- Email: [email protected]
High pressure zone formation during compressive ice failure
Engineering Fracture Mechanics 68 (2001) 1961±1974
www.elsevier.com/locate/engfracmech
High pressure zone formation during compressive ice failure J.P. Dempsey a,*, A.C. Palmer b, D.S. Sodhi c a
Department of Civil and Environmental Engineering, Clarkson University, 240B Rowley Laboratories, Potsdam, New York 13699-5710, USA b Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK c Ice Engineering Research Branch, US Army Cold Regions Research and Engineering Laboratory, 72 Lyme Road, Hannover, New Hampshire 03755-1290, USA Received 8 April 2000; received in revised form 13 October 2000; accepted 20 October 2000
Abstract An understanding of the mechanics and physics of the formation of the high pressure zones that form during ice± structure interactions is sought. The in¯uences of time, temperature and scale on the formation of these high pressure zones are explored in this paper. Line-like and localized high pressure contact zones are modeled via elastic-brittle hollow cylinder and hollow sphere idealizations, respectively. For both simultaneous and non-simultaneous contact, the critical lengths of stable cracking that may occur prior to ¯aking and ¯exural failure are strongly linked to the current level of speci®c pressure parameters for both line-like and localized high pressure zones. The stability aspects of the inplane cracking, and the link between the maximum possible crack lengths and the relative magnitudes of the local and far-®eld pressures help explain the transitions observed within the realms of ductile, intermittent, and brittle crushing. Ó 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction While ice exerts large forces on ships and oshore structures, the relative importance of the contact area, ice sheet velocity, and the structural width versus ice thickness ratio is not suciently well understood. An important question concerns the role of brittle failure processes in reducing eective pressures at relatively fast ice sheet velocities and for large contact areas and widths. Important related sub-problems include the nucleation of cleavage cracks, spalling and ¯aking, the size and failure of individual crushing zones, and the ductile-to-brittle transition speed and its dependence on temperature. The hypothesis that the eective pressure generated during ice±structure interaction depends primarily on the contact area is not universally accepted: the in¯uence of ice velocity (or indentation speed) and surrounding ice extent is perhaps just as important. Jordaan [9] and Sodhi [20] have presented a state-of-the-art overview of measurements and observations concerning the contact areas, ice pressures, and observed deformation mechanisms that accompany both
*
Corresponding author. Tel.: +1-315-268-6517; fax: +1-315-268-7636. E-mail address: [email protected] (J.P. Dempsey).
0013-7944/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 0 1 ) 0 0 0 3 3 - 9
1962
J.P. Dempsey et al. / Engineering Fracture Mechanics 68 (2001) 1961±1974
the small-scale and the full-scale indentation of ice features. These two papers include a representative survey of the associated literature. Many observations share a common feature: the presence of localized high pressure zones in which the contact force per unit area is much higher than elsewhere. Sometimes these zones are line-like (as ®rst observed more than a decade ago by Joensuu and Riska [8]), and sometimes they are quite localized (see Fig. 6 in Ref. [9], and Figs. 6 and 9 in Ref. [20]). A brief summary of the main features in¯uencing ice±structure interaction is now provided. Ice walls: During the indentation of ice walls (very thick ice features), contact apparently occurs through a layer of crushed ice, which is systematically extruded. Direct contact through high pressure zones shaped in the form of an `X' have been observed, these being associated with spalling. For rather rigid (noncompliant) indenters, ductile failure mechanisms were observed for indentation speeds below 3.16 mm/s, with brittle failure at faster speeds [11]. Ice sheets: During the edge indentation of ¯oating ice sheets, low speed indentation is often associated with cleavage cracking (cracks lying parallel to the ice surface, and often many times the ice sheet thickness in radius). Eventual buckling of the ice sheet is a common occurrence for low speed indentation. For slightly higher indentation speeds, brittle ¯aking occurs, and direct ice±structure contact occurs through a line-like high pressure zone distributed along the width of the indenter. If the rate of indentation is just under the transition speed at which brittle ¯aking occurs readily, a gradually enlarging width to the line-like high pressure zone is observed. This enlargement has been attributed to creep. For the case of constant speed indentation (again under rather non-compliant conditions), this transition speed lies between 0.3 and 3 mm/s. Con®nement: During the indentation of thin ice sheets, a case of low con®nement in the thickness direction, there is direct contact between intact ice and the structure. Whether this holds true for thick to very thick ice sheets is not known at the present time. Contact through a crushed ice layer has been observed during the indentation of ice walls, a case of signi®cant lateral con®nement. The degree of con®nement is not only a function of ice thickness, but also varies with the horizontal extent of ice indenting past the structure, the ice interface friction, and the ice temperature. Simultaneous contact: During the low speed edge indentation of ice sheets, the slow build up of contact pressure leads to full face contact; the resulting vertical expansion induces cleavage cracks, which can grow stably for some distance. The slow build up enhances the occurrence of symmetrical deformations, such that when unstable crack propagation takes place, the ®nal (much longer) cracks still lie parallel to the ice surface. Any vertical movement of the ice sheet immediately causes ¯exural failure and ¯aking. After contact resumes once more, a wedge shaped ice pro®le is encountered. If the ice edge indentation speed is low, the build up of ice pressure now occurs along a line, which increases in vertical width slowly (due to local cracking and crushing of asperities and the continual establishment of a ¯at contact line). The failure of these asperities causes the formation of local high pressure zones which quickly fade (apparently they exist for less than a hundredth of a second). While these high pressure zones are forming and fading on a continual basis, the failures are spatially well correlated because of an almost uniform level of activity across the whole width. In this paper, this situation will be characterized as a line-like high pressure zone. The individual high pressure zones are occurring along the line with no spatial preference across the width; this random occurrence of high pressure zones leads to an almost uniform distribution in the contact pressure and has been described in the ice literature as one of simultaneous contact. The ice failure for these low speeds is described as ductile crushing. Non-simultaneous contact: Once the edge indentation speed surpasses some critical speed, which is called a transition velocity, the contact pressure is much less uniform, and the overall average ice pressure is distinctly lower. While the high pressure zones are still forming and fading rapidly, the spatial correlation is low. Because of the low spatial correlation, the overall average is lower. In the ice literature, this contact scenario has been described as one of non-simultaneous contact. During the rapid build up of pressure along a rather non-uniform wedge-shaped `mountain range' of ice asperities, there is much less likelihood of
J.P. Dempsey et al. / Engineering Fracture Mechanics 68 (2001) 1961±1974
1963
the stable build up of a signi®cant ice force. A state of symmetrical deformations is equally unlikely. The ice failure for these fast speeds is described as brittle crushing. Structural compliance and mode switching: In the above discussion, the term `indentation speed' refers to the relative rate of indentation. It does not connote the actual rate of movement of an ice sheet past a structure or of an indenter into the edge of an ice sheet. During the load up phase, the compliance may `rob' the actual speed of most of the possible indentation. Given the signi®cant in¯uence of the structural or indenter compliance, there is an intermediate actual indentation speed range within which the load up phase is associated with simultaneous contact and a low speed of indentation, followed by an extrusion phase accompanying ice failure. During the extrusion phase, the structure or indenter rebounds and moves through the ice at a much higher indentation speed. The associated pressure distribution is non-uniform and the contact is non-simultaneous. This stable loading up followed by extrusion (and possibly an additional separation phase) is cyclic and leads to saw-tooth force±time plots. This failure sequence is called intermittent crushing. Rigid versus compliant: If the structure or indenter may be characterized as rigid, there is one transition velocity and the ice contact will switch from simultaneous to non-simultaneous (ductile-to-brittle crushing). If the structure or indenter is compliant, there are two transition velocities: one for the switch from ductile to intermittent crushing, and the other for the switch from intermittent to brittle crushing.
