum of principal stresses;
p0 is the reference stress, 100 kPa;
n is the porosity of the material;
nmax is the theoretical maximum value of porosity when Mr = 0;
A is a material parameter
Cu \(\approx\) 5.9; Cc \(\approx\) 1.2
This model is based on laboratory triaxial testing (constant confining pressure) [70].
The model is based on tests carried out in several types of coarse-grained granular materials typically used in unbound layers of road and railroad pavements.
The density was incorporated in the model by the porosity parameter.
Materials classified as natural gravels.
The study also includes the crushed gravel and crushed rock
Cu \(\approx\) 1.6; Cc \(\approx\) 1.0
Cu≈50; Cc≈1.0
QS3
SW
Well-graded sand
[79]
\({E}_{\mathrm{v}}=C{{\sigma }_{\text{v}}^{^{\prime}}}^{m}\)
C is a parameter that depends on the mineralogy of the particles, shape of the grains and particle size curve (in the case of granular materials. In the case of the cohesive materials; it also depends on the percentage of fines and Atterberg limits
D50 = 3.5 mm; Gs = 2.71;
Cu = 28; Cc \(\approx\) 1.0
The vertical Young’s modulus is independent of the lateral stress
Table 20
Models that describe resilient modulus of granular soils—gravels
UIC
ASTM classification
Ref.
Equation model
Variables and empirical constants
Parameters and characteristics
Observations
QS2
GP-GM
[67]
\({M}_{\mathrm{r}}={k}_{1}{p}_{\mathrm{a}}{\left(\frac{{\theta }_{\mathrm{net}}-3\Delta {u}_{\mathrm{w}-\mathrm{sat}}}{{p}_{a}}\right)}^{{k}_{2}}{\begin{array}{c} \\ \left(\frac{{\tau }_{\mathrm{oct}}}{{p}_{\mathrm{a}}}+1\right)\end{array}}^{{k}_{3}}{\begin{array}{c} \\ \left(\frac{{\psi }_{\mathrm{m}}-{\Delta \psi }_{\mathrm{m}}}{{p}_{\mathrm{a}}}+1\right)\end{array}}^{{k}_{4}}\)
θnet is the net bulk stress (θnet = θ−3ua);
Δuw-sat is the pore-water pressure build-up in a saturated condition (ψm = 0);
Δψm is the relative change in matric suction with respect to the initial matric suction in an unsaturated condition
(Δuw-sat = 0)
IP = NP; Gs = 2.71;
Passing sieve #200 = 7%;
Wopt = 7.2%
Maximum dry density
 = 22.16 kN/m3
The model is based on the universal model with an additional term that includes the matric suction effects into the resilient modulus
QS2
GP
Poor-graded gravel
[58]
\({M}_{\mathrm{r}}=A{\left(\frac{{p}_{\mathrm{m}}}{{p}_{\mathrm{u}}}\right)}^{B}{\left(\frac{{p}_{\mathrm{u}}}{p}\right)}^{C}\)
qm is the mean value of deviator stress;
pm is the mean value of mean normal stress;
p is the mean normal stress;
pu is unit pressure
(1 kPa);
A, B and C are constants
Gs = 2.69; Cu \(\approx\) 5.3;
Cc \(\approx\) 4.7
This model is based on the triaxial tests carried out in six aggregates.
The model is based on laboratory triaxial testing (constant and variable confining pressure) [70]
QS3
GW
Well-graded gravel
Gs = 2.69; Cu \(\approx\) 61; Cc \(\approx\) 2.11;
Aggregate abrasion value = 5.1;
QS2-QS3
GP
Poor-graded gravel
[79]
\({E}_{\mathrm{v}}=C{{\sigma }_{\text{v}}^{^{\prime}}}^{m}\)
C is a parameter that depends on the mineralogy of the particles, shape of the grains, particle size curve (in the case of the granular materials)
D50 = 8.5 mm;
Gs = 2.71;
Cu = 53; Cc \(\approx\) 4.5
The vertical Young’s modulus is independent of the lateral stress.
