Citation: Damian G. D'Souza. The Parathyroid Hormone Family of Ligands and Receptors[J]. AIMS Medical Science, 2015, 2(3): 118-130. doi: 10.3934/medsci.2015.3.118
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Magnetostrictive materials have been studied due to their great potential as sensor and actuator elements in a wide variety of applications that can benefit from their remote operation, high energy density and short response time [1]. Among giant magnetostrictive materials, Terbium Dysprosium Iron alloy (Terfenol-D) is the most attractive one because of its high saturation magnetostrain (1600 ppm), good coupling coefficient (as high as 60%), and high Young’s modulus [2]. On the other hand, piezoelectric ceramics, especially lead zircornate titanate (PZT), are widely used in smart materials and structures due to their high output force, compact size, and high power density. Magnetoelectric (ME) composites require giant magnetostrictive and piezoelectric materials, with a strong coupling between them, and many applications of these composites such as magnetic field sensing devices [3,4], coil-less transformers and read/write devices [5] are currently under investigation.
Recently, energy harvesting devices can be used as the power source for structural health monitoring sensors, tire pressure monitoring sensors, medical implants and other wireless sensors. The devices of energy harvesting from ambient sources, such as mechanical loads, provide a promising alternative to battery-powered systems. Gao et al. [6] discussed the effect of the length ratio between the substrate layer and the piezoelectric layer on the induced voltage of the PZT/stainless steel cantilevers due to a constant force. Hu et al. [7] designed and tested an optimal vibration-based energy harvesting system using magnetostrictive material. Dai et al. [8] investigated the performances of electric output due to vibration in three layered magnetostrictive/piezoelectric composite harvesters.
Energy harvesting magnetostrictive/piezoelectric laminates are subjected to high mechanical loads, and these loads cause high response levels that increase the generated power but induce the delamination and reduce the lifetime of the laminates. In this work, we study the detection and response characteristics of clamped-free giant magnetostrictive/piezoelectric laminates under concentrated loading in a combined numerical and experimental approach. Two and three layered laminates are fabricated by bonding Terfenol-D layers on one side and both sides of PZT layer, respectively. The tip deflection, stress, induced voltage and induced magnetic field for the laminates due to concentrated loads are calculated by the finite element analysis (FEA). The tip deflection, induced voltage and induced magnetic field are also measured, and the data produced by the experiment are then compared with analytical results.
The basic equations for magnetostrictive and piezoelectric materials are outlined here. The equilibrium equations in the rectangular Cartesian coordinate system O-x1x2x3 are given by
σji,j=0 | (1) |
Bi,i=0 | (2) |
Di,i=0 | (3) |
εij=sHijklσkl+d′kijHk | (4) |
Bi=d′iklσkl+μikHk | (5) |
εij=sEijklσkl+dkijEk | (6) |
Di=diklσkl+ϵTikEk | (7) |
sHijkl=sHjikl=sHijlk=sHklij, d′kij=d′kji, muij=μji | (8) |
sEijkl=sEjikl=sEijlk=sEklij, dkij=dkji, epsilonTij=ϵTji | (9) |
εij=12(uj,i+ui,j) | (10) |
Hi=φ,i | (11) |
Ei=−ϕ,i | (12) |
Two layered magnetostrictive/piezoelectric laminate is shown in Figure 1(a), in which a magnetostrictive layer, Terfenol-D, of length lm = 15 mm, width wm = 5 mm and thickness hm = 1 and 3 mm is perfectly bonded on the upper surface of a piezoelectric layer, PZT, of length lp = 20 mm, width wp = 5 mm and thickness hp = 0.5 mm. We will use subscripts m and p to refer to Terfenol-D and PZT layers, respectively. Dimensions hm(hp), wm(wp), lm(lp) are measured along the x1=x, x2=y and x3=z axis, respectively. The origin of the coordinate system is located at the center of the bottom left side of upper Terfenol-D layer, and the left end z=0 is clamped. Three layered laminate is also considered (see Figure 1(b)). Easy axis of the magnetization of Terfenol-D layer is the z-direction, while the polarization of PZT layer is the x-direction. The constitutive relations for Terfenol-D and PZT layers are given in Appendix A.