2. High pressure zones The distribution of the high pressure zones for a relatively thin ice sheet (less than 0.6 m) seems to be con®ned to the middle third of the ice thickness. The line-like contact is in reality caused by a rapidly ¯uctuating dense distribution of high pressure zones (see Fig. 1). If the ice thickness is large, the distribution of high pressure zones occurs over localized regions. As the ice thickness increases, the peak pressures observed in these high pressure zones increases. During the penetration of an indenter into an ice sheet, cracking on dierent scales is activated. These include the grain-scale grain boundary facet cracks, across column cracks, and transgranular cracks [19]. With increasing pressure, this cracking will progressively localize into partially and then fully developed process zones, followed by the stable growth of a system of macrocracks. The indentation pressure reaches a maximum when the crack growth becomes unstable, this happening because the cracks `feel' the presence of a remote boundary, in the sense that the boundary aects the strain energy release rate. A few of the cracks then propagate unstably towards the boundary. In the context of ice±structure indentation, this
Fig. 1. `Line-like' distribution of high pressure zones.
1964
J.P. Dempsey et al. / Engineering Fracture Mechanics 68 (2001) 1961±1974
remote boundary is most often the upper and lower surfaces of the ice sheet (for the indentation of ¯oating ice sheets). There is no doubt a number of complicated deformation mechanisms active in the contact zone during ice±structure interactions. The formation and spatial characteristics of the high pressure zones seem to be of central importance to the whole subject. The objective of this paper is to elucidate the factors that control both the size (`vertical width' if line-like, and `diameter' if localized) and peak pressures of the high pressure zones that form. For all of the creep-brittle results presented in this paper, the associated viscoelastic-brittle results are immediate if one remembers that the only change required is to replace the short-time modulus E by the associated secant modulus [18]. This observation allows one to consider the eect of moderate variations in penetration rate on the peak pressures observed. In addition, the formation of the crushed layer has not been included, and thus no extrusion of any crushed material has been modeled. Direct contact between the indenter and the ice has been assumed.
3. Line-like high pressure zones The `line-like high pressure zone' which occurs under simultaneous contact conditions across the full width of an indenter in the middle third of the ice sheet is idealized as an elastic-brittle hollow cylinder problem (compare Fig. 1 and Fig. 3). One of the questions to be explored is when does the internal pressure p acting over the vertical width 2a become aware of the scale at hand? In this model, the scale at hand is the outer radius b. In the case of edge indentation, the scale at hand is the shorter of the distance from the high pressure zone to the top and bottom surfaces. Consider now a long hollow elastic cylinder of internal and outer radii a and b, respectively, that is subjected to an internal pressure p and an external pressure q; radial cracks of length q a are assumed to emanate from the internal boundary (Fig. 3). These radial cracks (once formed), will stably grow under increasing pressure, but only as far as some critical radius qc ; the associated critical pressure is denoted here by pc . For p > pc , this radial crack growth becomes unstable, and rapid propagation of these cracks to the
Fig. 2. Localized high pressure zone.
J.P. Dempsey et al. / Engineering Fracture Mechanics 68 (2001) 1961±1974
1965
Fig. 3. The radially cracked cylinder or sphere.
outer surface of the cylinder ensues. A fracture mechanics analysis of this problem reveals the dependencies of the peak internal pressure on the cylinder dimensions, the crack radius, and the external pressure. For the present problem, with r, h, z as cylindrical coordinates, rrr
p;
rrh 0
r a
rrr
q;
rrh 0
r b
1
Within the radially cracked zone a < r < q, the pressure is assumed to be carried by radially-tapered columns. For a sucient number of cracks (greater than ®ve or so, [3]), disregarding any interaction between the columns, one can assume that a uniaxial radial state of stress exists. That is, within each tapered column rhh 0, rrh 0, while equilibrium in the radial direction requires that drrr =dr rrr =r 0 within each truncated wedge. Assuming no strain in the z-direction, rzz mrrr , and thus rrr E0 du=dr, in which E0 E= 1 m2 , E and m being the elastic modulus and Poisson's ratio, respectively, while u r is the displacement in the radial direction. Within the cracked region, therefore, the radial stress and displacement behave as rrr r
a p ; r
u r
p q a ln u q E0 r
a < r < q
2
in which u r signi®es the radial displacement within the intact or uncracked region. The uncracked region is a hollow cylinder of inner and outer radii q and b, respectively, that is subjected to an internal pressure p pa=q, as per the ®rst term in Eq. (2), and an external pressure q. The solution to the latter problem is [22] r b2 =r2 1 2m 1 2m q2 =r2 p u r q < r < b 3 q 2l b2 =q2 1 1 q2 =b2 in which l E=2 1 m. The displacement of the inner and outer boundaries of the cylinder now follows directly from the second term of Eq. (2) and Eq. (3) as
1966
J.P. Dempsey et al. / Engineering Fracture Mechanics 68 (2001) 1961±1974
pa q b2 Cbq 1 2m u a 0 ln 2 2 E a b 1 m q2 2 pa bq Cb C u b 0 2 2 a < q < b E 1 m b q2
4
in which C qb=pa. The energy release rate G for a system of n radial cracks of length q a follows from the relation G dP=dA, in which P is the potential energy, and A q an is the cracked area (per unit thickness in the long direction of the cylinder). Noting that the applied forces per unit thickness on the inner and outer boundaries are P 2ppa and Q 2pqb, respectively, the potential energy for prescribed pressures (dead loading) is given by P U Pu a Qu b, in which U is the strain energy. By Clapeyron's theorem, U Pu a=2 Qu b=2. Hence, 1 ou nG 2ppa a 2 oq
1 ou 2pqb b 2 oq
5
Alternatively, the energy release rate may be expressed in the form (see Ref. [3]) pq G 0 r2hh q En
6
The circumferential stress at r q is given, with n q=b, by rhh q
pa b2 q2 q b 2 q2
2qb2 pa 1 2Cn n2 b n 1 n2 b 2 q2
7
By Eqs. (4) and (5), or Eqs. (6) and (7), and with n q=b, G
p2 a2 p 1 2Cn n2 bE0 n n 1 n2 2
2
a=b < n < 1
8
The stability characteristics of crack growth under line-like contact is quickly revealed by evaluating the following stability index [2,6] b dG n4 6Cn3 8n2 2Cn 1 G dq n 1 n2 1 2Cn n2
a=b < n < 1
9
Were the far-®eld pressure to be zerop(with 0), the above expression predicts that stable crack growth C 1=2 would only occur for a < q < qc 17 4 b 0:351b. Under increasing far-®eld pressure q, C grows as qb=pa, and the feasible stable crack growth lengths are given in Fig. 4 for 0 < C 6 1. In Fig. 4, dG n4 8n2c 1 0)C c ; dq 2nc 1 3n2c 1 n2c G0)C ; 2nc
C 6 1 10
C P 1
in which nc qc =b. Note that both G 0 and rhh q 0 on the upper curve. The lower curve for C < 1 separates the stable region a=b < n < nc from the unstable cracking region; if the critical C-values are less than unity, the radial cracks will propagate unstably to the boundary once the length qc is reached. The upper curve for C > 1 portrays an upper bound to the critical crack lengths; at the critical C-values shown the cracks would be pinched shut. In reality, the maximum crack lengths would be slightly less than the lengths shown, never equal them. The associated behavior of the stability index is plotted in Fig. 5. Once C
J.P. Dempsey et al. / Engineering Fracture Mechanics 68 (2001) 1961±1974
1967
Fig. 4. In¯uence of the value of C qb=pa on the critical crack length.
Fig. 5. Stability plot for line-like contact.
increases past C 1, a compressive rhh pinches the crack shut at the critical crack radius. The plot in Fig. 5 for C 1:1 reveals the associated stability behavior.
