State-of-the-Art Constitutive Modelling of Frozen Soils

Abstract

In recent decades, the constitutive modelling for frozen soils has attracted remarkable attention from scholars and engineers due to the continuously growing constructions in cold regions. Frozen soils exhibit substantial differences in mechanical behaviours compared to unfrozen soils, due to the presence of ice and the complexity of phase changes. Accordingly, it is more difficult to establish constitutive models to reasonably capture the mechanical behaviours of frozen soils than unfrozen soils. This study attempts to present a comprehensive review of the state of the art of constitutive models for frozen soils, which is a focal topic in geotechnical engineering. Various constitutive models of frozen soils under static and dynamic loads are summarised based on their underlying theories. The advantages and limitations of the models are thoroughly discussed. On this basis, the challenges and potential future research possibilities in frozen soil modelling are outlined, including the development of open databases and unified constitutive models with the aid of advanced techniques. It is hoped that the review could facilitate research on describing the mechanical behaviours of frozen soils, and promote a deeper understanding of the thermo-hydro-mechanical (THM) coupled process occurring in cold regions.

1 Introduction

Frozen soils are frequently encountered in various engineering activities. On the one hand, permafrost covers 24% of the earth land surface, accounting for up to 35,760,000 km² [31]. Construction and maintenance of railways, highways as well as pipelines in permafrost regions are inevitably involved with frozen soils. On the other hand, artificial ground freezing (AGF) techniques are widely applied in underground constructions [72, 82, 188], where frozen soils are artificially created to serve as temporary support and groundwater sealing through enhancing the soil modulus and minimising its hydraulic conductivity. Reasonably capturing the mechanical behaviours of frozen soils through robust constitutive models is an urgent task faced by scientists and engineers but remains challenging.

Due to the sensitivity of frozen soil to climate and environmental conditions [104, 142], variations in the mechanical properties of frozen soils can give rise to multiple engineering and environmental issues, such as ground subsidence floor risk, slope instability, infrastructure damage, and climate change impacts (e.g., [12, 26, 92, 103, 109, 137, 175, 187]). The occurrence of these disasters associated with frozen soil has not only garnered significant attention from researchers across various engineering fields, but also prompted corresponding decision-making and research initiatives by government departments. Understanding mechanical behaviours and properties, such as strength, deformation, and creep of frozen soils, plays a pivotal role in the investigation, design, construction, and operation of engineering projects in cold regions undergoing a gradual degradation caused by climate change and human activities [175, 187].

In addition, a large number of significant infrastructure and energy pipeline projects are being implemented in cold regions [175], such as the Qinghai-Tibet Railway (1142 km, Golmud to Lhasa, China [14]), China-Russia Crude Oil Pipeline (953 km, from Skovorodina, Russia to Daqing, China [59]), Alaska Highway (2232 km, British Columbia, Canada to Alaska, USA [18]), and so on. These endeavours highlight the global scale and importance of construction initiatives in cold regions and emphasise the need for specialised engineering approaches and meticulous consideration of the unique and complex mechanical behaviours of frozen soils. Consequently, it is crucial to consider the mechanical behaviours of frozen soils when undertaking engineering activities in cold regions, particularly in light of the potential risks posed by deformation, to ensure safety and provide guidance for minimising these frozen soil engineering disasters and preventive measures for these crucial projects.

The mechanical behaviours of frozen soils are remarkably different from those of unfrozen soils owing to the existence of ice that can significantly alter the mechanical performance of frozen soils [30, 48, 65, 148]. Understanding the distinct mechanical responses of frozen soils is therefore essential for effectively addressing these engineering problems and ensuring the long-term performance and safety of infrastructure in cold regions. Accordingly, the constitutive model is an efficient and theoretical tool to quantitatively describe the complex behaviours of frozen soils. Up to now, tremendous constitutive models for unfrozen soils and other geomaterials have been established and applied in manifold engineering fields [15, 16, 144, 161, 185]. However, the constitutive models of frozen soils differ significantly from those applied to unfrozen soils due to the distinct features of frozen soils induced by ice, such as temperature dependency, time dependency, heterogeneity, and ice content. The unique mechanical behaviours of frozen soils are summarised as follows. (1) Frozen soils display a high level of temperature dependency; hence, the constitutive models for frozen soils should consider this temperature dependency, which is not usually necessary for unfrozen soils. (2) Frozen soils exhibit time-dependent behaviour (e.g., creep), which can substantially impact the deformation of frozen soils. (3) When water freezes into ice and expands, it induces volume change within the soil matrix, a phenomenon not commonly observed in unfrozen soils [107, 113]. Besides, the ice within frozen soils can significantly affect mechanical behaviours, which enhances the cohesive strength of soil solids so that the tensile strength of frozen soil is nonzero [53]. It has been reported that the enhanced strength of frozen soils stems from the intrinsic strength of the soil skeleton and the cementation of ice [9, 41, 55, 108]. (4) Frozen soils are prone to be highly heterogeneous and anisotropic owing to the impact of direction and connectivity of ice lenses [111].

Therefore, a reasonable constitutive model of frozen soils is critical for accurately describing and predicting the mechanical behaviours of frozen soils, which is capable of providing theoretical guidelines for the safety and stability of engineering projects in cold regions and is beneficial for the analysis of the thermo-hydro-mechanical (THM) coupled issues in geotechnical engineering. At present, numerous constitutive models have been proposed to capture and describe the complex mechanical behaviours of frozen soils based on different theories, but few of them can comprehensively account for these aforementioned unique characteristics of frozen soils. Given that numerous studies on constitutive models of frozen soils, herein, the distinguishing investigations in recent two decades have been checked according to the Web of Science (see Fig. 1, 13 articles before 2003). It can be noted that the publications on constitutive models of frozen soils are growing exponentially, while these constitutive models for frozen soils have not yet been adequately summarised.

Fig. 1

Number of publications related to constitutive models of frozen soils in recent two decades (from 2003 to 2023)

This study comprehensively reviews the current investigations on the constitutive models of frozen soils under static and dynamic loadings. A systematic review of constitutive models of frozen soils upon static and dynamic loadings are respectively presented in Sect. 2 and Sect. 3, according to their underlying theories. Subsequently, the advantages and limitations of each type of constitutive model are discussed in Sect. 4. The last section highlights the critical findings and outlines future research trends in the field of constitutive models for frozen soils. This study contributes to a better understanding of the mechanisms governing the multi-physical field coupling of frozen soils and offers valuable references for engineering design and maintenance.

2 Constitutive Models for Static Loading Condition

Herein, the constitutive models for frozen models under static loadings are reviewed, which are divided into ten categories based on their underlying principles.

2.1 Empirical Models

Empirical constitutive models are primarily derived by fitting experimental data of frozen soils at specific conditions, which are easy to compute and often used by engineers [124]. Early scholars proposed empirical constitutive models of frozen soils by fitting the experimental data from uniaxial compression and creep tests under static loadings (see Table 1). Vialov [120] established a power function relation between stress and strain based on uniaxial compression tests. However, Zhu et al. [192] indicated that the power function is not suitable for describing the constitutive relations for frozen soils since some factors (e.g., temperature and water content) could significantly affect the relations. Accordingly, Cai et al. [7] proposed a novel constitutive model for frozen soil by considering the impacts of temperature and loading rate. Besides, Liu and Peng [78] also established an empirical model to describe the stress–strain relation for thawing soils involving the effects of water content, cooling and thawing temperature.

Table 1 Typical investigations on empirical models of frozen soils under static loading

In addition, to describe the creep deformation of frozen soils, several empirical models were also developed. Fish [28] introduced an approximate equation to represent individual creep curves, illustrating the relationship between strain rate (ε̇) versus time (t). However, the applicability of this model was limited as it was only valid for extremely small values of t/t_m (where t_m represents the inflexion point of each creep curve, indicating the temporary minimum strain rate). Moreover, it failed to satisfy the condition where the strain rate reaches a minimum at t = t_m. To address these limitations, Gardner et al. [36] proposed a novel form of ε̇-t. Arenson and Springman [3] derived a straightforward formula to describe the creep behaviours of ice-rich frozen soils near the melting temperatures of ice based on a series of laboratory investigations. Besides, Yin et al. [162] developed a negative exponential nonlinear equation to capture the creep deformation of frozen clay under triaxial stress.

Table 1 summarises typical investigations of the empirical constitutive models of frozen soils. It can be noted that involving less underlying physical mechanics of frozen soils limits the accuracy and applicability of empirical constitutive models to various soil types and conditions. Therefore, empirical models of frozen soils are efficient tools to evaluate the mechanical behaviours of frozen soils, but they should be employed with more caution and validated with experimental data. In addition, frozen soils often suffer complex stress conditions, while the empirical models fail to reflect the mechanical characteristics of frozen soils under generalised stress conditions.

