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An engineering model for snow creep Olagne, Xavier 1989

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A N E N G I N E E R I N G M O D E L F O R S N O W C R E E P By Xavier Olagne Ingenieur Civil des Mines, Saint-Etienne (France), 1987 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF M A S T E R O F S C I E N C E in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF GEOPHYSICS AND ASTRONOMY We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA August 1989 © Xavier Olagne, 1989 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of G E O Y$ I C S The University of British Columbia Vancouver, Canada DE-6 (2/88) Abstract Snowcovers on slopes densify and deform continuously throughout the winter. These slow, mainly viscous deformations are known as snow creep and this thesis presents an attempt to model them by idealizing snow as a non-Newtonian fluid, where the bulk and shear viscosities depend upon both stress and density. A three-dimensional constitutive law is developed, based largely on analogy with the flow behavior of ice and soil materials. The model, primarily intended for engineering applications (design of structures erected in a deep snowpack), is tested for creep pressures on long rigid obstacles. Data recorded on two experimental sites are compared with numerical results obtained by the finite element method. In addition to predicting pressures in good agreement with the ones measured in the field, the constitutive law is flexible enough to accommodate the stiffness variations encountered at different locations and hence presents some improve-ment over the linear formulation. n Table of Contents Abstract ii List of Tables v List of Figures vi Acknowledgements vii 1 I N T R O D U C T I O N 1 2 T O W A R D A C O N S T I T U T I V E L A W F O R S N O W C R E E P 3 2.1 Framework of study 3 2.2 A three-dimensional law for snow creep 5 2.2.1 The deviatoric law 6 2.2.2 The volumetric law 9 2.2.3 The three-dimensional law 11 3 A P P L I C A T I O N T O T H E DESIGN OF S N O W BARRIERS 13 3.1 Description of the problem 13 3.1.1 Properties of the field data 13 3.1.2 The numerical calculations 15 3.2 Determination of the parameters 17 3.3 Results and discussion 24 3.3.1 Pressure results 24 iii 3.3.2 Non-Newtonian and density effects 29 3.3.3 Maximum versus average pressure 37 3.3.4 Use of the model 38 3.4 Discussion and conclusions 39 References 41 Appendices 44 A Reiner-Rivlin fluids with no dependence on the third strain rate invari-ant 44 B Brown's theory for the volumetric deformation of medium to high den-sity snow 48 iv List of Tables 3.1 Adjustment of Brown's law for the natural density data given by Mellor (1975) 18 3.2 Sets of possible parameters for the model for different M values 22 3.3 Values of the viscosities and of the resulting Poisson ratio at different densities for each set of parameters considered in Table 3.2 23 v List of Figures 2.1 Schematic deviatoric stress-strain rate response for ice and a material with a viscosity proportional to the bulk stress 8 2.2 Schematic volumetric stress-strain rate response 10 3.3 Definition of the coordinate system and of the velocities 14 3.4 Comparison of the pressures on the structure obtained experimentally and numerically for the Norwegian site 25 3.5 Comparison of the pressures on the structure obtained experimentally and numerically for the Swiss site 26 3.6 Comparison of the pressures on the structure obtained experimentally and numerically with the linear model for the Norwegian and Swiss sites. . . 28 3.7 Depth variations of the pressures on the structure calculated numerically, with M- 1 30 3.8 Depth variations of the pressures on the structure calculated numerically with M= 2 31 3.9 Depth profiles of the effective Poisson ratio obtained numerically in the neutral zone for different M values and constant depth-averaged densities. 33 3.10 Depth variations of the pressures on the structure obtained numerically for different density profiles 35 3.11 Depth variations of the effective Poisson ratio in the neutral zone obtained numerically for different density profiles 36 vi Acknowledgements First, I wish to thank my thesis supervisor D.M. McClung for his thoughtful guidance and steady availability during my two year stay as his student. I greatly appreciated the theoretical expertise and constructive criticism provided by G.K.C. Clarke. I am particularly indebted to Erik Blake whose assistance in the numerical part of my work saved me a substantial amount of time and effort. Special thanks as well to Simon O'Brien and David Dalton for their help in formatting this thesis. Finally and above all, I would like to extend my gratitude to all those who made my two years in Vancouver a fruitful and enjoyable experience. My stay in Vancouver has been supported financially by a Government of Canada Award administered by the World University Service of Canada and an NSERC Grant. vu Chapter 1 I N T R O D U C T I O N When subjected to a load, snow undergoes an immediate recoverable deformation (elastic part) as well as a time-dependent irreversible deformation, described as snow creep (vis-cous part). In the study of the mountain snowpack, scientists have long been interested in the latter phenomenon for its numerous engineering applications. In particular, the design of structures positioned in deep snow covers (such as barriers, powerline and ski lift towers) requires the calculation of snow creep pressures. For this purpose, the de-velopment of a constitutive relationship that can explain the physical patterns observed and recorded is the leading consideration. Indeed, a limited and imperfect flow law could result in high safety factors and hence, in prohibitive overdesign costs. As a first approximation, snow can be described as a compressible Newtonian fluid. Assuming constant bulk and shear viscosities, McClung (1982) following Haefeli (1948) derived analytical equations for snow pressures in the centre of a rigid barrier which are consistent with finite element results. Because of their simplicity, consultants still use them as guidelines for the design of snow structures. However, a comparison with exper-imental data (McClung and Larsen, in press) shows that the analytical model underesti-mates field measurements by about 20%. These discrepancies underline the presence of nonlinear deformation and give a rough quantitative estimate of non-Newtonian effects. In fact, nonlinear effects have long been suspected; Mellor (1975, 1977) and Salm (1982) reviewed some of the recent attempts to address the problem in a realistic way. 1 Chapter 1. INTRODUCTION 2 Though some of these models represent real progress in the understanding of the me-chanical behavior of snow, their complexity combined with a lack of extensive test data makes them often unsuitable for practical applications. Hence, there is a strong need to develop a constitutive law that includes the main features of snow deformation but which is simple enough to be usable for engineering purposes. This task is attempted in the present thesis using simple but physically plausible formulations for the effective shear and bulk viscosity of snow. Following the theoretical development, the model is checked using measurements of creep pressures on structures obtained in the field. This comparison shows that the model predicts pressures close to those measured in the field in a physically realistic manner and therefore it represents an improvement over the linear formulations. Chapter 2 T OWARD A C O N S T I T U T I V E LAW F O R SNOW C R E E P 2.1 Framework of study The prospects for a general constitutive relationship for snow deformation look discour-aging at present because of a combination of numerous difficulties. Seasonal alpine snow appears in a variety of configurations that require taking its microscopic structure and stress history into account. Under the action of gravity, it densifies continuously but unfortunately in a nonlinear, nonreversible and nonsteady manner. Moreover, grain re-arrangement causes deviatoric and volumetric deformations to be coupled intimately. Snow covers on slopes deform with strain rates as low as 1 0 _ 8 5 - 1 . When combined with the remoteness of certain study areas, this makes field measurements a strenuous and time-consuming process. Laboratory tests do not have these inconveniences, but the field conditions, and especially the low strain rates and the three-dimensional states of stresses are difficult to reproduce in the laboratory, so that the results obtained are often not applicable to field situations. For these reasons, the complexity of a model intended for practical purposes has to be compatible with the limits of the available data. Since a rigorous formulation of a creep flow law valid for every type and state of snow is beyond my reach, the following restrictions or framework of study will be considered. First of all, my approach will be largely driven by engineering motivations.