in Ice Crushing Failure

Transcription

1 Yield and Plastic Deformation in Ice Crushing Failure TERRANCE D. RALSTON ABSTRACT Plasticity theory has been widely used to describe the deformation and failure of metals, soils, rocks, and concrete. This approach also appears to be useful for describing the deformation and failure of ice. Yield functions that qualitatively possess the anisotropic and compression/tension strength differences of columnar-grained ice have been investigated within the context of rock mechanics. The solution procedures needed to implement this approach have been explored for some of these functions. The finite element method, finite difference procedures, the method of characteristics, and the upper and lower bound techniques of plastic limit analysis have all been used to some extent. The application to ice failure requires only the definition of appropriate strength information and the measurement of sufficient data to assess the adequacy of this approach. The example contained in this paper illustrates a plastic failure criterion and associated flow rule applied to the ductile deformation of columnar-grained ice. Unconfined ice strength values are used to define the yield function coefficients. The computed changes in ice strength that result from confinement applied both in the plane of the ice sheet and perpendicular to the plane agree with previously reported experimental observations. This description can probably be improved with the use of yield functions and flow rules that are supported more adequately by experimental measurements. The same type of calculation can be conducted for more complex situations, including the crashing failure of ice moving against offshore structures. INTRODUCTION The crushing failure of ice against offshore structures or structural models in a laboratory typically occurs with the ice sheet in a nonuniform, multidimensional 234

2 Yield and Plastic Deformation in Ice Crushing Failure 235 stress state. The state of stress in the ice is determined by the ice/structure configuration and environmental loading, together with the nonlinear and timedependent characteristics of ice deformation. The ice strength measurements that would be most useful for either understanding model behavior or establishing design criteria for structures should (ideally) duplicate the appropriate multidimensional stress states. Unfortunately these stress states are generally unknown. One possible approach to the description of ice failure under such conditions may be developed with the use of an analytic failure criterion that is determined by simple strength measurements and properly applied to the particular situation. The mechanical properties of ice are functions of the ice crystal structure, temperature, and the rate at which the deformation process occurs. The present analysis considers the anisotropy and two-dimensional pressure sensitivity of laboratory-grown, columnar-grained freshwater ice tested at one temperature and strain rate. A comprehensive discussion of other aspects of the mechanical behavior of ice can be found in the works by Weeks and Assur (1969), Carter and Michel (1971), Schwarz and Weeks (1977), and Vaudrey (1977). Relatively few basic property studies of multidimensional strength for ice of any type have been reported in the literature. The anisotropic structure of columnar-grained ice makes it of particular interest with regard to the design of ice-resistant offshore structures. The two-dimensional confined compression tests of Frederking (1977) and Croasdale et al. (1977) were conducted with this type of ice. Other ice types, which should be considered as mechanically different materials, have also been tested. Fine-grained, random polycrystalline ice, which is assumed to be isotropic, has been the subject of several brief experimental studies. Jones (1977) conducted triaxial compression tests on this type of ice at high confining pressures. Frederking (1977) presented confined compression data for both polycrystalline and columnar-grained ice. Haynes (1973) reported data for triaxial tension/compression tests, and Langford and Francis (1975) conducted biaxial tension/compression and compression/compression tests on polycrystalline ice. Each of these studies was motivated by separate objectives, and so differences exist in test temperature, strain rate, sample preparation, experimental technique, crystal size, and air bubble size. The present analysis uses anisotropic uniaxial strength data presented by Carter and Michel (1971) and biaxial test data reported by Frederking (1977). The crystallographic descriptions (columnar-grained, randomly oriented horizontal c-axis, 2-5 mm crystal diameter) of the ice are sufficiently similar to assume that the two sets of data are representative of the same material. The anisotropic and pressure sensitive strength characteristics of columnar-grained freshwater ice are evident from the strength tests indicated in Figure 1. The unconfined strength data of Carter and Michel (1971) show vertical to horizontal strength ratios of as much as a factor of 2. Frederking's (1977) biaxial confinement, or plane strain, tests imply that lateral confinement may increase ice strength by as much as a factor of 5. The effect of lateral confinement, however, is also anisotropic. The

