A non-linear constitutive model for describing the mechanical behaviour of frozen ground and permafrost

Cold Regions Science and Technology 133 (2017) 63–69

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A non-linear constitutive model for describing the mechanical behaviour of frozen ground and permafrost A.F. Rotta Loria* , B. Frigo, B. Chiaia Department of Structural, Geotechnical and Building Engineering, Politecnico di Torino, 9, Corso Duca degli Abruzzi, Torino 10129, Italy

A R T I C L E

I N F O

Article history: Received 25 June 2015 Received in revised form 2 April 2016 Accepted 27 October 2016 Available online 30 October 2016 Keywords: Frozen ground Permafrost Elasto-plastic model Associated flow rules

A B S T R A C T The mechanical behaviour of frozen ground and permafrost is changing under the increasing variation of environmental and anthropogenic boundary conditions. This phenomenon affects many civil structures and infrastructures built in Polar and Alpine areas. Mathematical formulations able to capture the mechanical behaviour of frozen ground and permafrost with adherence to reality and a limited employment of technical resources and time appear crucial for the engineering design and retrofit of these structures. To address this challenge, this study presents a relatively simple elasto-plastic constitutive model for capturing the non-linear mechanical behaviour of frozen silt. The model is based on associated flow rules. It employs an elliptical yield surface and a parabolic yield surface for describing the volumetric mechanisms that characterise the modelled material, together with a parabolic yield surface for describing the deviatoric mechanism. Comparisons with experimental triaxial test results available in the literature highlight the suitability of the model to capture the non-linear mechanical response of frozen silt subjected to both low and high confining pressures. This result, together with the doable implicit consideration in the model of the effects induced by environmental boundary conditions such as temperature on the mechanical behaviour of the material, makes this tool attractive for simplified yet thorough analyses of frozen ground and permafrost-related problems. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Frozen ground and permafrost in Polar (high latitudes, i.e., regions close to the North and South poles) and Alpine (high elevations in lower latitudes, i.e., 2200–3500 m a.s.l.) areas are more often negatively affected by a degradation of mechanical properties. Severe damages to buildings and infrastructures (Haeberli, 1992; Phillips et al., 2007; Phillips and Margreth, 2008; Arenson et al., 2009, Romanovsky and Osterkamp, 2001; Clarke et al., 2008; Instanes, 2003; Streletskiy et al., 2014; Duvillard et al., 2015) as well as phenomena of slope and flat ground instability (Davies et al., 2001; Gruber et al., 2004; Gruber and Haeberli, 2007; Haeberli, 1985; Huggel, 2009; Arenson and Springman, 2000; Nelson et al., 2003; Romanovsky et al., 2007; Arnold et al., 2005; Pei et al., 2014; Shan et al., 2014; Stoffel et al., 2014; Deline et al., 2015; Haeberli et al., 2016) are increasingly observed in such areas. These phenomena are generally induced by the combined action of climate change,

* Corresponding author. E-mail addresses: alessandro.rottaloria@epfl.ch (A. Loria), [email protected] (B. Frigo), [email protected] (B. Chiaia).

http://dx.doi.org/10.1016/j.coldregions.2016.10.010 0165-232X/© 2016 Elsevier B.V. All rights reserved.

