Constitutive Behaviour of Recycled Rubber-Involved Mixtures for Transportation Infrastructure

Abstract

The scarcity of natural aggregates and the growing accumulation of waste materials have driven the demand for sustainable and circular economy solutions in transportation infrastructure, and this has led to the utilization of waste materials in transport infrastructure, such as recycled rubber. Although numerous laboratory experiments have been conducted on granular mixtures mixed with rubber, predicting the complex stress–strain behaviour of these mixtures mathematically and capturing the influence of rubber on the geotechnical properties of waste mixtures are imperative. This paper presents a comprehensive review of the constitutive models developed to predict the stress–strain behaviour, dilatancy, and shear strength of rubber-mixed waste materials, including sand–rubber, coal wash–steel furnace slag–rubber crumbs, and coal wash–rubber crumbs in various transport infrastructure applications under static loading. This paper also highlights the innovations and limitations of these existing constitutive models on rubber-mixed materials. It was found that existing constitutive models based on hyperbolic, hypoplastic, critical state, and bounding surface plasticity approaches can capture the behaviour of these materials under static loading conditions. However, further developments are required to incorporate the influence of the type and size of the rubber, particle breakage, and damping properties and also account for train-induced cyclic loading in models developed for railway substructures. This paper contributes to advancing future research aimed at deepening the fundamental understanding of rubber-mixed materials used in transportation infrastructure.

1. Introduction

Roads and railways are the most commonly used modes of public transport due to their convenience and the continuous expansion of road and railway networks. Australia has one of the largest transport networks in the world with 873,573 km of roads; this makes it the 9th largest globally, with a railway network spanning 38,500 km that ranks 7th in the world [1]. However, factors such as population growth, industrial expansion, and urbanization have led to a rising demand for further development and expansion of our transport infrastructure. The growing demand for rail transport has led to higher train speeds and increased loading intensity, which in turn has contributed to track deterioration and more frequent maintenance. Every country spends significant amounts of money on the maintenance and development of their transport infrastructure, which means improving operational efficiency and passenger comfort and mitigating construction and maintenance costs are key priorities [2].

Transport infrastructure currently faces several major challenges. The depletion of natural aggregates such as sand and gravel is a pressing concern for many countries, as high-quality aggregates are becoming increasingly scarce due to stricter environmental regulations [3]. Essential materials that are widely used in road pavements and railway tracks, such as gravel and ballast, are typically produced through quarrying. This involves extracting large quantities of sand, gravel, and other raw materials from the earth, which then destroys habitats, depletes resources, and creates ecological degradation, which disrupts local communities and threatens wildlife. Roads and embankments often experience differential settlement [4,5,6], while railway tracks face additional challenges such as ballast breakage [7,8,9,10,11], excessive vibration [12,13,14], mud pumping [15,16,17], ballast fouling [18,19], and the lateral deflection of rails [20]. Moreover, while railway substructures are frequently subjected to damage from impact loads caused by rail irregularities and flat wheels [21,22], the high speeds of heavier trains cause the ballast and sub-ballast to undergo breakage and degradation after millions of loading cycles. Over time, inadequate lateral confinement leads to the spreading and settlement of ballast and sub-ballast, and ballast degradation occurs as the sharp edges and angular corners of particles break, allowing fine particles to infiltrate between the layers [23]. Furthermore, the growing demand for passenger and freight transport has led to the operation of heavier and faster trains, which increase the dynamic loads that intensify ground vibrations, heighten stress on the track substructure, and accelerate the deterioration of track components [13].

On the other hand, the rapid industrialization in the twenty-first century, particularly in Australia, has led to the accumulation of large volumes of waste by-products such as coal wash, coal tailings, steel furnace slag, demolition waste, fly ash, glass, waste tyres, and plastics. These materials pose serious environmental and socio-economic concerns worldwide because the stockpiles often occupy otherwise useable land in suburban and regional areas. Australia also faces challenges in managing the waste rubber from over 21 million registered vehicles, which generated approximately 537,000 tonnes of end-of-life tyres in 2023–2024. Although recycling and energy recovery processes handle around 66% of this waste, approximately 184,000 tonnes of tyres are still dumped illegally, stockpiled, or disposed of in landfills, posing significant environmental risks, particularly in rural and remote regions [24,25].

As a result, the growing demand for sustainable transport infrastructure has driven the search for more environmentally friendly materials with superior engineering properties. The key approach is to adopt “greener” alternatives to natural aggregates because they will reduce reliance on natural resources, support a circular economy, and promote the environmental sustainability of transport infrastructure [26,27]. Recycled waste materials such as crushed glass (i.e., [28,29,30,31,32,33]), plastics (i.e., [34,35,36]), coal mining waste (i.e., [37,38,39,40,41,42,43,44]), steel furnace slag (i.e., [45,46,47,48]), fly ash (i.e., [49,50,51]), and construction demolition waste (i.e., [52,53]) are widely utilised because they reduce reliance on scarce natural resources, divert waste from landfills, and lower carbon footprints better than traditional quarried materials, thus promoting environmental sustainability [53,54].

Among the various waste materials available, rubber stands out because its superior energy absorption and damping capacity enable it to mitigate the impact of repeated loads from trains and moving vehicles. For instance, when scrap rubber is mixed with asphalt, pavements are better able to resist rutting, permanent deformation decreases, and fatigue resistance improves [55,56]. In recent years, researchers have focused on incorporating different types of recycled rubber into railway tracks as they recognise their potential to reduce track deterioration, enhance track stability, and provide an eco-friendly solution that supports a circular economy. Mixing rubber crumbs with ballast and coal mining by-products can create energy-absorbing ballast and capping layers, which reduce track vibrations and ballast degradation [10,57,58,59,60,61,62]. Recycled tyre cells filled with recycled ballast can be incorporated into capping layers in railway tracks to provide substantial lateral confinement, which in turn enhances track stiffness and its energy-absorbing capacity, thereby reducing particle breakage and ballast settlement [63,64,65]. Moreover, installing rubber geogrids beneath the ballast layer has proven to reduce impact forces and reduce ballast deformation and degradation [66,67,68]. Rail pads are placed at the rail–sleeper interface to provide flexibility, prevent direct contact between rails and concrete sleepers, and limit vibrations from passing trains [61,69,70]. Under-sleeper pads (USP) positioned at the sleeper–ballast interface and under-ballast mats (UBM) installed beneath the ballast layer are used to enhance the contact area at the ballast–sub-ballast and ballast–sleeper interfaces, respectively, thereby reducing the concentration of stress and reducing ballast breakage and deformation [71,72,73,74,75,76]. Figure 1 shows the various positions of rubber inclusions within a ballasted railway track.

Unlike traditional materials, recycled materials exhibit distinct characteristics that make it imperative to study their fundamental geotechnical behaviour through constitutive relationships [77]. While previous research has investigated the geotechnical properties of waste material mixtures via laboratory experiments, a more comprehensive understanding can be achieved utilizing mathematical approaches. Despite the number of laboratory studies conducted on soil–rubber mixtures, only a limited number have explored their behaviour using theoretical models developed within a constitutive framework, hence the need for further advancements in this area.