2.2 Elastoplastic Modelling

Elastoplastic models of frozen soils have been proposed and developed in recent years since they can capture frozen soils’ elastic (reversible deformation) and plastic behaviours (irreversible deformation). The advancements of elastoplastic constitutive models of frozen soils are analysed herein.

The plastic relation involves three essential contents: yield function, flow rule, and hardening law of yield surface. The yield function represents the initiation of plastic deformation, while the flow rule, in conjunction with the plastic potential function, governs dilatancy, defining the evolution and direction of plastic deformation. The hardening law describes the magnitude of plastic deformation [83]. Three variables (i.e., shear strain, volumetric strain and plastic work) are often utilised as hardening parameters in the elastoplastic constitutive models of soil [46]. Since the plastic work involves two plasticity variables (volumetric strain and shear strain), adopting the plastic work as a hardening parameter is likely to reduce the number of parameters that need to be determined. Based on the classical elastoplastic theory, the total strain increment (dε) is the summation of elastic strain increment (dε^e) and plastic strain increment (dε^p).

$$d\varepsilon = d\varepsilon^{e} + d\varepsilon^{p}$$

(1)

Fish [29] initially determined a plastic yield surface as the critical state line (CSL) based on experimental results. However, this model did not consider the hardening and softening behaviours of frozen soils, leading to an inability to accurately predict volumetric strain in frozen soils. To solve this limitation, Lai et al. [53] proposed an elastoplastic constitutive model of frozen sandy soil by adopting plastic work as the hardening parameter where the plastic volumetric strain and shear strain were considered. In the work of Lai et al. [53], the K-G model was used to compute the elastic part of strain, and an ellipse yield function was proposed to describe the plastic potential loci based on Drucker's postulate. However, the associated flow rule employed in the model of Lai et al. [53] posed challenges in accurately capturing the volumetric deformation of frozen soil. To address this issue, Yang et al. [158] proposed an elastoplastic model for frozen silt soil that utilised a nonassociated follow rule to describe the stress–strain relationships under high confining pressure. An elastoplastic constitutive model of frozen silt was also developed by Lai et al. [54], which incorporates an elliptical yield surface for the compressive mechanism along with a nonassociated flow rule and two parabolic yield surfaces for the shear mechanism with nonassociated flow rules.

Based on experimental data from triaxial tests conducted by Lai et al. [54], Rotta Loria et al. [83] proposed a non-linear constitutive model for frozen silt by modifying the elastoplastic constitutive model from [54] and adopting the associated flow rule, whose model reduces the number of material parameters and facilitates the practical application of the constitutive model. Xu et al. [153] proposed an elastoplastic model to describe the strength and deformation behaviours of Helin loss under varying temperature and strain rates based on a series of uniaxial compression tests, and the material parameters were obtained by fitting the experimental data. To consider the effect of breakage, an elastoplastic constitutive model for frozen saline soil was established by Chang et al. [10]. Based on a series of triaxial tests with varying temperatures and confine pressures, Shastri et al. [110] derived a Hiss sub-loading model to address the mechanical behaviours of undistributed frozen soils. Yu et al. [165] extended the one-dimensional (1D) elastoplastic model (i.e., Clay and Sand Model, CASM [164]) to describe the mechanical behaviours of both unfrozen soils and frozen soils by considering the ice-bonding impact and using the conception of residual stress. Sun and Zhou [118] built an elastoplastic model for frozen soils by considering three yield surfaces (i.e., reference yield surface, subloading yield surface, and cryogenic suction yield surface). In their model, cryogenic suction and solid-phase stress were employed to reveal the impacts of confining pressure and temperature on the mechanical behaviours of frozen soil, and a nonlinear relationship between cryogenic cohesion and cryogenic suction was established to account for the phenomena of the strength increase with decreasing temperature. Furthermore, the validation of this model was implemented by using four triaxial tests on different frozen soils from other scholars.

Table 2 lists the typical elastoplastic models for frozen soils, where the primary features are also summarised. In these studies, the K-G model is widely used to calculate the elastic part of strain. Some scholars [53, 83] assumed that the bulk modulus (i.e., K) is a power function of σ₃, while G is the parabola function of σ₃. Yang et al. [158] employed the exponential function of σ₃ to calculate K and a parabola function of σ₃ to determine G. Chang et al. [10] adopted exponential functions to describe the relationships between K/G and σ₃. In contrast, Xu et al. [153] derived the empirical relationships between K/G and strain rate by fitting data from uniaxial compression tests. It can be noted from Table 2 that existing elastoplastic models for frozen soils are mainly obtained based on triaxial tests of artificial frozen soils and involve the impact of confining pressure, temperature and strain rate, in addition to the model of Shastri et al. [110] who proposed elastoplastic model for undisturbed natural frozen soil. Besides, the material parameters included in these models are generally obtained by fitting experimental data. When attempting to describe the mechanical behaviours of frozen soils under different external conditions, these constitutive models with complex mathematical formulations relying on many material parameters are adopted, which leads to the challenging and time-consuming analyses of practical engineering. Therefore, developing constitutive models with fewer material parameters considering various factors such as temperature, ice content, and salinity is more attractive, as they can capture the mechanical behaviours of frozen soils under complex conditions.

Table 2 Summary of existing elastoplastic models for frozen soils under static loading

2.3 Viscoplastic Modelling

Frozen soils exhibit significant creep deformation under complex external loadings due to the viscosity of ice and unfrozen water [35, 160]. Therefore, it is essential to incorporate viscous behaviour into constitutive models for frozen soils to accurately describe the relationship between stress, strain, and time.

The rheological model, as a common type of creep model, has been established and applied to predict the creep deformation of geomaterials, which consists of several rheologic elements, e.g., Hooke elastic body (spring), Newton viscous body (dashpot) and Saint–Venant rigid plastic body (slider). A compilation of typical rheological elements and their combinations is provided in Appendix Table A. Furthermore, Table 3 summarises typical creep models of frozen soils based on viscoplastic theory and their primary features. As a classical model, the Nishihara model [95] consists of the Hooke element, Kelvin and Bingham model in series, which has been widely used to describe the transient and steady creep process of frozen soils. However, the linear constitutive elements of Nishihara model are inadequate for capturing nonlinear rheological characteristics, such as accelerating creep and failure processes. To address these limitations, as shown in Table 3, researchers have improved Nishihara model [95] by adding rheological bodies or modifying specific elements (e.g., [43, 44, 60, 62, 64, 141, 159, 176, 196]), as well as combining typical rheological models (e.g., [40, 124, 193]). For example, Li et al. [60] modified Nishihara’s model by applying a parabolic yield function to define a viscoplastic body. Yang et al. [159] established a generalised Burgers model by combining Kelvin's body and Nishihara’s model to address models that fail to describe the accelerated creep stage. Hou et al. [43] developed a novel creep model involving a hardening parameter and a damage variable to account for the strengthening and weakening effects. Zhu et al. [196] developed a nonlinear creep model highlighting the impacts of temperature, dry density and grain size distribution. Specifically, they improved the elastic modulus of viscoelastic elements by considering the stress dependence, revised the coefficient of viscosity in the viscoelastic element to account for time and stress effects, and enhanced the viscoplastic element by incorporating a damage variable.

Table 3 Summary of viscoplastic modelling for frozen soils under static loading

Overall, these rheological models have relatively straightforward physical interpretations and are prone to be easily programmed. However, it is worth noting that combined elements were initially proposed to capture the rheological characteristics of other materials (e.g., metals and rubbers) at temperatures close to their melting points. Besides, when modelling the mechanical behaviours of frozen soils, the presence of heterogeneity and disordered internal structure poses challenges in determining the most appropriate combinations of elements. As a result, caution should be exercised when employing this approach. Simple combined models may struggle to accurately depict the actual mechanical behaviours of frozen soil, while complex combined models could introduce significant mathematical complexities.

Notably, the rheological elementary models derived from viscoplastic theory are primarily 1D models; therefore, the flow rule should be incorporated to describe frozen soils' mechanical behaviours under three-dimensional (3D) stress conditions. Accordingly, some scholars extended their proposed rheological elementary models into 3D models by employing the associated flow rules to reflect 3D stress state of frozen soils (e.g., [43, 44, 196]) or established novel models for frozen soils by combing viscoplastic theory and other principles. Specifically, Cai et al. [6] established a model to simulate the creep deformation of frozen soils under monotonic and cyclic loadings. In their model, the creep deformation was divided into attenuation creep (described by the linear viscoelastic model) and nonattenuation creep (represented by the overstress viscoplastic model). However, since this model assumed that the frozen soils obey the Mises yield criterion and ignored the influence of hydrostatic pressure, their predicted results are inconsistent with most of the experiment data and cannot describe the plastic volumetric strain of frozen soils. Ghoreishian Amiri et al. [37] developed a model based on the unsaturated soil modelling framework (i.e., Barcelona Basic Model, BBM), whose model involved two independent stress-state variables (i.e., cryogenic suction and solid phase stress). The calculated results from their model were compared with experimental data from triaxial, uniaxial, and creep tests, demonstrating its ability to describe mechanical behaviours from the frozen state to the unfrozen state using a set of parameters. However, this model has some limitations in describing the curves of σ-ε and volumetric changes under the combined impacts of temperature and confining pressure. For insurance, it would produce overestimations for the critical and peak strength of frozen soils due to the linear relations between cryogenic cohesion and suction [118].