As we know, a magnetic domain switching gives rise to the changes of the magnetoelastic constants [9], and the constants d′15,d′31andd′33 for Terfenol-D layer in the z-direction magnetic field are
d′15=dm15d′31=dm31+m31Hzd′33=dm33+m33Hz |
(13) |
We performed finite element calculations to obtain the tip deflection, stress, induced voltage and induced magnetic field for the magnetostrictive/piezoelectric laminates. The average induced magnetic field in the z-direction at the side surface (at z=lm plane) is calculated as
Bin=1A∫ABz(x,y,lm)dA | (14) |
Terfenol-D (Etrema Products, Inc., USA) of lm = 15 mm, wm = 5 mm, hm = 1 and 3 mm and PZT C- 91 (Fuji Ceramics, Co. Ltd., Japan) of lp = 20 mm, wp = 5 mm, hp = 0.5 mm were used to make giant magnetostrictive/piezoelectric laminates by epoxy bonding (EP-34B; Kyowa Electronic Instruments Co. Ltd., Japan). Owing to cost and time constrains, the number of specimens (one or two at each types and thicknesses) was limited. It is noted that Terfenol-D is a rare earth iron and very expensive. Table 1 and Table 2 list the material properties of Terfenol-D [13,14] and C-91 [15], respectively. The second-order magnetoelastic constants m33 of Terfenol-D layer with hm = 1 and 3 mm of two-layered laminate are 5.0 × 10-12 and 3.3 × 10-12 m2/A2 [11], and the constants m33 of hm = 1 and 3 mm of three-layered laminate are 5.2 × 10-12 and 2.3 × 10-12 m2/A2 [12], respectively.
Elastic compliance (×10-12m2/N) | Piezo-magnetic constant (×10-9m/A) | Magnetic permittivity (×10-6H/m) | ||||||||
sH11 | sH33 | sH44 | sH12 | sH13 | dm31 | dm33 | dm15 | μ11 | μ33 | |
Terfenol-D | 17.9 | 17.9 | 26.3 | -5.88 | -5.88 | -5.3 | 11 | 28 | 6.29 | 6.29 |
Elastic compliance (×10-12m2/N) | Direct piezoelectric constant (×10-12m/V) | Dielectric permittivity (×10-10C/Vm) | ||||||||
sE11 | sE33 | sE44 | sE12 | sE13 | dm31 | dm33 | dm15 | ϵT11 | ϵT33 | |
C-91 | 17.1 | 18.6 | 41.4 | -6.3 | -7.3 | -340 | 645 | 836 | 395 | 490 |
Consider magnetostrictive/piezoelectric laminates under concentrated loading. Figure 2 and 3, for example, shows the setup for the experiment of a two-layered laminate (Figure 1(a)). Concentrated load P0 was applied at x = y = 0, z = lp by the cantilever load cell [6], as shown in Figure 2. First, the displacement for the two-layered and three-layered laminates under concentrated loading was measured with a laser displacement meter (see Figure 3(a)). Next, the induced voltage of these laminates was measured using an oscilloscope (see Figure 3(b)). The x=hp plane was grounded. Then, the induced magnetic field over the total area on z=lm plane of Terfenol-D layer was measured using a Tesla meter (see Figure 3(c)). The hall probe was touched on the edge of Terfenol-D layer, and this set-up allowed a precision of induced magnetic field measurement of ± 0.01 mT.