1968
J.P. Dempsey et al. / Engineering Fracture Mechanics 68 (2001) 1961±1974
Fig. 4 reveals that there is a narrow, intermediate far-®eld pressure range which will allow the stable development of cracks over distinctly longer lengths. If one were to associate the current C value with the indentation speed, Fig. 4 portrays the fact that for a critical in-plane far-®eld pressure qc that gives C 1, the longest stable cracks form without ¯aking or spalling. 4. Localized high pressure zones Consider next a hollow sphere of internal and outer radii a and b, that is subjected to internal and external pressures p and q, respectively (the situation portrayed in Fig. 2 motivating the idealization in Fig. 3). Again, radial cracks of length q a are assumed to emanate from the internal boundary. With r, h and u as the spherical coordinates, rrr
p;
rrh rru 0 r a
rrr
q;
rrh rru 0 r b
11
Within the radially cracked zone a < r < q, the pressure is assumed to be carried by columns tapered both radially and circumferentially. For a sucient number of cracks (three to ®ve, perhaps), disregarding any interaction between the columns, one can assume that a purely radial state of stress exists. That is, within each bi-tapered column rhh ruu 0. In the absence of any shear stresses and body forces, the displacement equations of equilibrium reduce to 1=r2 d r2 u constant, while force equilibrium requires that 1=r2 d r2 rrr =dr 0 within each truncated column. Following from the fact that rrr Edu=dr, it is immediate that a2 p 2 1 1 rrr r p 2 ; u r a u q a < r < q 12 E r q r The uncracked portion is now a hollow sphere of inner and outer radii q and b, respectively, that is subjected to an internal pressure p pa2 =q2 , as per the ®rst term of Eq. (12), and an external pressure q. [22] provides the solution r b3 =2r3 1 2m= 1 m 1 2m= 1 m q3 =2r3 p u r ; q < r < b 13 q 2l b3 =q3 1 1 q3 =b3 The displacements of the inner and outer boundaries of the sphere now follow directly from the second term of Eq. (12) and Eq. (13) as p 2 1 2 1 m b3 Sbq2 3 1 m u a a E a q 2q b3 q3 14 2 2 3 pa b q Sb 3 1 m 3 u b S 1 m a < q < b 2bE b q3 in which S qb2 =pa2 . The energy release rate follows in a similar manner to the earlier calculation. In this case P 4ppa2 and Q 4pqb2 while the cracked area is given by A p q2 a2 n. Hence, 1 ou 2pqnG 4ppa2 a 2 oq
1 ou 4pqb2 b 2 oq
15
An alternative expression for the energy release rate is given by G
2 1 m 2 qrhh q En
16
J.P. Dempsey et al. / Engineering Fracture Mechanics 68 (2001) 1961±1974
1969
Given that rhh q
pa2 b3 2q3 2q2 b3 q3
3 qb3 pa2 1 3Sn2 2n3 2 b 3 q3 b2 2n2 1 n3
17
it follows from Eqs. (14) and (15), or from Eqs. (16) and (17), that G
p2 a4 1 m 1 3Sn2 2n3 2 2 Eb3 2n n3 1 n3
a=b < n < 1
18
The associated stability index is given by b dG n6 5Sn5 5n3 Sn2 1 3 G dq n 1 n3 1 3Sn2 2n3
a=b < n < 1
19
Were the far-®eld pressure to be zero, as assumed inpRef. [19], the above expression predicts that stable crack growth would only occur for a < q < qc 33 5=41=3 b 0:571b. Under increasing far-®eld pressure q, S grows as qb2 =pa2 , and the feasible stable crack growth lengths are given in Fig. 6 for 0 < S 6 1. In Fig. 6, dG 2n6 5n3c 1 0 ) S 2c ; dq nc 1 5n3c G0)S
1 2n3c ; 3n2c
S 6 1 20
S P 1
in which nc qc =b. Note that both G 0 and rhh q 0 on the upper curve. The lower curve for S < 1 separates the stable region a=b < n < nc from the unstable cracking region; if the critical S-values are less than unity, the radial cracks will propagate unstably to the boundary once the length qc is reached. The
Fig. 6. In¯uence of the value of S qb2 =pa2 on the critical crack length.
1970
J.P. Dempsey et al. / Engineering Fracture Mechanics 68 (2001) 1961±1974
Fig. 7. Stability plot for localized contact.
upper curve for S > 1 portrays an upper bound to the critical crack lengths; at the critical S-values shown the cracks would be pinched shut. In reality, they could approach the lengths shown, but never equal them. The associated behavior of the stability index is plotted in Fig. 7. Once S increases past S 1, a compressive rhh pinches the crack shut at the critical crack radius. The plot in Fig. 7 for S 1:1 reveals the associated stability behavior. 5. Scaling considerations The energy release rates associated with the radial cracking induced by either line-like or localized high pressure zones have been found to be as follows: G
p 2 a2 f C; n bE0
line-like;
G
p 2 a4 g S; n Eb3
localized
21
The dimensionless functions f C; n and g S; n are self-evident from Eqs. (8) and (18), respectively. The maximum pressure that may be observed under line-like and localized contact conditions therefore vary as q q b1=2 b3=2 line-like; p EGscale localized 22 E0 Gscale X pmax W max c c a a2 in which E is the short-time modulus and Gscale is the critical-energy-release-rate for the scale at hand. The c free-surface correction functions W and X have been introduced because one must modify the hollow cylinder and sphere idealizations and cut the sphere and cylinder in half (along the length in the latter case). Note that one could re-write Eq. (16) in the form pmax W
h1=2 KQ a
line-like;
pmax X
h3=2 KQ a2
23
J.P. Dempsey et al. / Engineering Fracture Mechanics 68 (2001) 1961±1974
1971 1=2
in which b has been loosely equated with the ice thickness h, and the equality EGscale KQ has been c assumed, KQ being the scale dependent fracture toughness. The force per unit length (along the width) associated with the line-like high pressure zone follows from the maximum pressure in the ®rst term of Eq. (22) multiplied by the internal diameter (2a) of the hollow cylinder in Fig. 3. The associated nominal pressure pnom follows by dividing the latter force by the ice thickness h. Thus q q F 2Wh1=2 E0 Gscale line-like; pnom 2Wh 1=2 E0 Gscale line-like 24 c c The force exerted via a localized high pressure zone follows from the maximum pressure in the second term of Eq. (22) multiplied by the eective loaded area (pa2 ) in a hollow sphere. Suppose that there are N active localized high pressure zones over a width equal to the ice thickness: the nominal pressure follows directly as NF =h2 . Thus, in this case, q q 1=2 F pXh3=2 EGscale localized; p EGscale localized 25 pXNh nom c c Note that the force per unit thickness in Eq. (24), and the force in Eq. (25), are both independent of the variable a. At this point, there is the need for the compilation of test data and measurements. Evidence suggests that Gscale should increase with ice thickness and the degree of stable crack growth [16]. This topic awaits c thorough experimental investigation. Suppose that Gscale were to vary as hb 0 < b < 1=2; in this case, the c 1=2 b nominal pressure would vary as h . In this regard, it is remarkable to note that the design ice force for large aspect ratios (aspect ratio being equal to the ratio of the structure width to the ice thickness) speci®es that pnom h 0:174 [13].
6. Cleavage fracture The above expressions for the peak pressures that may occur at high pressure zones assumes that these peaks are allowed to occur, that no other failure mechanism intervenes to limit the pressures observed. As discussed in the introduction, two common failure mechanisms are observed, namely, cleavage fracture followed by spalling, also called indentation spalling. Kry [10] discussed the formation of these cleavage cracks and commented that they seem to scale with the ice thickness, while being insensitive to the diameter of the indenter. Evans et al. [4] formulated a model for the formation of a cleavage crack which has as its basis an elastic±plastic through-the-thickness cavity expansion model. The model by Evans et al. [4] would seem to best suit ice sheet interactions with narrow structures, and then only if the ice is very warm (as in late season ice-lighthouse impacts, for instance) in which case the ice has more give and simultaneous contact is more likely. The horizontal cleavage of ice sheets subject to through-the-thickness indentation may be viewed as resulting directly from lateral self-con®nement [19]. Schulson has noted that once the con®ning pressure parallel to the width exceeds approximately 20% of the ice force pressure, cleavage is activated. The actual percentage varies with ice type and contact conditions. The nucleation cleavage crack provides a mechanism that can limit the peak pressures in the high pressure zones, in addition to the limitations imposed by scale (the ice thickness and stable crack growth lengths, qc a) discussed in the previous section. In a way, the situation is similar to the crack nucleation/ crack propagation transitional behavior that can happen in uniaxial tensile and compressive fracture [7,19]. If the mechanism to nucleate cracks is activated before crack propagation to the upper and lower ice surfaces occurs, cleavage cracking can take place. The conditions for this to happen needs to be explored further.