- Although scientifically disputable, I will in general adopt the simplest formulation consistent with the physics of the problem. My ultimate goal consists in explaining and predicting the 3 Chapter 2. TOWARD A CONSTITUTIVE LAW FOR SNOW CREEP 4 pressures recorded on structures in order that design specifications can be drawn up. Thus, I am particularly interested in the time period where the recorded pressures reach their peak value. This does not occur before the late winter or the early spring depending on the year and on the location: alpine snow is then a well-metamorphosed and a well-settled material displaying slowly evolving density profiles with a high average value. At the same time, the amount of snow accumulated throughout the whole winter is close to maximum, while the melting rate is still small. In these conditions, I will make several simplifying assumptions: (1) At small strain-rates, the deformations are nearly purely viscous and the elastic part can be dismissed. Therefore, snow will be idealized as a non-Newtonian, isotropic and homogeneous fluid without memory. (2) For simplicity, the influence of all structural parameters except for density will be neglected. (3) Since a deep and stiff snowpack is imperceptibly disturbed by an extra layer of low-density snow and the strain rates drop to slowly varying values, steady-state conditions will be assumed. Late in the season, the physical quantities change very slowly with time. Therefore, even if their variations have to be explored as time proceeds, they will be considered constant at a given instant of time. (4) In deep seasonal snowcovers, temperature gradients are usually small, rounded equilibrium forms prevail and the crystal growth rates are low. More-over, it is estimated that 90% of the deformation occurs by grain rearrange-ment. Hence, the influence of the temperature and of the thermodynamic Chapter 2. TOWARD A CONSTITUTIVE LAW FOR SNOW CREEP 5 pressure will be neglected. 2.2 A three-dimensional law for snow creep Assumption (1) allows expansion of the stress tensor <7,j as a power series of the strain rate tensor e;j (Reiner, 1945; Rivlin, 1948). Furthermore, by making use of the Cayley-Hamilton theorem, the constitutive law can be expressed as follows: o~ij = <f>\8ij + 4>2kj + faeikekj. (2.1) Equation (2.1) defines a Reiner-Rivlin fluid. The summation convention applies, and 8{j is the Kronecker delta. The fa are theoretically scalar functions of the density, temperature, time, snow structure and of the strain rate invariants Ely E2 and E3 defined as the coefficients of the characteristic equation (Malvern, 1969): Ei = en ' E2 = l/2(eyey - e«e#) In the last equation, | | is the determinant of the strain rate tensor. Similar expres-sions (with stress tensor components replacing strain rate components) are available for the invariants of the stress tensor S i , S2 and S3. For the simplest model (assumptions (2), (3) and (4)), the fa depend only on the density p and on the strain rate invariants. In the next step, I dismiss any E$ dependence on the scalar functions and on the second invariant of the stress tensor S2. This is a classical assumption postulated both in glaciology (Glen, 1958) and in soil mechanics (Vulliet and Hutter, 1988). Physically speaking, the volumetric strain rate E\ and the second invariant of the deviatoric strain Chapter 2. TOWARD A CONSTITUTIVE LAW FOR SNOW CREEP 6 rate tensor E'2 = E2 + (E%)/Z are quantitative representations for respectively the com-pression and shear deformations. By expanding the fa in power series of E\ and E2, it is possible to account for nonlinearities as well as for coupling features. This is why neglecting any dependence on E3 does not contradict in an obvious manner the physi-cal evidence. Assuming the fa and S 2 do not depend on Ez, it can then be shown (cf. Appendix A) that </>3=0 and the flow law becomes similar to the one proposed by Salm (1967) o-ij = fab~ij + fctij- (2.2) If I decompose the stress tensor into a deviatoric term and a hydrostatic pressure com-ponent p=— S i / 3 , I obtain °~H = <r'ij-p6ij, (2.3) J with oij = fafy = 2,1$ ( a n d p = -{fa/E1+falZ)E1 = -vEx. Herein p and n are respectively the effective shear and bulk viscosities. They both depend on the first and second strain rate invariants. However, either can be expressed in terms of the stress invariants. The minus sign arises by the sign convention: the pressure p is taken positive and the strain rates negative due to packing. 2.2.1 The deviatoric law Various authors (Haefeli, 1939; Martinelli, 1960; Frutiger and Martinelli, 1966; McClung, 1974) have noted that well-settled isothermal snowcovers in the neutral zone display ap-proximately linear velocity profiles as a function of depth (the neutral zone is defined as the region where longitudinal gradients in stresses or material properties vanish). The observations imply a constant shear strain rate with depth and hence that the shear Chapter 2. TOWARD A CONSTITUTIVE LAW FOR SNOW CREEP 7 viscosity varies linearly with shear stress if the assumptions above (Equation (2.3)) are correct. Furthermore, since shear and normal stress should be approximately propor-tional at a given slope angle, a linear dependence on bulk stress or mean pressure can be inferred for the shear viscosity (Mellor, 1968). Physically, this refers to the frictional properties of snow. The resistance to sliding at each contact point is proportional to the normal force at that point and hence, since local and overall pressures are related by a density scale factor (see Appendix B), an increase in the overall bulk stress causes a proportional increase in the resistance to shear and a decreasing deformation rate. At the same time, as shear motion proceeds, the degree of interlocking between the particles decreases so that further motion becomes easier. This stiffness decrease with increasing stress before failure is characteristic of soils (Lambe and Whitman, 1979). Comparing snow with ice, Mellor (1975) points out a power law behavior (called Glen's 1/2 law) by plotting the octahedral stress (S' 2 ) as a function of the octahedral strain rate 1/2 (E'2 )• In fact, a stress dependence of the power type has long been proposed for the shear viscosity and the value of the exponent is suspected to vary between 1 for low stress levels and 3 for high stress levels (Haefeli, 1967; Mellor and Smith, 1967) Figure 2.1 summarizes schematically both competing behaviors in triaxial compres-sion or simple shear conditions and obvious differences in shape and concavity can be noted. The conclusion is that a realistic shear stress-strain rate response is likely to lie between the two extreme curves. Mathematically, the connection between the pressure p and the deviatoric invariants S'2 and E'2 is examined for soils by Vyalov (1986). In the coordinate system formed by the three variables, he plots a surface representing their interdependence and shows the latter should have the following general form E'f = / a ( S ' f ) + p2(p) g(p) /2(S'5/2). (2.4) Chapter 2. TOWARD A CONSTITUTIVE LAW FOR SNOW CREEP 8 STRAIN RATE Figure 2.1:, Schematic deviatoric stress-strain rate response for ice and a material with a viscosity proportional to the bulk stress (pure shear or triatrial compression) The first term describes viscous flow in pure shear whereas the second term characterizes pressure effects, responsible for the differences observed in extension and in compression. If I apply these results to snow, assuming / x and fi are power laws with the same exponent M and the p—p relation is linear, I get £,5/2 = ( / ' i+/*aP)~ 1S , 2 a - (2-5) Given the definition of the shear viscosity, this is equivalent to 2p = (pl+ii2p)V~(Mfl\ (2.6) Chapter 2. TOWARD A CONSTITUTIVE LAW FOR SNOW CREEP 9 2.2.2 The volumetric law In hydrostatic compression or uniaxial strain, the stress-strain rate behavior is expected to be qualitatively described by Figure 2.2. This type of response, extensively encountered in soil mechanics (Lambe and