Therefore, this paper presents a comprehensive review of the various constitutive models developed to study the behaviour of recycled rubber-involved mixtures for transport infrastructure; this includes sand–rubber mixtures, coal wash–rubber crumb mixtures (CW + RC), and steel furnace slag–coal wash–rubber crumb mixtures (SFS + CW + RC). The effects of adding rubber to these mixtures on the stress–strain behaviour, critical state, shear strength, and dilatancy are critically reviewed, and the advantages and limitations of existing models are elaborated. Additionally, recommendations for future studies are included to address the gaps and unexplored aspects of past research.

2. Stress–Strain Characteristics of Rubber-Mixed Materials

2.1. Stress–Strain Behaviour

Previous studies have investigated the shear behaviour of rubber-mixed granular soils by incorporating different proportions of scrap tyres, measured by weight or volume, and utilizing various test equipment, such as direct shear apparatus and triaxial cells, on dry and fully saturated samples [51,58,78,79,80,81,82,83,84,85,86,87,88,89]. The stress–strain behaviour of materials mixed with rubber is influenced by the gradation of the material, the confining pressure, the initial void ratio, and the amount of rubber [41,44,90]. Of these, the gravimetric ratio of rubber and confining pressure are the primary factors that govern the shear behaviour of rubber-mixed materials. Indraratna et al. [42] demonstrated that incorporating rubber crumbs into SFS + CW + RC reduces the peak deviator stress because rubber has a lower shear strength than other stiffer components in mixtures like SFS and CW (Figure 2a). However, a higher ratio of SFS to CW compensates for this reduction because SFS provides a greater stiffness. Notably, contraction depends primarily on the amount of rubber: the more rubber, the more constructive the material, while the peak compression strain remains constant when the amount of rubber is the same when comparing SFS + CW + RC with CW + RC mixtures (Figure 2b). When the deviator stress reaches its peak, the mixtures contract and then dilate, while the larger pieces of rubber promote a shift from brittle to ductile behaviour (Figure 2). Similar results were reported by Tawk and Indraratna [58] for CW + RC mixtures used in railway capping layers (Figure 2) and by Arachchige et al. [91] for rubber–ballast mixtures in ballast layers; there, the increasing amount of rubber reduced the peak deviator stress but increased the ductility by raising the axial strain at peak stress. Similar findings for sand–rubber mixtures were reported by Sheikh et al. [86] when they investigated the shear behaviour of rubber crumbs and sand blends using a monotonic triaxial apparatus.

Studies on mixtures of tyre rubber–sand were initiated in the early 1990s. Ahmed [93] first investigated the shear behaviour of mixtures of sand–tyre chips using static and dynamic triaxial tests. He then analysed the size and ratio of the tyre chips, the type of soil, the confining pressure, and sample preparation. This study concluded that the shear behaviour depends mainly on the amount of tyre chips and the confining pressure. Lee et al. [83] investigated mixtures of rubber–sand with 40% rubber at confining pressures of 28, 97, and 193 kPa for use as lightweight backfill. With this amount of rubber, a clear peak stress could not be seen, even at an axial strain of 25%, thus indicating that the failure strain of these mixtures lies beyond the range of conventional testing equipment. Zornberg et al. [84] noted a similar trend in mixtures of sand–tyre shreds with up to 60% rubber. Lee et al. [83] observed a contractive phase in the volumetric strain followed by dilation at all confining pressures, which is consistent with the observations made by Youwai and Bergado [94] (Figure 3) and Anbazhagan et al. [95]. However, Zornberg et al. [84] only found contraction within the axial strain range for mixtures with less than 60% rubber, whereas adding 60% rubber caused a shift towards dilative behaviour. Mashiri et al. [85] observed that the deviator stress increased with an increasing amount of tyre chips in the blends of sand–tyre chips, but with more than 20% of chips, it declined slightly (Figure 3). Incorporating tyre chips reduced the dilation of sand, a trend that continued as the amount of chips and the confining pressure increased. Moreover, tyre chips also helped to control post-peak softening and maintain stable dilatancy under various confining pressures.

2.2. Shear Strength

Consistent with the behaviour of other granular materials, mixtures of tyre rubber and soil with a higher confining pressure and relative density were found to exhibit increased shear strength [93]. However, the impact that the amount of rubber had on the shear strength deviated considerably from findings in previous research, as shown in Table 1. Incorporating tyre shreds or chips into sand enhances the shear strength of the mixture [79,80,84,93] mainly due to the increase in the apparent cohesion as the sand and tyre shreds/chips interlock, which is similar to the effect seen in fibre reinforcement [81]. For instance, a study by Edil and Bosscher [80] demonstrated that randomly mixed tyre shreds increased the strength of sand and resulted in a stronger composite. Incorporating 10% of tyre shreds by volume significantly enhanced the strength under low to moderate normal stresses, as reported by Ahmed [93]. The optimal amount of tyre shreds/chips needed to improve the shear strength of the mixtures can vary depending on the source of the materials or the test conditions, as shown in Table 1. However, unlike the observations for sand–rubber chip/shred mixtures, adding tyre crumbs or granulated rubber into sand, ballast, or waste granulates (e.g., CW with or without SFS) actually reduced the shear strength [86,96]. This decrease in the shear strength occurs because the rubber crumbs cannot reinforce the mixture as random shreds/chips do, as observed by Tatlisoz et al. [81], and because in most scenarios, the rubber crumbs are not as strong as the host materials in the mixtures. Figure 4 shows the opposite effect of rubber crumbs and rubber shreds on the shear strength of their mixtures with sand.

Beyond the amount of tyre chips, factors such as the aspect ratio, chip width, normal stress, and the unit weight of the sand matrix may also influence the shear strength of mixtures of sand–tyre chips. However, the effect of particle size ratio on the shear strength of mixtures of sand–rubber shred mixtures is also not consistent. For instance, Lee et al. [97] indicated that the relatively higher size ratio increases its strength. Studies by Ghazavi and Sakhi [82], Zornberg et al. [84], and Rao and Dutta [98] using controlled rectangular-shaped tyre chips with specific dimensions and but varying thicknesses revealed that higher size ratios of tyre chips significantly improved the initial friction angle. On the contrary, Ahmed [93] found that the size of tyre chips had a negligible impact on the shear strength of sand–rubber mixtures.

Table 1. Effect of rubber content on the shear strength of rubber mixed soil.