We first present results for the two-layered magnetostirctive/piezoelectric laminates. Figure 4 shows the tip deflection wtip versus applied concentrated load P0 at x=hp,y=0,z=lp for the two-layered laminates with hm = 1 and 3 mm. The lines and plots denote the results of FEA and test. The experimental scatter is small, and the representative plots from the tests are shown. The tip deflection increases as the thickness of the Terfenol-D layer decreases. The FEA results are good agreement with experimental measurements. Figure 5 shows the induced voltage Vin versus applied concentrated load P0 at x=0 plane for the two-layered laminates with hm = 1 and 3 mm, obtained from the FEA and test. As the concentrated load increases, the induced voltage increases. The comparison between the numerical predictions and the experimental results for the two-layered laminate with hm = 3 mm yields a good agreement. For the laminate with hm = 1 mm, the trend is similar between the numerical predictions and the experimental results, though the experimental data are smaller than the predicted ones because of the voltage saturation under high mechanical loads. Figure 6 shows the induced magnetic field Bin versus applied concentrated load P0 for the two-layered laminates with hm = 1 and 3 mm, obtained from the FEA. Also shown are the measured data for hm = 3 mm. The comparison between the FEA and test is reasonable. As the concentrated load increases, the induced magnetic field increases. The induced magnetic field increases as the thickness of the Terfenol-D layers decreases. The variations of normal stress σzz along the thickness direction are calculated at the clamped end (y=0 mm and z=0 mm) for the two-layered laminates and the results are shown in Figure 7. All calculations are done at a fixed tip deflection of wtip = 1 μm. The applied loads of PZT layer for wtip = 1 μm are about P0 = 27.0, 62.7 mN for hm = 1 and 3 mm, and the corresponding induced voltage and induced magnetic field are about Vin = 0.338, 0.229 V and Bin = 0.024, 0.032 mT, respectively. Small stress gaps at the interface between Terfenol-D and PZT layers are observed. Figure 8 also shows the variations of normal stress σzz along the thickness direction near the free edge of Terfenol-D layer (y=0 mm and z=14.5 mm) for the two-layered laminates at the same condition. The normal stress in Terfenol-D layer is almost zero and the normal stress in PZT layer changes from tensile to compressive. There are some stress gaps at the interface between Terfenol-D and PZT layers. Figure 9 shows the variations of shear stress σzz along the length direction at the interface between Terfenol-D and PZT layers (x=0 mm and y=0 mm) for the two-layered laminates at the same condition. The maximum shear stress is observed near the free edge of Terfenol-D layer. Low shear stress is noted for small Terfenol-D layer thickness.
Next, the results of the three-layered magnetostrictive/piezoelectric laminates are presented. Figure 10 shows the tip deflection wtip versus applied concentrated load P0 at x=hp,y=0,z=lp for the three-layered laminates with hm = 1 and 3 mm. The lines and plots denote the results of FEA and test. The experimental scatter is small, and the representative plots from the tests are shown. The tip deflection for hm = 1 mm is larger than that for hm = 3 mm, and both the numerical predictions and the experimental results show a same tendency. Figure 11 shows the induced voltage Vin versus applied concentrated load P0 for the three-layered laminate with hm = 3 mm. Also shown is the induced magnetic field Bin. Only one datum for Bin is plotted due to the accuracy limit of the Tesla meter. As the concentrated load increases, both the induced voltage and the induced magnetic field increase. The induce voltage of the two-layered laminate was much larger than that of the three-layered laminate. In addition, the induced voltage increases with decrease in the thickness of magnetostrictive layers. It stems from the facts that two-layered laminate and thin magnetostrictive layers are more easily deformed than three-layered laminate and thick magnetostrictive layers. However, if the thickness of the Terfenol-D layer is reduced to several micrometers or more less, induced voltage decreases because the volume effect on the magnetization is more dominant than the magnetic field generation by deformation. The variations of normal stress σzz along the thickness direction are calculated near the free edge of Terfenol-D layer (y=0 mm and z=14.5 mm) for the three-layered laminates and the results are shown in Figure 12. All calculations are done at a fixed tip deflection of wtip = 1 μm. The applied loads of PZT layer for wtip = 1 μm are about P0 = 62.7, 78.3 mN for hm = 1 and 3 mm, and the corresponding induced voltage and induced magnetic field are about Vin = 2.64, 3.30 mV and Bin = 0.051, 0.026 mT, respectively. There are some stress gaps at the interface between Terfenol-D and PZT layers. At smaller Terfenol-D layer thickness, lower stress gap is found for the same tip deflection. The normal stress in Terfenol-D layer is almost zero and the normal stress in PZT layer changes from tensile to compressive.
These results are helpful in considering the energy harvesting from impact. And, it is important that the internal stress is evaluated for fracture and delamination of the laminates also. Our present study offers a method for aiding the design of new energy harvesting devices, and provides a rational basis for refining the real device designing in order to reduce fracture and increase efficiency of electric power generation. Work in this area is currently being pursued.