construction activity and existing structures, which involves variations of the temperature and stress fields characterising the substrate and results in detrimental changes of the mechanical behaviour of the ground. In fact, besides the strong dependence of the strength and stiffness of frozen ground and permafrost on intrinsic material properties such as moisture content, air bubbles, salts, organic matter and grain sizes (Parameswaran and Jones, 1998; Fukuo, 1966a,b; Anderson and Morgenstern, 1973; Sayles, 1988; Ladanyi, 1981, 1985; Andersland and Ladanyi, 2004; Arenson et al., 2007, 2014; Springman and Arenson, 2008; among others), and on different stress histories and strain rates/paths (Andersland and Ainouri, 1970; Baker, 1979; Fish, 1981; Zhu and Carbee, 1987; Wu and Ma, 1994; Guryanov and Ma, 1995; Arenson et al., 2004; Lai et al., 2014, among others), confining pressures and thermal loads are crucial for the variation of the mechanical behaviour of these materials (Chamberlain et al., 1972; Alkire and Andersland, 1973; Baker et al., 1982; Ting et al., 1983; Andersen et al., 1995; Cuccovillo and Coop, 1997; Da Re et al., 2003; Arenson and Springman, 2005a,b; Yang et al., 2010; Yamamoto and Springman, 2012, among others). The strength of frozen ground and permafrost generally increases with greater applied confining pressures, but may abruptly decrease under high confining pressures because of pressure melting and

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crushing phenomena. These phenomena may be enhanced by temperature variations, especially when changes of the phases that characterise the matrix constituents may occur. In an attempt to model the influence of the aforementioned variables on the mechanical behaviour of frozen ground and permafrost, mathematical formulations such as constitutive models and strength criteria are increasingly formulated and exploited, as they are the basis to describe the response of bodies under loading. Currently, besides a series of formulations that have been proposed for describing the mechanical behaviour of frozen ground and permafrost subjected to uniaxial loads (Zhu and Carbee, 1987; Zhu et al., 1992), long-term loads (Goughnour and Andersland, 1968; Ladanyi, 1972; Sayles, 1973; Fish, 1983, 1984; Zhu and Carbee, 1983; Gardner et al., 1984; Cai et al., 1990; Domaschuk et al., 1991; Wijeweera and Joshi, 1991; Miao et al., 1995; He et al., 2000; Yang et al., 2004; Wang et al., 2006, 2014), and evolutionary thermal fields (Arenson and Springman, 2005b; Li et al., 2009; Yang et al., 2010) with the consideration of a number of associated problems (Nishimura et al., 2009; Thomas et al., 2010; Zhang and Michalowski, 2015; Zhang et al., 2016), models for capturing the effects of confining loads have been presented (Ma et al., 1995; Lai et al., 2009, 2010, 2013, 2016; Yang et al., 2010; Li et al., 2010; Liao, 2016). However, when attempting to capture the highly nonlinear mechanical behaviour of frozen ground and permafrost under variations of confining pressure, there is the main issue of needing to use complex mathematical formulations that rely on many material parameters. This need often results in challenging and timeconsuming analyses of real engineering problems, with a number of technical and economical shortcomings for the engineering practice. The proposition of mathematical formulations based on fewer material parameters and able to capture with suitable adherence reality seems important in this scope. Looking at such challenge, this study presents an additional constitutive model for performing preliminary analyses of the mechanical behaviour of frozen ground and permafrost. In particular, this paper proposes a non-linear elasto-plastic constitutive model for describing the mechanical response of frozen silt subjected to different magnitudes of confining pressure, based on the groundwork performed by Lai et al. (2010). The goal of this work is to develop a mathematical tool that may be expediently and effectively applied for performing simplified yet thorough numerical analyses of frozen ground and permafrost-related problems. In the following, the elasto-plastic constitutive model is first formulated. The differences from the model presented by Lai et al. (2010) are discussed along the lines. Forethoughts that may allow to implicitly account through the proposed model for the effects of environmental boundary conditions such as temperature on the mechanical behaviour of the modelled material are also presented. Then, the constitutive model is applied for simulating the mechanical behaviour of frozen silt tested in triaxial conditions (Lai et al., 2010) under both low and high confining pressures, and is afterwards validated. Finally, comments on the capability of the constitutive model for capturing the non-linear mechanical behaviour of frozen ground and permafrost are summarised, and related concluding remarks are outlined.

of the material, including the definition of the mechanisms governing the development of plastic deformation and the direction of its evolution; and (iv) the expressions (hardening rules) describing the magnitude of plastic deformation. Aspects (i) and (ii–iv) characterise the elastic and plastic constitutive description of any modelled material, respectively. According to the theory of plasticity, the total strain tensor 4 = 4e + 4p