Effect Type	Reference	Mixtures	Rubber Content (%)	Test/Apparatus	Optimal Rubber Content Considering Shear Strength
Positive effect	Ahmed [93]	Sand–tyre rubber	0–100	Static and dynamic triaxial test	39% by mass
	Edil and Bosscher [80]	Sand–scrap tyre	5, 10, 25	Large-scale direct shear test	10% by volume (5% by mass)
	Tatlisoz et al. [81]	Silty sand–rubber chips/shreds	10, 20, 30	Large-scale direct shear test	At least 30% of rubber chip content
	Foose et al. [79]	Sand–tyre shreds	0, 10, 20, 30	Direct shear test	30% rubber content maximized shear strength
	Zornberg et al. [84]	Sand–tyre shreds	0, 5, 10, 15, 30, 38, 60, 100	Large-scale triaxial test	35% of tyre shreds by mass
	Ghazavi and Sakhi [82]	Sand–rubber shreds	15, 30, 50	Large-scale direct shear test	Friction angle peaks at 50% rubber shreds
	Rao and Dutta [98]	Sand–tyre chips	0, 5, 10, 15, 20, 30, 40, 60, 80, 100	Drained triaxial test; repetitive load test	20% by mass
	Mashiri et al. [85]	Sand–tyre chips	0, 10, 20, 30, 35, 40	Consolidated drained monotonic triaxial	35% by mass
	Anbazhagan et al. [95]	Sand–tyre chips	10, 15, 20, 25, 30, 35	Large-scale direct shear test	30% by volume
	Ahmed et al. [87]	Sand–tyre chips	0, 10, 20, 30, 40	Constant shear-drained stress path	20% by mass
Negative effect	Lee et al. [83]	Sand–tyre chips	40	Consolidated drained triaxial test	Shear strength falls between sand and pure tyre chips
	Youwai and Bergado [94]	Sand–tyre rubber shreds	0, 20, 30, 40, 50, 100%	Isotropic consolidated drained triaxial test	Shear strength decreases as tyre content increases
	Sheikh et al. [86]	Sand–tyre crumbs	0, 10, 20, 30, 40	Static triaxial test	Shear strength decreases as tyre content increases
	Indraratna et al. [42]	CW + SFS + rubber crumbs	0, 10, 20, 30, 40	Monotonic triaxial; drained cyclic triaxial	Shear strength decreases as crumb content increases
	Tawk and Indraratna [58]	CW + rubber crumbs	0, 5, 10, 15	Drained static triaxial	Shear strength decreases as crumb content increases

Table 1. Effect of rubber content on the shear strength of rubber mixed soil.

Effect Type	Reference	Mixtures	Rubber Content (%)	Test/Apparatus	Optimal Rubber Content Considering Shear Strength
Positive effect	Ahmed [93]	Sand–tyre rubber	0–100	Static and dynamic triaxial test	39% by mass
	Edil and Bosscher [80]	Sand–scrap tyre	5, 10, 25	Large-scale direct shear test	10% by volume (5% by mass)
	Tatlisoz et al. [81]	Silty sand–rubber chips/shreds	10, 20, 30	Large-scale direct shear test	At least 30% of rubber chip content
	Foose et al. [79]	Sand–tyre shreds	0, 10, 20, 30	Direct shear test	30% rubber content maximized shear strength
	Zornberg et al. [84]	Sand–tyre shreds	0, 5, 10, 15, 30, 38, 60, 100	Large-scale triaxial test	35% of tyre shreds by mass
	Ghazavi and Sakhi [82]	Sand–rubber shreds	15, 30, 50	Large-scale direct shear test	Friction angle peaks at 50% rubber shreds
	Rao and Dutta [98]	Sand–tyre chips	0, 5, 10, 15, 20, 30, 40, 60, 80, 100	Drained triaxial test; repetitive load test	20% by mass
	Mashiri et al. [85]	Sand–tyre chips	0, 10, 20, 30, 35, 40	Consolidated drained monotonic triaxial	35% by mass
	Anbazhagan et al. [95]	Sand–tyre chips	10, 15, 20, 25, 30, 35	Large-scale direct shear test	30% by volume
	Ahmed et al. [87]	Sand–tyre chips	0, 10, 20, 30, 40	Constant shear-drained stress path	20% by mass
Negative effect	Lee et al. [83]	Sand–tyre chips	40	Consolidated drained triaxial test	Shear strength falls between sand and pure tyre chips
	Youwai and Bergado [94]	Sand–tyre rubber shreds	0, 20, 30, 40, 50, 100%	Isotropic consolidated drained triaxial test	Shear strength decreases as tyre content increases
	Sheikh et al. [86]	Sand–tyre crumbs	0, 10, 20, 30, 40	Static triaxial test	Shear strength decreases as tyre content increases
	Indraratna et al. [42]	CW + SFS + rubber crumbs	0, 10, 20, 30, 40	Monotonic triaxial; drained cyclic triaxial	Shear strength decreases as crumb content increases
	Tawk and Indraratna [58]	CW + rubber crumbs	0, 5, 10, 15	Drained static triaxial	Shear strength decreases as crumb content increases

3. Constitutive Models for Materials Mixed with Rubber

The number of constitutive models from the literature for aggregates mixed with rubber is quite limited. The few existing ones, including those developed for mixtures of sand–rubber (i.e., [83,94,99,100,101]), mixtures of CW + RC (i.e., [77]), and mixtures of SFS + CW + RC (i.e., [102]) were developed for monotonic loading conditions inspired by existing models for granular soils, and they are reviewed in the sections below.

3.1. Constitutive Models for Sand–Rubber Mixtures

The first constitutive model for sand–rubber mixtures under monotonic loading for use as a lightweight backfill was proposed by Lee et al. [83], following the hyperbolic model proposed by Duncan et al. [103]. This hyperbolic model concept has also been applied in several other studies to predict the strength and deformation characteristics of sand–rubber blends (i.e., [104]). The relationship between deviatoric stress (q) and axial strain (${\epsilon }_1$) is defined by the hyperbolic function:

$$\mathrm{q}={\displaystyle \frac{{\epsilon }_1}{\frac{1}{{\mathrm{E}}_{\mathrm{i}}}+\frac{{\epsilon }_1}{{\mathrm{q}}_{\mathrm{u}\mathrm{l}\mathrm{t}}}}}$$

(1)

where E_i refers to the initial tangent of Young’s modulus, and q_ult refers to the asymptotic (ultimate) deviatoric stress that corresponds to the deviatoric stress at failure. The overall dilatancy characteristics cannot be fully captured by hyperbolic models, and neither can the dilative and contractive behaviour. This model can only represent contractive behaviour—it cannot predict the dilative response—and the post-peak strain softening behaviour of rubber–sand mixtures, as shown in Figure 5.

