3.1 STATE OF STRESS AT A POINT [LO1]

When a body is subjected to external forces, its behavior depends on the magnitude and distribution of forces and properties of the body material. Depending on these factors, the body may deform elastically or plastically, or it may fracture. The body may also fail by fatigue when subjected to repetitive loading. Here we are primarily interested in elastic deformation of materials.

In order to establish the concept of stress and stress at a point, let us consider a straight bar of uniform cross-section of area A and subjected to uniaxial force F as shown in Figure 3.1. Stress at a typical section A - A′ is normally given as σ = F/A. This is true only if the force is uniformly distributed over the area A, but this is rarely true. Therefore, definition of stress must be considered by progressively reducing the area until it is small enough such that the force may be considered to be uniformly distributed.

To understand this, consider a body subjected to external forces P1, P2, P3, and P4 as shown in Figure 3.2. If we now cut the body in two pieces,

Internal forces f1, f2, f3, etc. are developed to keep the pieces in equilibrium. Now consider an infinitesimal element of area ΔA Dat the cut section and let the resultant force on the element be Δf.

Superconductivity is a quantum state of matter that occurs through a phase transition driven by thermal fluctuations. In this state, materials show ideal electric conductivity and ideal diamagnetism to a very good approximation. Two main classes of superconductors, type I and type II, can be distinguished with regards to flux penetration under an applied magnetic field. The properties of these two types are first discussed in detail. Next, the Ginzburg–Landau theory is developed and it is shown that in the presence of a magnetic field, when the ratio of penetration and coherence lengths is smaller than 1⁄√2 the superconductor behaves as type I, while it behaves as type II when this ratio is larger than 1⁄√2. In this second case, the flux penetrates through vortices that form a hexagonal lattice. Finally, in the last part, the microscopic BCS theory is discussed in order to provide an understanding of the physical origin of superconductivity.

The chapter is an introduction to basic equilibrium aspects of phase transitions. It starts by reviewing thermodynamics and the thermodynamic description of phase transitions. Next, lattice models, such as the paradigmatic Ising model, are introduced as simple physical models that permit a mechano-statistical study of phase transitions from a more microscopic point of view. It is shown that the Ising model can quite faithfully describe many different systems after suitable interpretation of the lattice variables. Special emphasis is placed on the mean-field concept and the mean-field approximations. The deformable Ising model is then studied as an example that illustrates the interplay of different degrees of freedom. Subsequently, the Landau theory of phase transitions is introduced for continuous and first-order transitions, as well as critical and tricritical behaviour are analysed. Finally, scaling theories and the notion of universality within the framework of the renormalization group are briefly discussed.

The chapter starts by introducing the basic concepts of metastable and unstable states as well as time scales that control the occurrence of phase transitions. The limits for phase transitions taking place in equilibrium and out-of-equilibrium conditions are then established. In the latter case, thermally activated and athermal limits are distinguished associated with those situations where the transition is either driven or not driven by thermal fluctuations, respectively. Then the formal theory of the decay of metastable and unstable states in systems with conserved and non-conserved order parameters is developed. This general theory is in turn applied to the study of homogeneous and heterogeneous nucleation, spinodal decomposition and late stages of coarsening and domain growth.

The general concept of multiferroic materials as those with strong interplay between two or more ferroic properties is first introduced. Then, particular cases of materials with coupling magnetic and polar (magnetoelectric coupling), polar and structural (electrostructural coupling), and magnetic and structural (magnetostructural coupling) degrees of freedom are discussed in more detail. The physical origin of the interplay is analysed and symmetry-based considerations are used to determine the dominant coupling terms adequate to construct extended Ginzburg–Landau models that permit the determination of cross-response to multiple fields. The last part of the chapter is devoted to study morphotropic systems and morphotropic phase boundaries that separate crystallographic phases with different polar (magnetic) properties as examples of materials with electro(magneto)-structural interplay and that are expected to show giant cross-response to electric (magnetic) and mechanical fields.

Non-equilibrium phase transitions are non-thermal transitions that occur out-of-equilibrium. The chapter first discusses systems that are subjected and respond with hysteresis to an oscillating field due to a competition between driving and relaxation time scales. When the former is much shorter than the latter, a non-equilibrium transition occurs associated with the dynamical symmetry breaking due to hysteresis. A dynamical magnetic model is introduced and it is shown that the mean magnetization in a full cycle is the adequate order parameter for this transition. A mean-field solution predicting first-order, critical and tricritical behaviours is analysed in detail. The second example refers to externally driven disordered systems that respond intermittently through avalanches. The interesting aspect is that for a critical amount of disorder, avalanches occur with an absence of characteristic scales, which define avalanche criticality as reported in different ferroic materials. This behaviour can be accounted for by lattice models with disorder, driven by athermal dynamics.

The chapter starts with a unified view of glassy states in ferroic materials. Disorder and frustration are the main ingredients responsible for the glassy behaviour, which is identified as a strong frequency dependence of the ac-susceptibility in addition to the occurrence of memory effects detected in zero-field-cooling (ZFC) versus field-cooling (FC) measurements of the temperature dependence of the main ferroic property. Dilute magnetic alloys are taken as prototypical examples of materials displaying glassy behaviour. The physical origin is justified by considering the random distribution of the low concentration of magnetic atoms and their RKKY oscillating exchange interaction. This behaviour is used to inspire lattice models which are (extensions of the Ising model) adequate to study glassy behaviour at a microscopic scale. The particular case of spin glasses is considered in detail and mean-field solutions based on the replica symmetry approach are discussed. Finally, similar models for relaxor ferroelectric and strain glasses are also introduced and briefly described.