In the aforementioned investigations, it is evident that the majority of studies on the viscoplastic model of frozen soils have treated frozen soil as homogeneous solid material and primarily focused on the macroscopic deformation and evolution of damage. However, the creep models based on the micromechanics of creep deformation for the frozen soils are limited, particularly regarding the effect of ice content and temperature. Frozen soil is a typical multiphase material, and its macro responses stem from the collective contribution of each component and its individual properties [63]. On the one hand, the viscosity of ice, unfrozen water and soil fabric enhance the creep properties of frozen soils. On the other hand, the pore ice in the frozen soils is likely to be melted and crushed caused by stress concentration, which in turn results in the weakening of ice-cement and increment of unfrozen water content, facilitating the readjustment of solid particle displacement. In light of this, Wang et al. [130] established a macro–micro creep model for frozen soils based on the equivalent method for viscoelastic and viscoplastic models and discussed the influences of temperature and stress level, whose model was validated to be capable of simulating three stages of non-attenuation creep.

2.4 Hypoplastic Modelling

Hypoplastic models have been developed by many scholars (e.g., [49, 138, 139]) to describe the mechanical responses of granular materials and applied to manifold geotechnical issues [25, 99, 106] since it is an efficient approach in elucidating soil behaviours (e.g., nonlinear mechanical response, dependency of stress path and soil dilatation). Based on continuum mechanics, the hypoplastic models are formulated based on the representation theorem of isotropic tensor functions. They describe the relationship between stress rate, stress, and strain rate, which become hypoelastic when the nonlinear terms are omitted [149]. Table 4 lists the corresponding models based on the hypoplasticity theory.

Table 4 Typical investigations on hypoplastic modelling for frozen soils under static loading

Xu et al. [151] developed an extended hypoplastic model for granular materials, building upon the model proposed by Wu [139]. This extended model incorporates a backpressure term that accounts for the cohesion of frozen soils related to temperature, as well as a deformation-related function to capture strain softening. In the model of Xu et al. [151], eight parameters are included, including four parameters in the original model of Wu and Bauer [140], and four novel parameters where two parameters can be determined from the curves of cohesion-temperature and the other two parameters are relevant to the stress–strain curves that can be obtained by fitting the triaxial tests. The validation of this model was verified by two series of triaxial tests (i.e., varying temperature and constant confining pressure, varying confining pressure and constant temperature) where the simulated results are basically in accord with the experimental data. Nonetheless, it is worth noting that a distinct limitation in the extended hypoplastic model proposed by Xu et al. [151], i.e., the absence of consideration for the rheological properties (e.g., creep) of frozen soils. Accordingly, Xu et al. [152] presented a hypoplastic model to describe the three-stage creep deformation of frozen soils in a unified manner by coupling the extended hypoplastic model by Xu et al. [151] with the high-order model developed by Wu [141]. The model was verified by comparing the experimental data from uniaxial compression tests at different loading rates by Zhu and Carbee [190] as well as creep tests at different stress levels by Orth [98], which demonstrated that the model could efficiently capture the viscous characteristics of frozen soils, including the rate dependence of compressive strength, creep stain and creep rate of frozen soils. Xu et al. [154] also proposed a triaxial constitutive model with the framework of the hypoplastic model to characterise creep strains and strain rates in both axial and radial directions of frozen soil under various stress levels and temperatures. Unlike specialised models that only apply to specific stages of the creep process (e.g., [50, 121]), the proposed model can describe the entire three-stage creep process. Additionally, the proposed model eliminates the need to differentiate between high-stress and low-stress creep scenarios, where the specimen either fails or not.

2.5 Binary Element Models Based on Homogenization Theory

Homogenization theory is an efficient tool to establish the connection between microscale structure and macroscale behaviours of composite materials [66, 69, 76]. The concept of representative volume element (RVE) is often combined with homogenization theory [63]. The size of RVE should be carefully determined to ensure it is sufficiently large to encompass numerous internal particles at the microscale, while it should also be small enough to accurately represent the entirety of the composite material [69]. Based on the conception of RVE, the stress (${\varvec{\sigma}}_{ij}$) at any point within volume V for the macroscale RVE can be expressed as:

$$\sigma _{{ij}} = \frac{{\int_{V} {\sigma _{{_{{ij}} }}^{{local}} dV} }}{V}$$

(2)

where $\sigma_{{ij}}^{local}$ is the local stress within the RVE at microscale.

The frozen soil is considered a quasi-continuous medium comprising bonded blocks and weakened bands, respectively characterised as bonded and frictional elements. As shown in Fig. 2, the saturated frozen soil specimen is regarded as RVE at the macroscale, while it is composed of the bonded and frictional elements at the microscale that jointly withdraw the external loads. As for saturated frozen soils, the soil particles and ice crystals together constitute the bonded elements. In contrast, soil particles, broken ice crystals and unfrozen water are the frictional elements. The bonded elements can undergo irreversible breaking, transitioning into frictional elements when the external stress attains a certain strength threshold. The breakage ratio is an important parameter of binary materials such as frozen soils, which quantitatively describes the breakage degree of bonded elements transferring to the frictional elements and ranges from 0 to 1 with increasing external loads.

Fig. 2

Schematic diagram of bonded elements and frictional elements in binary element model

Therefore, within the concept of RVE and binary medium, the RVE, bonded element and frictional element volumes are defined as V, V_b and V_f (V = V_b + V_f). The conversion rate of the bonded element into the frictional element is expressed by the breakage rate (λ) that is equal to the ratio of V_f to V. At the macroscale level, the bonded and frictional elements are homogenous in their individual volume. Therefore, their average stress ($\sigma_{ij}^{average}$) can be computed as

$$\sigma_{ij}^{average} = \frac{{\int_{{V_{{^{b} }} + V_{{^{f} }} }} {\sigma_{{_{ij} }}^{local} dV} }}{V} = \frac{{\frac{1}{{V_{{^{b} }} }}\int_{{V_{{^{b} }} }} {\sigma_{{_{ij} }}^{b} dV + \frac{1}{{V_{{^{f} }} }}\int_{{V_{{^{f} }} }} {\sigma_{{_{ij} }}^{f} dV} } }}{V}$$

(3)

where $\sigma _{{ij}}^{b}$ and $\sigma _{{ij}}^{f}$ are local stresses of the bonded element and frictional element. Based on the definition of breakage ratio, Eq. (3) can be rewritten as

$$\sigma_{ij}^{average} = \left( {1 - \lambda } \right)\sigma_{{_{ij} }}^{b} + \lambda \sigma_{{_{ij} }}^{f}$$

(4)

Table 5 lists the existing binary element models based on homogenization theory for frozen soils. Chang et al. [10] developed a meso-macro constitutive model for frozen saline sandy soil based on homogenization theory, whose model can duplicate the strain softening/hardening, high dilatancy and pressure melting phenomenon well. In their mode, the saline frozen sandy soil was assumed to be composed of inclusion and matrix. The equivalent inclusion phase was described by an elastic-brittle model, while the sandy soil matrix was described by a critical state elastoplastic mode, and their expressions are derived from the three-phase sphere and Mori–Tanaka theory. Subsequently, Zhang and coworkers [167, 168, 170] proposed binary-medium constitutive models for frozen soils where the boned element and frictional elements were regarded as a matrix with elastic-brittle properties and inclusions with elastoplastic behaviours based on different constitutive relations for the bonded elements and frictional elements (see Table 5)

Table 5 Summary of existing binary element model based on homogenization theory for frozen soil under static loading