(1)

can be decomposed into a sum of two symmetric tensors, i.e., the elastic strain tensor, 4e , and the plastic strain tensor, 4p . It is assumed that the modelled material exhibits plastic flow, but suffers no damage. Therefore, the elastic stiffness of the material remains unchanged. The stress-strain relation is in this case: s = C e : 4e = C e : (4 − 4p )

(2)

where s is the total stress tensor and Ce is the elasticity matrix. To determine the complete stress-strain relation, the elastic and plastic strain tensors need to be determined. Attention is given to these aspects in the sections that follow. 2.2. Elastic constitutive description In this study, the elastic part of the strain is defined through the relation between the shear modulus, G, and the bulk modulus, K, which reads K=

2(1 + m) G 3(1 − 2m)

(3)

where m is the Poisson’s ratio, which can be determined through the formula proposed by Bardet (1997), for example. As it has been remarked in the literature (Andersland and Ladanyi, 2004; Lai et al., 2009, 2010; Yang et al., 2010), the shear modulus depends on factors such as atmospheric pressure, pa , and confining pressure, s 3 . The shear modulus generally increases at a first extent up to a maximum value with increasing confining pressure and then decreases for a further increase of confining pressure. Looking at a thorough description of this phenomenon, the empirical parabolic relation proposed by Lai et al. (2010) is employed in the following:   G = ag

s3 pa

2

 + bg

s3 pa



 + cg pa

(4)

where ag , bg and cg are material constants. Knowledge of the elastic parameters presented above allows calculating the elastic components of deformation (i.e., volumetric and deviatoric) in a conventional way. Decomposition of the total strain in the elastic and plastic components (Eq. (1)) allows next to face the problem of determining the plastic component of deformation.

2. Elasto-plastic constitutive modelling of frozen silt

2.3. Plastic constitutive description

2.1. Hypotheses and components of the model

2.3.1. Yield functions According to Yao et al. (2004), the yield surfaces of frozen soils and rocks are as follows: (i) smooth and convex in p-plane, i.e., the two-dimensional setting passing through the origin of the principal stress space where s 1 + s 2 + s 3 = const, and pq-plane, i.e., the two-dimensional setting where the mean stress p is paired with the generalised deviatoric stress q (Roscoe and Burland, 1968; Muir Wood, 1990); (ii) characterised by a critical state line that passes through the isotropic tensile point (r1 = r2 = r3 = fttt ) instead of the origin, differently from most geomaterials;

The following constitutive model is referred to an isotropic and homogeneous material that is characterised by an elastoplastic behaviour. The formulation of the model comprises the definition of four key components: (i) the relations governing the reversible behaviour of the material (elastic deformation); (ii) the criteria (yield functions) characterising the limit for which an irreversible behaviour may occur (plastic deformation); (iii) the relations (plastic potential functions) characterising the irreversible behaviour

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(iii) symmetric with regard to three principal stress axes in the pplane; (iv) characterised by an increasing circular shape with the increase of the mean pressure p in p-plane; and (v) temperaturedependent, generally highlighting increasing strength for increasing negative temperatures. For isotropic materials that follow an elasto-plastic hardening/ softening rule, the yield criterion can be written as follows

2.3.1.3. The yield function for the dilatant volumetric behaviour. To describe the dilatant volumetric behaviour for frozen silt during the strain softening stage, the following empirical parabolic yield function (Lai et al., 2010) is considered:

f (s, ai ) = f (I1 , I2 , I3 , ai ) = f (p, q, h, ai ) = fp−q (p)fp (h) = 0

In this formulation s 0 is the critical dilatant stress, whereas asv , bsv and cvs are material parameters. The parameters s 0 , asv , bsv and cvs all depend on both confining pressure and temperature and can be determined based on experimental data. The isotropic tensile stress, fttt , depends as well on temperature and can be determined based on experimental tests. In general, for the same ice content of any frozen soil matrix, the higher the negative temperature field is, the higher the isotropic tensile stress is because of the increasing cohesive strength between the ice and the soil particles (Ma et al., 1995; Lai et al., 2009, 2010; Yang et al., 2010; Li et al., 2010).

