The analysis of ice response to stress using finite elements is described, using multiaxial constitutive relationships, including damage, in a viscoelastic framework. The U-shaped relationship of compliance with pressure is part of this formulation. The results show that the layer of damaged ice adjacent to the indentor arises naturally through the formulation, giving rise to a peak load and subsequent decline. This shows that there can be “layer failure” in addition to failure due to fractures and spalling. Tests on extrusion of crushed ice are described together with a formulation of constitutive relationships based on special triaxial tests of crushed ice. The ice temperature measured during field indentation tests showed a drop in temperature during the upswings in load. This was attributed to localized pressure melting. Small scale indentor tests are described, which show clearly the difference between layer failure and spalling, as found using high-speed video and pressure-sensitive film. The question of scaling, as used in ice tanks, is addressed. Flexural failure can be scaled to some extent; scaling of high-pressure zones lies in the mechanics as developed in the book.

Viscoelastic theory is introduced, using ice as the material under consideration. Linear theory is first introduced, based on elasticity of the springs and on linear viscosity of the dashpots. The nonlinearity of the dashpots in modelling ice deformation is then introduced. The “crushed layer” and analysis by Kheisin and co-workers is outlined, based on linearly viscous modelling. Kelvin and Burgers models are introduced. Microstructural change is modelled using damage mechanics and state variables for material points. Stress and strain re-distribution arises from this aspect, as well as from nonlinearity with stress. Schaperys modified superposition principle is introduced.

Recent observations are summarized, in which it has been found that in compressive ice failure, zones of high-pressure form with pressures locally as high as 70 MPa. Various aspects of ice behaviour are summarized: creep, fracture, recrystallization, and the development of microstructurally modified layers of ice. Pressure melting is described, whereby the melting temperature decreases with accompanying hydrostatic pressure. The importance of fracture and spalling in the development of high-pressure zones is emphasized. The use of mechanics in analysis of ice failure is discussed.

The importance of field experimental results is emphasized, as well as the necessity to include consideration of time in ice mechanics

Measurements of pressure on fixed structures are reviewed including the Helsinki and JOIA test programmes. The Molikpaq experience and the Hans Island programmes are described in some detail. Loads tend to be concentrated in small areas, as was the case for ship structures (the high-pressure zones). Size effect of ice pressure with regard to ice thickness is discussed; average pressures decrease with ice thickness. The medium scale field indentation programmes are described, covering the Pond Inlet, Rae Point, and Hobsons Choice Ice Island test series. Ice-induced vibrations are introduced; these were observed in the Molikpaq structure and in many indentation tests. The vibrations tended to occur at certain speed ranges, associated with ice crushing. Results of field tests on iceberg failure are also reviewed, in which supporting evidence for layer failure was obtained.

The Appendix contains an outline of the development of Biot-Schapery theory based on the thermodynamics of irreversible processes. A brief biography of R. A. Schapery is followed by an exposition of the theory, the use of the modified superposition theory, and the use of J integral to deal with damage processes.

The states of stress in high-pressure zones involve a combinations of volumetric and deviatoric stresses. Modelling of ice behaviour under these states of stress is essential for developing proper mechanics of failure of high-pressure zones. Past triaxial tests are reviewed. There is a lack of information for higher confining pressures. The microstructural changes of microcracking and recrystallization needed to be studied in terms of past stress history. These were addressed in a special series of tests, which showed that microcracking at low confinements causes increase in compliance, which decreases with increasing confinement, but that at higher confinements, pressure softening, associated with melting, results in much increased compliance. Tests in which the activation energy at various confinements was measured using tests at a range of temperatures showed that the addition of pressure to ice resulted in behaviour similar to less confined ice at a higher temperature (pressure–temperature equivalence). Ice is prone to localize and small irregularities are sufficient to trigger this behaviour, as observed in some triaxial tests.

Methods used for sample preparation in the research are described, for triaxial testing and for small-scale indentation in the laboratory.

General principles of design are introduced. The consequences of local and global failure are discussed. The use of codes, in particular ISO 19906, is described. Probabilistic methods and limit-states design for ice loading are emphasized. The Titanic disaster is addressed, emphasizing the cause of failure as being the result of operational failures, not the rivets. High local pressures (high-pressure zones) are associated with the failure of the rivets and plating.

The crystal structure of ice is described, together with the concepts of elasticity and dissipation. The growth of ice on earth is analysed, including the effect of salinity on ice freezing. This leads to definitions of ice types on earth, and to definitions of first year and multiyear ice, as well as icebergs.

Problems involving mass, momentum and energy transport in one spatial direction in a Cartesian co-ordinate system are considered in this chapter. The concentration, velocity or temperature fields, here denoted field variables, vary along one spatial direction and in time. The ‘forcing’ for the field variables could be due to internal sources of mass, momentum or energy, or due to the fluxes/stresses at boundaries which are planes perpendicular to the spatial co-ordinate. Though the dependence on one spatial co-ordinate and time appears a gross simplification of practical situations, the solution methods developed here are applicable for problems involving transport in multiple directions as well.

There are two steps in the solution procedure. The first step is a ‘shell balance’ to derive a differential equation for the field variables. The procedure, discussed in Section 4.1, is easily extended to multiple dimensions and more complex geometries. The second step is the solution of the differential equation subject to boundary and initial conditions. Steady problems are considered in Section 4.2, where the field variable does not depend on time, and the conservation equation is an ordinary differential equation. For unsteady problems, the equation is a partial differential equation involving one spatial dimension and time. There is no general procedure for solving a partial differential equation; the procedure depends on the configuration and the kind of forcing, and physical insight is necessary to solve the problem. The procedures for different geometries and kinds of forcing are explained in Sections 4.4–4.7.

The conservation equations in Sections 4.2 and 4.4–4.7 are linear differential equations in the field variable—that is, the equations contain the field variable to the first power in addition to inhomogeneous terms independent of the field variable. For the special case of multicomponent diffusion in Section 4.3, the equations are non-linear in the field variable. This is because the diffusion of a molecular species generates a flow velocity, which contributes to the flux of the species. The conservation equation for the simple case of diffusion in a binary mixture is derived in Section 4.3, and some simple applications are discussed.

In Section 4.8, correlations for the average fluxes presented in Chapter 2 are used in the spatial or time evolution equations for the field variables.