By Jiashi Yang
This quantity is meant to supply researchers and graduate scholars with the fundamental facets of the continuum modeling of electroelastic interactions in solids. A concise therapy of linear, nonlinear, static and dynamic theories and difficulties is gifted. The emphasis is on formula and knowing of difficulties priceless in equipment purposes instead of resolution concepts of mathematical difficulties. the maths utilized in this booklet is minimum. This quantity is acceptable for a one-semester graduate direction on electroelasticity. it will probably even be used as a reference for graduate scholars and researchers in mechanics and acoustics.
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Extra resources for An introduction to the theory of piezoelectricity
Also define the second-rank four-tensor and the fourth-rank four-tensor and all other components of where Then and Therefore, yields the homogeneous equation of motion and the charge equation. 2 Vector Potential Formulation Consider the case when there is no body charge. Since the divergence of D vanishes, we can introduce a vector potential by 46 which satisfies the divergence-free condition on D. 5-8), yields Similarly, for the electric field E, we introduce an anti-symmetric tensor by Then the curl-free condition on E takes the following form: In summary, the equations for this formulation are Note that in this formulation the internal energy U is used, which is positive definite.
1-49) has been used. Use the divergence theorem 19 which implies that where and (a scalar) defined by are the reference or material electric displacement and body free change per unit undeformed volume. From Equation which implies that where defined by is the reference electric field. From the total mass of the material body is a constant, which is the mass in the reference state where is the mass density in the reference state. 1-56) have been used. Hence 21 In summary, the balance laws in material form are CONSTITUTIVE RELATIONS 5.
To analyze these devices, it is usually convenient to use cylindrical or spherical coordinates. 2 Spherical Coordinates The spherical coordinates are defined by In spherical coordinates we have the strain-displacement relation 48 The electric field-potential relation is The equations of motion are The electrostatic charge equation is 7. COMPACT MATRIX NOTATION We now introduce a compact matrix notation . 1-20) can also be written in matrix form. 7-6) are shown below. 1-19) which is rewritten in matrix form below: we identify 8.
An introduction to the theory of piezoelectricity by Jiashi Yang