In the preceding chapters, we have discussed definitions and identification techniques for observed material barriers to the transport of fluid particles, inertial particles and passive scalar fields. All these barriers are directly observable in flow visualizations based on their impact on tracers carried by the flow. However, the transport of several important physical quantities, such as the energy, momentum, angular momentum, vorticity and enstrophy, is also broadly studied but allows no direct experimental visualization. These important scalar and vector quantities are dynamically active fields, i.e., functions of the velocity field and its derivatives.We will collectively refer to barriers to the transport of such fieldsas dynamically active transport barriers. This chapter will be devoted to the development ofan objective notion of active barriers in 3D unsteady velocity data. Additionally, 2D velocity fields can also be handled via this approach by treating them as 3D flows with a symmetry.

Here, we elaborate in more detail on a few technical notions that we have used throughout this book. Among other things, the subjects include the implicit function theorem, the notion of a ridge, classic Lyapunov exponents, differentiable manifolds, Hamiltonian systems, the AVISO data set, surfaces normal to a vector field, calculus of variations, Beltrami flows and the Reynolds transport theorem.

If we accept that transport barriers should be material features for experimental verifiability, we must also remember a fundamental axiom of mechanics: material response of any moving continuum, including fluids, must be frame-indifferent.This means that the conclusions of different observers regarding material behavior must transform into each other by exactly the same rigid-body transformation that transforms the frames of the observers into each other. This requirement of the frame-indifference of material response is called objectivity in classical continuum mechanics. Its significance in fluid mechanics is often overlooked or forgotten, which prompts us to devote a whole chapter to this important physical axiom. We clarify some common misunderstandings of the principle of objectivity in fluid mechanics and discuss in detail the mathematical requirements imposed by objectivity on scalars, vectors and tensors to be used in describing transport barriers.

The chapter describes results of measurements during several ship trials, in which instrumented vessels were used to interact with ice. The main focus is the measurement of local ice pressures by strain-gauging of the ship hull. The results include ramming of ice features. A variety of results are analysed, including those from the Kigoriak, Polar Sea, Louis S. St.-Laurent, Oden, and Terry Fox. Analyses of high-pressure zones are presented and a novel method (the alpha-method) is presented for local design of vessels and structures.

The chapter commences with a description of various observations of time-dependent fractures in ice. In the medium scale tests, slow loading resulted in very large flaws, whereas fast loading resulted in many small fractures and spalls in the vicinity of the load application. Then, a summary of fracture toughness measurements on ice are summarized. The question of stress singularity at crack tips is raised, and to deal with this, Barenblatt’s analysis is introduced, based on linear elasticity. Schapery’s linear viscoelastic solution for this method is described, using the elastic-viscoelastic correspondence principle. The J integral forms the basis of the application to fracture, using the correspondence principle noted. A set of experiments on ice samples, beams with 4-point loading, was conducted. Tests with a range of loading rates, as well as constant-load tests, were conducted. Comparison of the results with theory was made. The results of Liu and Miller using the compact tension set-up were also considered. Good agreement with theory was found in all cases. Nonlinear viscoelastic theory of Schapery is also outlined.

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.