The Classical Theory of Plasticity applied to Permafrost Problems

Alessandro F. Rotta Loria1, Barbara Frigo1, Bernardino M. Chiaia11Department of Structural, Geotechnical and Building Engineering, Politecnico di Torino, ItalyE-mail: alessandrofrancesco.rottaloria@studenti.polito.it, barbara.frigo@polito.it,bernardino.chiaia@polito.it

Keywords: Permafrost, evolutionary behaviour, strength criterion, elasto-plastic model.

SUMMARY. From 1900 to 2100, the average global surface temperature on crustal Earth has beenprojected to increase from 1.4 to 5.8 oC [1]. This climate change will remarkably affect permafrostareas, probably leading to increasing natural hazards.

In this context, the evolutionary behaviour of permafrost seems to play a fundamental role: char-acterizing the physics of such grounds from smaller to bigger scales, this phenomenon is not oftentackled in the proper way and induces several engineering problems.

At current, albeit the topic has been investigated by several theoretical and experimental studiesin various scopes, it remains poorly characterized, especially in the structural engineering field. Theaim of this paper is to deepen the knowledge of the governing laws relating frozen soils within thecomplex soil-structure interaction. This is fundamental in order to properly design, build and retrofitstructures in permafrost areas.

1 INTRODUCTIONThe so-called ”permafrost” is a crust material (loose soil or rock) with included ice and organic

material that remains at or below 0 oC for at least two consecutive years [2]. It usually characterizeshigh latitudes (i.e. regions close to the North and South poles, highly diffused in Canada, Russiaand Greenland) for the cooling resulting from a decrease in solar energy supply, but exists also athigh altitudes (2.200 ÷ 3.500 m a.s.l.) in lower latitudes as Alpine or Mountainous permafrost [3].More generally, permafrost is considered to be a physical phenomenon occurring in a periglacialenvironment, therefore including a series of forms and processes connected with a cold, but notglacial, climate [4]. Indeed, this process is connected with an environment where water exists andoccurs in a liquid state for at least a short time. Freezing and thawing induce, what is referred to as,periglacial forms and processes. However, it is not water, but temperature an indispensable conditionfor permafrost to occur [3]. Permafrost occurrence is, in fact, an effect of the impact of a frostyclimate on the lithosphere in direct or indirect way. For example, permafrost originates when theextent of winter freezing of ground exceeds the extent of its summer thawing. Due to climatic cooling,an initial layer of frozen or cryotic ground is formed and remains there till next winter. If the coolingcontinues, there is an increase in the thickness of the layer year after year, leading to the growth of thepermafrost; conversely, if warming occurs, the active permafrost table reduces in thickness, usuallycausing settlements.

The temperature, through thawing and melting crushing phenomena, is also the factor that mainlycharacterizes the evolutionary behaviour of permafrost. From a mechanical point of view, even smalltemperature variations can induce remarkable changes in the strength of the considered ground, aswell as different responses with respect to external loads; yet, temperature variations can triggerseveral degradation processes, resulting in important environmental changes.

Looking at the recent climate warming, predicted to continue during the next century and amplifiedin Arctic and sub-Arctic regions, it seems hence fundamental to deepen the topic of evolutionary

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permafrost, in order to find theoretical and practical solutions for the engineering related problems.Many infrastructures, such as dams, bridges, pipelines and cable-transportation structures are

founded on permafrost, and therefore can be subjected to settlements and damage.

2 ENGINEERING PROBLEMS RELATED TO PERMAFROSTThe permafrost phenomenon has been investigated within the branches of geophysics, geomor-

phology, geology and biology, to geotechnical and structural engineering. In these two latter fields,several studies have analysed the warming of permafrost and the possible impacts on infrastructuresfrom global warming scenarios (i.e. [5, 6, 7]), and many others have deepened the topic of permafrostslope stability in relation to this phenomenon (i.e. [8, 9]). However, few researches have mergedthe topics dealing through a more theoretical approach with the key concept of the problem: themechanical modelization and the soil-structure interaction.

Geotechnical and structural engineering problems in permafrost areas are triggered by the climatewarming and are related to the resulting permafrost thawing. However, climate warming cannot bedirectly associated with infrastructure damage: degradation mechanisms due to thermal, chemicaland mechanical processes seem to be the damage causes. In most cases, climate warming may act asan accelerator for these processes, associated with construction activity and existing infrastructure(increased radiation and snow accumulation). Salts, surface water flows, ice inclusions, winter andsummertime precipitations are other factors that can potentially accelerate these phenomena. Theform and rate of these processes differ between regions, depending on geographical location, onspecific environmental settings, on the ground mechanical properties and stratigraphy.