(5)

where a i are a set of internal variables accounting for the material state that characterise the size and shape of the yield surface, I1 = s 1 + s 2 + s 3 , I2 = s 1 s 2 + s 2 s 3 + s 1 s 3 and I3 = s 1 s 2 s 3 are the first, second and third order invariants of the principal stress tensor, respectively, p = (s 1 + s 2 + s 3 )/3 is the mean stress, √  1/2 q = 1/ 2 (s1 − s2 )2 + (s2 − s3 )2 + (s3 − s1 )2 is the deviatoric stress, h is the Lode’s angle, and the product fp (h)fp−q (p) is a combination of the strength functions in the p-plane and in the p-q-plane, respectively. Because the material is assumed to be isotropic, the yield surface that results from the formulation of this function maintains its shape, center and orientation, and expands or contracts uniformly around its center (Yu, 2006). Based on experimental evidence available in the literature (Lai et al., 2010), it is considered that the mechanical behaviour of frozen silt can be described by an elliptical yield surface and a parabolic yield surface for capturing the volumetric mechanisms characterising the modelled material at both low and high confining pressures, respectively, and by a parabolic yield surface for capturing the deviatoric mechanism. These surfaces may allow to capture the strain hardening/softening, particle crushing and dilation phenomena characterising frozen silt subjected to different (e.g., notable) magnitudes of confining pressure. In this modelling approach, explicit consideration of aspects (i)–(iv) described above is made, although not of aspect (v), i.e., the temperature dependence of the strength of the material. Along with the capability of this model to explicitly account for complex phenomena such as those induced by varying confining pressures, implicit consideration of temperature effects is still possible through a variation of the input parameters based on reference experimental data. This approach appears in agreement with the isothermal formulation of the stress-strain relation proposed in Eq. (2). 2.3.1.1. The yield function for the contractive volumetric behaviour. To describe the contractive volumetric behaviour in the plastic region for frozen silt, the following empirical elliptical yield function (Lai et al., 2010) is considered: fv c (p, q, av ) =



p − dp0 av kp0



2 +

q wp0

2 − av2 = 0

(6)

In this formulation k, w and d are material parameters that determine the long and short axes, as well as the center of the elliptical yield function, p0 is the reference mean stress, and a v is a hardening term. The parameters k, w and d depend on temperature and can be determined based on experimental data. A variation in their quantity physically represents a variation of the elastic domain of the modelled material. 2.3.1.2. The yield function for the deviatoric behaviour. To describe the deviatoric behaviour in the plastic region for frozen silt, the following empirical parabolic yield function (Lai et al., 2010) is considered: fq (p, q, aq ) = q2 − pa aq p = 0 where a q is a hardening term.

(7)

 s0 p fv s p, q, 4v = G



(p − fttt ) q2

p

− asv − bsv 4v − cvs = 0

(8)

2.3.2. Plastic potential functions and hardening rules A key question for any elasto-plastic problem is how to determine plastic strains once the stress state is on the yield surface. To describe the mechanisms of plastic deformation and the direction of plastic strain for frozen silt the following flow rule is considered: 4˙p =