The parameter C, in the case of the cohesive materials, also depends on the percentage of fines and Atterberg limits
 ≥ QS2
(based on AASHTO classification)
 
[87]
\({M}_{\mathrm{r}}={k}_{1}{P}_{\mathrm{a}}{\left(1+\frac{{\sigma }_{\mathrm{d}}}{1+{\sigma }_{\mathrm{c}}}\right)}^{{k}_{2}}\)
Mr is the resilient modulus (MPa);
σd is the deviator stress (kPa);
σc is the confining pressure (kPa);
Pa is the atmospheric pressure (kPa);
k1 and k2 are model parameters
γd = 16.07 kN/m3
γdr = 1.02; wc = 16.3%;
wcr = 1.05;
Optimum wc = 15.5%;
Maximum dry density = 15.7 kN/m3;
Passing sieve #200 = 11%
k1 is dependent on γdr, wcr, #200 and Cu, where γdr and wcr are the density and moisture ratio, respectively.
k2 is dependent on γdr, wcr, #200 and Cu
γd = 18.04 kN/m3
γdr = 1.10; wcr = 1.15;
wc = 17.32%;
Optimum wc = 15.0%;
Maximum dry density = 16.7 kN/m3
Passing sieve #200
 = 34%
Table 21
Other models
Ref.
Equation model
Variables and empirical constants
Observations
[90]
\({M}_{\mathrm{r}}={k}_{1}{\left({\sigma }_{3}\right)}^{{k}_{2}}\)
σ3 is the confining pressure;
k1 and k2 are material constants
This model is based on the resilient modulus and Poisson’s ratio;
Model based on laboratory triaxial testing (constant confining pressure)
[70]
[77]
[91]
[78]
\({M}_{\mathrm{r}}={k}_{1}{\left(\theta \right)}^{{k}_{2}}\)
θ is the bulk stress (sum of the principal stresses σ1, σ2 and σ3);
ki are the regression parameters
Commonly known as the k–θ model; hyperbolic relationship.
This model assumes a constant Poisson’s ratio. However, several studies show that this parameter varies with the applied stress.
The effect of stress on the resilient modulus is only considered by the bulk stress (sum of the principal stresses).
It is a simplified version of the universal model.
This model is based on laboratory triaxial testing (constant confining pressure)[70].
The model fits well in the case of tests with constant confining pressure and shows poor results in the case of variable confining pressure [53]
[92]
\({M}_{\mathrm{r}}={K}_{1}+{K}_{2}{\sigma }_{\mathrm{d}}\), when σd < σdi
\({M}_{\mathrm{r}}={K}_{3}+{K}_{4}{\sigma }_{\mathrm{d}}\), when σd > σdi
K1, K2, K3, and K4 are model parameters dependent upon soil type and its physical state (normally, K2 and K4 are negative);
σdi is the deviator stress at which the slope of Mr versus σd changes
This is a bilinear model
[93]
\(\mathrm{log}{M}_{\mathrm{r}}={c}_{\mathrm{ld}}-{m}_{\mathrm{ld}}({\sigma }_{1}-{\sigma }_{3})\)
cld and mld are functions of matrix suction;
σ1 and σ3 are the major and minor principal stresses
The model was not fully validated with laboratory data.
This model is characterized as a semi-log model
[61]
\({\varepsilon }_{\mathrm{v}}=\frac{1}{{K}_{1}}{p}^{n}\left[1-\beta {\left(\frac{q}{p}\right)}^{2}\right]\)
\({\varepsilon }_{\mathrm{s}}=\frac{1}{{3G}_{1}}{p}^{n}\left(\frac{q}{p}\right)\)
\(K=\frac{p}{{\varepsilon }_{v}}\)
\(G=3\frac{q}{{\varepsilon }_{q}}\)
p is the mean normal stress;
q is the deviator stress;
εv is the volumetric strain and εs is the shear strain;
K is the bulk modulus;
G is the shear modulus
This model uses shear-volumetric approach.
This is a nonlinear elastic model that takes into account the effect of the stress path and it is expressed in terms of the bulk and shear modulus, that are stress-dependent.
The model is elastic and follows the application of the theorem of reciprocity.
The shear and volumetric components of stress and strain are treated separately.
The granular materials show an inelastic response and this elastic model can lead to inaccurate predictions.
The model is based on variable confining pressure.
Sweere [94] used this model but keep the relationship between shear and volumetric strains with stresses independent from each other
[47]
\({M}_{\mathrm{r}}={k}_{1}{\left({\sigma }_{\mathrm{d}}\right)}^{{k}_{2}}\)
k1 and k2 are constants dependent on the material;
σd is the deviator stress;
The model is characterized by a power model.
A modified elastic layered computer program was made to determine equivalent subgrade modulus values.
Multiple regression techniques were used to determine predictive equations for the equivalent subgrade modulus.
