effect of variation of temperature field on the … · key words: thermal consolidation,...

Studia Geotechnica et Mechanica, Vol. XXXVI, No. 2, 2014DOI: 10.2478/sgem-2014-0021

EFFECT OF VARIATION OF TEMPERATURE FIELDON THE PROCESS OF THERMAL CONSOLIDATION

OF TAILINGS POND “ŻELAZNY MOST”

MONIKA BARTLEWSKA-URBAN

Wrocław University of Technology, Institute of Mining, Wrocław, Poland,e-mail: [email protected]

TOMASZ STRZELECKI

Wrocław University of Technology, Institute of Geotechnics and Hydrotechnics, Wrocław, Poland,e-mail: [email protected]

Abstract: The following study presents numerical calculations for establishing an impact of temperature changes on the process of dis-tortion of bi-phase medium. The Biot consolidation equations with Kelvin–Voigt rheological skeleton were used for that purpose. Theprocess was exemplified by thermal consolidation of post floatation dump “Żelazny Most”. We analyzed the behavior of the landfill un-der the action of its own weight, forces of floating filtration and temperature gradient. Values of certain effective parameters of modelwere obtained during laboratory tests on material obtained from the landfill. The remaining data for mediums with similar characteristicswere taken from literature. The results obtained from the stress state in the landfill allow the magnitude of plasticity potential to be speci-fied based on known strength criteria. Change in the value sign of the plasticity potential clearly testifies to the emergence of an area ofplasticity of material from landfill, however, this does not indicate the loss of stability of this hydrotechnical structure.

Key words: thermal consolidation, poroelasticity theory, rheology, Kelvin–Voigt body, the theory of Biot

1. INTRODUCTION

For the selected cross-section, the article byStrzelecki and Bartlewska [1] presents the problem ofdistortion of Żelazny Most tailings pond under theaction of its own weight and forces of the floating ofwater which filters through the landfill. Subsequently,the article by Bartlewska and Strzelecki [2], based onthe numerical 3D model of landfill, analyzes the sen-sitivity of the hydraulic engineering structure to chan-ging value of thermal parameters and an impact ofperiodic temperature variation of landfill surroundingson the process of deformation, temperature and stressdistribution. In this paper, the authors attempt to iden-tify potential areas of plastification of waste dumpmaterial, based on the Drucker–Prager values of pla-sticity potential.

The “Żelazny Most”, being the largest post floata-tion waste landfill and one of the biggest hydraulicobjects in the world, plays an important role, both forKGHM POLSKA Miedź S.A. as well as general

evaluation of the influence of such objects on the en-vironment. Therefore, the usage of mine waste dumpto increase the “crown” of the landfill starts to gainsignificance especially in the context of stability of thehydrotechnical construction discussed. “Żelazny Most”plays a key role in the water-sludge economy of threeCopper Ore Enrichment Facilities which enrich themining output of copper ore using floatation method.Undoubtedly, it is a very specific and unique project.The perimeter of the reservoir is around 15 km and thecurrent production of the KGHM mines causes a con-stant expansion of the reservoir by overbuilding(raising) “the terraces” around the reservoir. In conse-quence, the most important elements of the object,that is, the intakes and drainages of water as well asthe dams are constantly under geotechnical monitor-ing. The data from the measurements are analyzed onregular basis, and the decision about the next “crown”formation is made by a group of experts after thor-ough evaluation. That is why, analyzing and predict-ing stress and distortion states are crucial preventionelements within the range of landfill creep and its

M. BARTLEWSKA-URBAN, T. STRZELECKI58

surrounding terrain. From this standpoint, it seemslegitimate to create more precise models of the landfillbehaviors under the influence of different boundaryconditions, including the temperature varying fromseason to season. Whereas the ability to point out thepotential areas of waste dump materials plasticizing,that is, its stability losses, may be an important ele-ment for predicting undesired incidents.