Youwai and Bergado [94] and Mashiri et al. [99] then modelled the static behaviour of rubber and sand mixtures by utilizing a hypoplasticity model, which captured the overall stress and strain behaviour and the contractive and dilative behaviour, thus addressing the drawbacks of the hyperbolic models. The hypoplastic model presented by Youwai and Bergado [94] is derived from the critical state framework previously developed for sand by Li and Dafalias [105]. The general equations for this model are derived from the plasticity theory introduced by Dafalias [106]:

$$\left\{{\displaystyle \frac{\partial \mathrm{q}}{\partial {\mathrm{p}}^{/}}}\right\}=\left[\left(\begin{array}{cc}3\mathrm{G}& 0\\ 0& \mathrm{K}\end{array}\right)-{\displaystyle \frac{1}{{\mathrm{K}}_{\mathrm{p}}+3\mathrm{G}-\mathrm{K}\mathsf{\eta }\mathrm{d}}}\left(\begin{array}{cc}9{\mathrm{G}}^2& -3\mathrm{K}\mathrm{G}\mathsf{\eta }\\ 3\mathrm{K}\mathrm{G}\mathrm{d}& -{\mathrm{K}}^2\mathsf{\eta }\mathrm{d}\end{array}\right)\right]\left({\displaystyle \frac{\partial {\epsilon }_{\mathrm{s}}}{\partial {\epsilon }_{\mathrm{v}}}}\right)$$

(2)

where ∂q and ∂p′ refer to the increment of deviatoric and mean effective stress, respectively, and G and K are the elastic shear and elastic bulk modulus, respectively. K_P, η, and d denote the plastic modulus, stress ratio, and dilatancy, and ${\partial \epsilon }_s \mathrm{a}\mathrm{n}\mathrm{d}{\partial \epsilon }_v$ are the increments of deviator strain and volumetric strain.

The state parameter (ψ), as established by Been and Jefferies [107], was also incorporated to account for state dependence. Meanwhile, Youwai and Bergado [94] observed that the initial dilatancy of mixtures of shredded rubber tyres and sand increased with the mean stress (p), leading to the following modification of the dilatancy equation:

$$\mathrm{d}={\mathrm{k}}_{\mathrm{d}}({\displaystyle \frac{\mathrm{p}}{{\mathrm{p}}_{\mathrm{a}}}})\left({\mathrm{e}}^{\mathrm{m}\mathsf{\psi }}-{\displaystyle \frac{\mathsf{\eta }}{{\mathrm{M}}_{\mathrm{c}\mathrm{s}}}}\right)$$

(3)

where k_d and m are the dilatancy parameters, and p_a and M_CS denote the atmospheric pressure and critical stress ratio, respectively.

Although this model accurately captures the overall dilatancy, strength, and deformation, determining the accurate critical state for sand and rubber mixtures was difficult because the mixture tended to dilate at large axial strains (25–100%); consequently, the authors assumed that the steady state was reached at 25% strain. In typical granular soils, the critical state is usually achieved after the axial strains exceed 10% [108]. However, for sand–tyre rubber mixtures, reaching a critical state within the strain limit in the laboratory condition proves to be challenging, particularly in mixtures with high tyre content [109]. To address this problem, Disfani et al. [110] allowed the shearing process to continue until an axial strain of approximately 25% was achieved, and this was considered to be the end-of-test state; in their study, this was assumed to represent the critical state. Moreover, Youwai and Bergado [94] confirmed that blends of sand and rubber with the rubber varying from 30% to 100% were unable to reach a fully developed critical state under laboratory conditions. Similarly, it was therefore assumed that the condition at the end of the test was taken as the critical state.

Mashiri et al. [99] presented a semi-empirical constitutive model to capture the monotonic shear behaviour of sand and tyre chip blends based on the critical state framework, plasticity theory, and the state parameter. Since defining the critical state for mixtures of rubber and tyre chips is challenging, the framework was modified into a constant stress ratio approach for the modified hardening and dilatancy functions and the modified state parameter. Shortly after softening, as the dilatancy and increments in the stress ratio decreased, the material reached a constant stress state; thus, the constant stress ratio proposed by Mashiri et al. [85] was used instead of the critical stress ratio. The dilatancy equation developed by Li and Dafalias [105] was adopted by incorporating the hardening parameter and eliminating its dependence on the shear modulus; a new framework was then introduced to identify the model parameters. Mashiri et al. [85] modified the state parameter introduced by Been and Jefferies [107] to quantify the vertical distance between the void ratio (e) of the sand and tyre chip mixture and the constant stress ratio state line (e_CSR); this modified state parameter (${\mathsf{\psi }}^{\mathrm{*}}$) was adopted in the model as follows:

$${\mathsf{\psi }}^{\mathrm{*}}=\mathrm{e}-{\mathrm{e}}_{\mathrm{C}\mathrm{S}\mathrm{R}}$$

(4)

The dilatancy function and the hardening modulus proposed by Li and Dafalias [105] were modified as follows:

$$\mathrm{d}={\mathrm{d}}_0({\mathrm{e}}^{\mathrm{m}{\mathsf{\psi }}^{\mathrm{*}}}-{\displaystyle \frac{{\mathrm{ŋ}}^{\mathrm{*}}}{{\mathrm{M}}_{\mathrm{C}\mathrm{S}\mathrm{R}}^{\mathrm{*}}}})$$

(5)

$${\mathrm{K}}_{\mathrm{p}}^{\mathrm{*}}={\mathrm{H}}^{\mathrm{*}}({\displaystyle \frac{{\mathrm{M}}_{\mathrm{C}\mathrm{S}\mathrm{R}}^{\mathrm{*}}}{{\mathrm{ŋ}}^{\mathrm{*}}}}-{\mathrm{e}}^{\mathrm{m}{\mathsf{\psi }}^{\mathrm{*}}})$$

(6)

where d₀ is the dilatancy parameters, ${ŋ}^{*}$ is the yield stress ratio, ${\mathrm{M}}_{\mathrm{C}\mathrm{S}\mathrm{R}}^{\mathrm{*}}$ is the equivalent stress ratio at constant stress state, ${\mathrm{K}}_{\mathrm{p}}^{\mathrm{*}}$ is the hardening modulus, and H* is the parameter of the hardening modulus.

This model successfully predicted the hardening and softening of the sand and tyre chip mixture and captured the experimental behaviour of sand mixed with tyres with different amounts of rubber chips. The constant stress ratio is influenced by the relative density of the sand and tyre chip mixtures, but it was not incorporated into the model. Further experimental studies are needed to establish the relationship between the constant stress ratio and relative density.

Cui et al. [101] recently developed an elastoplastic constitutive model within the critical-state framework to predict the dilatancy and stress–strain of sand and rubber with varying amounts of rubber. Cui et al. [101] considered the effect that the amount of rubber would have on the elastic, critical-state, and dilatancy parameters where the amount of rubber by volume (R_V) is a variable. A polynomial function was proposed to count for the nonlinear correlation between those model parameters and the amount of rubber:

$$\mathrm{y}=\mathrm{a}+\mathrm{b}{\mathrm{R}}_{\mathrm{v}}+\mathrm{c}{\mathrm{R}}_{\mathrm{v}}^2$$

(7)

where y represents the elastic, critical-state (slope, ${\mathsf{\lambda }}_{\mathrm{c}}$ and intercept ${\mathrm{e}}_{\Gamma }$ of the critical-state line, CSL), and the dilatancy model parameters, and a, b, and c are the fitting parameters. Figure 6 shows the polynomial relationship of the critical-state parameters changing with the amount of rubber. The influence of the amount of rubber is then incorporated into the model through the state parameter ψ:

$$\mathsf{\psi }=\mathrm{e}-{\mathrm{e}}_{\mathrm{c}\mathrm{s}}=e-{\mathrm{e}}_{\Gamma }+{\mathsf{\lambda }}_{\mathrm{c}}\ln {{\mathrm{p}}^{\prime }}_{\mathrm{c}\mathrm{s}}$$

(8)

where e_CS and p′_CS denote the void ratio and effective mean stress (kPa) at the critical state.