This model was widely used by other authors and good results were obtained (in agreement with test results)
[56]
\({M}_{\mathrm{r}}={k}_{1}{p}_{0}{\left(\frac{\theta }{{p}_{0}}\right)}^{{k}_{2}}{\left(\frac{q}{{p}_{0}}\right)}^{{k}_{3}}\) or
\({M}_{\mathrm{r}}={k}_{1}{p}_{0}{\left(\frac{\theta }{{p}_{0}}\right)}^{{k}_{2}}{\left(\frac{{\tau }_{\mathrm{oct}}}{{p}_{0}}\right)}^{{k}_{3}}\)
θ is the bulk stress (sum of the principal stresses σ1, σ2 and σ3);
τoct is the octahedral shear stress;
p0 is the atmospheric pressure;
ki are the material parameters
The model shows good results in the case of pre-failure stresses. However, in the case of stresses that go beyond the static failure, the predictions are poor.
This model is suitable for a large range of materials: from fine-grained soils to coarse-granular materials.
This model is based on laboratory triaxial testing (constant confining pressure) [70]
[95]
\({\varepsilon }_{\mathrm{v}}=\delta \left[{\left(\frac{p}{A}\right)}^{B}\left[1-C{\left(\frac{q}{p}\right)}^{2}\right]\right]\)
\({\varepsilon }_{\mathrm{s}}=D\delta \left[\frac{q}{p+E}\right]{\left[\frac{\sqrt{{p}_{\mathrm{r}}^{2}+{q}_{\mathrm{r}}^{2}}}{{p}_{\mathrm{m}}}\right]}^{F}\)
p is the mean normal stress;
q is the deviator stress;
A, B, C, D and E are material constants
This model uses shear-volumetric approach.
Two materials were considered: well-graded crushed limestone and uniformly graded limestone.
This model is also known as the contour model and takes into account effective mean and deviatoric stresses and stress path dependency.
This model predicts the shear strain (considering the length of the stress path corresponding to the load cycle).
The material parameters in the two parts of the contour model are independent of each other. In fact, independent material parameters mean that the model is inelastic (unlike the Boyce model)
[96]
\({M}_{\mathrm{r}}={k}_{1}{\left(\frac{{J}_{2}}{{\tau }_{\mathrm{oct}}}\right)}^{{k}_{2}}\)
k1 and k2 are constants;
J2 is the second stress invariant;
τoct is the octahedral shear stress
Model based on laboratory triaxial testing (constant confining pressure) [70]
[97]
\({M}_{\mathrm{r}}={k}_{1}{\left(\frac{p}{q}\right)}^{{k}_{2}}\)
k1 is a material constant;
p is the mean normal stress;
q is the deviator stress;
Model based on laboratory triaxial testing (constant confining pressure) [70]
[98]
\({\varepsilon }_{\mathrm{v}}=A{\left(\delta \mathrm{ln}p\right)}^{B}{(\delta p)}^{C}-D{\left[\delta {\left(\mathrm{ln}\frac{{\sigma }_{1}}{{\sigma }_{3}}\right)}^{2}\right]}^{E}\)
\({\varepsilon }_{\mathrm{s}}=E{\left[\delta {\left(\mathrm{ln}\frac{{\sigma }_{1}}{{\sigma }_{3}}\right)}^{2}\right]}^{F}\left({\delta }_{\mathrm{T}}+\frac{1}{3}\delta S\right)\)
σ1 and σ3 are the major and minor principal stresses
T and S are 2D stress invariants
Model developed for prediction of the results of triaxial and hollow cylinder tests.
The model separates the volumetric and shear components as the Boyce’s and contour models.
Karasahin [58] verified the model but found it very difficult to obtain the material parameters by nonlinear regression analysis
[99]
\({M}_{\mathrm{r}}={k}_{1}\frac{{\theta }^{{k}_{2}}}{{10}^{A_1}}\)
θ is the bulk stress (sum of the principal stresses σ1, σ2 and σ3);
k1 is a constant parameter;
A1 = mR3, m is a regression coefficient determined from laboratory data and R is the stress/strength ratio (deviator stress divided by the failure deviator stress)
k–θ model that includes a failure term (the impact is insignificant until the failure is approached.
The failure deviator stress is determined based on the confining pressure and the static triaxial shear stress.
This model is based on laboratory triaxial testing (constant confining pressure) [70]
Developed by
Mayhew [100]
and re-written by
Brown and Selig [101]
\({\varepsilon }_{\mathrm{v}}=\frac{1}{{K}_{1}}{p}^{A}\left[1-C{\left(\frac{q}{p}\right)}^{2}\right]\)
\({\varepsilon }_{\mathrm{s}}=\frac{1}{{3G}_{1}}{p}^{B}\left(\frac{q}{p}\right)\)
p is the mean normal stress;
q is the deviator stress;
εv;r is the recoverable volumetric strain and εs;r is the shear strain;
K is the bulk modulus;
G is the shear stress
This model is similar to the contour model.