Whitman, 1979) represents an acceptable behavior as long as the stress levels are small enough for particle crushing and fracturing to be neglected. This is the case for the deformation of natural snow. The curve, concave toward the stress axis, displays a stiffness increase with increasing stress. Indeed, closer packing of the snow particles makes the porous material stiffer. On a slope, volumetric deformations are induced not only by the effect of the mean normal stress, but also in combination with shearing causing intergranular glide. At high strain rates and low porosity, dilatancy and hence a decreasing stiffness might be expected. However, in the case of natural deformations snow consists mostly of air, and the ice grains accommodate permanently to the stresses by densifying toward a more stable configuration. Therefore, continuous settlement takes place and shearing intensifies the stiffening process. Using a pore collapse model (see Appendix B), Brown (1979) derived a constitutive relationship relating pressure loading to density. In the case of natural densification, the loads are due to the weight of the above snow layers. In steady-state conditions, the strain rates drop to values near 1 0 - 8 5 _ 1 and Brown's law can then be expressed as: p = - v E 1 = -Cpoe-*p°,pln(-^-)E1. (2.7) In this equation, p0 and p0 are the initial density and shear viscosity, C and $ constants depending on the snow type and pm the density of the ice. When the snow density approaches the ice density, the bulk viscosity becomes infinite which requires that Ei tends to zero (incompressibility condition). Generally speaking, the bulk viscosity increases as the volumetric strain rate decreases. Hence, indirectly Chapter 2. TOWARD A CONSTITUTIVE LAW FOR SNOW CREEP 10 CO CO UJ CC H co STRAIN RATE Figure 2.2: Schematic volumetric stress-strain rate response (uniaxial strain or hydro-static compression) through the density, pressure-related nonlinearities are included in the expression for the bulk viscosity (see Appendix B for further details). The expression for n (Equation 2.7) is not valid when significant shear stresses are present. As mentioned earlier, shearing effects are characterized by a stiffness increase. The power law is one of the most widely used nonlinear relations for viscous materials and analogy with the shear viscosity suggests it might be appropriate as well for the bulk viscosity. Therefore, I propose to account for the influence of shear with a law of the form <r—f(p) or o~=f(p) <r1_1/M e, assuming for simplicity that density and non-Newtonian effects can be expressed as a product. In three dimensions, this leads in terms of stresses Chapter 2. TOWARD A CONSTITUTIVE LAW FOR SNOW CREEP 11 to the relationship r, = Cp0e-*p"/p In (-EULJ] S / i /2(i- i /W) ( 2 g ) \Pm - P) where M is the exponent of the power law (if M = l, (2.8) reduces to Equation (2.7)). The same exponent ( M ) is assumed for both the bulk and shear viscosity but clearly this simplification could be relaxed if experimental data were available to relate both viscosities more precisely. 2.2.3 The three-dimensional law In some types of problems (e.g. plane strain), only the ratio of the viscosities is of primary importance in the prediction of the snow flow. It may be expressed in terms of the effective viscous Poisson ratio _ 3 i - 2 M 2(3T?-f/x) . ^ > Mellor (1975) examines the influence of the density on the Poisson ratio in compres-sion and gives an envelope of probable values. Physically, v should vary between 0.0 as p—»0 and 0.5 as >pm. The general aspect of the v—p relationship indicates that 77 and p, are about the same order of magnitude for most of the density range. There-fore, I propose to express p with the same exponential dependence as 77. Given that p=e~^p o / f p ln (p p"lp) / ( £ 2 , £ i ) , ^ n e g e n e r a l constitutive relationship can then be written as follows: try = (T/ - 2fi/3)E16ij + 2piij (2.10) where / A f - l \ 77 = Cu0e-^p In [-^) S V M 2u = ( ^ e - W ' + W ) S 2 2 } -Chapter 2. TOWARD A CONSTITUTIVE LAW FOR SNOW CREEP 12 In the last equation, pi and p2 are now constants that depend only on the initial con-ditions. The exponent M is likely to vary between 1 (no nonlinear effects due to shear) and 3 (value commonly accepted for ice), depending on the snow stiffness. In fact, when p tends to the density of ice pm, v is infinite and 2p ~ / / 1 e~* P 0 / ' P m S' 2 : this is the well-known expression for Glen's flow law (Glen, 1958). When p and 77 are constant, Equation (2.10) is equivalent to a compressible Newtonian viscous fluid with neglect of the static pressure term (Mc Clung and Larsen, in press). Chapter 3 A P P L I C A T I O N T O T H E DESIGN OF S N O W BARRIERS 3.1 Description of the problem 3.1.1 Properties of the field data Field data available for calibrating the model consist of two sets of measurements collected in Norway and Switzerland for several years. Only their main characteristics will be given here. For further details the reader is referred to Kiimmerli (1958) and Salm (1977) for the Swiss data and to McClung and others (1984, in press) and Larsen and others (1985) for the Norwegian data. In both cases, snow pressure measurements are performed in the centre of a rigid avalanche-defense structure erected perpendicular to the slope (37° for the Swiss site and 25° for the Norwegian one). In this configuration, the influence of side effects can be ignored and the problem will be approximated as plane strain. The structure itself consists of horizontal crossbeams fixed at equidistant intervals on vertical uprights. The geometry of the problem is shown in Figure 3.3. Different experimental techniques are used to record the forces on the structures on a regular basis as well as the density profiles and the snow depth. Other snowpack properties including temperature, stiffness (using a ram-sonde) and snowcrystal profiles are collected on the Norwegian test site. Maximum (rm and depth-averaged pressures are certainly the best quantities to compare field 13 Chapter 3. APPLICATION TO THE DESIGN OF SNOW BARRIERS 14 and numerical results (Mc Clung and Larsen, in press). They are denned as follows o~m - m a x(<7 f i ) 0 < 2 < H where crxx and rxz are stress components in the rectangular Cartesian coordinate system shown in Figure 3.3. Figure 3.3: Definition of the coordinate system and of the velocities. Determining the boundary conditions is essential for the full description of the prob-lem. On both sites, the slip of the entire snowpack over the ground or glide is found to be negligible so that a zero velocity will be assumed on the ground. On the structure Chapter 3. APPLICATION TO THE DESIGN OF SNOW BARRIERS 15 itself, the formulation of realistic boundary conditions is a very difficult task. As a first approximation I wiD regard the crossbeams as a perfect plane plate. The flow character-istics can then vary between two extreme physical situations (McClung and Larsen, in press): a smooth structure lubricated by free water is likely to prevent any shear stress to build up (U=Txz=0: traction free condition); at the other extreme a rough structure in a cold snowpack may offer so much factional resistance to the snow motion that no vertical deformation takes place {u—v=0: no slip condition). Both Swiss and Norwegian data display relatively important shear forces and this is a strong indication for the no slip condition to be the more appropriate of the two. Analyzing French data, Janet (1987) emphasizes important pressure fluctuations in the course of the day. He relates them to the diurnal variations in solar radiation, claiming that the heat energy accumulated in the beams is responsible for a melt-freeze mechanism close to the structure. Hence, boundary conditions can change with time and this might be an explanation for the scatter observed on the data at short time intervals. The snow conditions at the Norwegian site may differ from the Swiss one by noticeable variations in the local climate. At the site in Norway, strong windpacking makes the snowpack unusually stiff. In addition, the lower slope angle may produce a greater settlement component relatively to the shear which could accelerate the densification and stiffening processes. Indeed, despite a lower slope angle, the pressures recorded in Norway are nearly the same as those measured'in Switzerland. 