2  n=1

∂ fn k˙ n ∂s

(9)

where k˙ n are positive scalars that control the magnitude of plastic strain (related to the volumetric, kv , and deviatoric, kq , mechanisms that characterise frozen silt) and the gradients ∂ fn /∂ s are fixed tensors that specify the direction of plastic strain rate (for the volumetric, fv , and deviatoric, fq , mechanisms). In this plastic formulation, which expresses an associated (or normality) flow rule because the yield functions fn are the same as the plastic potential functions gn (fn = gn ), it is assumed that the principal axes of the plastic strain tensor coincide with those of the stress tensor. Thus, the directions of 4p and s are the same. In principle, non-associated (fn = gn ) constitutive equations should be preferred for geomaterials, which may be affected by density variations under either contractive or dilatative volume changes. In practice, especially in the framework of engineering numerical analyses, associated flow rules are usually employed because of their practical simplicity and adequate ability to describe reality. This choice can be considered particularly suitable when dealing with problems denoted by notable stress states, because the overestimation of the dilatant mechanisms governing the material behaviour that is associated with normality flow rules is limited. Based on this fact, and looking at a simplified version of the constitutive model proposed by Lai et al. (2010) – still capable of capturing with suitable adherence reality – in view of its advanced peculiarities, an associated flow rule is considered. The choice of associated flow rules instead of non-associated flow rules represents the crucial diversifying aspect characterising the formulation of the elasto-plastic constitutive model proposed in this study compared to that of Lai et al. (2010). This approach leads to need 6 material parameters less than the original mathematical formulation of Lai et al. (2010) for describing any considered problem. This feature allows for a more straightforward application of the constitutive model along with a still adherent simulation of reality. The loading-unloading conditions (or Karush-Kuhn-Trucker conditions (Karush, 1939; Kuhn and Tucker, 1951)) need then to be satisfied. These conditions mathematically specify that in the elastic region the yield function must remain negative and the rate of the plastic multiplier must be equal to zero (plastic strain remains constant), while in the plastic regime the yield function must be equal

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to zero (stress remains on the yield surface) and the rate of plastic multiplier must be positive (Jirásek and Bazant, 2002), i.e.,

The hardening term a q associated to the deviatoric mechanism is q related to the plastic deviatoric strains 4v through the relation

k˙ n ≥ 0, fn (s, ai ) ≤ 0, k˙ n fn (s, ai ) = 0

a q 4q aq =  2 p 4q + bq 4q + cq

(10)

The last condition expressed in Eq. (10) means that if k˙ k > 0 (plastic flow) then f(s, a i ) = 0 (stress on the yield surface), and if f(s, a i ) < 0 (stress inside the elastic domain) then k˙ k = 0. This is called condition of consistency and is exploited to determine the complete stress-strain relation through the definition of the plastic multipliers and the elasto-plasticity matrix. The functions related to the volumetric and deviatoric mechanisms can be defined in this context by referring to Eqs. (6)–(8) as fv (s, 4p ) = fv (p, q, av ) = 0

(11)

fq (s, 4 p ) = fq (p, q, aq ) = 0

(12)

Differentiating Eqs. (11)–(12) and referring to Eq. (9), it is found in compact form: 1 ∂ fv 1 ∂ fv k˙ v = p˙ + q˙ Av ∂ p Av ∂ q

(13)

1 ∂ fq 1 ∂ fq k˙ q = p˙ + q˙ Aq ∂ p Aq ∂ q

(14)

p

where aq , bq and cq are fitting parameters directly dependent on atmospheric pressure and confining pressure that can be determined at different temperatures based on experimental results. This formulation allows relating the hardening parameter with the hardening and softening behaviours of frozen silt observed experimentally (Lai et al., 2010) at different magnitudes of confining pressure. In particular, this formulation enables to reproduce stress-strain curves for the modelled material that change from a strain hardening behaviour to a strain softening behaviour with the increase of confining pressure. The various terms needed for deriving the plastic deviatoric strain are as follows: p a q 4q ∂ fq = −  2 ∂p p

(24)

∂ fq = 2q ∂q

(25)

(15)

Aq = −

∂ fq ∂ aq ∂ fq : : ∂ aq ∂ 4p ∂ s

(16)

and a v and a q two hardening parameters for the volumetric and deviatoric mechanisms, respectively. The hardening term a v associated to the volumetric contractive p mechanism is related to the plastic volumetric strains 4v through the relation  p av = acv + bcv ln 4v + cvc