The author plots the shear and volumetric strains in the p-q space.
The author concludes that the stress path length (included in the contour model) had little impact on the shear strain response. The model removes the stress path dependence and the equations of the contour model turn into the inelastic modified version of the Boyce model
[102]
\({\varepsilon }_{\mathrm{v}}={p}_{0}^{1-B}{p}^{B}\left[\frac{1}{A}-\frac{\left(1-B\right)}{6C}{\left(\frac{q}{p}\right)}^{2}-\frac{B}{D}\left(\frac{q}{p}\right)\right]\)
\({\varepsilon }_{\mathrm{s}}={p}_{0}^{1-B}{p}^{B}\left[\frac{1}{3C}\left(\frac{q}{p}\right)-\frac{1}{D}\right]\)
–
Model based on Boyce’s model; the author introduced the concept of anisotropy
[103]
[80]
\({M}_{\mathrm{r}}={N}_{1}{q}^{{N}_{2}}{\left({\sigma }_{3}\right)}^{{N}_{3}}\)
σ3 is the confining pressure;
q is the deviator stress
Model based on laboratory triaxial testing (constant confining pressure) [70]
[86]
\({R}_{\mathrm{m}1}=0.98-0.28\left(w-{w}_{\mathrm{opt}}\right)+\)
\(0.029{(w-{w}_{\mathrm{opt}})}^{2}\)
Rm1 = Mr/Mr;opt for the case of constant dry density; Mr is the resilient modulus at moisture content w (%) and the same dry density as Mr(opt); Mr(opt) is the resilient modulus at the maximum dry density and optimum moisture content wopt (%) for any compaction effort
This model includes the soil's physical state by two quantities: moisture content and dry density that are related to the soil-compaction curve.
The results are based on repeated load triaxial test results on fine-grained soils from literature (Seed, Chan [104]; Sauer and Monismith [105]; Culley [106]; Robnett and Thompson [107]; Fredlund, Began [93]; Edil and Moran [108]; Kirwan, Farrell [109]; Elfino and Davidson [110]).
The power-law model was selected for the representation of the relationship between resilient modulus and deviator stress
[86]
\({R}_{\mathrm{m}2}=0.96-0.18\left(w-{w}_{\mathrm{opt}}\right)+\)
\(0.0067{(w-{w}_{\mathrm{opt}})}^{2}\)
Rm2 = Mr/Mr;opt for constant compaction effort; Mr is the resilient modulus at moisture (%) and the same compaction effort as Mr;opt
This model includes the soil's physical state by two quantities: moisture content and dry density that are related to the soil-compaction curve.
The results are based on repeated load triaxial test results on fine-grained soils from literature (Seed, Chan [104]; Kallas and Riley [111]; Shifley and Monismith [112]; Tanimoto and Nishi [113]; Edris and Lytton [114]; Fredlund, Began [93]; Kirwan, Farrell [109]; Pezo, Kim [115].
The power-law model was selected for the representation of the relationship between resilient modulus and deviator stress
[116]
\({\varepsilon }_{\mathrm{v}}=\frac{{p}^{{*}^{A}}}{{p}_{0}^{A-1}}\left[\frac{\gamma +2}{3{K}_{1}}+\frac{A-1}{18{G}_{1}}\left(\gamma +2\right){\left(\frac{{q}^{*}}{{p}^{*}}\right)}^{2}\right.+\)
\(\left.\frac{\gamma -1}{3{G}_{1}}\left(\frac{{q}^{*}}{{p}^{*}}\right)\right]\)
\({\varepsilon }_{\mathrm{s}}=\frac{2}{3}\frac{{p}^{{*}^{A}}}{{p}_{0}^{A-1}}\left[\frac{\gamma -1}{3{K}_{1}}+\frac{A-1}{18{G}_{1}}\left(\gamma -1\right){\left(\frac{{q}^{*}}{{p}^{*}}\right)}^{2}+\frac{2\gamma +1}{6{G}_{1}}\left(\frac{{q}^{*}}{{p}^{*}}\right)\right]\)
\({p}^{*}=\frac{\gamma {G}_{1}+2{\sigma }_{3}}{3}\);\({q}^{*}=\gamma {\sigma }_{1}-{\sigma }_{3}\)
γ is the coefficient of anisotropy;
The parameter γ means that the material is isotropic (when γ = 1) and the material behaves more anisotropically as γ deviates further from γ = 1; when γ < 1, the material is stiffer vertically than horizontally;
This model is based on Boyce’s model and is also known as the anisotropic Boyce model