3.1.2 The numerical calculations Due to the additional complexity of a nonlinear flow law, numerical calculations are essential for its use in plane strain configuration. In this thesis, a plane strain finite element code is used to produce solutions with the constitutive law expressed in Equation (2.10). The snowpack is approximated by a rectangular mesh with quadrilateral elements. Chapter 3. APPLICATION TO THE DESIGN OF SNOW BARRIERS 16 The interruption of creep close to the barrier is examined with a fine grid. As the distance from the structure increases, the element density required to adequately describe the deformations diminishes. Elements are added until the stresses and creep velocities reach their neutral zone values (x—Xoo). In the calculations presented here, 22 x 10, 22 x 15 or 22 x 20 element grids are used depending on the complexity of the density profiles considered. I modified the original linear program to account for the non-Newtonian behavior, and use successive substitutions to achieve the solution. I start with the general equations of fluid flow written in Cartesian coordinates. Linear momentum equations - ^ + - / i - P 5 s i n V = 0. 3.11 ox oz ^ + ^ - ^ c o s V > = 0. (3.12) OX OZ These reduce to the equilibrium equations because steady-state conditions are assumed (assumption (3)) and the convection terms are small enough to be neglected. Plane strain constitutive law <rXx = (-7 + 4/i/3)era + (17 - 2p/Z)ezz °"yy = (V ~ 2/*/3)(e„. - izz) azz = (77 - 2u/3)kxx + (77 + 4 M / 3 ) e „ (3.13) <rxz = 2pkxz 0. The boundary conditions used to define the problem are specified as follows: (1) 2=0, 0<z<H: u=v=0 (no slip on the structure). Chapter 3. APPLICATION TO THE DESIGN OF SNOW BARRIERS 17 (2) 2=0, 0<x: u=0 (no glide over the ground). (3) z=H, 0<x: O-ZZ=Txz=0 (free surface at the top of the snowpack). (4) x=x00, 0<z<H: u—u00(z),v=v00(z) (neutral zone). The solution procedure begins with constant viscosities throughout the snowpack. At each iteration, the viscosities are computed in each element from the stress values, and if convergence is not achieved, new viscosities interpolated from their two previous values serve as input for the next calculations. The final solution (convergence) is obtained when the viscosities do not change enough to affect the stresses in the next iterations. Consistency between the results given by the linear and nonlinear models for different values of M and different nodes spacings strongly indicates that the technique leads to the correct solution. The number of iterations required for convergence increases with the value of the power law exponent M. When M is greater than 2, a better solution procedure (e.g. Newton-Raphson) seems desirable to improve the rate of convergence. 3.2 Determination of the parameters Though relatively simple, the formulations for the effective bulk and shear viscosities include parameters whose determination depends on the availability of reliable field data. Unfortunately extensive information on such fundamental quantities as the viscosities or the strain rates is scarce and the calibration of the model is accordingly affected. Data scatter due to the variations in alpine snow is a further complication. Nevertheless, it is possible to get plausible estimates of the parameters by assessing their relative importance on the results. Depending on the stress level and on the stiffness of the snowpack, the power law exponent M is taken to vary between 1 and 3. For each value of M, a full set of parameters (<f>,C,p\,p2) has to be determined theoretically. However, I will assume in Chapter 3. APPLICATION TO THE DESIGN OF SNOW BARRIERS 18 a first approximation that (j) changes only weakly with M so that it can be considered constant. First, the simplest case (M=l) is studied. Then for higher values of M, the quantities C, p\ and p2 are determined by estimating the stress invariants p and E 2 in the neutral zone. Brown (1979) compared his model ( M = l) with densiftcation data obtained in the laboratory by Abele and Gow (1975, 1976). He found a good fit for initial densities exceeding po = Z00kg/m3. Unfortunately, Abele and Gow's tests were only performed at large strain rates, which makes the data unusable for the present study. Consider instead the natural densification data provided by Mellor (1975). Since snow deforms continuously, the choice of an initial state is somewhat arbitrary. Table 3.1 summarizes the results obtained by adjusting the effective bulk viscosity with M=l to the field data for different initial densities. The conclusion is that <f> is relatively independent of initial density. Po -CpoEr kg/m 3 Pa 250 8.92 3.23 106 300 8.75 6.16 106 350 8.66 1.17 107 400 8.68 2.34 107 Table 3.1: Adjustment of Brown's law for the natural density data given by Mellor (1975). More generally, the overall model is weakly sensitive to the selection of an initial state for the following reason. The computation of the stresses is my main objective and in plane strain conditions these are essentially determined by the ratio of the effective bulk and shear viscosities. A shift in absolute value of viscosity causes only a proportional adjustment in the strain rates with no change in the stresses. Therefore, it is not necessary Chapter 3. APPLICATION TO THE DESIGN OF SNOW BARRIERS 19 to achieve great precision in the choice of the initial shear viscosity, as long as it is consistent through use of the Poisson ratio with the other parameters. Similarly, since the initial density appears with the same exponential dependence in both the viscosity expressions, it only slightly affects the stresses in the results. The range of the densities observed in the field in the late winter or early spring suggests that p0 — 300kg/m3 is a reasonable choice. Thus from Table 3.1, (f> — 8.75 and -Cp0E1=6.16 106. Several authors (Haefeli, 1939; McClung, 1974; MeUor, 1977) have reported vertical and shear strain rates, pointing out that they are both approximatively constant with depth in the neutral zone and at a given slope angle. In my case, E^ could well vary between 0.5 1 0 - 8 s - 1 and 5 10~ 8 5 _ 1 . By taking an average value of 2 10 _ 8.s _ 1, I obtain C/^0=3.1 1014. The next step is noticing that when M = l the bulk viscosity is constant at a given density and in particular when p=po- From the definition of the effective Poisson ratio, it follows that Mellor (1975) and Salm (1977) reviewed Poisson ratio data obtained by several authors in the field and in the laboratory. Generally speaking, the denser the snow, the higher the Poisson ratio. Meanwhile, the large data scatter for a constant density may suggest the presence of non-Newtonian effects, directly related to the value of the power law exponent (3.14) Substituting in the expression for the bulk viscosity, I get (3.15) M. When p = 300kg/m 3 , the medium-range value for v is 0.15 (data summarized by Chapter 3. APPLICATION TO THE DESIGN OF SNOW BARRIERS 20 Mellor) and I obtain C = 1.75 104 fio = 1.8 1010 Pa-s. For this choice, the initial shear viscosity is of the right order of magnitude (10loPa-s) according to data given by Haefeli (1967). The last unknown parameters are now p1 and p2. When M = l , 2p0 = p-ie~^' + u2p and there are several possible combinations for pi and pz- Facing this limitation, I will consider the widest choice of parameters sets (i.e. values that will cover a wide range of the probable Poisson ratio spectrum). The choice may be narrowed by noticing that the term / A 1 e _ * P o / ' p increases faster with density than p2p. I can reasonably assume that they should be comparable quantities for medium density snow (400-450%/m3) so that neither of the two terms dominates throughout the whole density range. If P2~^>\i\ for 300<£><550fc(7/m 3 , any direct dependence on the density disappears in the expression for the shear viscosity, resulting in unreasonably high values for the Poisson ratio. At the other extreme, if pi^>p2, the pressure dependence is lost and this is not physically realistic for a granular material like snow. In this limit, the constitutive relationship reduces to a linear law (at a given density). These considerations imply realistic bounds for the p\jpi ratio, especially when p can approximative^ be estimated. Therefore, the parameters p\ and p2 may be obtained by the following iterative process. Plausible values are first assigned to the pressure p for p = 300, 400 and 500kg/m3 and the relation 2tu0 = M i e _ * + enables initial guesses for / i i and pi- At each step, the finite element code is run with the latest parameters estimates. The resulting pressures in the neutral zone are recorded, depth-averaged