In general, permafrost alterations are exhibited through local ground movements, phenomena ofinstability that affect slopes or flat ground in a surface of hundreds to thousands of square meters,inducing horizontal downslope movements, due to the creep of permafrost bodies, and vertical settlingmovements, due to the melting of ice bodies and/or interstitial ice [10]. These movements maycause severe damages to buildings and infrastructures and more rarely represent hazards for persons.In the past, local ground movements were usually monitored and studied in low and middle-rangemountain environments, while in high mountain range were rarely investigated because of (i) logisticaldifficulties, (ii) scarce interaction with infrastructures and villages and (iii) the low level of knowledgeon the phenomena related with the cryosphere. However, the recent climate warming, the increasinglypresence of human tourism and infrastructures in high mountains, have pointed out the importance ofa more detailed study of these phenomena.

Especially for building foundations built on frozen grounds, roads and railway tracks, mountaincable structures, as well as hydraulic dams, the functionality and structural stability is often severelydamaged by evolutionary permafrost.

2.1 Damages to buildingsBuildings are often damaged by differential settling of the ground [10] through fissuring, tilting,

or even collapse (cf. Figure 1). Especially in permafrost discontinuous zones, differential settlementsare induced by thermokarst, a phenomenon that forms when ground ice melts, the resulting waterdrains and the remaining soil collapses into the space previously occupied by ice; this happens whenthe active layer of soil above the permafrost, which thaws during the summer, does not refreezecompletely even during the most severe winter. In this case year-round decomposition of organicmatter can occur and permafrost continues to thaw from the top down, causing subsidence.

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Figure 1: Effects of permafrost thawing on buildings (images courtesy of V. Romanovsky, NPS andA. Slater).

2.2 Damages to roads and railway tracksEspecially in the Arctic, where the permafrost is known to be very sensitive to any change in

surface conditions and in a very precarious equilibrium with climate, road, railways and pipe-lines areoften damaged by the settlements induced by thermokarst (cf. Figure 2). Even a local destabilizationof the permafrost layer beneath these structures can, in fact, induce severe functionality problems.

Figure 2: Effects of thermokarst on roads and railway tracks (images courtesy of J. Moore and USGeological Survey).

2.3 Damages to mountain cable structuresSkilifts, chairlifts, cabins and cable cars are very sensitive to the alignment of the pylons supporting

the cable. If small movements can be accommodated by technical adjustments, larger cumulatedmovements can lead to a bad alignment of pylons, causing the cable to jump out, severely damagingthe structure and maybe causing accidents.

Mountain cable structure supports can be therefore remarkably damaged by soil movements, aswell as by brittle mechanisms characterising rock masses. The ice segregation process is one of thesemechanisms: its action is catalysed by temperature gradient-induced suction, which involves watermigration in the matrix where lenses or layers of ice grow, maybe causing crack propagation. Thecrack propagation usually forms bedding planes into the rock mass and often causes the rock to fail.Thus, if the bearing layer of the structure support collapses, extremely dangerous consequences arise.

2.4 Damages to hazard protection systemsAvalanche defense structures and rock falls protection systems are frequently located at high

altitude sites, where unstable ice-bearing permafrost substrates often occur. Permafrost thawing

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particularly damages these structures, especially when they are directly loaded by snow or rockmasses and not equipped with proper anchor systems (cf. Figure 3).

Figure 3: Rock fall into snow nets and related technological problems [10].

The summary presented above about the engineering problems related to permafrost does notconsider cases induced by human errors. However, albeit structural damage is often blamed onclimate warming, several case histories indicate human design errors or exceeded design lifetimebeing the effective causes of these problems [7]. The aim of the present section, recognising theimportance of these latter triggering causes, was however to focus on the environmental causes ofstructural damage, for their increasingly importance with the future climate change.

3 PHYSICAL PROPERTIES OF PERMAFROSTIn general, permafrost soils are four-component systems constituted by solid grains, ice, unfrozen

water and gas or air. There are also cases in which permafrost contains an almost null percentage ofice or water (dry permafrost) or a very high percentage of ice (ice-rich permafrost).