[117]
\({E}_{\mathrm{v}}=\frac{\partial {\sigma }_{\mathrm{v}}}{\partial {\varepsilon }_{\mathrm{v}}}={E}_{\mathrm{v}0}\cdot {\left(\frac{{\sigma }_{v}}{{\sigma }_{0}}\right)}^{nv}\cdot F(e)/F({e}_{0})\)
\({E}_{\mathrm{h}}=\frac{\partial {\sigma }_{\mathrm{h}}}{\partial {\varepsilon }_{\mathrm{h}}}={E}_{\mathrm{h}0}\cdot {\left(\frac{{\sigma }_{\mathrm{h}}}{{\sigma }_{0}}\right)}^{nv}\cdot F(e)/F({e}_{0})\)
\(\frac{{E}_{V}}{{E}_{h}}=\frac{{E}_{\mathrm{v}0}}{{\left({E}_{h}\right)}_{0}}\cdot {R}^{n}=a{R}^{n}\)(when powers nv and nh are the same and equal to n)
The power n represents the stress-dependency of the quasi-elastic vertical Young’s modulus Ev and Eh;
a quantifies the inherent anisotropy;
\(R=\frac{{\sigma }_{\mathrm{h}}}{{\sigma }_{\mathrm{v}}}\) and quantifies the stress-induced anisotropy at the isotropic stress states;
Ev0 is the reference vertical stress
F(e) is the void ratio function;
Cross-anisotropic-hypo-quasi-elasticity model
This model depends on four material constants (Ev0, ν0, n and a).
The model can predict nonlinear and anisotropic resilient stress–strain properties for general stress paths under general stress conditions
[118]
\({M}_{\mathrm{r}}={10}^{A}{k}_{1}{p}_{\mathrm{a}}{\left(\frac{\theta }{{p}_{\mathrm{a}}}\right)}^{{k}_{2}}{\left(\frac{{\tau }_{\mathrm{oct}}}{{p}_{\mathrm{a}}}+1\right)}^{{k}_{3}}\)
A = \(a+\frac{b-a}{1+\mathrm{exp}\left(\beta +{k}_{\mathrm{s}}\left(S-{S}_{\mathrm{opt}}\right)\right)}\),
S is the degree of saturation and Sopt is the saturation at OPT conditions;
θ is the bulk stress;
τoct is the octahedral shear;
pa is atmospheric pressure;
k1, k2, k3, a, b and ks are fitting parameters; β = ln(-b/a)
The first term of the expression represents the environment related to the contribution of changes in moisture content
[59]
\(\mathrm{log }\frac{{M}_{\mathrm{r}}}{{M}_{\mathrm{ropt}}}=a+\)
\(\frac{b-a}{1+\mathrm{exp}[\mathrm{ln}\left(-b/a\right)+{k}_{\mathrm{m}}\left(S-{S}_{\mathrm{opt}}\right)]}\)
Mr/Mropt is the resilient modulus ratio, where Mr is the resilient modulus at optimum moisture content;
a is the minimum of
log (Mr/Mropt);
b is the maximum of
log (Mr/Mropt);
km is a regression parameter;
(S−Sopt) is the variation in the degree of saturation
This model can be used in the case of fine-grained and coarse-grained materials (the difference is expressed in terms of the parameters a, b and km).
The constants of the model are determined based on RLT testing data.
The model does not combine the effects of the stress state and the moisture content.
The model estimated the change of the resilient modulus at a certain moisture content and a certain known stress state.
Based on an intensive literature review study, it was developed (select) a model that analytically predicts changes in modulus due to changes in moisture
[81]
\({E}_{\text{r}}=49200+950{p}_{0}^{^{\prime}}-370{q}_{\mathrm{r}}-2400{w}_{\text{p}}\)
\({p}_{0}^{^{\prime}}\) is the initial mean normal effective stress (in kPa);
qr is the repeated deviator stress (in kPa);
wp is the plastic limit in %
This is an analysis based on repeated load triaxial testing carried out on recompacted soils
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Metadaten
Titel
A geomechanics classification for the rating of railroad subgrade performance
verfasst von
António Gomes Correia
Ana Ramos
Publikationsdatum
05.12.2021
Verlag
Springer Nature Singapore
Erschienen in
Railway Engineering Science / Ausgabe 3/2022
Print ISSN: 2662-4745
Elektronische ISSN: 2662-4753
DOI
https://doi.org/10.1007/s40534-021-00260-z

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