Introduction to Smart Materials - Civil Engineering (CE) PDF Download

Introduction to Smart Materials
Classification
Vibration control strategy using smart sensors and actuators largely relies on the appropriate choice of smart materials. Commonly used smart materials can be classified into four major groups based on the actuation/sensing mechanisms.

• Piezoelectric
• Magnetostrictive
• Phase-Transition dependent and
• Electro/Magneto Rheological Materials.

The existence of cross-coupling between mechanical fields (like stress/strain/viscosity) and other fields (like electric/magnetic/thermal) makes these materials very useful for vibration sensing and control.
In the table below, a group of five input-output pairs are shown along with the cross-coupling in material properties, which are commonly exploited for smart sensing and actuation.

Table 30.1: Smart sensing and actuation options​

Note: SS denotes smart sensors and SA denotes smart actuators


Piezoelectric Material
Piezoelectricity, meaning ‘Electricity from Pressure' was discovered by Pierre and Jacque Curie more than 100 years ago. The so called ‘direct effect' of charge generation in a class of crystals while subjected to mechanical stress was first observed in the crystals like zinc blend, sodium chlorate, tourmaline and quartz.

The ‘inverse effect' in piezoelectricity, signifying the deformation in crystals on the application of a voltage was theoretically predicted by Lippman showing it as a consequence of a reversible thermodynamics process and later experimentally validated by Jacque Curie. Voigt did the first rigorous analysis of correlating crystal structure symmetry to piezoelectricity.

Out of the possible 32 crystal groups (based on simple lattice geometry and symmetry operation on crystal faces), it was shown that only 11 groups could have piezoelectricity. The uniqueness of these groups of piezo-crystals is their non-centro- symmetric structure, and interestingly a subgroup of such crystals always falls to a state of spontaneous polarization. The phenomenon widely known as ‘Ferro electricity' occurs due to the presence of hydrogen bond dipoles in the elements.

Over the years, guided by these characteristics, material scientists have successfully identified/developed crystals with piezoelectric property. Most of them are found unfit for engineering applications due to their high temperature sensitivity and large hysteresis loop. It is only with the invention of Barium Titanate ceramic (around 1947) that building of efficient solid-state transducer has been made possible. However, within next ten years, Barium Titanate is substituted by Lead Titanate Zirconate (commonly known as PZT) as it could operate up to a much higher temperature and possesses stronger piezoelectric effect. Similar effect has also been discovered in a polymeric crystal called Polyvinylideneflouride (PVDF) by Kawai . The recent developments in smart structural technology are broadly centered on these two piezoelectric materials.


Figure 30.1: Barium Titanate in a tetrogonal symmetry below Curie Temperature

The fundamental equations of piezoelectricity are given by


where, the subscripts i , j , k , l = 1, 2, 3 denotes tensorial indices.

The stress tensor is represented by s , S is the strain tensor, E is the electric field intensity and D is the electric displacement field. The elastic stiffness matrix is denoted by the symbol C^E, where the superscript E denotes that the elastic constant is measured under constant electric field (henceforth, the superscript will not be shown explicitly for the sake of brevity); e is the piezoelectric stress-charge matrix and e the permittivity matrix, similar to C , is measured under constant strain-condition. The details of the axes are shown in Fig 30.2.

Figure 30.2: A piezoelastic plate with electrodes and poling direction

The crystal structure of common piezoelectric materials shows Dihexagonal symmetry. Hence, the following material symmetry conditions could be applied to the constitutive relationship:
                                                                                                       (30.3)

After experimentally obtaining the material constants and using the material symmetry for hexagonal crystal structure, Eqns. (31.1 and 31.2) could be simplified and
written as:

                                                                  (30.4)

The electro-mechanical coupling is shown inside the bordered boxes in Eqn. (30.4). It may be noted that axes 1, 2 and 3 used for the electrical system are identical with x, y and z, corresponding to the mechanical system.

Consider a thin slab of piezoelectric element shown in Fig. 30.1. On application of a high electric field (poling field), randomly oriented dipoles get aligned to the direction of the electric field. Consequently, the strain induced is magnified. The direction of alignment is called the poling direction. Ignoring the normal stress s_z and the shear stresses s_xz and s_yz for plane stress assumption, eqn. (30.4) could be further simplified as:
                                (30.5)

where E_p is the modulus of elasticity of the piezoelectric material, ν is the Poisson's ratio and dij are the piezoelectric strain-charge constants. The first three equations of (30.5) are generally used as the constitutive equations of piezoelectric actuators while the last equation is used to model piezoelectric sensors.

From eqn. (30.5), it is clear that if a piezoelectric thin slab is subjected to mechanical load, the total strain S developed in an active layer, would consist of two parts – the structural or elastic strain S_s and the piezoelectric strain S_a such that

S = S_s + S_a                                                                                                                   (30.6)

where, S_a = [-d₃₁ E₃ , -d₃₂ E₃ , 0] ^T .
The structural strain on the other hand will depend on the nature of loading and boundary condition applied to the piezo-plate.

It is also evident from eqn. (30.5) that for poling across the lamina the active strain is developed only in the normal plane. To generate strains along the direction of the thickness of the specimen, ceramics with different crystal-cuts are used which are commonly known as Piezo-stacks. The electro elastic coupling components in the 3-3 directions, like d₃₃ or e₃₃ , become important in such cases. Another important parameter used in piezoelectricity is the voltage constant g . The higher value of g signifying higher voltage sensitivity implies more suitability of the material for sensing application. The piezoelectric constants and their significance in active vibration control are discussed in the following section.

Example
A thin piezoelectric plate of size 25 mm x 10 mm and thickness 1.5 mm has electroding on top and bottom as shown in Fig. 30.3 (with an exaggerated view of the thickness). Assume the piezoelectric material to have elastic modulus of 65 GPa, Poisson's ratio 0.3 and electro-mechanical coupling coefficients, d₃₁ and d₃₂ to be -50x10^-12 m/V and 200x10 -12 m/V, respectively. Find out the strains in the plate when it is subjected to a voltage of 300V and a an axial force of 20 N.


Figure 30.3 : A typical piezoelectric plate with axial load and elastic field


Solution
Following Fig. 30.2 and eqn. (30.6), the active strains in the plate are:

since, d₃₁ = d₃₂ ,   strain . Also, since a tensile force of 20 N is acting on the plate along the x-direction, the structural strain along x-direction is   and the structural strain along y -direction is  Hence, the total strain along x -direction is 10.02 µ -strain and the total strain along y -direction is 9.99 µ -strain .