and if the pressures differ substantially from the previous ones, new values of px and p2 are calculated. Consistency with the strain rates results is checked. I repeated the Chapter 3. APPLICATION TO THE DESIGN OF SNOW BARRIERS 21 computations for different slope angles and snow depths so that the results are valid for a wide range of ip and H values. Even though numerous refinements are possible, Table 3.2 exhibits 3 different combinations for the parameters pi and u2 ranging from one probable extreme to the other. It represents by no means an extensive review of the different possibilities but I believe it indicates the main trends. Table 3.3 summarizes for each parameters set the values of the bulk and shear viscosities calculated from their general expression for ,0=300, 400 and 500%/m 3 (Equation (2.10)). Also given are the corresponding effective Poisson ratios. Their significant increase with density between 300 and 400%/m 3 is consistent with the envelope of probable values given by Mellor (1975). When M>1, the bulk viscosity becomes stress dependent and the full triplet (C,Ui,p2) has now to be evaluated, assuming the initial shear viscosity p0 to be independent of the choice of M. The same method of successive approximations is used again with both p and S'2 being estimated in the neutral zone. The power laws intensify the nonlinear effects and both viscosities vary widely with depth. Therefore, averaging the different quantities is complicated and results should be regarded with caution. As before, Tables 3.2 and 3.3 examine possible sets of parameters. Since the cases M = l and M>1 differ significantly (in particular in terms of the Poisson ratio variations), sets referenced with the same number are independent of each other. There are not enough experimental data available to make a unique choice of pa-rameter sets. Unless specified, I use the first sets in the numerical calculations because they represent plausible medium-range estimates. Nevertheless, results obtained with the other combinations will also be presented to study the effect of parameters variations. Chapter 3. APPLICATION TO THE DESIGN OF SNOW BARRIERS M = 1 C Mi Pa-s M2 s Set 1 Set 2 Set 3 1.75 104 1.75 lO 4 1.75 lO 4 5.0 1013 2.5 1013 1.25 1013 1.5 107 1.4 107 1.0 107 M = 2 C P a " 1 / 2 Set 1 350 Set 2 430 Set 3 280 Mi M2 Pa2s Pa-s 1.75 1017 3.5 l O 1 0 1.0 1017 1.4 l O 1 0 3.5 1017 3.0 l O 1 0 M = 3 C P a " 1 / 3 Set 1 100 Set 2 120 Set 3 75 Mi M2 Pa3s Pa2s 5.0 l O 2 0 2.0 1014 2.0 l O 2 0 2.2 1014 1.0 1021 1.5 1014 Table 3.2: Sets of possible parameters for the model for different M values. Chapter 3. APPLICATION TO THE DESIGN OF SNOW BARRIERS 23 M = l p = ZOOkg/m3 p = AOOkg/m3 p = 500%/m 3 V P Set 1 Set 2 Set 3 /* V i/ p V 0.2 2.0 0.18 0.15 0.16 0.18 0.2 0.13 2.5 4.5 0.69 0.37 0.49 0.40 1.11 0.31 13.0 7.0 1.84 0.43 1.15 0.46 3.63 0.37 s 2 P p = 300%/m 3 6.0 2.0 p = 400%/m 3 7.0 5.0 p = 500%/m 3 9.0 8.0 M = 2 Set 1 V V 0.19 0.2 0.12 0.24 2.6 0.8 0.35 3.13 14.1 2.0 0.43 17.4 Set 2 Sef 3 A* V V A* 0.2 0.17 0.16 0.23 0.01 0.64 0.4 2.1 1.21 0.26 1.41 0.46 11.3 3.47 0.36 p = 300%/m 3 p = 400%/m 3 p = 500kg/m3 M = 3 P Set 1 Set 2 Set 3 V /* V V /* V V /* V 6.0 2.0 0.2 0.4 -0.1 0.24 0.39 -0.03 0.15 0.25 -0.03 8.0 5.0 2.75 1.06 0.33 3.3 0.86 0.38 2.06 1.35 0.23 10.0 8.5 15.4 2.16 0.43 18.5 1.46 0.46 11.6 3.26 0.37 Table 3.3: Values of the viscosities and of the resulting Poisson ratio at different densities calculated from estimates of the stress invariants. Each set of parameter given in Table 3.2 is examined. The units are kPa for p, MP a2 for S'2 and 10nPa-s for the viscosities. Chapter 3. APPLICATION TO THE DESIGN OF SNOW BARRIERS 24 3 . 3 R e s u l t s a n d d i s c u s s i o n 3 . 3 . 1 P r e s s u r e r e s u l t s Unless specified, finite element calculations are performed with a constant depth-averaged density. Figures 3.4 and 3.5 compare numerical and experimental results for the Nor-wegian and Swiss sites respectively. In these figures, the average pressure is presented as a function of the product pgH (a gravity stress index). Constant density curves are shown for the extreme range of values encountered in the field. Thus, the influence of the snowpack depth can be closely examined. The large scatter band of the field data is mostly contained within the numerical predictions. Hence, the model can account to a large extent for the variety of snow conditions observed in the field. As expected, the 5-R—pgH relation depends on the value of M. When M — l, the numerical calculations reveal a strong linear dependence between <TR and pgH whereas for M>1 a nonlinear relationship prevails with a concavity directed upward. The curves corresponding to various M values cross each other (Figures 3.4-d and 3.5-d) and there is evidence for distinguishing low and high stress levels. For relatively thin snowpacks, the stresses remain small, the curves differ only slightly from each other and M = l represents a good approximation. On the other side, large snow accumulations may cause important non-Newtonian effects: M = 2 and M = 3 are then better approximations. This is in agreement with previous predictions for the nonlinear behavior of snow (Haefeli, 1967; Mellor and Smith, 1967). A comparison between the Norwegian and Swiss data sets may show the differences in the climatic conditions. The Norwegian snowpack appears to be stiffer and it displays a response closer to ice behavior (i.e. it is best described using M greater than 1). On the contrary, the Swiss data suggest that nonlinearity is limited for that data set so that M=l is a reasonable choice. Indeed, as mentioned earlier, despite a larger slope angle, Chapter 3. APPLICATION TO THE DESIGN OF SNOW BARRIERS 25 18 16 14 12 I 10 "I 8 6 4 2 0 18 16 14 12 1 10 8 6 4 2 0 4 18 16 14 12 a 550kg/m3 ^ ^ r ' JSOkg/nr5 i i i i i i i lb 8 6 4 2 0 18 16 14 12 S. 10 8 6 4 2 0 10 12 14 10 12 14 10 12 14 10 12 ftjH (KPa) 14 16 16 16 16 18 20 b 550kg/m3 - 450kg/m3 ^-<^<-T^ . ^SOkg/m3 1 1 1 1 1 1 1 18 20 c 550kg/mJ 450kg/m3 0 I • « • • 1  — i i i i 18 20 d M=3 - 550kg/m3 M=1 -^ ^ ^ ^ " M=3 eiZ^"' • • • — 3 5 0 k g / m 3 ' " M = 1 1 1 1 i _ i i i 18 20 Figure 3.4: Comparison of the pressures on the structure recorded on the Norwegian site (singular symbols) and the corresponding numerical results represented by constant density curves, (a), (b) and (c) are obtained with different values of the exponent M (1, 2 and 3 respectively) and (d) summarizes the influence of M. Chapter 3. APPLICATION TO THE DESIGN OF SNOW BARRIERS 26 -a 550kg/m3 -^^7_^ 450kg/m3 - ^ — — 350kg/m3 i^ -— . - l * — -• • ——• .. "~ 300kg/m3 1 1 1 4 6 8 10 12 14 16 18 20 I , 550kg/m3 b 450kg/m3 ^ ^ ^ ^ ^ 350kg/m3 ^ ^ ^ ^ ^ ^ • 300kg/m3 * " * ——~~" 1 _ _ l 1 1 1 1 1 4 6 8 10 12 14 16 18 20 oi 1 ' 1 1 1 1 1 1 4 6 8 10 12 14 16 18 20 20 15 d M=3 550kg/m3 o C L _ K • • . • _ ^ — 3 0 0 k g / m 3 _ M>1 5 0 1 I i i i I 1 4 6 8 10 12 14 16 18 20 pgH (kPo) Figure 3.5: Comparison of the pressures on the structure recorded on the Swiss site (sin-gular symbols) and the corresponding numerical results represented by constant density curves, (a), (b) and (c) are obtained with different values of the exponent M (1, 2 and 3 respectively) and (d) summarizes the influence of M. Chapter 3. APPLICATION TO THE DESIGN OF SNOW BARRIERS 27 the Swiss pressures are not higher than the Norwegian ones. Thus, the model proves flexible enough to accommodate the stiffness variations caused by the weather and the field conditions. Significant differences from a Newtonian flow law can be noted. In the linear case, the viscosities and hence the Poisson ratio are constant in the entire snowpack (assuming constant density). The resulting pressures are then given by the following simple relation <TR= pgH f{v,ip). (3.16) The OR—pgH relation is linear with a slope depending only on v at a given slope angle tp (Figure 3.6-a and 3.6-b). The Norwegian data can only be explained by using Poisson ratio values ranging from 0.0 up to 0.45 but most of the points lie between the 0.3 and 0.45 curves. On the other hand, the Swiss data require lower values, mainly between 0.2 and 0.35 and are in general more closely distributed. These large differences enlighten the difficulties to select an adequate value for the Poisson ratio; a 0.1 shift in v can result in changes in pressure up to 30%. Despite this shortcoming, the linear predictions are interesting as a basis of reference and comparison for any non-Newtonian constitutive relationship. In particular, they show that only high