1972
J.P. Dempsey et al. / Engineering Fracture Mechanics 68 (2001) 1961±1974
Consider now an isolated high pressure zone that is now forming. Just how this might occur will be left for discussion in the next section. As the radius increases, and the peak pressure grows, suppose that either the stably forming cracks `realize' the scale, and ¯akes form from the pre-cracked system, or the situation is such that a cleavage crack can form. The latter can be visualized to happen via scenarios discussed regarding the indentation fracture of ceramics [11]. Vickers microhardness indentation at loads high enough to produce well-developed half-penny surface cracks is used to assess the fracture toughness of ceramics. For ceramics it has been found that 1=2 E P ceramics KIc 26 H c3=2 in which H is the indentation hardness and c is the radius of the half-penny crack. Note that this expression applies to the nucleation of half-penny surface cracks in the surface of an elastic half-space. In ice sheet indentation, the nucleation of such cracks are occurring at the edge of a ®nite thickness plate. It is also well to remember that the E=H ratio is much higher (by a factor of 100 or more) for ceramics than for ice. Indentation fracture mechanics has been used to estimate both the fracture toughness of ice [5] and the size 1=2 of the damage zone beneath an indenter on ice [17]. On equating P with a constant times h3=2 EGscale , or c 3=2 equivalently, h KQ , both of which follow from the ®rst term in Eq. (25), and assuming both that this expression holds for ice, and that c is assumed to equate with the cleavage crack radius, it is immediate that the latter scales directly with ice thickness, as in c
E H
1=3 h
27
The indentation hardness of ice [1] is extremely dependent on the load duration and ice temperature. For instance, at load durations of the order of 1, 10, and 100 s and an ice temperature of 2:5°C, the associated hardnesses of ®ne-grained polycrystalline ice are given by 30, 15, and 7 MPa. The load duration depends on the ice velocity, but not in a simple manner, as it depends on the contact conditions at the interface (simultaneous or non-simultaneous contact, correlated or non-correlated spatial density of high pressure zones, ice thickness, degree of stable cracking). At the ice temperature just quoted, the pressure melting hardnesses are just a boundary layer away. If one were to assume associated secant moduli of 8, 6 and 4 GPa, then the cleavage fractures would size as c 6h, 7h and 8h, respectively. The link with pressure melting just mentioned would seem to be very important, as with even a slight amount of pressure melting a distinct shift in the frictional behavior at the contact interface would be expected, and an associated shift in the triaxiality would occur. If one considers that under typical indentation conditions there is always the background global pressure acting, ¯uctuating over a longer time scale than the individual high pressure zones, any sudden shift in the through-the-thickness tensile strain will aid in the formation of the cleavage fracture. Even without pressure melting, the link between friction and ice velocity merits more attention. The in¯uence of indentation rate (ice velocity) is inextricably linked to the fracture process. The increased crack opening displacements associated with viscoelasticity results in earlier crack growth compared to the elastic case, and stable growth of the physical crack occurs, which is not observed for the elastic case. One can therefore expect the stable development of longer cracks for longer load durations [15]. Major spalls in ®eld indentation tests are associated with low loading rates; ship captains during ramming `lean' on the ice, no doubt to generate a few long fractures that will aid the ramming process [9]. In the case of ice sheet indentation, the critical load duration regime, however, are linked to the ice temperature, ice thickness and con®nement. Attempts to de®ne an eective strain rate [14] for ice indentation problems are essentially trying to link strain rate with fracture processes.
J.P. Dempsey et al. / Engineering Fracture Mechanics 68 (2001) 1961±1974
1973
Before leaving this section, one should note that cleavage fractures occur for `slower' indentation speeds; within this `slower' regime, it is observed that longer cleavage cracks form for slow indentation speeds (but not many), while for faster indentation speeds, shorter cleavage cracks form (and many of them). These cleavage cracks ¯ake o because of ¯exure and out-of-plane movements.
7. Conclusions For both simultaneous and non-simultaneous contact, the critical lengths of stable cracking that may occur prior to ¯aking and ¯exural failure are strongly linked to the current level of the relevant pressure parameters C qb=pa for a line-like high pressure zone, and S qb2 =pa2 for a localized high pressure zone. The peak pressures in these zones scale as b1=2 a
q E0 Gscale c
line-like;
b3=2 a2
q EGscale localized c
28
The stability aspects of the in-plane cracking, and the link between the maximum possible crack lengths and the relative magnitudes of the local and far-®eld pressures help explain the transitions observed within the realms of ductile, intermittent, and brittle crushing.
Acknowledgements This work has been supported in part by the LOLEIF project in the framework of the EU-sponsored Marine Science and Technology (Mast III) Programme under contract number MAS3-CT97-0078 (JPO and ACP), in part by NSF grant CMS-9634846 (JPD) and in part by US Army grant DAAD 19-00-1-0479 (JPD).
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [13] [14] [15] [16] [17]
Barnes P, Tabor D, Walker JCF. The friction and creep of polycrystalline ice. Proc Roy Soc A 1974;324:127±55. DeFranco SJ, Dempsey JP. Crack propagation and fracture resistance in saline ice. J Glaciol 1994;40:451±62. Dempsey JP, Slepyan LI, Shekhtman II. Radial cracking with closure. Int J Fract 1995;73:233±61. Evans AG, Palmer AC, Goodman DJ, Ashby MF, Hutchinson JW, Ponter ARS, Williams GJ. Indentation spalling of edge loaded ice sheets. Proc Seventh IAHR Ice Symp 1984;1:113±21. Goodman DJ, Tabor D. Fracture toughness of ice: a preliminary account of some new experiments. J Glaciol 1978;21:371±80. Gurney C, Hunt J. Quasi-static crack propagation. Proc Roy Soc A 1967;299:508±24. Hallam SD. The role of fracture in limiting ice forces. Proc Eighth IAHR Symp Ice 1986;II:287±319. Joensuu A, Riska K. Contact between ice and a structure. Helsinki University of Technology, Ship Laboratory, Report M-88, 1989, 57p+App [in Finnish]. Jordaan IJ. Mechanics of ice±structure interaction. Engng Fract Mech 2001;68(17±18):1923±60. Kry PR. Scale eects in continuous crushing of ice. Int Symp on Ice, Quebec, Canada, 27±31 July 1981;2:565±580. Lawn B. Fracture of Brittle Solids., 2nd ed. Cambridge University Press, Cambridge, 1998. Masterson DM, Spencer PA. Ice force calculation for large and small aspect ratios. Proc IUTAM Symp: Scaling Laws in Ice Mechanics and Ice Dynamics, Fairbanks, Alaska, 13±16 June, 2000. Michel B, Toussaint N. Mechanisms and theory of indentation of ice plates. J Glaciol 1977;19:285±300. Mulmule SV, Dempsey JP. A viscoelastic ®ctitious crack model for the fracture of sea ice. Mech Time-Dependent Mater 1998;1:331±56. Mulmule SV, Dempsey JP. LEFM size requirements for the fracture testing of sea ice. Int J Fract 2000;102:85±98. Parsons BL. An estimate of the size of the damage zone beneath an indenter on ice. Cold Regions Sci Technol 1993;21:371±80.