(17)

where acv , bcv and cvc are fitting parameters directly dependent on atmospheric pressure and confining pressure that can be determined at different temperatures based on experimental results. The various terms needed for deriving the plastic volumetric strain are as follows:

∂ fq ∂ aq ∂ fq ∂ aq ∂ 4qp ∂ q

∂ fq = −pa p ∂ aq

where Av and Aq are hardening terms defined as

∂ fv ∂ av ∂ fv : : ∂ av ∂ 4p ∂ s

+ bq 4q + cq

4q

Aq = −

Av = −

(23)

∂ aq = ∂ 4qp

(26) (27)



 p 2 p p aq 4q + bq 4q + cq − aq 4q 24q + bq 

2 p 2 4q + bq 4q + cq

(28)

The various terms needed for deriving the plastic volumetric strain for a history in which only the dilatant plastic volumetric mechanism is activated are as follows:

∂ fv s s0 pa =  ∂p 2Gq (p − fttt ) pa

(29)

s0 (p − fttt ) pa ∂ fv s =− ∂q Gq2

(30)



∂ fv s ∂ fv s ∂ 4vp ∂ p

(31)

 ∂ fv s s p s p = − 2av 4v + bv ∂ 4v

(32)

Asv = −

2.4. The complete elasto-plastic stress-strain relation

∂ fv c 2 (p − dp0 av ) = ∂p k2 p20

(18)

According to Eq. (9), the stress-strain relation expressed by Eq. (2) is as follows:

2q ∂ fv c = ∂q w 2 p20

(19)

∂f ∂f ˙ = C e : (˙4 − 4˙ p1 − 4˙ p2 ) = C e : (˙4 − k˙v v − k˙q q ) s ∂s ∂s

∂ fv c ∂ av ∂ fv c ∂ av ∂ 4vp ∂ p

(20)

By multiplying both members of Eq. (33) for ∂ f/∂ s and referring to the condition of consistency, the plastic multipliers can be derived as

2d (dp0 av − p) ∂ fv c = − 2av ∂ av k2 p0

(21)

k˙ v =

Aq b 2 + b 2 b 4 − b 3 b 5 A v A q + A v b4 + A q b1 + b 1 b4 − b 3 b6

(34)

bcv ∂ av p = p ∂ 4v 4v + c v

(22)

k˙ q =

A v b5 + b 1 b5 − b 2 b6 A v A q + A v b4 + A q b1 + b 1 b4 − b 3 b6

(35)

Acv = −

(33)

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Table 1 Material parameters (Lai et al., 2010) for frozen silt exploited in the numerical simulations (the terms asv , bsv and cvs are referred to a confining pressure of s 3 = 2 MPa). fttt [MPa]

ag [-]

bg [-]

cg [-]

pa [MPa]

k [-]

w [-]

c [-]

asv [-]

bsv [-]

cvs [-]

s0 [MPa]

−2.4

−0.3997

67.123

−11.397

0.10133

0.941

0.7528

0.059

0.2488

−0.0148

0.0009

9.15

where:

b1 =

∂ fv e ∂ fv :C : ∂s ∂s

(36)

b2 =

∂ fv e :C :4 ∂s

(37)

b3 =

∂ fv e ∂ fq :C : ∂s ∂s

(38)

b4 =

∂ fq e ∂ fq :C : ∂s ∂s

(39)

b5 =

∂ fq e :C :4 ∂s

(40)

Fig. 1. Comparison between the numerical and experimental (Lai et al., 2010) q − 4a − 4v curves for frozen silt obtained at −6 ◦ C.