Besides, Wang and his coworkers (i.e., [125, 127, 130, 132, 134, 135]) have been devoted to developing the binary medium constitutive models for frozen soils according to homogenization theory in recent years. For example, given that the previous studies regarded the frozen soils as a whole solid material that cannot reflect the progressive breakage of ice-cement during the creep process, Wang et al. [125] initially analysed the breakage mechanism of creep deformation of frozen soils and developed a binary medium creep model to capture the creep deformation. By considering the effects of temperature and stress on the breakage evolution law, an exponential function for the breakage ratio was derived. To further reveal the mechanism of creep deformation of frozen soils under constant/varying temperatures, Wang et al. [132] proposed a unified macro-meso creep model to describe the creep process based on the creep tests of saturated frozen soils. In this model, the breakage ratio was derived by an empirical function of the plastic shear strain by referring to the determined methods of damage factors or plastic hardening parameters. Furthermore, some novel binary medium models to describe the stress–strain relations of frozen soils are also developed by Wang and his coworkers. Based on the experiment results from loading–unloading-loading tests, Wang et al. [127] observed that the elastic and plastic deformation would not generate simultaneously during the shear process. The mechanisms of these two types of deformation were analysed by a newly proposed binary medium model where the frozen soils were regarded as a binary medium consisting of bonded and frictional elements. Specifically, the boned elements have linear elastic properties described by generalised Hooke law, while the frictional elements exhibit plastic properties and are represented by the double-yield surface plasticity model. Since the breakage evolution is often empirically derived, reflecting the meso information within the constitutive models for frozen soils is difficult. Wang et al. [130] formulated a multi-scale incremental constitutive model to reflect this internal mechanical mechanism and introduced a new approach for determining the breakage ratio based on the thermodynamic theory that is more rational than previous empirical methods. Wang et al. [134] also established a binary-medium constitutive model combining the breakage mechanics and the homogenization method to describe the deformation mechanism of frozen soil with different coarse-grained contents. To address the limitation of the existing binary-medium-based models that are difficult to simulate the rate-dependent mechanical behaviours of frozen soils, Wang et al. [135] proposed a novel binary medium-based rate-dependent model by considering local bond breakage and the composition characteristics within the framework of breakage mechanics for geomaterials. Since the breaking process can be characterised as an irreversible thermodynamic process, a quantitative equation to describe the breaking ratio was also derived in the work of Wang et al. [135] by using a concept similar to that of plastic work. A traditional triaxial compression test of frozen soils under different strain rates was simulated by this model, whose results demonstrated that the model could well predict the nonlinearity, dilatancy and strain softening/hardening characteristics.

Based on the homogenization method, Chen et al. [13] also developed a meso-macro constitutive model for frozen saline sandy soil by considering the influence of salt concentration. In their model, the bonded elements exhibit elastic-brittle mechanical properties, while the frictional elements possess elastic–plastic properties. He et al. [40] proposed a creep constitutive model according to the binary medium concept and homogenization theory to describe the creep deformation of frozen soils, including decaying creep and non-decaying creep. In their model, the constitutive relations of the bonded and frictional elements were formulated by using the creep breakage mechanism and the theory of fractional calculus, and the breakage ratio was defined as a function of creep time. Specifically, the bonded elements comprise a Hooke body and a fractional viscoelastic body, while the frictional elements are simulated by modified fractional Bingham body.

These binary element models based on homogenization theory have been verified to duplicate the primary characteristics of frozen soils, which offers a novel perspective and tool to capture and describe the mechanical behaviours of frozen soils within a multi-scale framework.

2.6 Hyperplastic Modelling

The mechanical and strength of frozen soils are highly influenced by negative temperature, ice-water phase change, and moisture migration. The deformation process is an intricate thermodynamic process [56]. Hence, incorporating the thermodynamic theory into constitutive modelling is capable of making the coupled thermal–mechanical model more reasonable and systematic. The core of the framework is that the constitutive relations of materials (i.e., elastic law, yield function, flow rule and hardening rule relating to the traditional elastoplastic theory) are entirely represented by two thermodynamic functions (i.e., Gibbs free energy function and dissipation function), which breaks the theoretical framework of conventional plastic mechanics [182].

Based on the irreversible thermodynamic theory, Ziegler [200] introduced a framework for establishing constitutive models, along with its foundation known as Ziegler's orthogonality principle [45]. Nevertheless, this principle has encountered significant criticism from various perspectives. Srinivasa [115] contended that Ziegler’s orthogonality principle was based on an erroneous assumption. Nonetheless, Collins and Houlsby [17] argued that the principle is a weak assumption and can be employed to define a broad range of constitutive equations. Furthermore, using the convex analysis, Srinivasa [115] substantiated that the same orthogonality principle can be deduced from the maximum rate of dissipation criterion in both the rate-dependent and rate-independent scenarios (with the prerequisite of first-order homogeneity of the dissipation function). This method of constitutive modelling based on thermodynamics is also referred to as "hyperplasticity" [102], hereinafter referred to as "hyperplastic modelling". Hyperplasticity contains two types, i.e., single and multiple internal variable hyperplasticity that correspond to the single yield surface model and multiple yield surface model.

Establishing hyperplastic modelling possesses many advantages. Firstly, it ensures automatic adherence to thermodynamics laws. The second is the formulation exhibits a concise nature. It is crucial to establish competing constitutive models within a unified theoretical framework, facilitating convenient comparisons among different models and aiding in the selection of the most promising one. Thirdly, the constitutive modelling within the frame of hyperplasticity theory can seamlessly integrate with other differential equations, which makes the THM coupling model maintain a high level of consistency and systematicity [56].

Recently, some researchers have employed the hyperplasticity theory to describe the mechanical behaviours of frozen soils, as listed in Table 6. Fish [28] developed a hyperplastic model to describe the entire creep process (including primary, secondary, and tertiary creep) and failure for both constant stress tests and constant strain rate tests in the form of a unified constitutive equation and failure criteria. According to the second principle of thermodynamics and damage mechanics, He et al. [42] proposed a constitutive model for analysing the stress–strain relationship of frozen silt under complex stress conditions, whose model also explored damage development and the failure process of the internal structure of frozen soil. Sun et al. [116] established an elastoplastic anisotropic damage constitutive model by involving a plastic potential function considering the effect of volumetric deformation and anisotropic damage evolution. Lai et al. [56] adopted the orthogonality principle proposed by Srinicasa [115] to model Lanzhou frozen loess's mechanical behaviours. In their model, the specific expressions of the Gibbs free energy function and dissipation function for frozen loess were derived from the test results and the hyperplasticity theory. It is worth noting that the multiple internal variable hyperplasticity was employed since the test results showed that the direction of plastic strain increment was affected by the stress path. Based on the principles of hyperplasticity theory and using the non-associated flow rule, Lai et al. [57] established a double-yield surface model for frozen saline sandy soil, incorporating rotational hardening as well as initial anisotropy and loaded anisotropy to account for both plastic compression mechanisms and plastic shear mechanisms. The proposed model effectively captures the phenomena of pressure melting, crushing, and dilatancy in frozen saline sand. Furthermore, it accurately accounts for the influence of salt content on stress–strain curves. Based on a series of different types of tests (i.e., uniaxial, triaxial and hydrostatic tests) on frozen loess, Zhou et al. [186] developed a rate-dependent constitutive model to characterise the multiaxial creep deformation by using the hyperplasticity theory and incorporated multiple internal variables. Chang et al. [10] established a constitutive model considering particle breakage, compression and shear mechanisms. In their model, the yield surface functions and plastic potential functions of three mechanisms are derived based on data from a series of triaxial compression tests for frozen saline coarse sandy soils with varying salt content under a wide range of confining pressure and stress paths. Sun et al. [117] presented a comprehensive framework that integrates poromechanics, thermodynamics, and continuum damage mechanics to account for visco-elasto-plastic creep and anisotropic damage in saturated frozen soils. To derive this constitutive model, the soil skeleton’s free energy and the dissipation potential associated with the applied stress level are required. The decomposition of the first term includes four components: reversible free energy, strain energy stored in the soil skeleton, energy in the form of heat, and interfacial energy.

Table 6 Summary of hyperplastic modelling for frozen soils under static loading

It can be noted that conventional constitutive models for frozen soils relied on Drucker's stability postulate or plastic potential theory. In these models, yield functions, flow rules, and hardening laws are introduced independently and occasionally contradict each other, which increases the likelihood of violating the hyperplasticity theory. In contrast, hyperplastic modelling has recently been developed and applied for geotechnical engineering, which guarantees adherence to the principles of hyperplastic by incorporating dissipative incremental and free energy functions.

2.7 Thermo-Poromechanics-Based Constitutive Modelling

Previous studies on modelling the mechanical behaviours of frozen soils regarded the frozen soils as continuous solid materials, which failed to reflect the interaction of different components within frozen soils. Besides, since soil is a typical pore medium [71], the constitutive modelling of frozen soils based on thermo-poromechanics theory is more closely aligned with its fundamental nature. Coussy and his colleagues [19, 20] have conducted investigations on frozen porous materials from a poromechanics perspective. In recent years, some scholars have applied thermo-poromechanics to the research of frozen soils, as summarised in Table 7.