3 236 TERRANCE D. RALSTON restriction of motion perpendicular to the ice sheet plane had negligible effect on axial strengths, while the confinement of motion within the plane was very significant. Croasdale et al. (1977) also present plane strain ice strength data for laboratory-grown freshwater ice tested at 10 C C and 1.5 G C. These tests were of the type indicated by Frederking's test type A in Figure 1 and gave strength values similar to those reported by Frederking. The failure criteria discussed in this paper are yield criteria in the sense of plasticity theory (e.g., Prager and Hodge, 1968). The use of a yield criterion plus a flow rule, which describes post-yield deformation, enables one to describe the nonlinear behavior of some materials up to the point of fracture. This approach is commonly applied to metals, soils, rocks, and, more recently, concrete as well (Johnson and Mellor, 1962; Chen, 1975; Smith, 1974). The flow rule used in this paper is determined by the yield function through what is known as the normality principle. This is commonly called the associated flow rule. No direct experimental data exist that either confirm or contradict the applicability of this assumption for the ductile deformation of ice. Indirect evidence, in the form of comparisons of experimental data with the results of analyses that use this assumption, tends to support the associated flow rule. EXAMPLES OF YIELD FUNCTIONS Yield functions for elastic-perfectly plastic materials are usually presented in the form/fer) é 0, where the value of the function/ is determined by some algebraic combination of the stress components cr«. A stress state is said to be elastic if/(cr) < 0, whereas the material is at yield, or in the plastic state, if f(a) = 0. The criteria most commonly used to describe the initial yield behavior of metals are the von Mises yield function ACT) = Vi {(a r - a y y z + (o-j, - a z f + (o% - a x f] + 3[T xa 2 + Ty/ + T ZJ r} - (TO 1, and the Tresca yield function /(cr) = (o- max - o- min ) - (T 0, where cr 0 is the unconfined compressive or tensile strength of the metal, and <x max and <r min are the maximum and minimum principal stresses. Both of these functions describe isotropic materials that have equal tensile and compressive strengths. These functions also imply that the material strength would be unaffected by the presence of any hydrostatic stress (cr,,. = cr u = <J Z = ±p). The Tresca function depends only on the extreme principal stresses and is therefore independent of the intermediate principal stress. This implies that strength would be independent of biaxial confinement and thus the plane strain strength of such a

4 Yield and Plastic Deformation in Ice Crushing Failure 237 CARTER AND MICHEL'S UNCONFINED STRENGTH TESTS FREDERKING'S PtANE STRAIN STRENGTH TESTS TYPE A TYPÉ B Figure 1. Unconfined and plane strain ice strength tests for columnar-grained ice. material would be equal to its unconfined compressive strength. The strength of a von Mises material is somewhat dependent on the intermediate principal stress; however, its plane strain strength is only about 15% greater than its confined strength. Both of these descriptions require only a single strength measurement, <r 0, to determine all values of the yield function. The strengths of soils and rocks are known to increase in the presence of a hydrostatic pressure. A Mohr-Coulomb yield function of the form /(cr) = cr max (1 - sin $) - cr min (1 + sin < ) - 2c cos d> is often used to describe a linear strength increase. The two strength parameters, c and <, are called the cohesion and internal friction angle, respectively. This particular function can be viewed as a linear, pressure-dependent extension of the Tresca criterion. Kivisild and Iyer (1976) have used the extended Tresca criterion as a theoretical basis for the interpretation of in situ ice strength tests in columnar-grained ice. That criterion is both isotropic and independent of the intermediate principal stress. Hence it would not describe either the anisotropy observed by Carter and Michel (1971) or the strong effect of lateral confinement measured by Frederking (1977). Yield functions that describe anisotropic materials have not been as widely used, although Smith (1974) has noted that numerous materials appear to be suitable for such a description. For ice applications, two functional forms appear to be particularly useful. The first is a special case of the «-type yield functions presented by Pariseau (1968) and discussed further by Reinicke and Ralston (1977). /(cr) = a,(o- w - <r~) 2 + a 2 (a z - cr x f + a 3 (a x - a,j- + a 4 T Z Z + «s T,/ + a 6 Tj.,/ + a 7 crj. + a H a y + a B <j z 1. (1)