For describing rheological properties of the soilsand rocks, authors of various publications use therheological model of mechanics for one or bi-phasemediums. According to numerous publications dealingwith this subject, accepting the model of multiphasemedium as initial one, it is assumed that the solid phasehas hydraulically connected pores. The links betweenpores, or micro slots, allowing the filtration flowof fluid and/or gas allow the consolidation process.Taking into consideration the temperature aspect in thisprocess makes it possible to conduct precise analy-sis of the deformation process of the soil medium.A mathematical creeping model of the porous mediumdefined as a multiphase body was for the first timeintroduced by Maurice Biot in [3], [4]. This model hasbeen the subject of numerous publications. Using thefundamental theorem of thermodynamics of irreversi-ble processes Derski derived constitutive relations forbi-phase porous body for isothermal processes [5].Using the method of asymptotic homogenizationfor periodic mediums (Bensoussan, Lions and Papa-nicolaou [6]) a collective system of equations of iso-thermal theory of consolidation was obtained in theworks ofAuriault [7], Auriault and Sanchez Palencia[8], and the same was achieved with the use of statisti-cal methods by Kröner [9], Rubinstein and Torquato[10]. A mathematical model for the case where gas fillsthe pores of the medium, based on asymptomatichomogenization method, was presented by Auriaultet al. [11]. The experimental research conducted showsa significant role of appearance of a double layer ofwaters bound by the forces of electric field in the model

soils. This fact combined with considerable variation intemperature causes the solid particles of the skeleton toactually connect to each other via these waters. Thus, inthe description of the creep process of this type of soils,in addition to shear and volume compressibility, theviscosity of the skeleton needs to be considered as pre-sented in the works of Bartlewska and Strzelecki [12]and Bartlewska [1]. The problem of the impact of thetemperature field for adiabatic processes was re-searched by Caussy, who created a mathematical modelof thermal consolidation [13], Kowalski et al. [14],Strzelecki et al. [15], and Bartlewska and Strzelecki [2].This study is a continuation of the numerical researchof thermal consolidation process of floatation wastedumps “Żelazny Most”. The model proposed includesgeneralization of Biot’s equations for different non-isothermal processes including rheological features ofthe skeleton. The crown deformations process of the“Żelazny Most” tailings pond was analyzed over timein the context of appearance of changes in the value ofthe Coulomb–Mohr plasticity potential, which maytestify to the appearance of areas of landfill materialsplasticizing.

The analysis was conducted in two variants: de-pending on temperature changeability based on theannual temperature distribution around the construc-tion, presented using the sine function. Second is thebenchmarking variant that assumed constant tem-perature of the environment. A comparison of theresults from the two experiments allowed us to for-mulate conclusions about the impact of seasonal tem-perature changes on the process of creation of plastic-ity zones of the “Żelazny Most” materials.

1.1. GEOGRAPHICAL LOCATIONOF “ŻELAZNY MOST” TAILINGS POND

Administratively, the tailings pond “Żelazny Most”is located in Lower Silesia in the Legnicko-Głogowski

Fig. 1. Satellite image of area occupied by the “Żelazny Most”

Effect of variation of temperature field on the process of thermal consolidation of tailings pond “Żelazny Most” 59

Copper Territory (LGOM), in two counties: Lubinand Polkowice. This floatation waste landfill wasbuilt in 1974, and its simultaneous operation andexpansion has lasted since 1977. Currently the totallength of dams surrounding the dump amounts to14.3 km and the total area 1394 ha. The annual vol-ume of waste deposited from the floatation ranges from20 to 26 million tones, whereof almost 75% is used tooverbuild the structure and only 25% undergoes theprocess of neutralization. Location and satellite view ofthe dump based on information from the KGHM PolishCopper SA (www.kghm.pl) is shown in Fig. 1.