Table 7 Summary of thermo-poromechanics-based modelling for frozen soils under static loading

Based on thermo-poromechanics theory, Liu et al. [79] formulated a constitutive model for poro-elastoplastic frozen soils to simulate the cryogenic triaxial compression of frozen soils by considering the Lagrangian saturation and solid–fluid interface interactions. In their model, the frozen soil was regarded as an open system, and both Lagrangian and Eulerian formulations presented the associated governing equations. Compared to other models, this model exhibits a more rigorous structure and derivation process, which is crucial for establishing a constitutive model for frozen soils. Subsequently, given the fact that the existing poromechanics-based models ignored the impact of rate dependence on the mechanical behaviours of frozen soils, Liu and Lai [81] established a framework to consider the time-dependent influence of frozen soils based on the extending of the poro-elastoplasticity to poro-elasto-viscoplasticity, whose model can be applied for the frost heave of freezing ground in the cold region. However, the impact of the water–ice phase change was not considered in their model. Therefore, as the temperature is close to the temperature of the water–ice phase change, the phase change between unfrozen water and ice crystals should be taken into account, especially for warm frozen soil. Accordingly, Ma et al. [88] proposed a poromechanics-based constitutive model for warm frozen soil by considering the interaction between soil particles, ice crystals and unfrozen water caused by pressure melting, which can predict the unfrozen water and ice pressure during the loading process. Furthermore, a comparison between experimental data and theoretical results has demonstrated that the model proposed by Ma et al. [88] can accurately simulate the mechanical behaviours of warm frozen soil, particularly the volumetric dilatancy behaviours and strain softening observed during both isotropic loading and axial stress loading stages. The aforementioned three thermo-poromechanics-based models introduce a novel way to construct frozen soil's constitutive model.

2.8 Constitutive Models Considering Damage Mechanics

The theory of damage mechanics describes the damage initiation, propagation, and eventual fracture of materials. Over the past few decades, damage mechanics theory has been widely utilised to develop constitutive models for unfrozen soils and rocks (e.g., [61, 80, 129]). More recently, damage mechanics has also been employed to develop constitutive models for frozen soils. The deterioration of macroscopic mechanical performance and the progressive failure of microscopic structures are the core of damage theory. By incorporating a "damage variable", the variation of mechanical responses caused by micro-defects can be quantified. This damage variable can be subsequently added to the constitutive model, also known as the damage model, to account for the impact of damage on the material's mechanical behaviour.

To construct a damage model for frozen soils, three aspects relating to the damage behaviour should be determined. The first is the definition of the damage variable, which is crucial in measuring the changes in macroscopic response resulting from the development of damage. Various variables have been adopted as the damage variable, such as particle orientation and area reduction [89], the deterioration level of elastic modulus [150], strength [52, 58], creep time [77] and so on. The second is the damage threshold value, which serves as a damage criterion to determine whether damage has occurred, given that the process is irreversible and often accompanied by irreversible deformation. The third aspect is the damage evolution law, which quantitatively describes the change in damage with irreversible deformation. Once these three aspects are established, a damage model can be used to describe the frozen soil's damage behaviours. Therefore, investigating these aspects is crucial in understanding the damage characteristics of and developing reliable damage models for frozen soils. Accordingly, the damage evolution laws of several important mechanical parameters (i.e., strength, stiffness, and viscosity) have received considerable attention. Four typical damage functions are employed to serve as damage evolution laws, including statistical functions, empirical functions, functions based on the energy dissipation principle, and functions based on elasticity theory.

• (1) In the statistical damage model, some statistical distribution functions are applied to quantify the damage degree. Many different parameters are adopted to describe the damage variables. Lai et al. [52] indicated that the Weibull distribution is more suitable for describing the strength distribution law than normal and lognormal distribution. However, the axial strain was considered a random variable in the work of Lai et al. [52], and the frozen soil would be damaged when the axial strain reached a certain threshold. Accordingly, Li et al. [58] regarded the strength of soil elements as a random variable and improved the original statistical damage model of Lai et al. [52] for warm frozen clay and warm ice-rich frozen clay.

  $$D = 1 - \frac{{N_{t} }}{N} = 1 - \exp \left( { - \left( {\frac{\varepsilon }{\eta }} \right)^{\beta } } \right)$$

  (5)

  where D is the damage variable; N_t is damage element numbers; N represents all the elements; ε is strain; β is the shape parameter; η is scale parameter.

• (2) As for the empirical damage theory, the damage degree obeys some functional relationships with the damage variables. For instance, Liao et al. [77] established an exponential damage function for ice-rich frozen soil during the creep process.

  $${\text{D}} = 1 - \exp \left( { - \theta {\text{t}}} \right)$$

  (6)

  where D is the damage variable; θ is a parameter relating the damage evolution with time, which varies with the stress level; t is creep time. Besides, Hou et al. [44] defined a damage variable as

  $$D\left( {\sigma ,t} \right) = \left\{ \begin{gathered} 0 \ , \sigma < \sigma_{\infty } \, \hfill \\ 1 - \exp ( - Ct) \ ,\sigma \ge \sigma_{\infty } \, \hfill \\ \end{gathered} \right.$$

  (7)

  where C is parameter reflecting the impact of damage; σ_∞ is long-term strength.

• (3) Liu and Lai [81] regarded that the damage originated from the gradual breaking of ice bonding under external loading and determined the evolution of damage based on the assumption that the dissipation energy caused by damage is only dependent on the damage variable.

• (4) According to Sidoroff's postulation [112], the form of the complementary elastic energy for a damaged material is identical to that of an undamaged material. Accordingly, Lai et al. [53] defined a cross-anisotropic damage variable as follows.

  $$\begin{gathered} {\text{D}}_{1} = 1 - \sqrt {\frac{{{\tilde{\text{ E}}}}}{{\text{E}}}} \hfill \\ {\text{D}}_{2} = {\text{D}}_{3} = 1 - \frac{{\text{v}}}{{{\tilde{\text{v}}}}}\left( {1 - {\text{D}}_{1} } \right) \hfill \\ \end{gathered}$$

  (8)

  where D₁, D₂ and D₃ are the damage variables in the corresponding principal axes (D₁ ≠ D₂ = D₃); E is elastic modulus; v is Poisson’s ratio; ${\tilde{\text{E}}}$ and ${\tilde{\text{v}}}$ are the effective elastic modulus and Poisson ratio of damaged materials.

Table 8 summarises typical investigation on modelling the mechanical behaviours of frozen soils by damage theory. It is noted that the damage within frozen soil is 3D due to the anisotropic characteristics of ice crystals under loading. Therefore, assuming an isotropic hypothesis would not provide an accurate representation of the damage process in frozen soil. In this regard, Lai et al. [53] proposed cross-anisotropic damage variables and incorporated these into a novel elastoplastic constitutive model. The advantage of constitutive models based on damage theory is that they can elucidate the macroscopic behaviours of frozen soils from a microscopic perspective. However, describing the damage evolution for frozen soils in a reliable and efficient manner remains a significant challenge.

Table 8 Summary of four typical damage functions used in static constitutive models for frozen soils

2.9 DEM-Based Constitutive Modelling

The microstructure of frozen soil plays a pivotal role in influencing the macro-mechanical characteristics of the material. However, limited by the nondestructive testing and monitoring techniques, most existing studies were conducted to explore the macroscopic mechanical behaviours of frozen soils. Some researchers have performed scanning technologies (e.g., computed tomography, CT) on the microstructures of frozen soils, and they noticed that the structures of frozen soils were continuously reorganised throughout the loading and deformation process, and the triaxial tests exerted simultaneous effects of structural reinforcement and weakening. Additionally, the failure of frozen soils was attributed to the slippage occurring between soil particles and bonded ice particles. Although these microscopic studies can observe the development of the structure of frozen soils, it is still a challenge to quantify the variations of microscopic information (e.g., article contact force and cementation of ice) within frozen soils under external loadings. Consequently, a noticeable research gap persists in understanding the underlying micromechanics governing the failure of frozen soils.

Initially proposed by Cundall and Strack [21], the discrete element method (DEM) has gradually become an alternative approach to explore the mechanical behaviours of geomaterials, especially the unfrozen soils (e.g., [5, 93, 100, 133, 156]). However, only a few studies have employed DEM to simulate the mechanical behaviours of frozen soils (see Table 9). Zhou and Lai [184] adopted a two-dimensional (2D) DEM method to simulate the triaxial test of frozen sandy soil where the contact bond model was used to replicate the cementation of ice within the frozen sand clay and a slip model to account for friction. Yin et al. [63] established a 3D model to simulate a series of triaxial tests on frozen clay and compared the numerical results with laboratory results. Wang and Calvetti [126, 131] presented a DEM model to assess the influence of ice content on the behaviours of the mixture of ice and granular soils in triaxial compression and shear tests. Xu and Wang [155] used a 2D DEM method to explore the creep deformation of cemented methane hydrate-bearing sand where the methane hydrate was simulated as contact bonds between particles, and proposed a parallel bar creep model to describe the evolution of damage with time. Their simulations showed that the methane hydrate-bearing sand displays considerable creep, similar to frozen sand. Considering the size effect in the conventional small-size direct shear test, Chang et al. [11] established a DEM model to explore the microscopic mechanical properties of frozen sand in direct shear tests, whose simulation results demonstrated that the DEM model was capable of elucidating the microscopic mechanisms underlying the macroscopic strength and deformation behaviour of frozen sandy soils. Zhang et al. [179] proposed a creep contact model for frozen soils to explore the bond effect on the mechanical behaviours of frozen soils by incorporating shear strength and tensile strength into the generalised Kelvin body. Based on this model, a series of triaxial creep DEM simulations for frozen soils were conducted to investigate macro- and meso-mechanical behaviours of frozen soils.