1974
J.P. Dempsey et al. / Engineering Fracture Mechanics 68 (2001) 1961±1974
[18] Schapery RA. A theory of crack initiation and growth in viscoelastic media. Part II. Approximate method of analysis. Int J Fract 1993;11:369±87. [19] Schulson EM. The brittle failure of ice. Engng Fract Mech 2001;68(17±18):1839±87. [20] Sodhi DS. Crushing failure during ice±structure interaction. Engng Fract Mech 2001;68(17±18):1889±921. [22] Westergaard HM. Theory of Elasticity and Plasticity. Harvard University Press, Cambridge, 1952.
www.elsevier.com/locate/engfracmech
High pressure zone formation during compressive ice failure J.P. Dempsey a,*, A.C. Palmer b, D.S. Sodhi c a
Department of Civil and Environmental Engineering, Clarkson University, 240B Rowley Laboratories, Potsdam, New York 13699-5710, USA b Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK c Ice Engineering Research Branch, US Army Cold Regions Research and Engineering Laboratory, 72 Lyme Road, Hannover, New Hampshire 03755-1290, USA Received 8 April 2000; received in revised form 13 October 2000; accepted 20 October 2000
Abstract An understanding of the mechanics and physics of the formation of the high pressure zones that form during ice± structure interactions is sought. The in¯uences of time, temperature and scale on the formation of these high pressure zones are explored in this paper. Line-like and localized high pressure contact zones are modeled via elastic-brittle hollow cylinder and hollow sphere idealizations, respectively. For both simultaneous and non-simultaneous contact, the critical lengths of stable cracking that may occur prior to ¯aking and ¯exural failure are strongly linked to the current level of speci®c pressure parameters for both line-like and localized high pressure zones. The stability aspects of the inplane cracking, and the link between the maximum possible crack lengths and the relative magnitudes of the local and far-®eld pressures help explain the transitions observed within the realms of ductile, intermittent, and brittle crushing. Ó 2001 Elsevier Science Ltd. All rights reserved.
1. Introduction While ice exerts large forces on ships and oshore structures, the relative importance of the contact area, ice sheet velocity, and the structural width versus ice thickness ratio is not suciently well understood. An important question concerns the role of brittle failure processes in reducing eective pressures at relatively fast ice sheet velocities and for large contact areas and widths. Important related sub-problems include the nucleation of cleavage cracks, spalling and ¯aking, the size and failure of individual crushing zones, and the ductile-to-brittle transition speed and its dependence on temperature. The hypothesis that the eective pressure generated during ice±structure interaction depends primarily on the contact area is not universally accepted: the in¯uence of ice velocity (or indentation speed) and surrounding ice extent is perhaps just as important. Jordaan [9] and Sodhi [20] have presented a state-of-the-art overview of measurements and observations concerning the contact areas, ice pressures, and observed deformation mechanisms that accompany both
*
Corresponding author. Tel.: +1-315-268-6517; fax: +1-315-268-7636. E-mail address: [email protected] (J.P. Dempsey).
0013-7944/01/$ - see front matter Ó 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 0 1 ) 0 0 0 3 3 - 9
1962
J.P. Dempsey et al. / Engineering Fracture Mechanics 68 (2001) 1961±1974
the small-scale and the full-scale indentation of ice features. These two papers include a representative survey of the associated literature. Many observations share a common feature: the presence of localized high pressure zones in which the contact force per unit area is much higher than elsewhere. Sometimes these zones are line-like (as ®rst observed more than a decade ago by Joensuu and Riska [8]), and sometimes they are quite localized (see Fig. 6 in Ref. [9], and Figs. 6 and 9 in Ref. [20]). A brief summary of the main features in¯uencing ice±structure interaction is now provided. Ice walls: During the indentation of ice walls (very thick ice features), contact apparently occurs through a layer of crushed ice, which is systematically extruded. Direct contact through high pressure zones shaped in the form of an `X' have been observed, these being associated with spalling. For rather rigid (noncompliant) indenters, ductile failure mechanisms were observed for indentation speeds below 3.16 mm/s, with brittle failure at faster speeds [11]. Ice sheets: During the edge indentation of ¯oating ice sheets, low speed indentation is often associated with cleavage cracking (cracks lying parallel to the ice surface, and often many times the ice sheet thickness in radius). Eventual buckling of the ice sheet is a common occurrence for low speed indentation. For slightly higher indentation speeds, brittle ¯aking occurs, and direct ice±structure contact occurs through a line-like high pressure zone distributed along the width of the indenter. If the rate of indentation is just under the transition speed at which brittle ¯aking occurs readily, a gradually enlarging width to the line-like high pressure zone is observed. This enlargement has been attributed to creep. For the case of constant speed indentation (again under rather non-compliant conditions), this transition speed lies between 0.3 and 3 mm/s. Con®nement: During the indentation of thin ice sheets, a case of low con®nement in the thickness direction, there is direct contact between intact ice and the structure. Whether this holds true for thick to very thick ice sheets is not known at the present time. Contact through a crushed ice layer has been observed during the indentation of ice walls, a case of signi®cant lateral con®nement. The degree of con®nement is not only a function of ice thickness, but also varies with the horizontal extent of ice indenting past the structure, the ice interface friction, and the ice temperature. Simultaneous contact: During the low speed edge indentation of ice sheets, the slow build up of contact pressure leads to full face contact; the resulting vertical expansion induces cleavage cracks, which can grow stably for some distance. The slow build up enhances the occurrence of symmetrical deformations, such that when unstable crack propagation takes place, the ®nal (much longer) cracks still lie parallel to the ice surface. Any vertical movement of the ice sheet immediately causes ¯exural failure and ¯aking. After contact resumes once more, a wedge shaped ice pro®le is encountered. If the ice edge indentation speed is low, the build up of ice pressure now occurs along a line, which increases in vertical width slowly (due to local cracking and crushing of asperities and the continual establishment of a ¯at contact line). The failure of these asperities causes the formation of local high pressure zones which quickly fade (apparently they exist for less than a hundredth of a second). While these high pressure zones are forming and fading on a continual basis, the failures are spatially well correlated because of an almost uniform level of activity across the whole width. In this paper, this situation will be characterized as a line-like high pressure zone. The individual high pressure zones are occurring along the line with no spatial preference across the width; this random occurrence of high pressure zones leads to an almost uniform distribution in the contact pressure and has been described in the ice literature as one of simultaneous contact. The ice failure for these low speeds is described as ductile crushing. Non-simultaneous contact: Once the edge indentation speed surpasses some critical speed, which is called a transition velocity, the contact pressure is much less uniform, and the overall average ice pressure is distinctly lower. While the high pressure zones are still forming and fading rapidly, the spatial correlation is low. Because of the low spatial correlation, the overall average is lower. In the ice literature, this contact scenario has been described as one of non-simultaneous contact. During the rapid build up of pressure along a rather non-uniform wedge-shaped `mountain range' of ice asperities, there is much less likelihood of
J.P. Dempsey et al. / Engineering Fracture Mechanics 68 (2001) 1961±1974
1963
the stable build up of a signi®cant ice force. A state of symmetrical deformations is equally unlikely. The ice failure for these fast speeds is described as brittle crushing. Structural compliance and mode switching: In the above discussion, the term `indentation speed' refers to the relative rate of indentation. It does not connote the actual rate of movement of an ice sheet past a structure or of an indenter into the edge of an ice sheet. During the load up phase, the compliance may `rob' the actual speed of most of the possible indentation. Given the signi®cant in¯uence of the structural or indenter compliance, there is an intermediate actual indentation speed range within which the load up phase is associated with simultaneous contact and a low speed of indentation, followed by an extrusion phase accompanying ice failure. During the extrusion phase, the structure or indenter rebounds and moves through the ice at a much higher indentation speed. The associated pressure distribution is non-uniform and the contact is non-simultaneous. This stable loading up followed by extrusion (and possibly an additional separation phase) is cyclic and leads to saw-tooth force±time plots. This failure sequence is called intermittent crushing. Rigid versus compliant: If the structure or indenter may be characterized as rigid, there is one transition velocity and the ice contact will switch from simultaneous to non-simultaneous (ductile-to-brittle crushing). If the structure or indenter is compliant, there are two transition velocities: one for the switch from ductile to intermittent crushing, and the other for the switch from intermittent to brittle crushing.
