Phase Change in Mechanics

Frontmatter

Chapter 1. Introduction

Abstract

A model is a theory which predicts the evolution of a structure. It does not explain the underlying physical phenomenon, it only predicts what is going to happen: physics explains, mechanics predicts the motion. A model is schematic and limited to some aspects of the phenomenon which occurs. In this point of view, the scientist or the engineer has an important and major role.

Michel Frémond

Chapter 2. The State Quantities and the Quantities Describing the Evolution

Abstract

As already said, the state quantities are the quantities chosen by the scientist or the engineer to describe a phenomenon he wants to predict. The state quantities define the equilibrium: when the state quantities remain constant with respect to the time, the phenomenon is at an equilibrium. The notion of equilibrium is subjective and depends on the sophistication of the modelling, i.e., on the choice of the state variables.

Michel Frémond

Chapter 3. The Basic Laws of Mechanics

Abstract

The laws we need to build models for phase change problems are the principle of virtual power to get the equations of motion and the laws of thermodynamics to deal with thermal actions. For the sake of simplicity, we assume the small perturbation assumption (see Appendix B) after deriving the equations of motion. Within this assumption, the mass balance which describes the density evolution has not a major role, except for phase change with voids (Sect. 4.5), phase change with different densities (Sect. 4.6).

Michel Frémond

Chapter 4. Solid–Liquid Phase Change

Abstract

Besides the classical ice–water phase change, the solid–liquid phase change exhibits many different behaviours: diffuse with respect to time and space phase change, phase change with voids and bubbles, phase change with different densities, dissipative phase change, irreversible phase change, phase change with thermal memory,... The different predictive theories result from the choices of the free energies and pseudopotentials of dissipation. Examples are given: soil freezing and the insulation of an house in winter, liquid gas underground storage, fossil permafrost.

Michel Frémond

Chapter 5. Shape Memory Alloys

Abstract

Shape memory alloys are mixtures of many martensites and of austenite. The composition of the mixture varies: the martensites and the austenite transform into one another. These phase changes can be produced either by thermal actions or by mechanical actions. The striking properties of shape memory alloys result from interactions between mechanical and thermal actions [29, 136, 183].

Michel Frémond

Chapter 6. Damage

Abstract

It is known that damage results from microscopic motions in the structures, as it is caused by microfractures and microcavities resulting in the decreasing of the material stiffness. There is wide literature on this topic and it is difficult to be exhaustive. Let us mention [145,151,152] modelling damage within the framework of continuum mechanics. The damage quantities are internal quantities which are involved in the free energy of materials [104, 129, 138, 151, 152].

Michel Frémond

Chapter 7. Contact with Adhesion

Abstract

Let us consider two solidsΩ₁ and Ω₂ glued on one another on their contact surface Г. In order to take into account the adhesive properties of the glue which result from fibers connecting the contact surfaces and breaking progressively, Fig. 7.1, [57, 78], we choose as state quantity the surface fraction of active glue fibers, β (x, t), [31,58, 66,75,81,104,106–108,168,180,187,204,206].When β (x, t) = 0 all the glue fibers are broken and Signorini contact properties are valid: the interactions of the two solids result only from the impenetrability condition, [11, 97, 161, 182, 199].

Michel Frémond

Chapter 8. Damage of Solids Glued on One Another: Coupling of Volume and Surface Damages

Abstract

Let us consider two pieces of concrete glued on one another. Both the concrete 4 and the glue responsible for the adhesion, can damage due to external actions. The 5 volume and surface damages are coupled. The interesting part of the predictive 6 theory is the set of equations on the contact surface, [99,100]. Mathematical results 7 are reported in [43].

Michel Frémond

Chapter 9. Phase Change with Discontinuity of Temperature: Warm Water in Contact with Cold Ice

Abstract

Let us consider warm water on cold ice, for instance water flowing on a frozen lake. On the surface of the lake, there is a discontinuity of temperature between the temperature T ₂ of the liquid water which is larger than the water freezing temperature T ₀ and the temperature T ₁ of the ice of the lake which is lower than T ₀. It is to be known if the water freezes or if the ice thaws, [111, 153].

Michel Frémond

Chapter 10. Phase Change and Collisions

Abstract

Let us consider warm rain falling on a frozen ground. On the surface of the ground, there is a discontinuity of temperature between the temperature T ₂ of the rain which is larger than the water freezing temperature T ₀ and the temperature T ₁ of the frozen ground which is lower than T ₀. It is to be known if the rain freezes or if the frozen ground thaws. In this problem, the temperatures are discontinuous both with respect to space and with respect to time. Let us also note that the velocities are also discontinuous. In order to investigate this problem, we recall the theory of collisions without thermal effects and the theory of collision with thermal effects. For the sake of simplicity, we consider first collisions of balls schematized by points then in next chapter, collisions of continuous media either solid or liquid. This theory is developed in [114].

Michel Frémond

Chapter 11. Collisions of Deformable Bodies and Phase Change

Abstract

Let us consider again a piece of ice colliding with a warm soil. Work is mostly dissipated on the contact surface and in its neighbourhood. Thus the temperature discontinuity is more important in this zone and some melting may occur there and not elsewhere. In this chapter, collisions of deformable solids with volume dis- continuities of velocity are investigated together with the resulting phase changes. We introduce percussions related to volume velocity discontinuities and to phase volume fraction discontinuities. We assume the contact surface of the colliding solids is conformal, i.e., it has a nonzero Lebesgue measure.

Michel Frémond

Chapter 12. Phase Change Depending on a State Quantity: Liquid–Vapor Phase Change

Abstract

Water may be solid, liquid and gaseous. We investigate in this chapter the liquid 4 vapor phase change. Compared to the solid liquid phase change, there is a major 5 difference: the phase change temperature depends on a parameter, for instance on 6 the pressure. It is well known that water boils at the Mont Blanc summit at a lower 7 temperature than at sea level.

Michel Frémond

Chapter 13. Clouds: Mixture of Air, Vapor and Liquid Water

Abstract

Clouds are mixtures of air, vapor, liquid water and even of ice. For the sake of simplicity, we assume there is no ice phase. At each point of the mixture there is air and water either gaseous or liquid. Different indices are used: v for vapor, l for liquid, a for air, w for water. The clouds we see in the sky have a structure evolving with time. We think that the clouds cohesion results from local interactions: at a point x, the physical quantities, for instance, the velocities, the vapor content and liquid water content,.. do depend on their values in the neighbourhood of point x. A way to take into account these interactions is to introduce space derivatives which clearly depend on the neighbourhood values.

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Chapter 14. Conclusion

Abstract

Continuum and discontinuum mechanics give predictive theories to investigate at the engineering level, i.e., the macroscopic level, numerous phase change phenomena occurring during smooth or violent evolutions. The important elements of the theories are the choice of:

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