5 238 TERRANCE D. RALSTON This function can describe materials with differing tensile and compressive strengths and predicts a nonlinear (parabolic) increase in strength with confining pressure. This type of behavior is consistent with recent triaxial tests of polycrystalline ice by Jones (1977). If the material is completely anisotropic, nine independent strength measurements would be required to determine the coefficients of (1). Any isotropy, such as transverse isotropy, will reduce the number of required tests. The application of this function to ice strength will be discussed in greater detail in the next section. A generalization of (1) has been applied to rock deformation by Smith (1974). This generalization is given by /(or) = {A(cr x -(T u f + B(u u - a z f + C(a z cr x ) 2 + DTj.y ET IIZ + FTXZ 2 } - (G(T X + Ha u + Ka z ) - (La,,. + Mcr y + NaJ 1-1. (2) Note that 12 independent strength measurements would be required to determine all of the coefficients in this function for a completely anisotropic material. This function has all of the capabilities of (1) plus some additional advantages that result from the extra terms. When this yield function is used with the associated flow rule, it can describe materials that compact (decrease in volume) when plastic deformation occurs in the presence of sufficient hydrostatic pressure. A frequent criticism of the use of the associated flow rule with the Mohr-Coulomb yield function is that it always predicts volumetric expansion with plastic deformation at any hydrostatic pressure. The anisotropic function (1) also predicts expansion with plastic deformation, but the amount of expansion decreases with increasing pressure. Thus far we have presented yield criteria as a function of stress. Wu (1973) and Chang ( 1974) have explored the possibility of developing failure criteria for concrete by expanding a function of the scalar strain invariants in a Taylor series. Terms of order 3 and higher were truncated to give a quadratic function of strain for the failure criterion. Elastic behavior was then assumed to apply prior to yield and the elastic stress-strain law was used to express the criterion in terms of stress. The end result was a stress failure criterion that is the isotropic restriction of (2). PLASTIC ANALYSIS OF ICE CRUSHING FAILURE The use of an ice yield criterion is best illustrated by an example calculation. Frederking's (1977) plane strain compression tests and Carter and Michel's (1971) uniaxial strength tests for columnar-grained ice provide the data for such a calculation. The coordinate system, ice sheet geometry, and Frederking's two plane strain test types are illustrated in Figure 1 along with Carter and Michel's horizontal and vertical tensile and compressive strength tests. The sheet is assumed to

6 Yield and Plastic Deformation in Ice Crushing Failure 239 consist of horizontal c-axis ice, with the c-axis direction being random within the horizontal plane. The crystal size is assumed to be sufficiently small, relative to the test specimen dimensions, that ice strength measurements are homogeneous and isotropic within the horizontal plane. The yield function (1) will be used in the present analysis. A similar description could be developed using (2); however, additional ice strength data would be required to define the appropriate coefficients. The isotropy of strength within the horizontal plane must be represented by the yield function. This implies that the coefficients in (1) are not independent, but are subject to the restrictions a 1 = a 2, a4 = a 5, a 1 = a H, a 6 = 2(a, + 2a 3 ). The values of the coefficients and ÛQ can be determined from compressive and tensile strength measurements as follows: 1 _ 1 1 2C Z T Z ' T t.c x. 2C Z T Z (3) a =.L -JL a = - 7 T x Cj 9 T z C z where T x, T z, Ç,., C z are the absolute values of the horizontal and vertical tensile and compressive strengths, respectively. The value of a 4 could be determined from either a shear test or a compression test on a sample inclined from the vertical direction; however, its value will not be needed in the present analysis. The ice strength data reported by Carter and Michel are sufficient to determine the values of the yield function coefficients needed for the present analysis. At 10 C they found the maximum value of C x to occur at a reported strain rate of about 1.3 x 10~ :i sec -1. For this rate, their strength values are, approximately, T J: = 1.01 MPa (146 psi), T z = 1.21 MPa (175 psi), C,. = 7.11 MPa (1030 psi), C z = 13.5 MPa (1960 psi). (4) The vertical compressive strength, C z, did not attain its maximum value at the 1.3 x 10~ :l sec~' strain rate. At a slower strain rate of 6 x 10~' sec" 1 the maximum for C z was approximately 16.5 MPa (2390 psi). If this value were used in the following analysis, somewhat greater plane strain strengths would be calculated. The values of the yield function coefficients that correspond to (4) are a 1 = 3.06 x 10-- MPa" 2, a 3 = 10.9 x 1Q- 2 MPa" 2, a 7 = 84.9 X 10-2 MPa" 1, a g = 75.2 x 10" 2 MPa~'. (5) The assumption of the associated flow rule implies that the yield function is a potential function for the plastic strain rate, i.e.,