2. High pressure zones The distribution of the high pressure zones for a relatively thin ice sheet (less than 0.6 m) seems to be con®ned to the middle third of the ice thickness. The line-like contact is in reality caused by a rapidly ¯uctuating dense distribution of high pressure zones (see Fig. 1). If the ice thickness is large, the distribution of high pressure zones occurs over localized regions. As the ice thickness increases, the peak pressures observed in these high pressure zones increases. During the penetration of an indenter into an ice sheet, cracking on dierent scales is activated. These include the grain-scale grain boundary facet cracks, across column cracks, and transgranular cracks [19]. With increasing pressure, this cracking will progressively localize into partially and then fully developed process zones, followed by the stable growth of a system of macrocracks. The indentation pressure reaches a maximum when the crack growth becomes unstable, this happening because the cracks `feel' the presence of a remote boundary, in the sense that the boundary aects the strain energy release rate. A few of the cracks then propagate unstably towards the boundary. In the context of ice±structure indentation, this
Fig. 1. `Line-like' distribution of high pressure zones.
1964
J.P. Dempsey et al. / Engineering Fracture Mechanics 68 (2001) 1961±1974
remote boundary is most often the upper and lower surfaces of the ice sheet (for the indentation of ¯oating ice sheets). There is no doubt a number of complicated deformation mechanisms active in the contact zone during ice±structure interactions. The formation and spatial characteristics of the high pressure zones seem to be of central importance to the whole subject. The objective of this paper is to elucidate the factors that control both the size (`vertical width' if line-like, and `diameter' if localized) and peak pressures of the high pressure zones that form. For all of the creep-brittle results presented in this paper, the associated viscoelastic-brittle results are immediate if one remembers that the only change required is to replace the short-time modulus E by the associated secant modulus [18]. This observation allows one to consider the eect of moderate variations in penetration rate on the peak pressures observed. In addition, the formation of the crushed layer has not been included, and thus no extrusion of any crushed material has been modeled. Direct contact between the indenter and the ice has been assumed.
3. Line-like high pressure zones The `line-like high pressure zone' which occurs under simultaneous contact conditions across the full width of an indenter in the middle third of the ice sheet is idealized as an elastic-brittle hollow cylinder problem (compare Fig. 1 and Fig. 3). One of the questions to be explored is when does the internal pressure p acting over the vertical width 2a become aware of the scale at hand? In this model, the scale at hand is the outer radius b. In the case of edge indentation, the scale at hand is the shorter of the distance from the high pressure zone to the top and bottom surfaces. Consider now a long hollow elastic cylinder of internal and outer radii a and b, respectively, that is subjected to an internal pressure p and an external pressure q; radial cracks of length q a are assumed to emanate from the internal boundary (Fig. 3). These radial cracks (once formed), will stably grow under increasing pressure, but only as far as some critical radius qc ; the associated critical pressure is denoted here by pc . For p > pc , this radial crack growth becomes unstable, and rapid propagation of these cracks to the
Fig. 2. Localized high pressure zone.
J.P. Dempsey et al. / Engineering Fracture Mechanics 68 (2001) 1961±1974
1965
Fig. 3. The radially cracked cylinder or sphere.
outer surface of the cylinder ensues. A fracture mechanics analysis of this problem reveals the dependencies of the peak internal pressure on the cylinder dimensions, the crack radius, and the external pressure. For the present problem, with r, h, z as cylindrical coordinates, rrr
p;
rrh 0
r a
rrr
q;
rrh 0
r b
1
Within the radially cracked zone a < r < q, the pressure is assumed to be carried by radially-tapered columns. For a sucient number of cracks (greater than ®ve or so, [3]), disregarding any interaction between the columns, one can assume that a uniaxial radial state of stress exists. That is, within each tapered column rhh 0, rrh 0, while equilibrium in the radial direction requires that drrr =dr rrr =r 0 within each truncated wedge. Assuming no strain in the z-direction, rzz mrrr , and thus rrr E0 du=dr, in which E0 E= 1 m2 , E and m being the elastic modulus and Poisson's ratio, respectively, while u r is the displacement in the radial direction. Within the cracked region, therefore, the radial stress and displacement behave as rrr r
a p ; r
u r
p q a ln u q E0 r
a < r < q
2
in which u r signi®es the radial displacement within the intact or uncracked region. The uncracked region is a hollow cylinder of inner and outer radii q and b, respectively, that is subjected to an internal pressure p pa=q, as per the ®rst term in Eq. (2), and an external pressure q. The solution to the latter problem is [22] r b2 =r2 1 2m 1 2m q2 =r2 p u r q < r < b 3 q 2l b2 =q2 1 1 q2 =b2 in which l E=2 1 m. The displacement of the inner and outer boundaries of the cylinder now follows directly from the second term of Eq. (2) and Eq. (3) as
1966
J.P. Dempsey et al. / Engineering Fracture Mechanics 68 (2001) 1961±1974
pa q b2 Cbq 1 2m u a 0 ln 2 2 E a b 1 m q2 2 pa bq Cb C u b 0 2 2 a < q < b E 1 m b q2
4
in which C qb=pa. The energy release rate G for a system of n radial cracks of length q a follows from the relation G dP=dA, in which P is the potential energy, and A q an is the cracked area (per unit thickness in the long direction of the cylinder). Noting that the applied forces per unit thickness on the inner and outer boundaries are P 2ppa and Q 2pqb, respectively, the potential energy for prescribed pressures (dead loading) is given by P U Pu a Qu b, in which U is the strain energy. By Clapeyron's theorem, U Pu a=2 Qu b=2. Hence, 1 ou nG 2ppa a 2 oq
1 ou 2pqb b 2 oq
5
Alternatively, the energy release rate may be expressed in the form (see Ref. [3]) pq G 0 r2hh q En
6
The circumferential stress at r q is given, with n q=b, by rhh q
pa b2 q2 q b 2 q2
2qb2 pa 1 2Cn n2 b n 1 n2 b 2 q2
7
By Eqs. (4) and (5), or Eqs. (6) and (7), and with n q=b, G
p2 a2 p 1 2Cn n2 bE0 n n 1 n2 2
2
a=b < n < 1
8
The stability characteristics of crack growth under line-like contact is quickly revealed by evaluating the following stability index [2,6] b dG n4 6Cn3 8n2 2Cn 1 G dq n 1 n2 1 2Cn n2
a=b < n < 1
9
Were the far-®eld pressure to be zerop(with 0), the above expression predicts that stable crack growth C 1=2 would only occur for a < q < qc 17 4 b 0:351b. Under increasing far-®eld pressure q, C grows as qb=pa, and the feasible stable crack growth lengths are given in Fig. 4 for 0 < C 6 1. In Fig. 4, dG n4 8n2c 1 0)C c ; dq 2nc 1 3n2c 1 n2c G0)C ; 2nc
C 6 1 10
C P 1
in which nc qc =b. Note that both G 0 and rhh q 0 on the upper curve. The lower curve for C < 1 separates the stable region a=b < n < nc from the unstable cracking region; if the critical C-values are less than unity, the radial cracks will propagate unstably to the boundary once the length qc is reached. The upper curve for C > 1 portrays an upper bound to the critical crack lengths; at the critical C-values shown the cracks would be pinched shut. In reality, the maximum crack lengths would be slightly less than the lengths shown, never equal them. The associated behavior of the stability index is plotted in Fig. 5. Once C
J.P. Dempsey et al. / Engineering Fracture Mechanics 68 (2001) 1961±1974
1967
Fig. 4. In¯uence of the value of C qb=pa on the critical crack length.
Fig. 5. Stability plot for line-like contact.
increases past C 1, a compressive rhh pinches the crack shut at the critical crack radius. The plot in Fig. 5 for C 1:1 reveals the associated stability behavior.