The dilatancy function was then modified from Li and Dafalias [105] to reflect the dilatancy coefficient of rubber and sand mixtures as follows:

$$\mathrm{d}={\mathrm{d}}_0{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}({\mathrm{e}}^{\mathrm{m}\mathsf{\psi }}-{\displaystyle \frac{\mathsf{\eta }}{{\mathrm{M}}_{\mathrm{c}\mathrm{s}}}}+1)$$

(9)

where $\mathsf{\eta }=\mathrm{q}/{\mathrm{p}}^{\prime }$ is the stress ratio, while the dilatancy model parameters d₀ and m are defined to fit a polynomial function based on the amount of rubber, and they are expressed as follows:

$${\mathrm{d}}_0=1.79455-0.02645{\mathrm{R}}_{\mathrm{V}}+9.09091\times {10}^{-5}{\mathrm{R}}_{\mathrm{V}}^2$$

(10)

$$\mathrm{m}=1.50055+0.02115{\mathrm{R}}_{\mathrm{V}}-3.09091 \times {10}^{-4}{\mathrm{R}}_{\mathrm{V}}^2$$

(11)

In addition, the elastic model parameter G₀ is also defined as a polynomial function of R_V and is expressed as follows:

$${\mathrm{G}}_0=85-4.5{\mathrm{R}}_{\mathrm{V}}+0.1{\mathrm{R}}_{\mathrm{V}}^2$$

(12)

By incorporating these modifications, the model effectively captures the critical-state characteristics, the strain softening behaviour, and the dilatancy characteristics of sand and rubber mixtures. This study further highlights how the confining pressure and amount of rubber influence the peak deviatoric stress, the dilatancy, and the volumetric strain in mixtures of rubber and sand.

In addition to implicit constitutive models utilized to predict the behaviour of rubber and sand mixtures, Li et al. [111] developed an empirical model to determine the maximum dynamic shear modulus of sand and rubber blends by considering the effect of the confining pressure, the size of rubber particle, and the volume of rubber. The maximum shear modulus (G_dmax) of a sand and rubber mixture is described in terms of the confining pressure (${\sigma }_3$) and the rubber content by volume (R_V). G_dmax was determined by analysing the experimental data using the multiple regression analysis method below:

$${\mathrm{G}}_{\mathrm{d}\mathrm{m}\mathrm{a}\mathrm{x}}={\mathrm{G}}_1({\mathsf{\alpha }}_{\mathrm{i}}{\mathrm{R}}_{\mathrm{V}}+{\mathsf{\lambda }}_{\mathrm{i}})\left({\mathrm{ϴ}}_{\mathrm{i}}{\mathrm{P}}_{\mathrm{a}}{\left({\displaystyle \frac{{\mathsf{\sigma }}_3}{{\mathrm{P}}_{\mathrm{a}}}}\right)}^{{\mathsf{\beta }}_{\mathrm{i}}}\right)$$

(13)

where G₁ represents the model parameter; ${\mathsf{\alpha }}_{\mathrm{i}}$,${\mathsf{\beta }}_i$, ${\mathsf{\lambda }}_{\mathrm{i}}$, and ${\mathrm{ϴ}}_{\mathrm{i}}$ are the fitting parameters; and i denotes the size of the rubber particles. P_a also refers to the reference stress, which is considered to be equivalent to the atmospheric pressure. This empirical model accurately represents the attenuation of the shear modulus, which is essential for modelling the dynamic behaviour of sand and rubber mixtures.

3.2. Dilatancy Model for SFS + CW + RC Mixtures

Qi et al. [102] developed a bounding surface model within the critical-state framework for mixtures of steel furnace slag (SFS), coal wash (CW), and rubber crumbs (RC) for railway capping layers, i.e., SFS + CW + RC mixtures. Note that only those mixtures with an optimal mixing ratio of SFS:CW = 7:3 were considered in the model [42,112]. The highlight of this model is that it can capture the energy absorbing property of the mixtures and the influence of the amount of rubber and accurately predict the dilatancy of the mixtures via the semi-empirical relationships of critical-state parameters. Unlike conventional granular materials, the critical-state parameters for SFS+CW+RC mixtures can change with the amount of rubber due to its varying energy absorbing capacity [41,113]. Hence, an empirical relationship was established between the critical-state stress ratio $M_{cs}^{\ast }$ and the total work input W_total [41], as shown in Figure 7a.

$$M_{cs}^{*}={\mathrm{M}}_0\times {\left({\displaystyle \frac{{\mathrm{W}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}}{{\mathrm{W}}_0}}\right)}^{\mathsf{\alpha }}$$

(14)

where M₀ is the critical-state ratio if the total work input is equal to 1 kPa, W₀ is the unit work input to ensure the units at both sides of the equation are the same, and α is a material constant that can be obtained through curve regression fitting. The value of M₀ and α is shown in Figure 7a. W_total here indicates the total work input when the stress–strain curves reach the peak point, and it can be calculated with the equation below:

$$\mathrm{d}\mathrm{W}={\mathrm{p}}^{\prime }\mathrm{d}{\epsilon }_{\mathrm{v}}+\mathrm{q}\mathrm{d}{\epsilon }_{\mathrm{q}}$$

(15)

where $\mathrm{d}{\epsilon }_{\mathrm{v}}$ and $\mathrm{d}{\epsilon }_{\mathrm{q}}$ are the total volumetric and deviatoric strain increments. The critical-state lines for SFS + CW + RC mixtures are linear in the space of $e-\ln p^{\prime }$, represented by Equation (16), and are found to rotate clockwise as the amount of rubber increases in the mixtures, as seen in Figure 7b. Hence, the parameters ${\mathsf{\Gamma }}^{*}$ and ${\lambda }^{*}$ are linked with the dosage of RC by mass, R_b(%):

$${\mathrm{e}}_{\mathrm{c}\mathrm{s}}={\mathsf{\Gamma }}^{\mathrm{*}}-{\mathsf{\lambda }}^{\mathrm{*}}\ln {{\mathrm{p}}^{\prime }}_{\mathrm{c}\mathrm{s}}$$