7 240 TERRANCE D. RALSTON e,/=\-^, (6) d(t U where A is a positive scalar function, whose value generally depends on stress and/or strain rate. When applied to (1) the flow rule becomes six equations that relate the stress components to the components of the plastic strain rate. Test Type A Frederking's type A test is indicated in Figure 1. In this test, the upper and lower faces of the sample are stress free and we assume that a state of plane stress exists in the plane of the ice sheet. That is, 0"j = 7 V z = T xz = 0. With this assumption, the isotropic restriction of (1) reduces to l a x (u x + dy z ) + a 3 (a x - (T y ) (a, + 2a 3 ) T JH + a 7 (a x + a u ) = 1. If x, y are the principal stress directions, then T XU vanishes and we arrive at a^a/ + cry 2 ) + a 3 (a x - a u )- + a 7 (a x + cr ) =1. (7) Frederking noticed an appreciable side friction force in some of his tests; however, the nominal shear stress, T,,,,, required to produce that force was quite small compared with a x. Thus our assumption that x and y are principal stress directions should not introduce significant errors. The yield surface given by (7) with the coefficients (5) is plotted in Figure 2. This yield surface is a long narrow ellipse symmetric about the line a x = cr. If we assume rigid lateral restraint by the confining mechanism, the elastic loading path will follow the line cr u = ya x, where y is Poisson 's ratio. When the loading path reaches the yield surface, plastic deformation begins to occur and the stress trajectory follows the yield surface. Our prior experience with numerical calculations for similar problems has indicated that the apparent stiffness of the ice sample will not change significantly until later in the loading when the plastic limit (fracture) load is approached. The stress-strain curves presented by Frederking are consistent with our experience. Next, we must determine how far the stress trajectory will travel along the yield surface before failure occurs. The total strain in an elastic-plastic material is usually represented as the sum of an elastic strain and a plastic strain, i.e., e = e e + e". At the limit load, one of the classical results of limit analysis states that the rate of change of elastic strain must vanish. Thus the total strain rate is equal to the

8 Yield and Plastic Deformation in Ice Crushing Failure 241 Figure 2. Loading path in stress space for the Type A plane strain test. plastic strain rate (at limit load). It is reasonable to assume that the confining apparatus may deform elastically but does not yield during the loading process. It follows that when the limit load is achieved, the ice strain satisfies é = èj> = 0, Thus (6) and (7) imply * = 2 f l 3 ^ -"7. (g) 2(a, + a 3 ) If this expression is substituted for cr u in (7) we arrive at a quadratic equation for a,,.. The two solutions of that equation represent the two points in Figure 2 where the normal to the yield surface is parallel to thex axis. The point in the compression quadrant is the stress state for the limit load of the plane strain test type A. This calculated value is a m A = 29.4 MPa (4260 psi). This value is somewhat higher than the results of Frederking shown in Figure 3. However, it is plausible that the calculated value could be experimentally achieved if the plane strain tests were conducted at higher strain rates.

