Ferroelastic materials

At high temperatures, ferroelectric materials transform to the paraelectric state (where dipoles are randomly oriented), ferromagnetic materials to the paramagnetic state, and ferroelastic materials to the twin-free normal state. The transitions are characterized through order parameters (Rao Rao, 1978). These order parameters are characteristic properties parametrized in such a way that the resulting quantity is unity for the ferroic state at a temperature sufficiently below the transition temperature, and is zero in the nonferroic phase beyond the transition temperature. Polarization, magnetization and strain are the proper order parameters for the ferroelectric. 

The size of the ferroelastic hysteresis depends sensitively on thermodynamic parameters such a temperature, T, pressure, P, or chemical composition, N. Most ferroelastic materials show phase transitions between a ferroelastic phase and a paraelastic phase. As it is rather a major experimental undertaking to measure ferroelastic hysteresis with any acceptable degree of accuracy, it has become customary to call a material ferroelastic if a phase transition occurs (or may be thought to occur) which may... 

Needle twin walls with exponential trajectories were observed in many ferroelastic materials such as Pb3(P04)2 and KSCN and for twin walls in BaTiOs. Typical length scales, X, are 85 pm (KSCN), 55 nm to 3pm [Pb3(P04)2] and 270 nm (BaTiOs). Furthermore, although the parabolic and linear cases show a correlation between the tip angle and needle width, this is less apparent for the exponential needles. The tip angle for... 

Twin domains and their boundaries can form complex microstructural patterns in ferroelastic materials. It has been shown theoretically and experimentally that diffusion of impurity ions or vacancies can be enhanced within twin domain walls [1-4]. Furthermore, impurities [5] or oxygen vacancies [6-8] may be pinned in the wall or in its close vicinity. The fact that walls and bulk material show different dielectric transport and diffusion rates could explain the high ionic conductivity in doped heavy twinned lanthanum gallate, which is of interest... 

Rychetsky, I. Deformation of crystal surfaces in ferroelastic materials caused by antiphase domain boundaries. J. Rhys. Condens. Matter 9,4583-4592 (1997)... 

The nanoferroics are known as low-dimensional objects of different morphology with dimensions from zero to three, i.e., quantum dots, nanoparticles, nanorods, thin films and bulk solids. As stated above, the properties of ferroelectric, ferromagnetic and ferroelastic materials are highly dependent on their dimensionality and symmetry of stressed state. In particular, above consideration shows that the phase transitions have the same nature in the particles, films and 3D polycrystals. 

Koimga-Njiwa, A.B., Lupascu, D.C., and Rddel, J. (2004) Crack tip switching zone in ferroelectric ferroelastic materials. 

Landis, C. (2003) On the fracture toughness of ferroelastic materials. J. Mech. Phys. Solids., 50, 1347-1369. 

If the adaptronic structure requires temperature stability, active functional materials must be used since they can have a flat temperature response away from the phase transition and are controllable with external fields. Most materials in this category are ferroic materials, i. e., ferroelectric, ferromagnetic and ferroelastic materials. 

The remainder of the chapter is organized as follows. Section 2 presents the constitutive law used to describe polycrystalline ferroelastic materials. Section 3 presents the fracture model, including the finite element method implemented to determine the fields near a steadily growing crack and the crack tip energy release rate. Results for the toughness enhancement predicted by the model will also be presented in this section. Finally, Section 4 will be used to discuss the results and their comparison to experimental observations. 

The back stress tensor leads to kinematic hardening and must be used to enforce the remanent strain saturation conditions. The approach used to determine the back stresses is based on the assumption that the internal state of the ferroelastic material is completely characterized by the components of the remanent strain tensor. Cocks and McMeeking (1999) and Landis (2002). This assumption leads to the identification of a remanent potential,, such that the back stresses are derived from the... 

Within this model for fracture in ferroelastic materials, dimensional analysis dictates that all field quantities, for example the stresses, will be of the form... 

Due to this strain saturation effect, the stresses near the crack tip in the ferroelastic material increase severely. In fact, the numerical results suggest that very close to the crack tip the stresses have a 1 / /r radial dependence. Hence the crack tip stress intensity factor Kiup can be defined such that on the plane ahead of the crack tip... 

Figure 3. The distribution of effective remanent strain near a growing crack in a ferroelastic material. The active switching zone, elastically unloaded wake, and unloaded elastic sector behind the crack tip are each denoted on the illustration. The material law is given in Section 2, and the material parameters for this computation are s if/cro 5, ffo oq, m 0.01 and v 0.25.

The primary goal of this fracture model is to determine how the steady state toughness enhancement in ferroelastic materials, Ggs/Go, depends on the material properties. Eq. (3.9) identifies the material properties in question and ranks them in order of significance. Poisson s ratio v will be shown to have a very weak influence over the toughness enhancement. The two hardening parameters, Hq/o-q and m, have a much stronger influence on Gss/Gq, and it will be shown that as the hardness of the... 

Figure 5. The toughness enhancement, G /Go, during steady crack growth in a ferroelastic material as a function of the saturation strain level, ScElao, for a range of initial hardening values, Ho/cto.

In summary, results from numerical computations of the stress and strain fields and the toughness enhancement during steady crack growth in ferroelastic materials have been presented. The computations illustrate a few interesting features and confirm some intuitive hypotheses about the solution. First, the near tip stresses appear to recover a 1/v r singular form, however the radial dependence of these stresses is not the same as those for an isotropic elastic material. It was also shown that the distributions of remanent strain are not trivial and do reorient as the crack tip passes. Lastly, as would be expected, the steady state toughness of the material increases as the relative saturation strain increases, and decreases as the hardness of the material increases. 

Reece, M.J. and Guiu, F., 2002, Toughening produced by crack tip stress induced domain reorientation in ferroelectric and/or ferroelastic materials. Philosophical Magazine A82 29 38. 

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