(16)

$${\mathsf{\Gamma }}^{\mathrm{*}}={\mathsf{\Gamma }}_1+{\mathsf{\Gamma }}_2{\mathrm{R}}_{\mathrm{b}}$$

(17)

$${\mathsf{\lambda }}^{\mathrm{*}}={\mathsf{\lambda }}_1+{\mathsf{\lambda }}_2{\mathrm{R}}_{\mathrm{b}}$$

(18)

where ${\mathsf{\Gamma }}_{\mathrm{1,2}}$ and ${\mathsf{\lambda }}_{\mathrm{1,2}}$ are the regression parameters obtained via Figure 7b, and their values are 0.64, 0.01, 0.069, and 0.003, respectively. By substituting Equations (17) and (18) into Equation (16), a critical-state surface for the SFS + CW + RC mixtures that change with the amount of RC can be established, as shown in Figure 8:

$${\mathrm{e}}_{\mathrm{c}\mathrm{s}}={\mathsf{\Gamma }}_1+{\mathsf{\Gamma }}_2{\mathrm{R}}_{\mathrm{b}}-({\mathsf{\lambda }}_1+{\mathsf{\lambda }}_2{\mathrm{R}}_{\mathrm{b}})\ln {{\mathrm{p}}^{\prime }}_{\mathrm{c}\mathrm{s}}$$

(19)

Therefore, the state parameter ${\mathsf{\psi }}_0=\mathrm{e}-{\mathrm{e}}_{\mathrm{c}\mathrm{s}}$ is updated to reflect the relationship with the amount of RC:

$${\mathsf{\psi }}_0=\mathrm{e}-{\mathsf{\Gamma }}_1+{\mathsf{\Gamma }}_2{\mathrm{R}}_{\mathrm{b}}-({\mathsf{\lambda }}_1+{\mathsf{\lambda }}_2{\mathrm{R}}_{\mathrm{b}})\ln {{\mathrm{p}}^{\prime }}_{\mathrm{c}\mathrm{s}}$$

(20)

The dilatancy response of the mixtures is calculated following [105] and can be expressed as follows:

$$\mathrm{d}={\displaystyle \frac{\mathrm{d}{\epsilon }_{\mathrm{v}}^{\mathrm{p}}}{\mathrm{d}{\epsilon }_{\mathrm{q}}^{\mathrm{p}}}}={\mathrm{d}}_0({\mathrm{e}}^{\mathrm{m}{\mathsf{\psi }}_0}-{\displaystyle \frac{\mathsf{\eta }}{M_{cs}^{*}}})$$

(21)

where $\mathrm{d}{\epsilon }_{\mathrm{v}}^{\mathrm{p}}$ and $\mathrm{d}{\epsilon }_{\mathrm{q}}^{\mathrm{p}}$ are the increments in plastic volumetric and deviatoric strains, respectively. $M_{cs}^{*}$ and ${\mathsf{\psi }}_0$ indicate the modified critical-state stress ratio and state parameter as they relate to the total work input and amount of RC represented by Equation (14) and Equation (20), respectively. Figure 9 is a comparison between the experimental data and the model prediction, and this comparison indicates a very good match.

3.3. Constitutive Model for CW + RC Mixtures

Riyad et al. [77] developed a new constitutive model for waste granular mixtures for railway capping layers based on the bounding surface concept, it includes a modified void ratio (e*) to better capture the compressibility of rubber. Experimental data from consolidated drained triaxial tests on mixtures of coal wash and rubber crumbs (CW+RC) with varying rubber crumbs contents by mass (R_b), as reported by Indraratna et al. [114] was utilised to calibrate and validate the model.

Inspired by Jefferies and Been [115], this study considered a bullet-shaped bounding surface that encompasses the triaxial compression part. The shape of the bounding surface is assumed to be identical to the loading surface, i.e., where the bounding surface ${\mathrm{F}}_{\mathrm{b}}\left(\overline{{\mathrm{p}}^{\prime }},\overline{\mathrm{q}},\overline{{\mathrm{p}}_{\mathrm{c}}^{\prime }}\right)=0$ and for the loading surface ${\mathrm{f}}_{\mathrm{L}}\left({\mathrm{p}}^{\prime },\mathrm{q},{\mathrm{p}}_{\mathrm{c}}^{\prime }\right)=0$ are in densely compacted conditions (Figure 10).

The bounding surface and the loading surface are defined as shown in Equation (22) and Equation (23) [77].

$${\mathrm{F}}_{\mathrm{b}}\left(\overline{{\mathrm{p}}^{\prime }},\overline{\mathrm{q}},\overline{{\mathrm{p}}_{\mathrm{c}}^{\prime }}\right)=\left\{\overline{\mathrm{q}}-{(\mathrm{M}}_{\mathrm{c}\mathrm{s}}+\mathrm{N}{\mathsf{\psi }}_{\mathrm{i}}{\mathsf{\chi }}_{\mathrm{i}})\overline{{\mathrm{p}}^{\prime }}\left[1+\ln \left({\displaystyle \frac{\mathrm{R}\overline{{\mathrm{p}}_{\mathrm{c}}^{\prime }}}{\overline{{\mathrm{p}}^{\prime }}}}\right)\right]\right\}=0, \mathrm{f}\mathrm{o}\mathrm{r} {\mathsf{\psi }}_{\mathrm{i}}<0$$

(22)

$${\mathrm{f}}_{\mathrm{L}}\left({\mathrm{p}}^{\prime },\mathrm{q},{\mathrm{p}}_{\mathrm{c}}^{\prime }\right)=\left\{\mathrm{q}-{(\mathrm{M}}_{\mathrm{c}\mathrm{s}}+\mathrm{N}{\mathsf{\psi }}_{\mathrm{i}}{\mathsf{\chi }}_{\mathrm{i}}){\mathrm{p}}^{\prime }\left[1+\ln \left({\displaystyle \frac{\mathrm{R}{\mathrm{p}}_{\mathrm{c}}^{\prime }}{{\mathrm{p}}^{\prime }}}\right)\right]\right\}=0, \mathrm{f}\mathrm{o}\mathrm{r} {\mathsf{\psi }}_{\mathrm{i}}<0$$

(23)

$$\mathrm{where} \overline{{\mathrm{p}}_{\mathrm{c}}^{\prime }}={\displaystyle \frac{{\mathrm{p}}_{\mathrm{r}}}{\mathrm{R}}}\mathrm{e}\mathrm{x}\mathrm{p}\left({\displaystyle \frac{\Gamma -{\mathrm{e}}^{\mathrm{*}}-\mathsf{\kappa }\mathrm{l}\mathrm{n}{\mathrm{p}}^{\prime }}{\mathsf{\lambda }-\mathrm{k}}}\right)$$