1968
J.P. Dempsey et al. / Engineering Fracture Mechanics 68 (2001) 1961±1974
Fig. 4 reveals that there is a narrow, intermediate far-®eld pressure range which will allow the stable development of cracks over distinctly longer lengths. If one were to associate the current C value with the indentation speed, Fig. 4 portrays the fact that for a critical in-plane far-®eld pressure qc that gives C 1, the longest stable cracks form without ¯aking or spalling. 4. Localized high pressure zones Consider next a hollow sphere of internal and outer radii a and b, that is subjected to internal and external pressures p and q, respectively (the situation portrayed in Fig. 2 motivating the idealization in Fig. 3). Again, radial cracks of length q a are assumed to emanate from the internal boundary. With r, h and u as the spherical coordinates, rrr
p;
rrh rru 0 r a
rrr
q;
rrh rru 0 r b
11
Within the radially cracked zone a < r < q, the pressure is assumed to be carried by columns tapered both radially and circumferentially. For a sucient number of cracks (three to ®ve, perhaps), disregarding any interaction between the columns, one can assume that a purely radial state of stress exists. That is, within each bi-tapered column rhh ruu 0. In the absence of any shear stresses and body forces, the displacement equations of equilibrium reduce to 1=r2 d r2 u constant, while force equilibrium requires that 1=r2 d r2 rrr =dr 0 within each truncated column. Following from the fact that rrr Edu=dr, it is immediate that a2 p 2 1 1 rrr r p 2 ; u r a u q a < r < q 12 E r q r The uncracked portion is now a hollow sphere of inner and outer radii q and b, respectively, that is subjected to an internal pressure p pa2 =q2 , as per the ®rst term of Eq. (12), and an external pressure q. [22] provides the solution r b3 =2r3 1 2m= 1 m 1 2m= 1 m q3 =2r3 p u r ; q < r < b 13 q 2l b3 =q3 1 1 q3 =b3 The displacements of the inner and outer boundaries of the sphere now follow directly from the second term of Eq. (12) and Eq. (13) as p 2 1 2 1 m b3 Sbq2 3 1 m u a a E a q 2q b3 q3 14 2 2 3 pa b q Sb 3 1 m 3 u b S 1 m a < q < b 2bE b q3 in which S qb2 =pa2 . The energy release rate follows in a similar manner to the earlier calculation. In this case P 4ppa2 and Q 4pqb2 while the cracked area is given by A p q2 a2 n. Hence, 1 ou 2pqnG 4ppa2 a 2 oq
1 ou 4pqb2 b 2 oq
15
An alternative expression for the energy release rate is given by G
2 1 m 2 qrhh q En
16
J.P. Dempsey et al. / Engineering Fracture Mechanics 68 (2001) 1961±1974
1969
Given that rhh q
pa2 b3 2q3 2q2 b3 q3
3 qb3 pa2 1 3Sn2 2n3 2 b 3 q3 b2 2n2 1 n3
17
it follows from Eqs. (14) and (15), or from Eqs. (16) and (17), that G
p2 a4 1 m 1 3Sn2 2n3 2 2 Eb3 2n n3 1 n3
a=b < n < 1
18
The associated stability index is given by b dG n6 5Sn5 5n3 Sn2 1 3 G dq n 1 n3 1 3Sn2 2n3
a=b < n < 1
19
Were the far-®eld pressure to be zero, as assumed inpRef. [19], the above expression predicts that stable crack growth would only occur for a < q < qc 33 5=41=3 b 0:571b. Under increasing far-®eld pressure q, S grows as qb2 =pa2 , and the feasible stable crack growth lengths are given in Fig. 6 for 0 < S 6 1. In Fig. 6, dG 2n6 5n3c 1 0 ) S 2c ; dq nc 1 5n3c G0)S
1 2n3c ; 3n2c
S 6 1 20
S P 1
in which nc qc =b. Note that both G 0 and rhh q 0 on the upper curve. The lower curve for S < 1 separates the stable region a=b < n < nc from the unstable cracking region; if the critical S-values are less than unity, the radial cracks will propagate unstably to the boundary once the length qc is reached. The
Fig. 6. In¯uence of the value of S qb2 =pa2 on the critical crack length.
1970
J.P. Dempsey et al. / Engineering Fracture Mechanics 68 (2001) 1961±1974
Fig. 7. Stability plot for localized contact.
upper curve for S > 1 portrays an upper bound to the critical crack lengths; at the critical S-values shown the cracks would be pinched shut. In reality, they could approach the lengths shown, but never equal them. The associated behavior of the stability index is plotted in Fig. 7. Once S increases past S 1, a compressive rhh pinches the crack shut at the critical crack radius. The plot in Fig. 7 for S 1:1 reveals the associated stability behavior. 5. Scaling considerations The energy release rates associated with the radial cracking induced by either line-like or localized high pressure zones have been found to be as follows: G
p 2 a2 f C; n bE0
line-like;
G
p 2 a4 g S; n Eb3
localized
21
The dimensionless functions f C; n and g S; n are self-evident from Eqs. (8) and (18), respectively. The maximum pressure that may be observed under line-like and localized contact conditions therefore vary as q q b1=2 b3=2 line-like; p EGscale localized 22 E0 Gscale X pmax W max c c a a2 in which E is the short-time modulus and Gscale is the critical-energy-release-rate for the scale at hand. The c free-surface correction functions W and X have been introduced because one must modify the hollow cylinder and sphere idealizations and cut the sphere and cylinder in half (along the length in the latter case). Note that one could re-write Eq. (16) in the form pmax W
h1=2 KQ a
line-like;
pmax X
h3=2 KQ a2
23
J.P. Dempsey et al. / Engineering Fracture Mechanics 68 (2001) 1961±1974
1971 1=2
in which b has been loosely equated with the ice thickness h, and the equality EGscale KQ has been c assumed, KQ being the scale dependent fracture toughness. The force per unit length (along the width) associated with the line-like high pressure zone follows from the maximum pressure in the ®rst term of Eq. (22) multiplied by the internal diameter (2a) of the hollow cylinder in Fig. 3. The associated nominal pressure pnom follows by dividing the latter force by the ice thickness h. Thus q q F 2Wh1=2 E0 Gscale line-like; pnom 2Wh 1=2 E0 Gscale line-like 24 c c The force exerted via a localized high pressure zone follows from the maximum pressure in the second term of Eq. (22) multiplied by the eective loaded area (pa2 ) in a hollow sphere. Suppose that there are N active localized high pressure zones over a width equal to the ice thickness: the nominal pressure follows directly as NF =h2 . Thus, in this case, q q 1=2 F pXh3=2 EGscale localized; p EGscale localized 25 pXNh nom c c Note that the force per unit thickness in Eq. (24), and the force in Eq. (25), are both independent of the variable a. At this point, there is the need for the compilation of test data and measurements. Evidence suggests that Gscale should increase with ice thickness and the degree of stable crack growth [16]. This topic awaits c thorough experimental investigation. Suppose that Gscale were to vary as hb 0 < b < 1=2; in this case, the c 1=2 b nominal pressure would vary as h . In this regard, it is remarkable to note that the design ice force for large aspect ratios (aspect ratio being equal to the ratio of the structure width to the ice thickness) speci®es that pnom h 0:174 [13].
6. Cleavage fracture The above expressions for the peak pressures that may occur at high pressure zones assumes that these peaks are allowed to occur, that no other failure mechanism intervenes to limit the pressures observed. As discussed in the introduction, two common failure mechanisms are observed, namely, cleavage fracture followed by spalling, also called indentation spalling. Kry [10] discussed the formation of these cleavage cracks and commented that they seem to scale with the ice thickness, while being insensitive to the diameter of the indenter. Evans et al. [4] formulated a model for the formation of a cleavage crack which has as its basis an elastic±plastic through-the-thickness cavity expansion model. The model by Evans et al. [4] would seem to best suit ice sheet interactions with narrow structures, and then only if the ice is very warm (as in late season ice-lighthouse impacts, for instance) in which case the ice has more give and simultaneous contact is more likely. The horizontal cleavage of ice sheets subject to through-the-thickness indentation may be viewed as resulting directly from lateral self-con®nement [19]. Schulson has noted that once the con®ning pressure parallel to the width exceeds approximately 20% of the ice force pressure, cleavage is activated. The actual percentage varies with ice type and contact conditions. The nucleation cleavage crack provides a mechanism that can limit the peak pressures in the high pressure zones, in addition to the limitations imposed by scale (the ice thickness and stable crack growth lengths, qc a) discussed in the previous section. In a way, the situation is similar to the crack nucleation/ crack propagation transitional behavior that can happen in uniaxial tensile and compressive fracture [7,19]. If the mechanism to nucleate cracks is activated before crack propagation to the upper and lower ice surfaces occurs, cleavage cracking can take place. The conditions for this to happen needs to be explored further.