9 242 TERRANCE D. RALSTON I H a z 40.0, 1! 1! IMIIj! II! M i I I I I! 111! TTTTTTT 20 - h SIDE FRICTION TESTS »/ f - _ _-4>J. i CALCULATED PLASTIC LIMIT LOADS LEGEND A-Type Specimen B-Type Specimen Simple Compression o D Piane Strain Compression j i M i ml i i i M ml i i i i 11 i i M nil i 10" 10" 10- STRAIN RATE, s~ 1 Figure 3. Preliminary results of strain rate dependence of failure stress; columnar-grained ice, -10 C (after Frederking, 1977). m m The maximum lateral confining stress that develops as a result of plastic deformation is given by (8) to be ay 4 = 25.9 MPa (3760 psi). This value is approximately three times as great as what would be expected if the deformation were elastic and Poisson's ratio were taken to be 0.3. This result is in accord with Frederking's observation that his confining apparatus deformed approximately three times as much as expected for elastic ice deformation. He noted the possibility of non-elastic ice behavior as one possible explanation for this effect. Frederking also suggested that the crushing failure of an ice sheet in uniform contact with a long straight wall was an example of a plane strain process. It is not obvious that the lateral pressure that develops in this situation would be sufficiently great to support the plane strain strength. It seems reasonable to question whether the ice boundary conditions and the total stress distribution in the ice sheet would allow lateral pressures of this magnitude to be developed. Test Type B Frederking's test type B confined the ice sample in the vertical direction as shown in Figure 1. An analysis of this test, similar to that for test type A, leads to a calculated failure stress of <T PS B = 12.0 MPa (1740 psi). This value is much less than cr ps A, where the confinement is applied in the plane

10 Yield and Plastic Deformation in Ice Crushing Failure 243 of the ice sheet. This difference is consistent with Frederking's findings; however, the calculated value of <j ps B is significantly greater than the measured maximum. Plane Stress/Plane Strain Yield Criteria Since many practical applications are concerned with two-dimensional stress states in the ice sheet plane, the in-plane yield criteria corresponding to plane stress (a z = r xz = r yx = 0) and plane strain (e z = e zx = e zy = 0) are of particular interest. The ellipse shown in Figure 2 and repeated in Figure 4 is the calculated plane stress yield surface for equation (1). The plane strain yield surface can be computed by imposing the restriction e z = e zy = e zy =0 on equation (1). The calculated plane strain yield function that results is given by a e ( * Ça.) 2 + (2a 7 + ae ) o"x r JL = 1 + «Si 8a, This function is represented by the parabola in Figure 4. The compression intercepts of this parabola with the coordinate axes occur at the calculated failure Figure 4. Two-dimensional yield criteria for in-plane ice sheet loading.

11 244 TERRANCE D. RALSTON stress for test type B. The high strengths implied by this parabola in the tension/ tension quadrant are probably unrealistic; however, no data exist to improve the definition of the yield function in this region. Frederking's data from test type B imply that the plane strain and plane stress yield surfaces should intersect on the compression axes at the point corresponding to the unconfined strength test. The hypothetical plane strain yield surface indicated by the dashed curve in Figure 4 is suggested by the test type B data, together with an assumption that tension behavior in plane strain is not much different from that in plane stress. If this curve were experimentally justified, it would imply that only in the compression/compression quadrant would the plane stress and plane strain strength behavior of sheet ice differ significantly under in-plane loading. CONCLUSIONS Plasticity theory appears to be a reasonable approach to describing the crushing failure of ice that occurs in complex stress states. The two essential ingredients in this approach are the definitions of an appropriate yield function and a flow rule. Both of these should be determined from experimental data. The yield function that was illustrated in the preceding section was selected because it qualitatively possesses some of the properties (anisotropy, compression/tension strength difference) of columnar-grained ice. Other functions, such as equation (2), also possess these characteristics and may provide an improved description of ice failure. In addition to uniaxial tests, biaxial and triaxial strength tests are particularly well suited for selecting the types of yield functions that are most suitable for describing ice strength and determining the appropriate values of the yield function coefficients. This analysis provides a definitive correlation between ice strength data from two independent studies where no correlation was previously known to exist. Plastic analysis provides a unified framework for understanding the anisotropy and confining effects that have been observed in ice strength tests. Confinement that restricts motion in the plane of an ice sheet can significantly increase ice strength. This increase has been predicted using data from unconfined strength tests and has been observed experimentally. The lateral confining stress required to produce the maximum axial strength is nearly equal to the axial stress and is approximately three times as great as that which would be predicted for elastic-brittle failure. Confinement that restricts motion perpendicular to the ice sheet plane does not significantly increase uniaxial ice strength in the plane. The yield function used in this analysis predicted a strength increase for this configuration that was small compared with that predicted for in-plane confinement. Experimental data confirm that little, if any, strength increase results from the restriction of motion perpendicular to the sheet.