(24)

$${\mathrm{e}}^{\mathrm{*}}=\left\{{\mathrm{e}}_0+{\epsilon }_{\mathrm{v}}^{\mathrm{*}}(1+{\mathrm{e}}_0)\right\}\left(1+{\displaystyle \frac{{\mathrm{R}}_{\mathrm{b}}}{100}}\times {\displaystyle \frac{{\mathrm{G}}_{\mathrm{s},\mathrm{C}\mathrm{W}}}{{\mathrm{G}}_{\mathrm{s},\mathrm{R}\mathrm{C}}}}\right)$$

(25)

$${\mathsf{\psi }}_{\mathrm{i}}={\mathrm{e}}^{\mathrm{*}}-{\mathrm{e}}_{\mathrm{c}\mathrm{s},\mathrm{i}}$$

(26)

where ${\mathrm{p}}_{\mathrm{c}}^{\prime }$ and $\overline{{\mathrm{p}}_{\mathrm{c}}^{\prime }}$ represents the intercept between the loading and bounding surfaces with the q = 0 axis, respectively (Figure 10). The stress ratio at the image state ${(\mathrm{M}}_{\mathrm{c}\mathrm{s}}+\mathrm{N}{\mathsf{\psi }}_{\mathrm{i}}{\mathsf{\chi }}_{\mathrm{i}})$ is denoted as M_i in Figure 10. The material constant R relates the intercepts of the bounding and loading surfaces with the M_i line. p_r is a reference stress (typically 1 kPa) used for dimensional consistency. $\mathsf{\lambda }$ and $\mathsf{\kappa }$ are the gradients of the critical-state void ratio and swelling lines. Γ is the void ratio at p_r, and ${\mathsf{\psi }}_{\mathrm{i}}$ represents the state parameter at the image state. e* is the modified void ratio, which incorporates the change in volume of the rubber crumbs. G_S,CW and G_S,RC are the specific gravities of coal wash and rubber crumbs, respectively. ${\epsilon }_{\mathrm{v}}^{\mathrm{*}}$ is the void volumetric strain and only accounts for the change in volume of stiff particles (i.e., CW). N is the volumetric coupling parameter, which is proposed to lessen the effect of dilatancy on the shear strength, inspired by Nova and Wood [116], as expressed in the below equation:

$$\mathsf{\eta }=\left(\mathrm{N}-1\right){\mathrm{D}}^{\mathrm{p}}+{\mathrm{M}}_{\mathrm{c}\mathrm{s}}$$

(27)

${\mathrm{D}}^{\mathrm{p}}={\displaystyle \frac{\mathrm{d}{\epsilon }_{\mathrm{v}}^{\mathrm{p}}}{\mathrm{d}{\epsilon }_{\mathrm{q}}^{\mathrm{p}}}}$ is the plastic dilatancy. The dilatancy at the peak deviator stress ${\mathrm{D}}_{\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}}^{\mathrm{p}}$ is typically linear and correlated with the state parameter at the peak deviator stress ${\mathsf{\psi }}_{\mathrm{ŋ}\mathrm{m}\mathrm{a}\mathrm{x}}$ [77]. Since there is no elastic strain at peak stress, ${\mathrm{D}}_{\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}}^{\mathrm{p}}={\mathrm{D}}_{\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}}$.

$${{\mathrm{D}}_{\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}}=\mathrm{D}}_{\mathrm{p}\mathrm{e}\mathrm{a}\mathrm{k}}^{\mathrm{p}}=\mathsf{\chi }{\mathsf{\psi }}_{\mathrm{ŋ}\mathrm{m}\mathrm{a}\mathrm{x}}$$

(28)

where $\mathsf{\chi }$ is the dilatancy constant at the image state condition and is given by Equation (26):

$${\mathsf{\chi }}_{\mathrm{i}}={\displaystyle \frac{\mathsf{\chi }{\mathrm{M}}_{\mathrm{c}\mathrm{s}}}{{\mathrm{M}}_{\mathrm{c}\mathrm{s}}-\mathsf{\lambda }\mathsf{\chi }}}$$

(29)

The plastic dilatancy D^p is related to the plastic potential and is obtained by substituting Equation (27) into Equation (22):

$${\mathrm{D}}^{\mathrm{p}}={\mathrm{M}}_{\mathrm{i}}\mathrm{l}\mathrm{n}\left({\displaystyle \frac{\overline{{\mathrm{p}}^{\prime }}}{\mathrm{R}\overline{{\mathrm{p}}_{\mathrm{c}}^{\prime }}}}\right)$$

(30)

The hardening rule in this study is expressed mathematically by Equation (31):

$${\displaystyle \frac{\mathrm{d}\left(\mathrm{R}\overline{{\mathrm{p}}_{\mathrm{c}}^{\prime }}\right)}{\mathrm{R}\overline{{\mathrm{p}}_{\mathrm{c}}^{\prime }}}}=\mathrm{H}\times {\left({\displaystyle \frac{\mathrm{R}\overline{{\mathrm{p}}_{\mathrm{c}}^{\prime }}}{\overline{{\mathrm{p}}^{\prime }}}}\right)}^{-2}\times \left({\left(\overline{{\mathrm{p}}_{\mathrm{c}}^{\prime }}\right)}_{\mathrm{m}\mathrm{a}\mathrm{x}}-\overline{{\mathrm{p}}_{\mathrm{c}}^{\prime }}\right)\times {\displaystyle \frac{\mathrm{d}{\epsilon }_{\mathrm{q}}^{\mathrm{p}}}{\overline{{\mathrm{p}}_{\mathrm{c}}^{\prime }}}}$$

(31)

$$\mathrm{H}={\mathrm{H}}_{\mathrm{m}\mathrm{i}\mathrm{n}}{\left({\mathsf{\sigma }}_3^{\prime }\right)}^{-\mathsf{\beta }}\mathrm{e}\mathrm{x}\mathrm{p}(-{\mathsf{\delta }}_{\mathrm{H}}{\mathrm{R}}_{\mathrm{b}})$$

(32)

where H_min is the minimum hardening modulus at p_r, β relates to changes in the confining pressure $\left({\mathsf{\sigma }}_3^{\prime }\right),$ and ${\mathsf{\delta }}_{\mathrm{H}}$ denotes the diminution of H with the amount of rubber.

Figure 11 shows the predicted stress–strain and volumetric strain curves for CW + RC mixtures; they match the test data from [114]. The model thus accurately captures the overall volumetric response and stress–strain relationship, with strain hardening and the peak deviator stress increasing with the confining pressure. As the amount of rubber increases, the peak deviator stress decreases, and strain softening shifts to strain hardening; this indicates a transition from brittle to ductile behaviour.

4. Model Limitations and Applications

Table 2. Highlights and limitations of the constitutive models for rubber-mixed materials.