Poisson ratio values, in the upper range of the data reviewed by Mellor (1975) and Salm (1977), can explain the higher pressures recorded on the field. The accuracy of the boundary conditions used in the calculations can be checked by looking at the ratio of the shear stress to normal stress on the structure tane = ^ . (3.17) °~xx The Swiss data yield for tan e values between 0.36 and 0.62 with most of them close to 0.5. No density dependence is really implicit in the data. Numerical calculations indicate Chapter 3. APPLICATION TO THE DESIGN OF SNOW BARRIERS 28 18 16 14 12 o i b 6 4 2 0 20 18 16 14 " o 12 GL e 10 8 6 4 2 0 _ a - i/=0.45 i/=0.4 • i/=0.3 -—' * ^—-"""^ — * •-" **"" i/=0.0 — """"jfc--*** i i i i i I l 8 10 12 14 16 18 20 pgH (kPa) i/=0.45 j/=0.35 i/=0.2 y=0.0 8 10 12 14 16 18 20 pgH (kPa) Figure 3.6: Comparison of the pressures on the structure recorded in the field (singular symbols) with the numerical results obtained with the linear model for different values of the effective Poisson ratio, (a): Norwegian site ( ^ = 2 5 ° ) . (b): Swiss site (•0=37°). Chapter 3. APPLICATION TO THE DESIGN OF SNOW BARRIERS 29 a strong influence of the Poisson ratio and depth-averaged values of tan e range from 0.4 for i/=0.2 to 0.1 for u=0A5. These variations are expected since tan e is a measure of the relative importance of compression and shear terms. The field data may imply that the idealization of the snow barrier as a plane plate is not realistic. Therefore, new boundary conditions including the effects due to the beam geometry of the structure may need to be specified but this is beyond the scope of the present thesis. However limited, the no slip condition is certainly more appropriate than the traction free condition for the tan e values resulting in the latter case would differ even more from the field data (for traction free, tan e=0). 3 . 3 . 2 N o n - N e w t o n i a n a n d d e n s i t y e f f e c t s In the calculation of the stresses, the input variables are in two categories. The depth of the snowpack H, directly affecting the stress invariants p and S'2, is responsible for the non-Newtonian effects whereas p refers primarily to the densification mechanism. This distinction is somewhat artificial because the mechanical pressure p and the rate of density change p are related through the bulk viscosity and the mass conservation equation. Nevertheless, since the density is assumed constant at the instant of time considered (i.e. when the numerical computations are performed), these effects will be examined separately. Figures 3.7 and 3.8 display the stress variations with depth obtained numerically for different values of M. These figures also exhibit the influence of the different (C,p1,p2) combinations summarized in Table 3.2. All the curves shown have some similarities. The maximum values of pressure are reached for z/H between 0.38 and 0.44. The differences in magnitude ( ± 1 0 % for the average stress) are due to different values of the effective Poisson ratio. There is no strong rationale for choosing one set of parameters over another and the question may remain open until more extensive field data are available. Yet, the Chapter 3. APPLICATION TO THE DESIGN OF SNOW BARRIERS 30 r j R (kPa) 0 2 4 6 8 10 12 Figure 3.7: Depth variations of the pressures on the structure calculated numerically with M = l and a constant depth-averaged density. p=400kg/m3, •0=25° and H=3 m. The profiles obtained with the different sets of parameters (Table 3.2) are compared with the results given by the linear model with i/=0.33. Chapter 3. APPLICATION TO THE DESIGN OF SNOW BARRIERS 31 crR (kPa) 0 2 4 6 8 10 12 Figure 3.8: Depth variations of the pressures on the structure calculated numerically with M=2 and a constant depth-averaged density. p—A00kg/m3,ip=25o,H=3m. The profiles obtained with the different sets of parameters (Table 3.2) are compared with the results given by the linear model with *v=0.33. Chapter 3. APPLICATION TO THE DESIGN OF SNOW BARRIERS 32 selection of the first sets is quite reasonable. Sets 2 and 3 produce pressures on the structure representing respectively the upper and lower edges of the plausible scatter band. Indeed, any further significant shift in the values of the parameters would result in Poisson ratio estimates that cannot explain the field data, as the linear predictions show. Therefore, even though better resolution of the parameters is desirable, the small differences in the results given by the different sets make the first sets a plausible choice. For comparison, Figures 3.7 and 3.8 also exhibit linear predictions with v equal to the average value computed with the first sets of parameters. Depth profiles of the Poisson ratio in the neutral zone are represented in Figure 3.9 for -0=25° (Norwegian site). As expected (see previous section), they tend to be in the upper range of the data reviewed by Mellor (1975) and Salm (1977). The large variations obtained, especially when M=2, are indeed a good illustration of the problems encoun-tered to estimate v. They reveal fundamental differences in patterns depending on M. When M = l the shear viscosity is maximum at the bottom of the snowpack where the bulk stress p is also maximum. Since the bulk viscosity is stress independent, v decreases with depth. This seems to be physically unrealistic. However, these results are obtained with a constant density with depth. Layering (examined later) mitigates substantially this defect. When M>1 , the power law terms in both the viscosity expressions cause the effective Poisson ratio to increase with depth as Figure 3.9 shows. For M = 2, the wide range of Poisson ratio variations result in a relatively low average value for v. Fur-thermore, high peak values at the bottom produce large stresses there and imply a small amount of densification is taking place close to the ground (i>=0.5 is the incompressibility condition). Both these features become more pronounced as M increases from 1 to 3. Density effects can be examined from Figure 9. As the density increases, v approaches 0.5 and snow becomes less compressible. This result was certainly expected on physical grounds. When p=550kg/m3, v is greater than 0.45 in the largest part of the snowpack Chapter 3. APPLICATION TO THE DESIGN OF SNOW BARRIERS 33 Poisson ratio v Figure 3.9: Depth profiles of the Poisson ratio obtained numerically in the neutral zone for different M values and constant depth-averaged densities. •0=25°, pgH=10.3kPa. Chapter 3. APPLICATION TO THE DESIGN OF SNOW BARRIERS 34 and this indicates that the densification process is dramatically reduced. Field obser-vations support this evidence: densities over Q00kg/m3 are not normally encountered in seasonal snow. At this density, higher pressures are required for a further packing but seasonal snowpacks are usually not deep enough to produce such pressures. It is certainly not a coincidence that 5$0kg/m3 is the density of a randomly closely packed group of ice spheres. A look at the expression for the bulk and shear viscosities provides the explanations for the variations of v with density. v = e - ^ l n ( ^ ) / ( S ' 2 ) 2/* = e - ^ I / i i - M i C ^ i l n ^ ) ] ^ ) . As p increases, the volumetric strain rate E\ drops dramatically so that the product —Ei ln ( p p " l p ) in the shear viscosity p actually decreases. This phenomenon combines with the logarithmic effects in the expression of 7/ to make the bulk viscosity increase with density much faster than the shear viscosity. For example, if iv=0.15 when p=300kg/m3, the Poisson ratio approaches 0.4 when p=500kg/m3 due solely to density effects. So far, I have assumed the snow cover to be homogeneous. Figures 3.10 and 3.11 explore the influence of layering on the numerical results. Two different density profiles representative of the field conditions are examined. In Figure 3.10, the resulting pressures are compared with the constant density model. As expected, layering affects the pressure distributions. The peaks and step patterns strongly reflect the density variations while the average pressures are lower than with a constant density. Such a behavior is in good agreement with the field data (Salm, 1977) but since the numerical calculations accommodate only imperfectly to any jump in density, some caution is necessary in the interpretation of the results. Figure 3.11 depicts the density influence on the effective Poisson ratio in the neutral zone for M = l . In this case, v increases with density but it also decreases with depth