Static and Dynamic Coupled Fields in Bodies with Piezoeffects or Polarization Gradient (eBook)

Table of Contents 
7 
Introduction 10
Part I Two Types of Electro-Elastic Coupling and Their Role in Properties of Line Defects in Unbounded Anisotropic Media 
16 
1 Piezoelectricity and Piezomagnetism: Basic Concepts and Equations 
17 
1.1 Thermodynamic potentials 
17 
1.2 Linear piezoelectricity 20
1.3 Piezomagnetic effects of linear and linearized origins 28
2 Mindlin's Electro-Elastic Coupling due to Polarization Gradient 
31 
2.1 Toupin's alternative formulation of the theory of piezoelectricity 
31 
2.2 Mindlin's extension of the classical theory of piezoelectricity 
34 
2.3 Mindlin's equations for centrosymmetric and isotropic media 
40 
2.4 Thermoelasticity of dielectrics with polarization gradient 
45 
3 General Electro-Elastic Line Defec tin Unbounded Piezoelectric Body 
51 
3.1 Dislocation and its electrostatic analog as a combined line defect 
51 
3.2 Fundamental properties of 4D dislocations of arbitrary shape 
56 
3.3 Electro-elastic fields of straight 4D dislocations 64
4 Thermo-Electro-Elastic Fields Accompanying Plastic Deformation 
76 
4.1 Thermodynamics of plastic deformation in thermoelastic piezoelectrics 
76 
4.2 Coupled fields produced by a moving straight dislocation 
80 
4.3 The temperature distribution around a moving edge dislocation 
84 
5 Some Extensions for Dislocation in Materials with Polarization Gradient 
93 
5.1 Linear and surface definitions of a dislocation 93
5.2 The field of a straight dislocation 98
Part II 1D and 2D Electro-Elastic Coupled Fields in Media with Surfaces and Interfaces 
105 
6 Coupled Fields of Thermal Inclusion in Media with Polarization Gradient 
106 
6.1 Plain thermal inclusion in unbounded dielectric medium 
106 
6.2 Thermal inclusion at the surface of a half-infinite dielectric 
112 
6.3 Plain thermal inclusion in the middle of a dielectric plate 
115 
6.4 Plain thermal inclusion at the surface of a dielectric plate 
120 
6.5 Cylindrical thermal inclusion in an infinite dielectric medium 
125 
6.6 Conclusions 
127 
7 Green's Functions for Piezoelectric Strip with a General Line Source 
129 
7.1 Statement of the problem and basic equations 129
7.2 Boundary conditions and their block representations 134
7.3 Green's function for the plate with free surfaces 137
7.4 Green's functions for plates with other boundary conditions 
143 
7.5 Conclusions 145
8 Green's Functions for Piezoelectric Strip with Line Surface Sources 
146 
8.1 Basic equations and boundary conditions 146
8.2 Strips with prescribed loads at both surfaces 150
8.3 Strips with prescribed displacements at both surfaces 158
8.4 Strips with prescribed displacements at one surface and loads at another 
163 
8.5 Conclusions 169
9 2D Electro-Elastic Fields in a Piezoelectric Layer-Substrate Structure 
170 
9.1 Basic equations 170
9.2 Boundary conditions at the surface and at the interface 
173 
9.3 General and partial solutions for n(k,y) and n(x,y) 
175 
9.4 Electro-elastic fields excited by the line source in the interior of the structure 
178 
9.5 Electro-elastic fields excited at the surface of the structure 
184 
9.6 Conclusions 
187 
10 Acoustic Waves in Piezomagnetic and Piezoelectric Structures 
189 
10.1 Bluestien-Gulyaev type surface waves in piezomagnetics 
189 
10.2 Magnetoelastic SH waves in a bicrystalline gap structure 
192 
10.3 Resonance excitation of Bluestein-Gulyaev waves 
196 
10.4 Waveguide Sh acoustic modes in a piezoelictric plate 
200 
References 205

"7 GREENS FUNCTIONS FOR PIEZOELECTRIC STRIP WITH A GENERAL LINE SOURCE (p. 125-126)

In this chapter we are going to derive the static Green function describing the 2D coupled fields in arbitrary piezoelectric strip excited by a general line defect parallel to the surfaces and consisting of the four line sources introduced in Sec. 3.3 (c): the line of forces f, the line of charge q, the straight dislocation with the Burgers vector b, and its electrostatic analog of the strength A(p. As we have seen in Ch. 3, the latter line defect is completely determined by the discontinuity Aip in the electrical potential across the arbitrary plane cut ending on the line, which in our case determines the coinciding positions of all the mentioned sources.

The first studies in this field were accomplished for a particular case of straight dislocations in isotropic elastic plates [55-59]. These results were later extended for anisotropic [53] and possibly inhomogeneous [54] purely elastic (nonpiezoelectric) infinite strips. The further extension of the theory for the case of arbitrary piezoelectric strips has been developed in our paper [28]. The content of Chapter 7 is based on this paper.

7.1 Statement of the problem and basic equations

(a) Some preliminary remarks

The aim of this chapter is a derivation of the electro-elastic fields excited in a piezoelectric plate by the general line defect defined above. Of course, such an object, as the general line defect first introduced by Lothe & Barnett [63], is only conventionally a "defect", because among its four constituent line sources there is only one lattice defect - the dislocation. So, perhaps, it would be more correct to talk about the general line source rather than about a defect.

Anyway, independently of the terminology introduced and accepted before us, below we are going to find the physical fields excited by such a combined source. In our analysis it will be convenient to follow the method developed in [54] for purely elastic anisotropic continuously layered media, specifying the problem to a homogeneous medium and simultaneously extending it for a piezoelectric case. Such an extension turns out to be not quite straightforward because of the electrical boundary conditions, which are determined by a continuity of the normal component of electric displacement on the interfaces between the strip and the adjoined isotropic dielectric media. Our consideration will be based on the Stroh-like approach presented in details in Chapter 3 (Sec. 3.3 (c)) where the problem was already formulated for the same general line defect in unbounded medium."