2. NUMERICAL CALCULATIONOF THERMAL CONSOLIDATION

OF “ŻELAZNY MOST” TAILINGS POND

Unquestionable advantage of using the finite ele-ment method in the modeling process is the possibilityof conducting comprehensive analysis of specificengineering issue in separate building stages. Thecomprehensive approach used in our work consists insimultaneous analysis of deformations and stresses inthe landfill area and the ground as well as the bound-ary conditions, (i.e., the temperature impact on thedistortion process) in one calculation process. How-ever, it is important to remember that the quality andprecision of each analysis is limited to the amount ofinitial information.

2.1. CRITICAL STATE CONDITIONS

General conditions of critical state were used forbrittle materials and granules such as soil. The startingpoint of unrestricted malleable flow is defined by thecritical state condition for the model of perfect elas-ticity. Depending on the prevailing conditions of hy-drostatic pressure, the same material may behave asmalleable or brittle. Thus, the dependence of the criti-cal point surface of the first constant of the strain ten-sor is necessary. The computation model allows theprocess of distortion of viscoelastic skeleton to beanalyzed in the context of theory of elasticity. A com-plete analysis would require a model which considersboth elastic and plastic proporties. Such a scope ofwork, however, goes beyond the simulation coveredin this publication. Nevertheless, calculations of stressstate allow us to verify if the elastic limit has not beenexceeded (using the knowledge of strength propertiesof the material obtained from landfills). In practice,obtaining parameters for viscoelastic-plastic models is

difficult. In most cases, these models are calibratedbased on approximate empirical correlations, directresults from laboratory tests, or based on field meas-urements of strain tests [16]. Literature concerningsoil mechanics extensively describes different shear-ing resistance criteria. The present study is limited tothe criteria of resistance, isotropic criteria – describedusing the constants of the stress state.

2.1.1. APPLIED STRENGTH CRITERIA

The Coulomb–Mohr criterion is the fundamentalstrength criterion for the shearing of the soils. Thiscriterion is a starting point for the construction ofmore advanced constitutive models. This criterion isoften used as a benchmarking criterion in alternativeproposals of resistance criterion for the shearingthrough comparison of the contours on the deviatoricplane ( p = const) as well as the uniaxial one (σ2 = σ3)in the space of main components of the effectivestress, as presented in Fig. 2.

Fig. 2. (a) The Coulomb–Mohr criterion in the spaceof the main component of the effective stress,

(b) comparison of the Coulomb–Mohrand Drucker–Prager criterion – two respective variantsof compliance with the assumption of equal resistance

to the uniaxial compression Mc and the resistanceto the uniaxial tensile strength Me [16]

The plasticity condition resulting from the Coulomb–Mohr criterion written using the main stress compo-nent becomes

,0cossin)(21

)(21

minmax

minmax

=−++

−=

φφσσ

σσ

c

FMC

where σmin, σmax – the smallest and the biggest strain,respectively, assuming a convention where the com-pressing strain takes negative values. This conven-tion also applies to each formula taken from thepublication [16]. Plasticity condition formulated insuch a way describes six planes forming a specific


pyramid in space of the main components of stresstensor, as presented in Fig. 2a. On the deviatoric planethe Coulomb–Mohr plasticity criterion representsa hexagonal contour connecting the points wherethe value of the internal friction angle is constant(ϕ = const) on which the ratios of side lengths and thespecific angles of the hexagon are dependent. An im-portant aspect of applying the Coulomb–Mohr criterionemphasized by the authors of the above mentioned pub-lication is the undeniable difficulty in the implementa-tion of the constitutive models. The main difficulty iscaused by the presence of sharp edges of the surfaceplasticity in the space of the main stress components,where the direct result is difficulty in correct and unam-biguous determination of the gradient of the plasticityfunction in those places. More about this issue can befound in paper [17]. Therefore, the Drucker–Pragerstrength criterion is often used for the solutions of themodels where the plasticity function is a complicatedhigher-order function. The plasticity conditions for thiscriterion look as presented in the formula below

,23,0 ijijqDP ssqcMpqF ==−−=

where q is a constant of the stress deviator s, parame-ter M is the slope of plasticity surface FDP = 0 plane,while p–q is the limit constant of invariant q withp = 0 and is related indirectly with the soil cohesion.