References	Mixtures	Model Concepts	Highlights	Limitations
Lee et al. [83]	Sand–rubber crumbs	Hyperbolic model	A simple model to capture the strain hardening response of sand–rubber mixtures	Cannot capture the strain-softening response and the dilative behaviour
Youwai and Bergado [94]	Sand–rubber shreds	Hypoplastic model; critical-state framework	Assumed that the condition at the end of the test was taken as the critical state	Could not determine the accurate critical-state parameters
Mashiri et al. [99]	Sand–rubber chips	Hypoplastic model; critical-state framework	Developed the constant stress ratio to replace the critical state ratio	The constant stress ratio cannot exactly represent the critical-state ratio; hence, it still induces the divergence between the model prediction and test results
Cui et al. [101]	Sand–rubber crumbs	Elastoplastic constitutive model; critical-state framework	Established polynomial functions to incorporate the influence of rubber on the elastic, critical state, and dilatancy parameters	A lot of curve-fitting relationships may not be suitable for other rubber-mixed materials
Li et al. [111]	Sand-rubber crumbs	Empirical model	Developed an exponential relationship between the maximum shear modulus and the rubber content by volume and confining pressures	Concerning only the maximum shear modulus rather than the stress–strain relationship
Qi et al. [102]	Steel furnace slag–coal wash–rubber crumbs	Bounding surface model; critical-state framework	Extrapolation methods were used to determine the critical state; developed a relationship of energy with the critical-state ratio; modified critical-state and dilatancy parameters with the confining pressures and rubber contents	The influence of particle breakage of coal wash and the deformation of rubber particles was not considered
Riyad et al. [77]	Coal wash–rubber crumbs	Bounding surface model; critical-state framework	Used the revised void ratio to incorporate the deformation of rubber particles	The energy absorbing property of rubber and the breakage of coal wash were not considered

provides a summary of the highlights and limitations of the aforementioned constitutive models. In addition to the inherent limitations of the model itself, as outlined in Table 2, most existing models (with the exception of Riyad et al. [77]) do not account for the deformable nature of rubber, which is a key characteristic that should be incorporated into future research. Moreover, more complex conditions, such as cyclic loading and unsaturated conditions that have yet to be addressed in current models, should also be considered.

It is noteworthy that in all the hypoplastic, elastoplastic, and bounding surface models developed for rubber-mixed materials, the critical-state framework has been consistently adopted. The primary focus of these models has been on modifying the critical-state parameters and determining the critical state. This emphasis stems from the fact that rubber significantly enhances the ductility of the mixtures, which in turn can delay the occurrence of the critical state in the mixtures having small rubber particles or make it more challenging to achieve for the mixtures having larger tyre chips or shreds. The impact of rubber on the critical state varies with the type of host material and rubber particle size, leading to diverse modifications in the critical-state parameters. Consequently, the methods for identifying the critical state also differ depending on the rubber content and rubber size. For instance, a constant stress state or the end-of-the-test state has been considered as the critical state for sand–rubber chips/shreds mixtures [94,99], while extrapolation was used for SFS + CW + RC mixtures having 20–40% rubber by mass [102].

Although the hyperbolic model has seldom been used for rubber-modified materials since Lee et al. [83], it remains a promising approach for loosely packed rubber-mixed materials, as it effectively captures the behaviour of strain hardening. For mixtures containing rubber chips or shreds, the hypoplastic models [94,99] may be more appropriate, as they offer a viable alternative to the critical-state parameters, particularly where the critical state cannot be defined due to the continuous deformation associated with the larger rubber particle sizes. While the elastoplastic constitutive model [101] serves as an excellent reference for sand–rubber mixtures, it should be applied with caution to other rubber-modified materials, as the fitted relationships may not hold due to variations in material composition. Similarly, the bounding surface models [77,102] developed for the mixtures of coal wash–rubber with or without steel furnace slag show strong predictive performance and may be extended to other rubber–waste mixtures with confidence. Finally, the experimental model for maximum shear modulus [111] presents a valuable reference point for developing constitutive models under dynamic loading conditions, contributing to a more accurate representation of rubber-mixed materials in such scenarios.

5. Conclusions and Recommendations

This paper reviews the existing constitutive models developed for materials mixed with rubber, i.e., sand mixed with rubber, mixtures of steel furnace slag–coal wash–rubber crumbs (SFS + CW + RC), and mixtures of coal wash and rubber crumbs (CW + RC). The key findings of this study are outlined below:

• The shear behaviour of rubber and granular mixtures is primarily governed by the confining pressure and the amount of (shredded) rubber tyres. The inclusion of rubber in granular soil mixtures significantly influences the stress–strain response, improves its energy absorption ability, increases its ductility, and reduces its dilative behaviour. Moreover, the shear strength of materials mixed with rubber depends on the type and amount of rubber and the confining pressure. While including rubber shreds and chips in the mixtures enhances the shear strength through particle interlocking, the addition of rubber crumbs tends to reduce the shear strength due to its lack of structural reinforcement;
• The constitutive models developed for mixtures of sand and rubber lightweight backfill material highlight the significant advancements in predicting stress–strain behaviour. The early hyperbolic model struggled with post-peak softening and dilatancy. To overcome these limitations, the hypoplastic model was developed by incorporating state-dependent behaviour to enable the prediction of dilative and compressive behaviour. Critical state and bounding surface plasticity models further refined these predictions by accounting for variations in the amount of rubber and capturing the hardening and softening in mixtures of sand and rubber. An empirical model was also developed to estimate the dynamic shear modulus;
• The dilatancy model developed for mixtures of SFS + CW + RC to be used as a railway capping layer within the critical-state framework also captured the energy absorbing properties of the mixture due to inclusion of rubber and its influence on dilatancy behaviour. Unlike conventional granular materials the critical-state parameters varied with the amount of rubber due to changes in its energy absorption capacity. The predicted stress–strain behaviour of the mixture aligned well with the laboratory observations;
• The constitutive model for mixtures of CW + RC predicted the stress–strain behaviour of mining waste and rubber crumbs using a bounding surface plasticity approach with a compressibility-dependent void ratio to consider the effect of rubber inclusion on the volumetric deformation of the mixture and an image of the critical state. An image-state ratio-based plastic flow rule and a hardening modulus that depends on the amount of rubber and the confining pressure were used to model the stress–strain behaviour. The model remains suitable for capping layers involved with rubber particles, though further refinements are needed to better account for the internal deformation of rubber within the granular matrix.

Existing constitutive models for waste materials mixed with rubber were developed primarily for static loading, whereas the railway capping layers experience train-induced cyclic loading. Future models should capture the cyclic behaviour by incorporating energy absorbed by the rubber through its deformation, the reduced particle breakage of the host materials (e.g., sand, CW, etc.), damping effects, and evolving fabric arrangement under loading (e.g., how the skeleton changes with increasing rubber contents or upon deformation under various loading conditions). Furthermore, the effect of the size and type of rubber (e.g., rubber crumbs, rubber fibre, or rubber chips) should be considered when developing models to ensure more accurate predictions of material behaviour. Finally, while rubber is used in pavements, its mechanical behaviour lacks mathematical analysis (e.g., long-term stress–strain response under repeated loading), which is why developing constitutive models for pavements modified by rubber is essential to predict their performance and optimise their design.