Granular solid particles can be organic or inorganic and are usually characterized by differentsizes and shapes; voids are in general filled with ice or unfrozen water and almost always with air.Ice particles may be distributed more or less uniformly, depending on their own type of formation,constituting ice masses (of different shapes and dimensions) or small ice inclusions [11]. Thepercentage of these components and their physical properties remarkably affect the thermal andmechanical properties of permafrost.

3.1 Thermal propertiesPermafrost thermal behaviour mainly depends on the consolidation state of the soil and on the

dimensions of its grains, in relation to the ice, moisture, water and organic matter content of the matrix.Moreover, different climatic conditions, geothermal gradient, air and ground surface temperatureof the area considered, in addition to construction activity, can induce remarkable variations of theground latent heat, thermal conductivity, and heat or insulation capacity, characterising the thermalbehaviour of permafrost as extremely sensitive to boundary conditions.

3.2 Mechanical propertiesPermafrost mechanical behaviour is governed by the intrinsic material properties such as moisture

content, air bubbles, salts, organic matter and grain sizes [12]; nonetheless, it may differ a lot fordifferent stress and strain histories, strain rates, confining pressures and load types. This variablebehaviour is also highly affected by the environmental boundary conditions; the temperature trendis one of the most influencing factors for the evolution of the mechanical behaviour of permafrost,

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indeed. Even small increase in temperature values can lead to very different soil resulting strengthsand, therefore, to unexpected conditions for which the building or the infrastructure built into oron the ground considered could not be properly designed. With respect to all these parameters andboundary conditions, the mechanical behaviour of permafrost can range from brittle to plastic, withmore or less marked viscous components.

4 THE CLASSICAL THEORY OF PLASTICITY APPLIED TO PERMAFROSTBased on recent experimental and theoretical achievements in the applied mechanics of frozen

soils, and on the work performed by Yuanming for frozen silts [13], a strength criterion and aconstitutive elasto-plastic model to describe the evolutionary behaviour of ice-rich frozen sands arepresented. The strength criterion is referred to an isotropic equivalent material and is based on theLade-Duncan model; thus, the rupture surface is smooth, not containing any singularities that requirespecial numerical treatments, conversely to other frequently used criteria (i.e. the Mohr-Coulombmodel) whose exhaustive 3D-resolution may be harsh. The constitutive model takes into account,through a non associated flow rule, the two basic plastic mechanisms of shear deformation andvolumetric deformation (i.e. dilatancy), in order to well describe the non linear mechanical behaviourof permafrost under high confining pressures.

4.1 A generalised strength criterion for a frictional frozen soilAccording to Yao et al. [14], strength criteria of frozen soils and rock materials own several

important characteristics: (i) the strength function curves in π-plane and p-q-plane are smooth andconvex, (ii) the critical state line passes through the isotropic tensile point (r1 = r2 = r3 = fttt)instead of the origin, (iii) the strength function curve in π-plane is symmetric on three principal stressaxes, and (iv) the shapes of strength curves in π-plane of friction materials become more circular withthe further increase of mean pressure p.

For isotropic materials that follow an elasto-plastic rule, it is generally accepted that the strengthfunction can be written as:

f(σij , κi) = f(I1, I2, I3, κi) = f(p, q, θ, κi) = 0 (1)

where σij is a second rank symmetric tensor with six independent variables, κi are a set of internalvariables accounting for the material state that characterize the size and shape of the yield surface, I1,I2 and I3 are the first, second and third principal stress invariants, p is the hydrostatic pressure, q theoctahedral shear stress and θ the Lode angle.

From equation (1) it can be inferred that f can be expressed by a combination of the strengthfunctions fπ(θ) in π-plane and fp−q(p) in p-q-plane, that is:

f = fπ(θ)fp−q(p) (2)

In order to describe the non linear behaviour of permafrost, the strength function f has beendefined through a combination of the extended Lade-Duncan model in π-plane, with the empiricalparabolic formulation of the strength function in p-q-plane presented by Yuanming [13].

The strength function expressed by the Lade-Duncan criterion is given by:

f = I31 − κI3 = 0 (3)

where κ is a material parameter that determines the shape of strength curves in π-plane anddepends on the shear strength of the material. For unfrozen soils the value of κ is constant; this means

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that strength curves in π-plane have the same shapes under different pressures p, both for loading(f >0) and unloading (f <0) processes . On the contrary, for frozen soils κ depends on the firststress invariant I1, and equation (3) must be rewritten as:

f(I1, I3) = I31 − κ(I1)I3 = 0 (4)

The extended Lade-Duncan criterion describes a function f which crosses the origin: equation(3) shows that if one of the three principal stresses is equal to zero, the strength function f will bezero. However, in frozen soils, the cohesive strength between ice and soil particles could stand tensilestress, hence the strength criterion will pass through the isotropic tensile stress point instead of theorigin. For this reason, equation (4) becomes:

f(I ′1, I′3) = I

′31 − κ(I ′1)I ′3 = 0 (5)

where σ′1 = σ1 − fttt, σ′2 = σ2 − fttt, σ′3 = σ3 − fttt, I ′1 = σ′1 + σ′2 + σ′3 and I ′3 = σ′1σ′2σ′3.

