ic repose of the particulate material fully dry or

of qualitative significance and awaits further ex-

immersed in liquid. For a dry, angular cohesion-

less material,

cate the range of ice-particle behavior. Its use is

analogous to the use of a Reynolds number in

π φ

characterizing the drag coefficients of bodies in

(20)

4 2

moving fluids. In modeling, the aim would be to

ensure that the model-scale value of *B*o remains

within a range of values for which the sticky forces

,

(21)

are scaled in correct proportion with the inertial

and gravitational forces. Order-of-magnitude esti-

and

mates can be made for *B*o, however. As it ap-

ξ = tanφ.

proaches zero, an ice-piece accumulation behaves

(22)

as if it were a cohesionless assemblage of discrete

When φ increases beyond about 45, such that

pieces; buoyancy dominates. When *B*o exceeds

about 106, the accumulation behaves as a fused

extensive interlocking of ice pieces takes place, eq

structure of connected particles. At the extreme,

21 and 22 become less appropriate for estimating

the accumulation becomes monolithic ice. The im-

portant message here is that it may not be possible

cover, Zufelt (1992) shows that the value of an

alternate parameter, , the internal resistance coef-

to use ice to model ice, because fine-sized ice piec-

ficient of an ice cover, is related to *k*1, *k*0, and ξ by

es do not behave like large ice pieces, just as clay

does not behave like gravel.

= *k*1ξ*k*0 .

(23)

The geometric factors affecting accumulation

strength are fairly straightforward to identify and

Beltaos (1993) has reported values of for natural

to scale. They include accumulation thickness,

ice jams in the range of 0.8 to 1.6, but these were

accumulation porosity, size and size distribution

back-calculated from estimates of jam thickness

of ice pieces comprising the accumulation, and

assuming equilibrium thickness theory. Little

shape and roughness of constituent ice pieces. It is

work has been accomplished on the interparticle

much less straightforward to scale the material fac-

friction of particulate ice pieces, ξ, or the coeffi-

tors, which include strength and deformation

cient of lateral pressure of wetted particulate

properties of constituent ice pieces and the tem-

perature of ice pieces. All of the aforementioned

masses, *k*0.

variables affect the angle of internal resistance of

an ice accumulation. Ice-piece size also affects the

strength and deformation properties of ice pieces.

The lateral distribution of stress through an ac-

Practical considerations often make it necessary

cumulation, and the friction of accumulated ice

to relax similitude criteria in order to ensure that

pieces against other surfaces, are additional prop-

the model adequately replicates the dominant pro-

erties to be taken into account. Forces attributable

cess under investigation. Limitations in modeling

to lateral stress and ice friction are important for

area or flow capacity commonly constrain the hor-

structures or ships flanked by ice accumulations

izontal space, and thereby the horizontal scale, of a

and for ice-jam formation. The shear force at a slip

model. The necessity for small scale may result in

plane (e.g., along the side of a structure or a river

very small model depths, to the extent that viscous

and surface tension effects become significant. The

bank) depends on the coefficient of lateral pres-

sure, *k*0 (akin to a Poisson ratio), and the coefficient

remedy is to resort to geometric, or vertical, distor-

of friction of ice rubbing against itself, ξ, or against

tion.

A vertical length scale, βL, is chosen to keep the

some other material forming one side of the plane.

The passive pressure coefficient, *k*1, relates the

viscous and surface tension forces in the model at

maximum internal resistance to the average verti-

negligible levels. The resulting model distortion,

cal stress within a material. The parameters *k*1, *k*0,

and ξ can be expressed in terms of the angle of in-

open-water hydrodynamic models, although no

ternal friction, φ, of the accumulation. In turn, φ is

strict limit is set. Care must be taken in determin-

related to the shape and size distribution of the ice

ing the appropriate scales for horizontal or vertical

pieces constituting the accumulation. As a lower-

forces acting on horizontal or vertical planes. The

bound estimate, φ can be taken as the angle of stat-

scales are not the same. Consequently, whereas

6