68

b6 =

A. Loria et al. / Cold Regions Science and Technology 133 (2017) 63–69

∂ fq e ∂ fv :C : ∂s ∂s

(41)

Eqs. (34)–(35) can be then substituted into Eq. (33) to get the formulation of the elasto-plasticity matrix Dep , which is a symmetric matrix because the flow rule is associated. Finally, the complete elasto-plastic stress-strain relation in rate form can be written as ˙ = Dep : 4˙ s

(42)

3. Validation of the elasto-plastic constitutive model In order to validate the presented elasto-plastic constitutive model based on associated flow, triaxial tests on frozen silt (Lai et al., 2010) are numerically simulated. The triaxial tests were performed on specimens taken from the Qinghai-Tibet Railway constructions site. After preparation, these specimens were placed into a triaxial pressure cell, kept at a given constant negative temperature for 24 h, subjected to confining and axial pressures for 5 min, and tested. The testing temperatures were −2, −4 and −6 ◦ C with a precision of ±0.1 ◦ C. The confining pressures ranged from 0 to 14 MPa and the shear strain rate was 1.67 × 10 −4 s −1 . The numerical simulations consider a series of results of the experimental tests performed at −6 ◦ C, for which material parameters were available (cf., Table 1), with reference to the increasing values of confining pressure applied of s 3 = 2, 7 and 12 MPa. A comparison between the numerical and experimental results is presented in the following. Panels a, c, and e in Fig. 1 show the comparison of the numerical and experimental deviatoric stress-axial deformation curves for the tests denoted by s 3 = 2, 7 and 12 MPa, respectively. Panels b, d, and f in Fig. 1 present the comparison of the numerical and experimental deviatoric stress-volumetric deformation curves obtained for the same considered tests. A close agreement between the curves can be remarked in all cases. The analyses show that the mechanical behaviour of frozen silt is markedly affected by confining pressure. The higher the confining pressure is, the higher the shear strength of the material is. Nonetheless, the higher the confining pressure is, the weaker the strain softening phenomenon and the stronger the strain hardening of the material are. Under the lower value of confining pressure of s 3 = 2 MPa, the presented yield criteria, coupled with the associated flow rules, enable to accurately describe the plastic deformation mechanisms that characterise frozen silt, i.e., compressive and dilatant. Under the higher values of confining pressure of s 3 = 7 and 12 MPa, the presented formulations enable as well to describe with close adherence the plastic deformation mechanisms that characterise frozen silt, i.e., compressive and contractive. Therefore, the employment of associated flow rules allows in all cases the behaviour of the modelled material to be captured with acceptable discrepancies. This result appears to make the proposed constitutive model a potential simplified alternative to that originally presented by Lai et al. (2010), still suitable for capturing the non-linear mechanical behaviour of frozen silt in foreseeable numerical analyses. 4. Conclusions This study proposes a mathematical tool to analyse the nonlinear mechanical behaviour of frozen ground and permafrost, and to tackle the engineering issues that are increasingly affecting structures and infrastructures built in cold regions as a consequence of the combined action of construction activity, existing structures and climate change. In particular, the work proposes an elasto-plastic constitutive model based on associated flow rules for describing the non-linear mechanical behaviour of frozen silt subjected to different magnitudes of confining pressure. Based on a comparison

with a series of triaxial test results, the constitutive model shows the capability of capturing the fundamental mechanisms denoting the mechanical response of frozen silt subjected to notable confining pressures, i.e., strain hardening/softening, particle crushing and dilation. The model is characterised by an empirical nature and thus needs to be calibrated based on experimental test results. However, the model does allow for explicit consideration of the effects of a complex variable such as confining pressure and still for implicit consideration of the effects of other variables such as temperature. The capability of the model to be calibrated with results of experimental tests performed at different temperatures allows in fact to implicitly account for the effects of a variation of this environmental variable on the mechanical behaviour of the modelled material. This feature may allow performing soil-structure interaction analyses with reference to different thermal fields. In view of all of these features, it is considered that the proposed constitutive model represents a relatively simple mathematical tool that may be applied in an effective way for performing preliminary yet thorough analyses of frozen ground and permafrost-related problems.

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