Table 9 Summary of DEM-based modelling for frozen soils under static loading

Nevertheless, the contact models utilised in these previous investigations neglect the impact of temperature on the ice’s strength and stiffness. Accordingly, Sun et al. [119] extended an existing 3D contact model by involving the effect of temperature on ice cementation’s strength and stiffness. By utilising this novel model based on DEM, they investigated the bond breakage mechanism during the loading process, which is not readily observable in conventional laboratory tests. They also qualitatively examined the influences of temperature on the macro–micro mechanical responses of frozen soils. These aforementioned investigations (as summarised in Table 9) offer a novel research avenue for comprehending the mechanical behaviours of frozen soils from a microscopic perspective and present opportunities for exploring other crucial properties of frozen soils in future studies, such as anisotropic and rheological behaviours, as well as the nonlinearity of strength. It is noted that modelling mechanical behaviours of frozen soils based on DEM is an efficient tool to explicitly simulate the interaction between ice and soil particles rather than the constitutive model based on a continuum approach for reproducing the macroscopic behaviours of frozen soils.

2.9.1 Machine Learning-Based Modelling

With the advancements in computation methods, some scholars have utilised machine learning (ML) methods to tackle geotechnical engineering issues [136, 173]. Machine learning methods have proven to be effective in predicting soil properties [67, 68, 73] and behaviours [74, 174, 180], which are helpful since direct measurements are relatively time-consuming and costly.

Therefore, some researchers have been devoted to modelling the mechanical behaviours of frozen soils in recent years (see Table 10). Nassr et al. [91] first presented the evolutionary polynomial regression (EPR) to model the thermos-mechanical behaviours of frozen soils with the consideration of the effects of strain rate, confining pressure and temperature. A comprehensive dataset of unconsolidated undrained triaxial tests on frozen soils from Iran was compiled and used to train the EPR model. This data-mining method was validated by comparing their predictions and experimental data. Pham et al. [101] also modelled the mechanical responses of frozen sand under varying confining pressure and temperature using the Bayesian Neural Network. However, these two models only focused on the evolution of deviatoric stress of frozen soils under external loadings and the volumetric variation was not considered. Accordingly, Li et al. [74] proposed a novel data-driven method (i.e., long short-term memory, LSTM) to capture and describe stress–strain relationships. Given the fact that uncertainty in modelling is one of the crucial sources of uncertainty in geotechnical engineering, their model combined with the Monte Carlo dropout scheme to provide reliability evaluations for the deviatoric stress and volumetric strain of frozen soils. Besides, Ren et al. [47] proposed a prediction model for unfrozen water based on a genetic algorithm-back propagation (GA-BP) neural network and analysed the mechanical behaviours of frozen soils under extremely-low-temperature conditions based on a series of laboratory experiments on frozen clay. The work of Ren et al. [47] indicated that the compressive strength of frozen soil exhibits a direct proportionality with the absolute value of temperature, while the unfrozen water content follows a power function relationship with compressive strength.

Table 10 Summary of machine learning-based modelling for frozen soils under static loading

It can be noted from Table 10 that the applicability of the ML method in modelling the mechanical behaviours of frozen soils, which exhibits considerable performance in capturing and forecasting the stress–strain relations of frozen soils as well as the strain softening/hardening phenomenon. Notably, these ML-based models require fewer parameters compared to conventional rheological models since these models based on the ML method are implemented on the available database from laboratory measurements (e.g., triaxial tests).

3 Constitutive Models for Dynamic Loading Condition

As summarised in Sect. 2, extensive research efforts have been dedicated to developing constitutive models for frozen soils under static loading conditions. However, it is essential to note that construction projects in cold regions are often subjected to dynamic loading rather than quasi-static loading conditions. The mechanical behaviours of frozen soils under dynamic loads are more sophisticated than those under static conditions. For example, owing to the short duration of impact loads, the constitutive model developed for quasi-static loading conditions is unsuitable for analysing frozen soils' mechanical responses at impact loading scenarios. Accordingly, investigations into dynamic constitutive models for frozen soils have achieved significant progress in recent decades. This section provides a brief overview and discussion of these dynamic models. Furthermore, Table 11 presents a compilation of constitutive models for frozen soils under dynamic loads, which is categorised into eight distinct types based on their underlying fundamental principles.

Table 11 Summary of existing dynamic constitutive models for frozen soils

3.1 Empirical Models

Empirical dynamic constitutive models for frozen soil are mathematical formulations that aim to capture the dynamic behaviour of frozen soil through empirical functions and experimental observations. These models are developed based on experimental data obtained from dynamic tests of frozen soils, which often incorporate various parameters (i.e., strain rate, temperature, and stress) to describe the mechanical response of frozen soils under dynamic loading conditions. These empirical models provide an efficient approach to numerically predict the mechanical behaviour of frozen soils and are easily applied for engineering applications.

Zhang et al. [166] conducted the uniaxial stress and nearly uniaxial strain conditions to explore the dynamic behaviours of frozen soils and formulated a model to describe the temperature and strain-rate-dependent behaviour of frozen soils. However, their model exhibits limitations in capturing stress–strain curves when the strain approaches the final strain, and it fails to simulate the dynamic behaviours of frozen soils under general conditions, such as 3D scenarios. Since it is difficult to establish a dynamic constitutive model for frozen soils with precise physical meanings based on experimental data, researchers have devised alternative approaches to modelling the dynamic behaviours of frozen soils.

3.2 Elastoplastic Modelling

Based on a series of dynamic experiments on frozen soils using the SHPB method, Ning et al. [94] proposed an isotropic hardening constitutive model with a modified Drucker-Prager yield function considering the effect of strain rate to describe the dynamic behaviours of frozen soils. Considering the adiabatic temperature increase and interfacial debonding, Xie et al. [147] developed an elastoplastic model for frozen soils where a modified D-P criterion was used to describe the strain hardening before peak stress and deboning damage of ice particles obeying the Weibull distribution was consider for reproduce the phenomenon of softening. Qiao et al. [105] proposed a rate-dependent yield surface model within the framework of elastoplastic theory by considering the matrix suction impact to describe the dynamic responses of unsaturated frozen soils, where the damage variable also obeyed the Weibull distribution.

3.3 Viscoelastic Modelling

The Zhu Wang Tang (ZWT) model is a typical nonlinear viscoelastic model that was derived based on dynamic impact tests on high-polymer materials [122], which includes two parts, i.e., a nonlinear transient response component independent of time and a linear transient response component closely associated with time. Figure 3 plots the schematic diagram of the ZWT model.

Fig. 3

Schematic diagram of ZWT nonlinear viscoelastic model and its formulation

The ZWT model has been widely used for the dynamic behaviours of frozen soils. Since frozen soils are rate-dependent materials under impact loadings, Zhang et al. [171] noted that the low-frequency Maxwell element no longer affects the dynamical responses of frozen soils. Some scholars have devoted themselves to improving the ZWT model to better capture and describe frozen soils' dynamic behaviours.

3.4 Viscoplastic Modelling

Under impact loadings, the damage evolution within frozen soils can be regarded as a rate-dependent viscoplastic deformation process [8]. Some scholars have established viscoplastic models to replicate the mechanical behaviours of frozen soils under impact loading conditions. Zhu et al. [197] established a viscoplastic model to depict the dynamic mechanical behaviours, whose model was developed by integrating the Chaboche unified viscoplastic constitutive theory, the Drucker–Prager yield criterion and associated flow rule. Zhang et al. [172] derived a damage dynamic model for frozen soil based on the unified viscoplastic theory. This model incorporates an inelastic multiplier equation that accounts for the characteristic thermal softening observed during impact loading, while the phenomenon of adiabatic heating induces damage. Based on a series of dynamic tests of frozen soil under F-T loading and impact loadings, Li et al. [70] derived a viscoplastic model by an improved ZWT model and D-P yield criterion. Besides, their model also considered two damage mechanisms, i.e., F-T damage and impact damage, which were described by wave impedance and Weibull distribution function, respectively. Recognizing the influence of soil particle size on the dynamic responses of frozen soils, Zhang et al. [177] proposed a dynamic viscoplastic constitutive model by incorporating the average strain gradient of the soil cell model into a multiplicative hardening law that includes a work hardening function, a hardening deterioration function and a strain rate-temperature sensitive function.

