THE SCIENCE AND APPLICATIONS OF ACOUSTICS - H. H. Arnold ...

16.5 Transducers 463If no piezoelectric effect is present, the term d vanishes from Equations (16.20)and (16.23), which yields the familiar relationshipsands = aσ = σ/Y (16.24)D = εE (16.25)where Y is the elastic constant (or Young’s modulus) for the material and ε is thecorresponding electrical permittivityUnder short-circuit conditions for the crystal, Equations (16.21) and (16.24)lead toP = es (for short-circuit conditions) (16.26)where e = d/Y.When a compressive stress is applied to the crystal, Equation (16.23) becomess =−aσ + dE (16.27)When the crystal is clamped to keep strain zero and when stress is applied, wesee from Equations (16.26) and (16.27) thatσ = eE (for the constraint s = 0) (16.28)where e is the piezoelectric stress constant which is expressed on C/m 2 or N/V m.It must be realized at this stage that the piezoelectric phenomenon is a threedimensionalone. Not only we have to consider the changes in voltage, stress,strain, and dielectric polarization in the thickness direction of the crystal, we mustalso take into account the effects in any direction. A stress applied to a solid in agiven direction may be resolved into six components: three tensile stresses σ x , σ y ,σ z along the principal axes x, y, and z, respectively, and three shear stresses τ yz ,τ xz , and τ yz about axes x, y, and z. In the notation for shear, the subscripts indicatethe action plane of the shear—thus yz denotes a shear in the yz plane acting aboutthe x-axis. Also, we note that τ yz = τ zy , and so on. The corresponding componentsof strain are ε x , ε y , ε z , ε yz , ε xz , and ε yx . In general, the following stress–strainrelationship of Equation (5.2) can be generalized as followsσ ji = c jk ε ij (16.29)where c jk denotes the elastic modulus or stiffness coefficient. This yields 36 valuesof c jk :⎤σ xx = σ x = c 11 ε x + c 12 ε y + c 13 ε z + c 14 ε yz + c 15 ε xz + c 16 ε xyσ yy = σ y = c 21 ε x + c 22 ε y + c 23 ε z + c 24 ε yz + c 25 ε xz + c 26 ε xyσ xx = σ z = c 31 ε x + c 32 ε y + c 33 ε z + c 34 ε yz + c 35 ε xz + c 36 ε xyτ yz = c 41 ε x + c 42 ε y + c 43 ε z + c 44 ε yz + c 45 ε xz + c 46 ε xy(16.30)⎥τ xz = c 51 ε x + c 52 ε y + c 53 ε z + c 54 ε yz + c 55 ε xz + c 56 ε xy⎦τ xy = c 61 ε x + c 62 ε y + c 63 ε z + c 64 ε yz + c 65 ε xz + c 66 ε xy