In order to find the general solution of fπ(θ), the problem has been tackled by expressing f as afunction of the hydrostatic pressure, p′, of the shear stress, q′, and of the Lode angle, θσ′ , by using therevised definitions of the stress invariants. Defining the function fp−q(p) through material parameters,for θ = −π/6 the general expression of fπ(θσ) has been found, finally determining the completeformulation of the yield function f .

4.2 A constitutive model for a frozen soil in generalized plastic mechanicsFor the elasto-plasticity principle, the total strain increment can be split into the elastic strain

increment and in the plastic one:

dεij = dεeij + dεpij (6)

The theory of multi-mechanism plasticity allows to express the plastic strain increment as:

dεpij =

3∑κ=1

dλκ∂gκ∂σij

(7)

where dλκ are positive scalars related to the yield function and gκ are plastic potential functions.For geo-materials, non associated constitutive equations (f 6= g) are in principle preferred; this is dueto considerable volumetric changes that these materials undergo either in compression or dilation,which could induce significant changes in density. However, especially in numerical analyses forengineering, non associated flow rule are usually employed, because of their practical simplicity. Inorder to theoretically deal with the problem in the most possible analytical way, the following studyis referred to a non associated flow rule, but for numerical simplicity it is finally applied consideringan associated flow rule.

In order to determine the plastic strain increment, the volumetric yield function, the shear yieldfunction in q-direction and the shear yield function in θσ-direction are firstly presented as follows:

fv(σij , εpij) = fv(p, q, θσ, hv) = 0 (8)

fq(σij , εpij) = fq(p, q, θσ, hq) = 0 (9)

fθ(σij , εpij) = fθ(p, q, θσ, hθ) = 0 (10)

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By using equation (7) and neglecting θσ since it is a small term, the positive scalars are calculatedthrough differentiation as follows:

dλv =1

Av

∂fv∂p

dp+1

Av

∂fv∂q

dq (11)

dλq =1

Aq

∂fq∂p

dp+1

Aq

∂fq∂q

dq (12)

where Av and Aq are hardening parameters.Defining the strain elastic part through the K-G model, the total strain increment is hence:

{dε} = {dεe}+{dεp1

}+{dεp2

}(13)

From the last equation, the following incremental stress-strain relationship is obtained:

{dσ} = [D] {dε} − [D] dλv

{∂gv∂σ

}− [D] dλq

{∂gq∂σ

}(14)

In view of equations (11), (12) and (14), dλv and dλq can be rewritten in terms of the hardeningparameters, the stress and strain components, the elastic stiffness matrix and of the yield and potentialfunction related to the considered mechanisms:

dλv =Aqβ2 + β2β4 − β3β5

AvAq +Avβ4 +Aqβ1 + β1β4 − β3β6(15)

dλq =Avβ5 + β1β5 − β2β6

AvAq +Avβ4 +Aqβ1 + β1β4 − β3β6(16)

Finally, reformulating equations (14), (15) and (16), the general formula of elasto-plastic incre-mental stress-strain relationship can be written as:

{dσ} = [Dep] {dε} (17)

where {dσ} and {dε} are expressed in terms of volumetric and shear stress components, and[Dep] is an asymmetric matrix, since the flow rule is non associated.

5 THE EVOLUTIONARY BEHAVIOUR OF PERMAFROST5.1 Validation of the extended Lade-Duncan criterion

In order to validate the reformulation of the Lade-Duncan criterion, a triaxial test has beennumerically modeled by using the Mohr-Coulomb model.

The developed tests have simulated a frozen sand specimen subjected to different confinementpressures (3, 5, 7 and 10 MPa) and to an axial prescribed displacement (34 mm), at different temper-ature values (-10, -5, -2 and -1 oC). The simulations referred to quasi-static controlled conditions,through a strain rate of 2·10−4 s−1.