3.5 Energy-Based Models

Energy-based methods are often adopted to indirectly establish the dynamic constitutive models of frozen soils. For instance, based on a series of impact compressive loading tests, Xie et al. [145] defined the energy absorbed per unit volume of frozen soil as the integral of the stress–strain curve and used a GuassAmp peak function to describe the absorbed energy. Their energy-based dynamical model can reasonably reflect the positive strain-rate sensitivity and negative temperature dependence of the artificial frozen soil, but it fails to simulate the failure process of frozen soils owing to the path independence of the energy-based method. Similarly, Ma et al. [87] also employed the GuassAmp peak function to describe the energy absorption curves and derived a dynamic model to reflect the strength and deformation of frozen clay with pre-existing cracks under impact loading conditions. The comparison between theoretical results and measurements showed that the predictive error could be constrained to 1.35 MPa.

3.6 Two Surface Plasticity Models

To improve the capacity of model for cyclic loadings, the concept of two surface plasticity theory is often introduced. Two surface plastic models, i.e., the bounding surface model (initially proposed by Dafalias and Popov [22, 23]) and the subloading surface model (firstly developed by Hashiguchi and Ueno [39]), have been acknowledged as powerful tools for simulating the deformation characteristics of soils under cyclic loading, owing to their clear concepts and straightforward numerical implementations. The boundary surface plasticity theory defines the loading surface and bounding surface (referring to the maximum yield surface that corresponds to the maximum loading stress experienced during the loading process). In addition to the bounding surface and plastic potential surface, the boundary surface plasticity theory also contains the hardening rule, mapping origin and radial mapping rule, as well as plastic modulus. The subloading surface model possesses a similar structure as the bounding surface model, consisting of a normal yield surface and a subloading yield surface.

In practice, the frozen soils are often subjected to cyclic loading conditions (e.g., traffic load and seismic action). The dynamic deformation of frozen soils under reciprocating dynamic loadings exhibits nonlinearity, hysteresis and plastic strain accumulation, which are more complex than unfrozen soils since the ice crystals considerably influence the soil cohesion. Therefore, it is not appropriate to directly apply the two surface plasticity model developed for unfrozen soils to frozen soils. To reproduce the dynamic behaviours of frozen saline silty clay under cyclic loading, Zhao et al. [181] formulated an anisotropic boundary surface elastoplastic constitutive model based on boundary surface plasticity theory. In their model, the rate dependence and visco impact were neglected. The comparison between experimental data and predicted deformation from the proposed model illustrates that the dynamic model of Zhao et al. [181] can reasonably simulate the deviator strain of saline frozen soils while some differences between measured volumetric strain and corresponding predictions can be observed. Their dynamic model comprehensively accounts for the impact of salt content by incorporating appropriate model parameters, which can accurately simulate the intricate deformation characteristics of frozen saline soil and unfrozen/frozen salt-free soils.

3.7 Micromechanics-Based Models

The frozen soil is a typical porous, multiphase and heterogeneous medium, and its macroscopic behaviours are affected by the internal microstructure. Therefore, it is reasonable and beneficial to construct the dynamic constitutive models for frozen soils based on micromechanics due to its unique features, such as the ability to shed light on the underlying mechanisms behind the effects of influencing factors (e.g., strain rate and temperature), elucidate the softening impact of adiabatic temperature rise that is initially reported by Zhu et al. [194] and model the microscopic constitutive relationships within frozen soils. For example, based on the representative volume element (RVE) concept, Xie et al. [146] assumed that the frozen soil could be represented as a binary medium with ice particles serving as spherical inclusions and soil particles acting as the matrix. Zhu et al. [193, 194] developed dynamic constitutive models for frozen soils by combining damage theory and equivalent inclusion methods. Using the Mori–Tanaka method, Zhu et al. [197] developed a dynamic constitutive model for frozen soils under a multiaxial state, whose model incorporates the debonding damage of ice particles and the damage evolution of the soil matrix based on thermal activation theory. Fu et al. [34] developed a dynamic damage constitutive model for frozen soil using Mori–Tanaka and Eshelby’s equivalent theory, which can simulate the dynamical responses of frozen soils, such as rate-dependence, temperature sensitivity and dynamic deformation. Besides, in their model, the strength degradation caused by microcrack damage evolution was considered by the complementary strain energy equivalence principle. Wang et al. [201] formulated a dynamic binary medium model based on homogenization and breakage theory, whose model can reflect the influences of coarse-grain contents, axial deviatoric stress amplitude, and confining pressures on the dynamic behaviours of frozen soils under cyclic loading conditions. Zhang et al. [178] developed a microscopic dynamic model for unsaturated frozen soils to elucidate the underlying physical mechanism of local crack frictional sliding and damage evolution by incorporating a Drucker–Prager type yielding criterion and a damage criterion based on the damage energy release rate. The comparison between experimental data and model computation results showed that this model could capture crucial aspects such as the effect of strain rate and temperature as well as the nonlinear hardening and softening behaviour of unsaturated frozen soil.

3.8 Constitutive Models Considering Damage Mechanics

Based on the microcosmic mechanics mixing theory and damage theory, Zhu et al. [193] proposed an elastic constitutive model assuming that the damage variable was a Weibull distribution function of strain. By considering frozen soil as a particle-reinforced composite material consisting of clay soil (matrix) and ice particles, Xie et al. [146] formulated a micromechanical constitutive mode to characterise the dynamic compressive deformation of frozen soil. Based on the debonding damage theory, this model assumes that the damage variable follows a Weibull distribution of strain and accounts for strain rate and temperature's influence on frozen soil's dynamic compressive behaviour. However, the model of Xie et al. [146]) is unsuitable for warm frozen soils since it neglects the existence of unfrozen water. Zhu et al. [194] initially introduced the concept of temperature damage (i.e., strength reduction caused by adiabatic temperature increase), and developed a constitutive model for frozen soils to describe the heat damage and softening behaviour under dynamic loading conditions based on the equivalent inclusion method and damage theory.

The previously developed ZWT model [122] has demonstrated its capability to effectively capture and characterise the viscoelastic behaviour of solid materials, while the low-frequency Maxwell element in the ZWT model would lose its effectiveness due to the microcrack propagation. Accordingly, many scholars have improved the ZWT model by considering the damage stemming from macro- and microdamages. Considering the impacts of confining pressure, strain rate and temperature, Ma et al. [84] modified the ZWT model by considering the influences of these three factors on the dynamic behaviours of frozen sandy clay, whose model included a damage variable that also obeyed the Weibull distribution. Fu et al. [33] introduced the damage evolution and temperature damage into the original ZWT model to formulate a novel viscoelastic constitutive model to reflect the intricate interplay of strain rate effect, temperature effect, strain convergence effect, damage softening impact on dynamic behaviours of frozen soils. Ma et al. [86] improved the ZWT model by considering the damage of frozen soils where the Weibull distribution was used to describe the damage evolution within frozen silty soil, whose model can effectively capture the strain rate enhancement and the weakening influence of prefabricated crack number on the dynamic strength of frozen soils. Besides, Zhang et al. [171] employed the longitudinal wave velocity to represent the damage variable and developed a modified ZWT model by introducing this damage variable. They also investigated the macroscopic physical quantity and mesoscopic parameters to explain the changes in damage in frozen soil caused by internal microcracks.

Previous research studies have neglected the process of damage evolution in frozen soil and relied on the assumption that the strength of microcells within frozen soil follows a probability distribution (i.e., Weibull distribution). Although this approach effectively reflects damage state at the macroscopic level, it fails to account for the temporal evolution of damage. To enhance the accuracy of describing the constitutive relation of frozen soil, it becomes imperative to explore the damage evolution law of frozen soils under impact loading. Cao et al. [8] incorporated the strain rate item into the damage evolution formula based on thermal activation theory [4] and derived a dynamic constitutive model with rate and temperature effects. Zhu et al. [199] modified the Ottsen model, a nonlinear elastic model, and introduced damage based on thermal activation theory to formulate a dynamic model. Zhu et al. [199] proposed a damage viscoelastic model by considering the debonding damage and temperature softening to describe the dynamic behaviours of frozen soils under high strain rates. Specifically, an exponential equation was derived for the debonding effect based on thermal activation theory, and a temperature softening coefficient was incorporated into the Johnson–Cook model to reflect the softening phenomenon.