1972
J.P. Dempsey et al. / Engineering Fracture Mechanics 68 (2001) 1961±1974
Consider now an isolated high pressure zone that is now forming. Just how this might occur will be left for discussion in the next section. As the radius increases, and the peak pressure grows, suppose that either the stably forming cracks `realize' the scale, and ¯akes form from the pre-cracked system, or the situation is such that a cleavage crack can form. The latter can be visualized to happen via scenarios discussed regarding the indentation fracture of ceramics [11]. Vickers microhardness indentation at loads high enough to produce well-developed half-penny surface cracks is used to assess the fracture toughness of ceramics. For ceramics it has been found that 1=2 E P ceramics KIc 26 H c3=2 in which H is the indentation hardness and c is the radius of the half-penny crack. Note that this expression applies to the nucleation of half-penny surface cracks in the surface of an elastic half-space. In ice sheet indentation, the nucleation of such cracks are occurring at the edge of a ®nite thickness plate. It is also well to remember that the E=H ratio is much higher (by a factor of 100 or more) for ceramics than for ice. Indentation fracture mechanics has been used to estimate both the fracture toughness of ice [5] and the size 1=2 of the damage zone beneath an indenter on ice [17]. On equating P with a constant times h3=2 EGscale , or c 3=2 equivalently, h KQ , both of which follow from the ®rst term in Eq. (25), and assuming both that this expression holds for ice, and that c is assumed to equate with the cleavage crack radius, it is immediate that the latter scales directly with ice thickness, as in c
E H
1=3 h
27
The indentation hardness of ice [1] is extremely dependent on the load duration and ice temperature. For instance, at load durations of the order of 1, 10, and 100 s and an ice temperature of 2:5°C, the associated hardnesses of ®ne-grained polycrystalline ice are given by 30, 15, and 7 MPa. The load duration depends on the ice velocity, but not in a simple manner, as it depends on the contact conditions at the interface (simultaneous or non-simultaneous contact, correlated or non-correlated spatial density of high pressure zones, ice thickness, degree of stable cracking). At the ice temperature just quoted, the pressure melting hardnesses are just a boundary layer away. If one were to assume associated secant moduli of 8, 6 and 4 GPa, then the cleavage fractures would size as c 6h, 7h and 8h, respectively. The link with pressure melting just mentioned would seem to be very important, as with even a slight amount of pressure melting a distinct shift in the frictional behavior at the contact interface would be expected, and an associated shift in the triaxiality would occur. If one considers that under typical indentation conditions there is always the background global pressure acting, ¯uctuating over a longer time scale than the individual high pressure zones, any sudden shift in the through-the-thickness tensile strain will aid in the formation of the cleavage fracture. Even without pressure melting, the link between friction and ice velocity merits more attention. The in¯uence of indentation rate (ice velocity) is inextricably linked to the fracture process. The increased crack opening displacements associated with viscoelasticity results in earlier crack growth compared to the elastic case, and stable growth of the physical crack occurs, which is not observed for the elastic case. One can therefore expect the stable development of longer cracks for longer load durations [15]. Major spalls in ®eld indentation tests are associated with low loading rates; ship captains during ramming `lean' on the ice, no doubt to generate a few long fractures that will aid the ramming process [9]. In the case of ice sheet indentation, the critical load duration regime, however, are linked to the ice temperature, ice thickness and con®nement. Attempts to de®ne an eective strain rate [14] for ice indentation problems are essentially trying to link strain rate with fracture processes.
J.P. Dempsey et al. / Engineering Fracture Mechanics 68 (2001) 1961±1974
1973
Before leaving this section, one should note that cleavage fractures occur for `slower' indentation speeds; within this `slower' regime, it is observed that longer cleavage cracks form for slow indentation speeds (but not many), while for faster indentation speeds, shorter cleavage cracks form (and many of them). These cleavage cracks ¯ake o because of ¯exure and out-of-plane movements.
7. Conclusions For both simultaneous and non-simultaneous contact, the critical lengths of stable cracking that may occur prior to ¯aking and ¯exural failure are strongly linked to the current level of the relevant pressure parameters C qb=pa for a line-like high pressure zone, and S qb2 =pa2 for a localized high pressure zone. The peak pressures in these zones scale as b1=2 a
q E0 Gscale c
line-like;
b3=2 a2
q EGscale localized c
28
The stability aspects of the in-plane cracking, and the link between the maximum possible crack lengths and the relative magnitudes of the local and far-®eld pressures help explain the transitions observed within the realms of ductile, intermittent, and brittle crushing.
Acknowledgements This work has been supported in part by the LOLEIF project in the framework of the EU-sponsored Marine Science and Technology (Mast III) Programme under contract number MAS3-CT97-0078 (JPO and ACP), in part by NSF grant CMS-9634846 (JPD) and in part by US Army grant DAAD 19-00-1-0479 (JPD).
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [13] [14] [15] [16] [17]
Barnes P, Tabor D, Walker JCF. The friction and creep of polycrystalline ice. Proc Roy Soc A 1974;324:127±55. DeFranco SJ, Dempsey JP. Crack propagation and fracture resistance in saline ice. J Glaciol 1994;40:451±62. Dempsey JP, Slepyan LI, Shekhtman II. Radial cracking with closure. Int J Fract 1995;73:233±61. Evans AG, Palmer AC, Goodman DJ, Ashby MF, Hutchinson JW, Ponter ARS, Williams GJ. Indentation spalling of edge loaded ice sheets. Proc Seventh IAHR Ice Symp 1984;1:113±21. Goodman DJ, Tabor D. Fracture toughness of ice: a preliminary account of some new experiments. J Glaciol 1978;21:371±80. Gurney C, Hunt J. Quasi-static crack propagation. Proc Roy Soc A 1967;299:508±24. Hallam SD. The role of fracture in limiting ice forces. Proc Eighth IAHR Symp Ice 1986;II:287±319. Joensuu A, Riska K. Contact between ice and a structure. Helsinki University of Technology, Ship Laboratory, Report M-88, 1989, 57p+App [in Finnish]. Jordaan IJ. Mechanics of ice±structure interaction. Engng Fract Mech 2001;68(17±18):1923±60. Kry PR. Scale eects in continuous crushing of ice. Int Symp on Ice, Quebec, Canada, 27±31 July 1981;2:565±580. Lawn B. Fracture of Brittle Solids., 2nd ed. Cambridge University Press, Cambridge, 1998. Masterson DM, Spencer PA. Ice force calculation for large and small aspect ratios. Proc IUTAM Symp: Scaling Laws in Ice Mechanics and Ice Dynamics, Fairbanks, Alaska, 13±16 June, 2000. Michel B, Toussaint N. Mechanisms and theory of indentation of ice plates. J Glaciol 1977;19:285±300. Mulmule SV, Dempsey JP. A viscoelastic ®ctitious crack model for the fracture of sea ice. Mech Time-Dependent Mater 1998;1:331±56. Mulmule SV, Dempsey JP. LEFM size requirements for the fracture testing of sea ice. Int J Fract 2000;102:85±98. Parsons BL. An estimate of the size of the damage zone beneath an indenter on ice. Cold Regions Sci Technol 1993;21:371±80.
1974
J.P. Dempsey et al. / Engineering Fracture Mechanics 68 (2001) 1961±1974
[18] Schapery RA. A theory of crack initiation and growth in viscoelastic media. Part II. Approximate method of analysis. Int J Fract 1993;11:369±87. [19] Schulson EM. The brittle failure of ice. Engng Fract Mech 2001;68(17±18):1839±87. [20] Sodhi DS. Crushing failure during ice±structure interaction. Engng Fract Mech 2001;68(17±18):1889±921. [22] Westergaard HM. Theory of Elasticity and Plasticity. Harvard University Press, Cambridge, 1952.