Figure 4 shows the comparison in the shear stress-axial deformation plane between the resultsobtained through the application of the two criteria, for σ3 = 10 MPa and σ3 = 3 MPa, at T = -10 oC.As it can be noticed, the trends of the curves described by the two models are well comparable. Themost remarkable differences arise after the end of the elastic branch and at failure, where the curvesdiverge. In particular, for higher confining pressures than 5 MPa, the Mohr-Coulomb model accuracy

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Figure 4: Comparison with the extended Lade-Duncan and the Mohr-Coulomb criteria (T = -10 oC).

decreases with respect to extended Lade-Duncan criterion, especially at failure. This implies that theMohr-Coulomb linear model cannot describe the complex behaviour of frozen sandy soils under highconfining pressures; thus, its application for the mechanical characterization of frozen soils is notsuitable, whereas the proposed constitutive model can better describe the problem.

5.2 Ice-rich frozen sand mechanical behaviourOnce the reliability of the extended Lade-Duncan model was verified, the mechanical behaviour

of ice-rich frozen sand has been analysed employing the criterion.Similarly to the experimental evidences observed for frozen silt [13], the stress-strain behaviour

of frozen sand approximately experiences three stages: (i) the initial linear elastic stage, where thestress increases linearly with axial strain, with a little plastic component; (ii) the plastic stage, whereplastic deformation dominates over the specimen deformation process and the elastic deformation isrelatively subordinate; (iii) the softening stage, where the stress decreases with increasing axial strain.

The numerical results show that the mechanical behaviour of ice-rich frozen sand is highly affectedby confining pressures: the higher the confining pressures, the higher the strength of the material.Nonetheless, temperature plays an important role on the evolution of the mechanical properties ofice-rich frozen sand: the lower the temperature, the higher the matrix strength for the same values ofconfining pressure, phenomenon that increases for increasing confining pressures.

Numerical tests have in particular pointed out that for increasing confining pressures, the maximumshear strength of the material increases from 4 to 6 MPa, whereas for increasing temperatures itincreases from 6 to 8 MPa. Therefore, for these load cases, the temperature effect seems to be largerthan the confinement pressure effect on the strength evolution of ice-rich frozen sand.

5.3 Evolutionary settlements behaviour for a flexible footing on ice-rich frozen sandIn order to deepen the topic of the temperature effect on ice-rich frozen sand, the settlement

evolution of a flexible footing (3 m width) built on a uniform layer of ice-rich frozen sand, subjectedto a constant building load of 500 kPa, has been numerically analysed for the temperatures of -10, -5,-2 and -1 oC.

The analyses, performed in plane strain conditions, have pointed out that the induced settlements

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Figure 5: Ice-rich frozen sand behaviour under different confining pressures (T = -10 and -1 oC).

Figure 6: Evolutionary behaviour of settlements with respect to temperature variations.

vary from 0.9 mm at -10 oC to 1.71 mm at -1 oC (cf. Figure 6), therefore increasing by a factor 1.9.Since, in this case, the building dead load was particularly low if related to the high mechanical

properties of the soil, the observed settlements are low and not considered dangerous from a structuralpoint of view; however, the factor 1.9 for a ∆T = 9 oC can be considered as extremely significant. Infact, for weaker soils and higher loads the induced settlements would be probably higher, resultingparticularly dangerous from an engineering point of view.

6 CONCLUSIONSThrough the study of permafrost physical properties and the evolutionary mechanical behaviour

of ice-rich frozen sands, the paper proposes the analysis of the evolution of short-term settlementsdepending on temperature for a flexible footing, pointing out that the evolutionary behaviour ofpermafrost remarkably depends on the temperature effect. The analysis is based on the extendedLade-Duncan model presented and numerically validated in this paper for ice-rich frozen sand. The

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next goal is to deeply consider this phenomenon referring to long term conditions, possibly expandingthe proposed strength criterion and the constitutive model through a law coupled with temperature.

This is fundamental in order to exhaustively deal with the permafrost-structure interaction that, ifnot tackled in the proper way, will cause severe engineering problems with the future climate change.

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[11] Andersland, O. B., Ladanyi, B., Frozen ground engineering, Second Edition, Hoboken, NewJersey: John Wiley and Sons, Inc., (2004).

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[13] Yuanming, L., Yugui, Y., Xiaoxiao, C., Shuangyang, L., “Strength criterion and elastoplasticconstitutive model of frozen silt in generalized plastic mechanics,” in: International Journal ofPlasticity, 26, 1461-1484 (2010).

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