12 Yield and Plastic Deformation in Ice Crushing Failure 245 ACKNOWLEDGMENT I would like to thank Exxon Production Research Company for permission to publish this paper. REFERENCES Carter, D., and B. Michel Lois et méchanismes de l'apparent fracture fragile de la glace de rivière et de lac, Rapport S-22, 393 pp. Dépt. de Génie Civil, Université Laval. Chang, K Compression of columnar-grained ice and some further aspects of brittle fracture. PhD thesis, 109 pp., University of Iowa, Ames. Chen, W.-F Limit Analysis and Soil Plasticity, 638 pp., Elsevier, New York. Croasdale, K. R., N. R. Morgenstern, and J. B. Nuttall Indentation tests to investigate ice pressures on vertical piers. Journal of Glaciology, 19(81), Drucker, D. C, W. Prager, and H. J. Greenberg Extended limit design theorems for continuous media. Quarterly of Applied Mathematics, 9, Frederking, R Plane strain compressive strength of columnar-grained and granular snow ice. Journal of Glaciology, 15(80), Haynes, F. D Tensile strength of ice under triaxial stresses. Research Report 312, 21 pp., U.S. Army CRREL, Hanover, N.H. Johnson, W., and P. B. Mellor Plasticity for Mechanical Engineers, 412 pp., Van Nostrand, New York. Jones, S. J Triaxial testing of polycrystalline ice, Technical Memorandum 121, Workshop on the Mechanical Properties of Ice, pp , National Research Council of Canada, Ottawa. Kivisild, H. R., and S. H. Iyer In situ tests for ice strength measurements. Ocean Engineering, 3, Lankford, J., and P. H. Francis Strength of ice under multiaxial loading, Report No , 10 pp., Southwest Research Institute, San Antonio, Texas. Pariseau, W. G Plasticity theory for anisotropic rocks and soils. In Proceedings of the Tenth Symposium on Rock Mechanics (ed. K. E. Gray), pp , University of Texas, Austin. Prager, W., and P. G. Hodge Theory of Perfectly Plastic Solids, 264 pp., Dover, New York. Reinicke, K. M., and T. D. Ralston Plastic limit analysis with an anisotropic, parabolic yield function. International Journal of Rock Mechanics, Mining Sciences and Geomechanical Abstracts, 14, Schwarz, J., and W. F. Weeks Engineering properties of sea ice, Journal of Glaciology, 19(81), Smith, M. B A parabolic yield condition for anisotropic rocks and soils, PhD thesis, 190 pp., Rice University, Houston, Texas. Vaudrey, K. D Ice Engineering: Study of related properties of floating sea-ice sheets and summary of elastic and viscoelastic analyses, Technical Report R860, 81 pp., Civil Engineering Laboratory, Naval Construction Battalion Center, Port Hueneme, California. Weeks, W. F., and A. Assur Fracture of lake and sea ice, Research Report 269, U.S. Army CRREL, Hanover, N.H. Wu, H. C Failure criterion for plain concrete under short-time load, IIHR Report No. 149, 25 pp., University of Iowa, Ames.