A numerical and experimental investigation of the magnetostrictive/piezoelectric laminates under concentrated loading was conducted. The tip deflection, induced voltage and induced magnetic field are predicted using finite element simulations, and comparison with the measured data shows that current predictions are reasonable. It was found that smaller magnetostrictive layer thickness gives larger tip deflection, induced voltage and induced magnetic field. Also, the induce voltage of the twolayered laminate was much larger than that of the three-layered laminate. In addition, the stress gap at the interface is small when the magnetostrictive layer thickness is small. This study may be useful in designing advanced magnetostrictive/piezoelectric laminates with energy harvesting capabilities.
This work was supported by Grant-in-Aid for JSPS Fellows (23-3402).
All authors declare no conflicts of interest in this paper.
For Terfenol-D, the constitutive relations can be written in the following form:
{εxxεyyεzzεyzεzxεxy}=[sH11sH12sH13000sH12sH11sH13000sH13sH13sH33000000sH44/2000000sH44/2000000sH66/2]{σxxσyyσzzσyzσzxσxy}+[00d′3100d′3100d′330d′15/20d′15/200000]{HxHyHz} |
(A.1) |
{BxByBz}=[0000d′150000d′1500d′31d′31d′33000]{σxxσyyσzzσyzσzxσxy}+[μ11000μ11000μ33]{HxHyHz} |
(A.2) |
sH11=sH1111=sH2222, sH12=sH1122, sH13=sH1133=sH2233, sH33=sH3333sH44=4sH2323=4sH3131, sH66=4sH1212=2(sH11−sH12)} |
(A.3) |
d′15=2d′131=2d′223, d′31=d′311=d′322, d′33=d′333 |
(A.4) |
{εxxεyyεzzεyzεzxεxy}=[sE33sE13sE13000sE13sE11sE12000sE13sE12sE11000000sE66/2000000sE44/2000000sE44/2]{σxxσyyσzzσyzσzxσxy}+[d3300d3100d310000000d15/20d15/20]{ExEyEz} |
(A.5) |
{DxDyDz}=[d33d31d3100000000d150000d150]{σxxσyyσzzσyzσzxσxy}+[ϵT33000ϵT11000ϵT11]{ExEyEz} |
(A.6) |
sE11=sE2222=sE3333, sE12=sE2233, sE13=sE1122=sE1133, sE33=sE1111sE44=4sE1212=4sE1313, sE66=4sE2323=2(sE11−sE12)} |
(A.7) |
d15=2d313=2d212, d31=d122=d133, d33=d111 |
(A.8) |
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Elastic compliance (×10-12m2/N) | Piezo-magnetic constant (×10-9m/A) | Magnetic permittivity (×10-6H/m) | ||||||||
sH11 | sH33 | sH44 | sH12 | sH13 | dm31 | dm33 | dm15 | μ11 | μ33 | |
Terfenol-D | 17.9 | 17.9 | 26.3 | -5.88 | -5.88 | -5.3 | 11 | 28 | 6.29 | 6.29 |
Elastic compliance (×10-12m2/N) | Direct piezoelectric constant (×10-12m/V) | Dielectric permittivity (×10-10C/Vm) | ||||||||
sE11 | sE33 | sE44 | sE12 | sE13 | dm31 | dm33 | dm15 | ϵT11 | ϵT33 | |
C-91 | 17.1 | 18.6 | 41.4 | -6.3 | -7.3 | -340 | 645 | 836 | 395 | 490 |
Elastic compliance (×10-12m2/N) | Piezo-magnetic constant (×10-9m/A) | Magnetic permittivity (×10-6H/m) | ||||||||
sH11 | sH33 | sH44 | sH12 | sH13 | dm31 | dm33 | dm15 | μ11 | μ33 | |
Terfenol-D | 17.9 | 17.9 | 26.3 | -5.88 | -5.88 | -5.3 | 11 | 28 | 6.29 | 6.29 |
Elastic compliance (×10-12m2/N) | Direct piezoelectric constant (×10-12m/V) | Dielectric permittivity (×10-10C/Vm) | ||||||||
sE11 | sE33 | sE44 | sE12 | sE13 | dm31 | dm33 | dm15 | ϵT11 | ϵT33 | |
C-91 | 17.1 | 18.6 | 41.4 | -6.3 | -7.3 | -340 | 645 | 836 | 395 | 490 |