In the case of the Drucker–Prager criterion, theplasticity plane is cone-shaped (in the plane of maincomponents) however in the deviatoric plane, thecriterion presents a sphere regardless of the value ofparameter M connected to the internal friction angle.Therefore, the Coulomb–Mohr and Drucker–Pragercriteria may be presented as two extreme cases todescribe shearing resistance (understood as a constantvalue of the stress deviator invariant (q = qf), Fig. 2b.

Although there is no clear correlation between theinternal friction angle and resistance parameters,which according to Cudny and Binder [16] restrict theuse of the Drucker–Prager criterion for practical solu-tions to basic research related to determination ofpotential areas of soil yielding described in this paper,this criterion was used because of the aforementionedease of its implementation.

2.1. MATHEMATICAL MODELOF THE BIOT THERMAL CONSOLIDATION

WITH RHEOLOGICAL SKELETON

Mathematical equations of thermo-consolidationprocess for Biot’s body with a rheological skeleton of

Kelvin–Voigt are presented below. Those equationswere derived based on fundamental laws of Newto-nian mechanics for continuous mediums and thermo-dynamics of irreversible processes. The starting pointis the initial assumption of the theory of bi-phasemediums composed of elastic-viscous skeleton anda viscous compressible fluid filling the pores of thismedium. The assumptions as well as the course of thespecific stages of building an analytical model aredescribed in paper [2]. According to Bartlewska andStrzelecki [1], the physical relations for the Biot sub-stance with a Kelvin–Voigt skeleton for any non-isothermal process can be expressed by the followingequation

⎪⎪

⎩

⎪⎪

⎨

⎧

−++=

−++

++++=

).(

,)(

)(22

0

01

TTdRQQ

TTPRQ

NTATNTMN

ijij

ijabijaijijij

εσ

δσδ

δεεεδεσ

(1)

In the Biot–Willis (1957) paper the constants ap-pearing in constitutive relations were interpreted asfollows:• N is a the module of shape deformation of the

skeleton.• A is a volumetric distortion module of the skeleton

filled with liquid.• Q is a volumetric distortion coefficient of the

liquid for the stress in the skeleton or on thecontrary – a coefficient of the impact of thevolumetric distortion of the skeleton for stress inthe liquid.

• R is a volumetric distortion module of the liquidfilling pores of the Biot substance.

• The parameter M is expressed by the formula

RQAM

2

−= . (2)

Constant d is expressed by the formula

]3[ RrQrd ls +−= (3)

where rs and rl present the linear expansion of theskeleton and the volumetric expansion of the liquid,respectively, constant 1P is calculated using the for-mula

λ)3(

1

ls QrKrTP +−= (4)


where λ is the soil heat transfer coefficient, Ta and Tbare the parameters of the skeleton expressed with theformulas

NT

s

aη

= (5)

and

,A

Ts

bλ

= (6)

η s, λ s are the shear and volume viscosity of soil ske-leton.

The system of equations of linear theory of ther-mal consolidation in the shifts of the skeleton and thefunction of stress in the liquid σ, the filtration flowequation and the heat conduction equation for the Biotsubstance with the Kelvin–Voigt rheological skeletonconsists of five differential equations

⎪⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪⎪

⎨

⎧

+−=∇

+−=∇

−=−+

⎟⎟⎠

⎞⎜⎜⎝

⎛Ψ+−Ψ+∇Ψ

][

][

,,

,

53202

402

2

13

22

TPPPTT

TPHTgf

kR

TPgRH

NR

QAuN

o

iii

ikLik

σελ

εσσρ

δρσ

ε

(7)

where t

Tt

T bLak ∂∂

+=Ψ∂∂

+=Ψ 1,1 are differential

operators, k is the coefficient of filtration of fluidthrough a porous medium, g – gravitational accelera-tion, and the coefficients P2, P3, P4 and P5 are givenby the equations

ls RrR

HQKrP −⎟⎠⎞

⎜⎝⎛ −= 32 , (8)

,33ls r

RQrP += (9)

34 RPP = , (10)

Tc

RRrQrP vws

ls )()3( 2

5ρρ +

++

= . (11)

cv is the specific heat at constant volume. The aboveset of equations is a starting point for the issue re-solved in this study.