Based on the Drucker–Prager failure criterion and coupled damage-plasticity, Zhu et al. [195] developed a new constitutive model to describe the dynamic mechanical behaviours of frozen soil with different particle sizes. Their equations involved two damage mechanisms (microcrack-induced damage and microvoid collapse-included damage). Fu et al. [32] also proposed a dynamic model where the Drucker–Prager yield function was modified by adding the strain rate enhancement impact to capture the mechanical behaviours of frozen soils before reaching the peak stress. Besides, since the impact loading process exhibits a pronounced strain-rate effect that becomes more unstable after the peak stress, a cohesive crack model based on fracture mechanics theory was established by Fu et al. [32] to describe this nonlinear process after peak stress.

Frozen soils are often subjected to static stresses resulting from geostatic forces, frozen-heave effects, and dynamic loads induced by impact stress waves. Therefore, frozen soil's mechanical behaviour differs significantly from materials subjected individually to static stress or dynamic loading. Referring to the test results for rocks (e.g., [27, 38]) that axial pre-stress significantly impacts materials' dynamic strength, failure mode, and energy dissipation features of geomaterials, Ma et al. [85] conducted a series of laboratory tests on artificial frozen silty clay (i.e., static uniaxial compression test, the dynamic uniaxial compression test, and the coupled static and dynamic loading test), and established a dynamic constitutive model for artificial frozen silty clay under coupled static and dynamic loads based on modified ZWT model and damage theory.

4 Discussion

The aforementioned studies are the latest developments in the constitutive model for frozen soils, presenting significant advancements in exploring the mechanical behaviours of frozen soils under static and dynamic loadings.

Various types of constitutive models of frozen soils under static loads have been proposed, which can be divided into ten categories: empirical models, elastoplastic modelling, viscoplastic modelling, hypoplastic modelling, binary element models based on homogenization theory, hyperplastic modelling, thermo-poromechanics-based constitutive modelling, constitutive models considering damage mechanics, DEM-based constitutive modelling, machine learning-based modelling. As for constitutive models under dynamic loads, eight types of models have also been developed. The primary advantages and limitations of these constitutive models under static and dynamic loadings are summarised in Tables 12 ans 13 to provide references for engineers and researchers.

Table 12 Typical characteristics of constitutive models for frozen soils under static loads

Table 13 Typical characteristics of constitutive models for frozen soils under dynamic loadings

It can be seen that empirical models, viscoplastic models, constitutive models considering damage mechanics, energy-based models, and micromechanics-based models have been developed for both static and dynamic analyses of frozen soils. Among the static models, the hypoplastic model exhibits distinct advantages compared to empirical models and conventional elastoplastic models, which is a powerful tool to capture and describe the mechanical behaviours of frozen soils. Besides, models based on damage mechanics and homogenization theory can explore the macro mechanical behaviours of frozen soils from a microscopic perspective, which can shed light on the impact of heterogeneous microstructures and provide a comprehensive understanding of internal damage evolution law. Incorporating thermodynamic theory or thermo-poromechanics into constitutive models for frozen soils can make the coupled thermo-mechanical model more rational and systematic, which enhances the model's capability to represent the mechanical behaviour of frozen soils accurately. In contrast, constitutive models for frozen soils based on DEM or ML method are appealing in describing the mechanical behaviours of frozen soils, although research in these areas is relatively scarce. DEM-based models offer a granular-level understanding of frozen soil behaviour by simulating the interactions between individual particles, representing micro-scale phenomena such as ice particle rearrangement, contact forces, and particle breakage. In addition, ML-based models display the potential to capture complex relationships and patterns in frozen soil behaviour using data-driven approaches, which enables the development of robust predictive models for frozen soil properties and behaviours, even in cases where traditional constitutive models may be limited. Both DEM and ML methods provide advantages in capturing the intricate nature of frozen soil behaviours and offer new avenues for enhanced understanding and predictive capabilities. However, it is essential to note that these methods also present their challenges, including the need for accurate calibration, rigorous validation, and substantial computational resources in the case of DEM. Careful consideration and extensive verification of these models are necessary before their widespread application in engineering practice.

The dynamic models for frozen soils under dynamic loadings are similar to the static constitutive models, including empirical models, viscoplastic models, and constitutive models considering damage mechanics. In addition, inspired by the behaviour of other composite materials, some researchers have utilised the conventional viscoelastic model, such as ZWT model, to capture the time-dependent characteristics of frozen soils. However, it is important to note that this viscoelastic model loses its capability to accurately predict nonlinearity and align with experimental data at lower strain rates. Energy-based models, incorporating the Gaussian peak function to describe absorbed energy, have been proposed to capture the dynamic behaviours of frozen soils subjected to uniaxial compressive impact loadings. Additionally, two surface plasticity model has gained popularity in modelling the deformation characteristics of soils under cyclic dynamic loading due to its clear conceptual framework and ease of numerical implementation. It is crucial to acknowledge that developing dynamic constitutive models for frozen soils is an ongoing area of research, and different models have been established based on various principles. Therefore, careful consideration should be given to the specific requirements of the intended application and the availability of reliable data for calibration and validation purposes.

5 Conclusions and Prospects

Developing a reliable and accurate constitutive model for frozen soil is crucial for understanding and predicting its mechanical behaviour, which plays a significant role in foundation design and engineering damage prevention. However, accurately capturing the mechanical behaviours of frozen soil is challenging due to its unique properties and complex nature. In this study, an extensive review was undertaken to examine the existing constitutive models for frozen soils under both static and dynamic loadings according to their theoretical foundations.

The static models were divided into ten categories, while the dynamic models consisted of eight types, and their primary advantages and limitations are also summarised. Up to now, significant advancements have been achieved in the development of constitutive models for frozen soil, which are verified to be efficient in describing the mechanical responses of frozen soils and providing theoretical and engineering references for construction projects associated with frozen soils. However, due to the complexity and randomness of frozen soils, as well as the variability of external conditions, some issues need to be addressed.

• (1) Previous constitutive models for frozen soils have been derived from different theories and verified using certain soil types and test conditions. These models possess complex mathematical formulations with model parameters obtained through fitting experimental data, typically considering individual factors while neglecting their combined effects. However, existing models have insufficiently addressed the random distribution of defects within frozen soils, which plays a crucial role in the formation of ice lenses and contributes to the heterogeneous and anisotropic nature of frozen soils. Additionally, uncertainties in the mechanical behaviours of frozen soils associated with these defects have not been adequately discussed. Therefore, it is necessary to establish comprehensive constitutive models that consider the joint influences of factors and account for the unique characteristics of frozen soils (i.e., temperature/time dependency, heterogeneity, and ice lenses). Moreover, these models should involve as few parameters as possible to facilitate their practical applications and numerical implementation. Besides, conducting reliability evaluations based on advanced constitutive models is beneficial to provide valuable guidelines for practical engineering.

• (2) Existing models often treat the strength criterion and creep model separately, leading to a fragmented representation of the stress–strain relationship and rate-dependent behaviours. In addition, creep deformation is often divided into two separate types (i.e., attenuation creep and nonattenuation creep), which hampers the overall applicability of these models. Therefore, there is an urgent need to establish a unified model that comprehensively captures and represents the diverse aspects of mechanical behaviours of frozen soils under complex conditions.

• (3) The validation of constitutive models for frozen soil should not be limited to specific soil types. Future investigations should aim to validate the models using a wider range of soil samples and conditions to ensure their reliability and generalisability. Moreover, the determination of model parameters should not solely rely on fitting experimental data but also incorporate intelligent algorithms to enhance their accuracy and robustness. To facilitate this validation process and promote further research in the field, it would be beneficial to establish a comprehensive database that consolidates information on the mechanical behaviours of frozen soils. This database can be constructed by incorporating data from scientific publications and government reports, thereby providing a valuable resource for the constitutive modelling of frozen soils.

• (4) The advancements in discrete element method (DEM) and machine learning (ML) techniques provide a promising approach to expanding the capabilities of constitutive modelling for frozen soils. These two powerful tools can be leveraged to explore the micro-mechanisms underlying the behaviour of frozen soils and enhance the capacity for data mining and analysis. With the ability to investigate the micromechanical properties and strong capacity in data mining, incorporating DEM or ML methods into constitutive modelling can provide a more comprehensive understanding of the complex interactions within frozen soils and unlock new possibilities for predictive modelling and analysis.

  Overall, the development of an advanced and comprehensive constitutive model for frozen soil remains an ongoing research endeavour, with the potential to significantly enhance engineering practices in cold regions and improve the safety of infrastructure built on frozen ground. Further research and in-depth investigations should be conducted to advance the development of constitutive models for frozen soils, particularly those recently developed based on machine learning and DEM, and seamlessly integrate these models with computer simulations (such as the finite element method) to tackle the intractable thermo-hydro-mechanical problems associated with frozen soils.

Appendix

See Table

Table 14 Summary of typical rheological elements and combinations

14.