in each layer. Both effects compete simultaneously so that Chapter 3. APPLICATION TO THE DESIGN OF SNOW BARRIERS 35 1.0 0.8 0.6 0.4 0.2 PROFILE 1 PROFILE 2 0.0 1.0 0.8 0.6 0.4 0.2 0 200 400 P (kg/™ 3 ) 0.0 0 200 400 P (kg/m3) Figure 3.10: D e p t h variations of the pressures on the structure obta ined numerical ly for different density profiles. M = l , V>=37°, H=3m, p=4kl4.kg/m3. Figure 3.11: Depth variations of the effective Poisson ratio in the neutral zone obtained numerically for different density profiles. M = l , rp=37°, H—3m, p=350kg/rn3. Chapter 3. APPLICATION TO THE DESIGN OF SNOW BARRIERS 37 the general distribution of v depends on how strongly and smoothly the density varies. Hence, when M = l , my constitutive law can still predict peak v values at the bottom of the snowpack as physically expected. The relative influence of the non-Newtonian and density effects can be summarized as follows. For low density snow (300-350fc<7/m3), the stress invariants, directly related to the snow depth, dominate. The results obtained with M—l and M>\ differ widely as Figures 3.4 and 3.5 suggest. For high density snow (550%/m3), densification effects become predominant and the high values of the effective Poisson ratio are mainly due to rapid increase in bulk viscosity with density. Depth plays only a secondary role in that higher snow depths are normally needed to produce the densification. 3.3.3 Maximum versus average pressure Estimates of the ratio c r m / a j i on the structure are of primary interest in design. Again, both Swiss and Norwegian data sets display a large scatter band of the ratio with values ranging from 1.2 to 2.0 with a 1.5 average. A careful study of the data reveals the following tendency on the d-R—pgH representations (Figures 3.4 and 3.5). The points corresponding to lower values of <rm/o-fl he on the higher pressure range and inversely for high (Trn/crji ratios. Numerical calculations performed with a constant density show a dependence on the density p, the depth of the snowpack H and the slope angle ip. Generally speaking, (Tm/aji decreases as p and ip increase whereas it increases with H. All the numerical results for the ratio He between 1.10 and 1.25, well under the field values. These discrepancies can be addressed by taking layering effects into account. For example, Profiles 1 and 2 in Figure 3.10 generate cXm/vR values of 1.49 and 1.31 respectively, in comparison to 1.17 for the constant density case. The conclusion is that stronger varying density profiles will result in high peak "pressure values as well as high Cm/vR ratios but at the same time in relatively low average pressures. These results are Chapter 3. APPLICATION TO THE DESIGN OF SNOW BARRIERS 38 also expected for the linear model. Field data support this evidence: the a-m/a-R values drop as time proceeds i.e. as density homogenizes throughout the whole snowpack. 3.3.4 U s e o f t h e m o d e l The model has been developed for engineering applications and the initial assumptions greatly limit its domain of validity. Since the viscosities vary exponentially with density, large discrepencies can arise outside of the range considered (300 - 550 (or 600)%/m3). Brown's law is valid for po>300kg/m3] choosing an initial density lower than this value is not desirable. Higher initial densities might be considered and through use of Table 3.1 and appropriate estimates of Ei and v at this density, new values can be assigned to p0 and subsequently to C , /ij and p2- Since this narrows the possible variation range for the density (po<p<Q00kglrn3), the pressure results on the structure will be accordingly constrained and will certainly fall within the upper and lower pressure bounds predicted by Sets 2 and 3 of Table 3.2 when p0—300kg/m3. Therefore, it is possible to a large extent to use p0 = 300kgjm3 as a standard value without losing the specifics of the problem examined. Furthermore, however arbitrary it may seem, Set 1 (Table 3.2) explains the field data relatively well and it is likely to give plausible pressure estimates on structures in most cases encountered. This does not imply that better estimates cannot be made. Finally, unless important nonlinearities are expected (stiff or deep snowpacks), M = l may be used to model accurately the snow deformations. In this limit, the model reduces to one in which the viscosities are only pressure dependent with respect to stress invariants. Chapter 3. APPLICATION TO THE DESIGN OF SNOW BARRIERS 39 3.4 Discussion and conclusions A new nonlinear constitutive law has been proposed to model snow creep in a physically realistic manner. Its development was largely motivated by the importance of the en-gineering applications and hence simplicity was a leading consideration. Nevertheless, the expressions for the bulk and shear viscosities are based on fairly general concepts known to describe the stress-strain rate response of geotechnical granular materials. Fur-thermore, they match in the limits Brown's theory for volumetric snow deformation and Glen's flow law for ice. The model is intended for well-settled, well-metamorphosed sea-sonal snow with a density between 300kg/m3 and 600kg/m3 and therefore it does not apply for newly fallen snow. It is also restricted to the slow viscous deformation of the snowpack, which requires the strain rates to not exceed 10~7s~1. Finally, the viscosities are expressed in a simple and explicit form, particularly suitable for numerical applica-tions. When M<2, calculations can be easily performed with a finite element code run iteratively and convergence is achieved rapidly. Comparisons of numerical calculations with field measurements show that the law is capable of predicting creep pressures on a rigid structure in a physically plausible manner. Through the choice of an adequate exponent M , it is flexible enough to accommodate variations in the snowpack stiffness for the density range of interest in deep snow covers. The values of the effective Poisson ratio obtained numerically are consistent with the evidence that limited densification occurs for densities in excess of 550kg/m3 and in layers close to the ground. Acccounting for layering effects results in fairly realistic predictions of pressure as a function of depth. For example, density variations with depth allow good estimates of the ratio of the maximum to the average pressure on the structure; this is of primary interest for design specifications. This result is also true for a linear model (constant viscosity) but the physical interpretation is less realistic in the linear case. Chapter 3. APPLICATION TO THE DESIGN OF SNOW BARRIERS 40 In addition to being physically more plausible, the nonlinear model has a further attractive feature that the linear formulation does not have. The predictions of the linear law are mainly controlled by a single parameter, the viscous Poisson ratio v, whose value at present can only be estimated from a limited data set. In particular, the variations of v with snow cover thickness are not well known. Although my model also depends on an appropriate choice for the parameters, the effective viscosities can adjust to the local stress levels. Thus, for example, where the snowpack is abnormally deep or the pressures high, at the edges of structures, the linear theory may be too simplistic to explain the pressure patterns observed. The main limitation of the model lies in the lack of extensive data (field or laboratory); this makes its verification and calibration difficult. In particular, relating such fundamen-tal quantities as the strain rates or the viscosities with other snowpack properties as well as density appears highly desirable. Any further refinement in the expressions for the viscosities will certainly prove superfluous unless more data are available to calibrate the model or determine the forms of the viscosity expressions with greater precision. There-fore, future research should be concentrated on the experimental aspect of the problem. Finally, other structure geometries such as a cylindrical pole deserve further consideration since the results are expected to differ substantially from the plane strain case. R E F E R E N C E S Abele, G. and Gow, A.J. 1975. Compressibility characteristics of undisturbed snow. CRREL Res. Rep. 336. Abele, G. and Gow, A.J. 1976. Compressibility characteristics of compacted snow. CRREL Rep. 76-21. Brown, R.L. 1979. A volumetric constitutive law for snow subjected to large strains and strain rates. CRREL Rep. 79-20. Brown, R.L. 1980. A volumetric constitutive law for snow based on a neck growth model. J. Appl. Phys., 51(1), 161-165. Carrol, M . M . and Holt, A .C . 