3. CONSTRUCTIONOF NUMERICAL MODEL

FOR THERMAL CONSOLIDATIONOF THREE-DIMENSIONAL ISSUES

A three-dimensional numerical model of geologi-cal structure and the system of equations of thermalconsolidation (7) constituted the basis for numericalexperiments aimed at simulating the distortion of thepost floatation reservoir “Żelazny Most”, includingperiodic temperature changes in the environment ofthe landfill and presenting probable areas of landfillmaterial plasticizing. Calculations were conductedusing the Flex PDE V.6 Professional. Based on theanalytical model of the Biot rheological skeleton,a three-dimensional numerical model of consolidationof the construction described was generated on thebasis of the system of equations (7). The physical andstrength parameters applied in the experiment wereobtained in part from our own research, the scope ofwhich included the standard analysis of soil, andpartly from the related literature. Effective Biot’s pa-rameters were obtained based on statistical methodspresented in paper [18]. The entire area subjected tocomputer simulation of the consolidation process con-sists of smaller areas with different parameters of thesediments stored within. For this reason the entire areawas divided into smaller ones with well known pa-rameters. Five sub-regions were distinguished as maybe noticed in Fig. 4 showing initial network of finiteelements. Changes of the value sign of potential plas-ticity and shift of each layer of the landfill under itsown weight and temperature gradient between the sur-face of the bottom of the landfill and the surroundingswere analyzed taking into account existing landfilldrainage systems. The experiment covered the timespan of 30 years and the results for the last two years ofthe simulation are being presented. The lower surfaceof the landfill was assumed to be heat permeable;T = 27 °C. For the side area boundaries it was as-sumed in the calculations that the temperature dis-tribution depends on the process of heat flow ina landfill.

On the upper surface of hydro technical structuresit was assumed that shifts are undefined, and the waterpressure is equal to atmospheric pressure. The aim ofthe simulation was to analyse potential areas of plasti-cizing of the material under the influence of tempera-ture which varies seasonally. By simple optimizationa sine curve was matched based on minimizing thesquare error to the average monthly temperature data


(Fig. 3). Equation (12) describes temperature variationfor the duration of the experiment.

( ) 18.9825.0*71992sin*42.11)( +−−= tetT . (12)

The results were generated for the first year ofthe experiment and then after 10, 20 and 30 years.Observed was the effect of temperature variation onthe process of distortion of the landfill over time(Fig. 3).

Fig. 3. Matching sine graph (series 2)to the average temperature (series 1)

4. CALCULATION RESULTSOF NUMERICAL THERMAL

CONSOLIDATION

The results of numerical calculations performedusing the finite element software Flex PDE 6 for the3D model in the form of 2D graph charts in cross-sections plane are given below. The geometry of thereservoir along with the generated finite element gridis shown in Fig. 4.

Figure 5 is the cross section of tailing pond. Themaximum subsidence magnitude obtained after a timeequal to 30 years occurring in the central part of thewaste dump is about 2.3 m. The graphs (Fig. 6a and 6b)confirm subsidence of each layer of the central part ofthe waste dump.

It should be noted, however, that individual layerssubside with varying intensity. The largest dis-placement was observed on the surface of the dump(Fig. 6b), where we may also observe (with the ap-plied strength parameters) sine dependence betweensubsidence of landfill and ambient temperature. Thisrelation is represented by a sine graph. This relation isevident, however, on the edges of the landfill (Fig. 7a

and 7b). Subsidence is highly dependent on the vari-able temperature during the year. It is greater duringwarmer months. In addition, a slight displacement ofthe edges of landfill to the opposite direction may beobserved, which effect is caused by temperature anddepends on the ambient temperature. The higher thetemperature (summer months), the greater the effectof expansion. The graph illustrating shift area showsthe direction of displacement of two selected layers ofthe landfill: surface area (Fig. 8a) and a layer at heightz = 108 m (Fig. 8b). The direction of the shift is “out”of the landfill.

Fig. 4. Finite element grid generated by the software

Fig. 5. Vertical displacement in m (side view ZY)

Figure 9 is a horizontal displacement graph after thetime span of thermal consolidation equal to 30 years.We can read from the graph that the maximum hori-zontal displacement occurs in the middle of the dumpon the border regions and in the surrounding wastedump bank and amount to 0.10 m.


a) b)

Fig. 6. (a) The displacement changes at defined heights of the waste dump for the point x = 2500, y = 2500 in the last two years,(b) distortion of the upper layer of the waste dump for the point x = 2500, y = 2500

a) b)

Fig. 7. (a) The displacement changes at defined heights of the waste dump for the point x = 2500, y = 4000 in the last two years,(b) distortion of the upper layer of the waste dump for the point x = 2500, y = 4000

a) b)

Fig. 8. Displacement vector field a) the surface area b) tailing s pond at height z = 108

Displacements

Displacements


Figure 10 shows the stress distribution of liquid(pore stress) at the beginning (a) and at the end of theconsolidation process (b). An increase of stress in theliquid is clearly visible. At the beginning of the ex-periment the highest compressive stresses were at thebottom of the dump, the lowest at the top. At the endof the experiment, the stress distribution was even.The stresses in the entire landfill are much lower thanat the beginning of the experiment and slightly higherin the central part and the outer banks of the landfill.

Figure 11 shows the distribution of shearing stressin the dump. We may observe that the highest valuesof shearing stress are in the central section of thedump and overlap the areas of the greatest shifts in thehorizontal direction.

Figure 12 shows Drucker’s plastic potential. Nega-tive value of the potential defines areas which are is inthe range of elasticity. Where the potential is positivethe value exceeds the shearing strength, within theaccepted values of friction angle and cohesion. The

graph shows that there are areas in which the mediummay be in a state of plasticity.

Fig. 11. Shear stress in Pa

Displacements

Fig. 9. Horizontal displacement in [m] (side view ZY )

a) b)

Fig. 10. The stresses in the liquid in Pa: (a) beginning of the experiment, (b) after 30 years


Fig. 12. Results for the 3D model – Drucker’s plastic potential

3. CONCLUSIONS

The example presented shows that the current stateof knowledge in engineering allows us to use calcula-tions of more complex models of stress distortion proc-esses in porous mediums. The results concerning thestress state for the dump give an answer to whether ornot the linear model of the viscoelastic medium leads tochanges in the sign of Drucker’s plastic potential,which might indicate that the area of dump materialplasticity appears. The calculation process includesa significant change in the mechanical and hydro-geological parameters of the landfill and its base aswell as their anisotropy. Most of the results are consis-tent with intuitive predictions. The largest vertical shiftsoccur in the central part of the landfill and depend onthe temperature of the environment. Also the effect ofthe displacement in the opposite direction is observed(expansion). This phenomenon is the effect of tem-perature and will be the subject of another publication.The largest horizontal shifts are in the central part ofthe landfill in the border zone, between the regions, atthe same time creating a zone with the largest shearstresses. However, Fig. 12 indicates the existence ofareas where the Drucker criterion was exceeded, whichmeans that probable yielding zones exist in the dumpsubareas. This article is a continuation of the discussionabout an impact of temperature on the consolidationprocess. The proposed model indicates the changingnature of the deformation process depending on tem-perature, but also on the plastic potential areas of thesoil, which from the practical point of view could con-tribute to the prevention of the tailings pond stability.