1972. Static and dynamic pore-collapse relations for ductile porous materials. J. Appl. Phys., 43(4), 1626-1635. Frutiger, H. and Martinelli, M. 1966. A manual for planning structural control of avalanches. Rocky Mountain Forest and Range Experiment Station, Research Paper RM-19, US Forest Service. Glen, J.W. 1958. The flow law of ice. IASH Publ, 47, 171-183. Haefeli, R. 1939. Snow mechanics with references to soil mechanics. In Bader and others, Der Schnee und seine Metamorphose, Beitrage zur Geologie der Schweiz. Geotechnische Serie. Hydrologie, Lief.3. (English translation, U.S. Snow, Ice and Permafrost Research Establishment. Translation 14, 1954). Haefeli, R. 1948. Schnee, Lawinen, Firn und Gletscher. In Bendel, L., ed. Ingenieur-Geologie. 2Bd. Wien, Springer-Verlag, 663-735. Haefeli, R. 1967. Some mechanical aspects on the formation of avalanches. In Physics of Snow and Ice, Vol. 1, Part 2. 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McClung, D.M., Larsen, J.O. and Hansen, S.B. 1984 Comparison of snow pressure mea-surements and theoretical predictions. Can. Geotech. J., 21(2), 250-258. McClung D.M. and Larsen, J.O. In press. Snow creep pressures: effects of structure boundary conditions and snowpack properties compared with field data. Cold Reg. Sci. Technol. Malvern L.E. 1969. Introduction to the mechanics of a continuous medium, Prenctice-Hall, Inc, New-Jersey. Martinelli, M. 1960. Creep and settlement in an alpine snowpack. Rocky Mountain Forest and Range Experiment Station, Research Note J^S, US Forest Service. Mellor, M. and Smith, J.H. 1967. Creep of snow and ice. In Physics of Snow and Ice, Vol. 1, Part 2. Oura, H., ed., Proceedings of International Conference on Low Temperature Science, The Institute of Low Temperature Science, Sapporo, 867-874. Mellor, M. 1968. Avalanches. CRREL Monograph, Part III, Section A3d. Mellor, M. 1975. A review of basic snow mechanics. IASH Publ, 114, 251-291. Mellor, M. 1977. Engineering properties of snow. J. Glaciol.., 19(81), 15-66. Salm, B. 1967. A n attempt to clarify triatrial creep mechanics of snow. In Physics of Snow and Ice, Vol. 1, Part 2. Oura, H., ed., Proceedings of International Conference on Low Temperature Science, The Institute of Low Temperature Science, Sapporo, 867-874. Salm, B. 1977. Snow forces. J. Glaciol. 19(81), 67-100. Salm, B. 1982. Mechanical properties of snow. Rev. Geophys. Space Phys., 20(1), 1-19. REFERENCES 43 Reiner, M . 1945. A mathematical theory of dilatancy. Am. J. Math., 67, 350-362. Rivlin, R.S. 1948. The hydrodynamics of non-Newtonian fluids, /. Proc. R. Soc. London, Ser. A, 193, 260-281. Vulliet, L. and Hutter, K. 1988. Set of constitutive models for soils under slow movement. J. Geotech. Engng Div., ASCE, 114(9), 1022-1041. Vyalov, S.S. 1986 Rheological fundamentals of soil mechanics. Developments in Geotech-nical Engineering, 36, Elsevier, Amsterdam. Appendix A Reiner-Rivlin fluids with no dependence on the third strain rate invariant Consider the constitutive equation of a Reiner-Rivlin fluid <Tij = faSij + fceij - f faeikekj. (A - I) The fa depend theoretically on all the strain rate invariants Ei, E2 and E3. Let me dismiss any E3 dependence on the fa and on the second stress invariant Ti2 = l/2(aij<Tij — <THO~JJ). The consequences of such an assumption can be further explored by expressing Ti2 in terms of the strain rate invariants. For notation simplicity, the strain rate tensor will be written as e in what follows: = ^ + f t r { k 2 ) + f i r ( ' 4 )  + fo <h trW + ^ ^ ( e 2 ) + fafatr(e3). (A - 2) From the definition of the tensor invariants, it can be easily shown tr{e2) = El + 2E2 < tr(e3) = 3 £ 3 + ZEiE2 + E\. The Cayley-Hamilton theorem allows us expressing the fourth power of the strain rate tensor e4 in terms of its lower powers e4 = Eje3 + E2e2 + £ 3 c (A - 3) 44 Appendix A. A SPECIAL CLASS OF REINER-RIVLIN FLUIDS 45 Taking the trace of all the tensors yields • tr(e4) = AEyEz + AE2E2 + 1E\ + E\. (A - 4) I can then further expand (A-2) and obtain 1 3<T32 S2 - a ^ = -|1 + + fa4>2)(E2 + 2E2) + 0 2 0 3 (3£ 3 + ZErE2 + E3) 12 +^(AE1E3 + AEl2E2 + 2E\ + £ 4 ) + M2E,. (A - 5) Similarly, since cr;;=£1=30i + 02-Ei + 4>z{E\ + 2£ 2 ) , I get 1 90? 0? . - < r ^ = -p + ^  + ^ (£4 + 4E*E2 + 4 £ 2 ) + 30 10 3(£ 1 2 + 2£ 2 ) +30x02^ + 0 2 0 3 ( £ 3 + 2 £ 1 E 2 ) . (-4-6) Combining (A-5) and (A-6) gives finally S 2 = (-30? + <t>\E2 - 20 1 0 2 £ 1 ) - 0 3 ( 0 3 £ 2 + 20 1(£ 1 2 + 2E2) - 0 2 £ i £ 2 ) + 0 3 £ 3 (20 3 £l + 302). (A - 7) Hence, if 0i, 02 and 03 do not depend on E3, I have £2 = f(Eu E2) + 0 3 £ 3 ( 2 S i 0 3 + 302). (A - 8) Appendix A. A SPECIAL CLASS OF REINER-RIVLIN FLUIDS 46 Furthermore, I assumed that E 2 is not a function of E3. This is obviously true in plane deformation problems where £'3=0. However, a constitutive law is assumed to be valid for every possible deformation pattern and this implies 03 = 0 or 03 = - T j j j - -If i?i=0, the second implication still holds provided 02=O. In this case, 0 3 can be anything. In reality, every snow deformation causes a density change so that E\ is never zero. The remaining ambiguity can be removed by considering the second law of thermo-dynamics. The strong form of the Clausius-Duhem inequality (Malvern, 1969) rules that the rate of dissipative energy crf^k{j should be non-negative. Herein, <jf^ is the dissipative stress. It is often equated (Glen, 1958) to the deviatoric stress given by cr'- = aij — ^-Sij-Substituting the constitutive equation (A-l) in the expression for the rate of strain working c-jCij, I eventually get o[.ei5 = 2 0 2 ( £ 2 + El/3) + 0 3 ( 3 £ 3 + + ^ ) - (A - 9) Let me assume that 0 3 = I ^ 2 is the correct possibihty. Equation (A-9) then becomes <* = -*{-f-T + Wj (A-10) where E'2—E2-\-(E\)/Z, the second invariant of the deviatoric strain rate tensor is always a positive quantity. From (A-10), it appears that 0 2 and — £ - f ||£ should always have the opposite sign if the Clausius-Duhem inequality is to be verified for every possible motion. This is Appendix A. A SPECIAL CLASS OF REINER-RIVLIN FLUIDS 47 not feasible unless fa has a. very special form. Therefore, c/>3=0 is highly probable and in this case, the expression for the rate of strain working reduces to fffa = 2faE'2. (A - 11) Hence, provided </>2>0, the rate of strain working always remains positive as required. Appendix B Brown's theory for the volumetric deformation of medium to high density snow Brown (1979) uses a pore collapse model to develop a relationship between material porosity and pressure loading. Though intergranular glide and inelastic deformation of intergranular necks are other possible mechanisms, he claims (Brown, 1980) that the latter ones are of secondary importance for medium to high density snow, at least in unconfined compression (equivalent to a flat snowfield). Snow is idealized as an assembly of thick-walled hollow spheres of polycrystalline ice containing air voids. The geometry of the model implies that pores do not interact with one another. This is obviously an unreasonable assumption but Brown addresses this problem by introducing two mitigating factors. First, due to the interconnection of pores, stresses tend to intensify in the skeleton and the pressure in the matrix material has to be divided by a density scale factor a=pm/p to yield the average pressure in the porous material (Carrol and Holt, 1972). Herein, pm and p are the densities of the ice and the snow respectively. This reduction of the overall stiffness is compensated by a work-hardening phenomenon probably caused by grain rearrangement and represented by the term J e - ^ / " 0 , where J and a are snow constants and a0 is the initial value of the density ratio a. When only viscous deformations are considered, Brown's constitutive law can be 48 BROWN'S THEORY 49 written as follows: _ j e - 0 < * / a o In a(a — 1) ( B - l ) P = 3a Herein A, B and So are ice constants, d is the rate of change of a. This equation is valid for strain rates in excess of l C T 5 ^ 1 . For low strain rates (< l C T 6 ^ 1 ) , the term 2(S0-B) can be neglected and the second term can be replaced by a linear representation in a (Brown, personal communication). Furthermore, the mass conservation equation states E\—aja.——pjp and \Pm ~ P) where /z0 is the initial shear viscosity and C a constant depending on the snow type. 

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