REFERENCES

[1] BARTLEWSKA M., STRZELECKI T., Equations of Biots con-solidation with Kelvin–Voight rheological frame, StudiaGeotechnica et Mechanica, Vol. XXXI, 2009, No. 2,3–15.

[2] BARTLEWSKA-URBAN M., STRZELECKI T., Thermal consoli-dation of porous medium with a rheological Kelvin–Voigtskeleton, Studia Geotechnica et Mechanica, 2012, 34, 3,17–35.

[3] BIOT M.A., General theory of three-dimensional consolida-tion, J. Appl. Phys., 1941, 12, 155.

[4] BIOT M.A., General Solutions of the Equations of Elasticityand Consolidation of a Porous Material, J. Appl. Mech.,1956, 23.

[5] DERSKI W., Some Contributions to the Theory of Flow ofFluids through porous Deformable Media, Acta Mechanica,1967, IV/1.

[6] BENSOUSSAN A., LIONS J.L., PAPANICOLAU G., Asymptoticanalysis for periodic structures, North Holland PublishingCompany, Amsterdam 1978.

[7] AURIAULT J.L., Dynamic behaviour of porous media, Trans-port Processes in Porous Media, Kluver Academic Publish-ers, 1991, 471–519.

[8] AURIAULT J.L., SANCHEZ PALENCIA E., Etude de comporte-ment macroscopique d’un milieu poreuxsature deformable,Journal de Mecanique, 1977, 16 (4), 575–603.

[9] KRÖNER E., Effective elastic moduli of periodic and randommedia: a unification, Mechanics Research Communication,1980, 7 (5), 323–327.

[10] RUBINSTEIN J., TORQUATO S., Flow in random porous media:mathematical formulation, variational principles and rigor-ous bounds, J. Fluid Mech., 1989, 206, 25–46.

[11] AURIAULT J.L., STRZELECKI T., BAUER J. HE S., Porousdeformable media by a very compressible fluid, Eur. J. Mech.a. Solid, 1990, 9, 4, 373–392.

[12] BARTLEWSKA-URBAN M., STRZELECKI T., One-dimensionalconsolidation of the porous medium with the rheologicalKelvin–Voigt skeleton, Studia Geotechnica et Mechanica,2008, 30, 1/2, 115–122.

[13] COUSSY O., Mechanics and physics of porous solids, JohnWiley & Sons, 2011.

[14] KOWALSKI S.J., MUSIELAK G., RYBICKI A., Drying Processesin Context of the Theory of Fluid Saturated Porous Materi-als, J. Theoretical and Applied Mechanics, 1998, 36, 3.

[15] STRZELECKI T., KOSTECKI S., ŻAK S., Modelowanie przepły-wów przez ośrodki porowate, Dolnośląskie WydawnictwoEdukacyjne, Wrocław 2008.

[16] CUDNY M., BINDER K. Kryteria wytrzymałości gruntu naścinanie w zagadnieniach geotechniki (On shear strength crite-ria for soils in geotechnics), Inżynieria Morska i Geotechnika,2005, 6, 456–465 (PDF).

[17] NIEMUNIS A., Über die Anwendung der Kontinuumstheorieauf bodenmechanische Probleme, Eine Vorlesung für Grund-bau- und Tunnelbauvertiefer, Uniwersytet w Bochum, on line:www.gub.ruhr-uni-bochum.de/mitarbeiter/andrzej_niemunis.htm,2003.

[18] BARTLEWSKA M., The doctoral dissertation on the theme:Określenie parametrów efektywnych modeli reologicznychgruntów spoistych, Politechnika Wrocławska, Faculty ofGeoengineering, Mining and Geology, Wrocław 2009.

effect of variation of temperature field on the … · key words: thermal consolidation,...

Documents