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Smart Material Systems and MEMS - Vijay K. Varadan Part 8

Tham khảo tài liệu 'smart material systems and mems - vijay k. varadan part 8', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 206 Smart Material Systems and MEMS o 0.5 0 2 10 Figure 8.13 Magnetostriction-magnetic field curves for different stress levels in Terfenol-D. 4 6 8 Magnetic field x104 amp m --- 6 MPa --- 8 MPa -e- 10 MPa 12 MPa H 14 MPa 16 MPa -B 18 MPa 20 MPa 22 MPa 24 MPa 12 Here u x y t v x y z t and w x y z t are the displacement components in the three coordinate directions Nm is the shape functions associated with mechanical degrees of freedom and U e is the nodal displacement vector. If an isoparametric formulation is used then the conventional isoparametric shape functions in natural coordinate could be adopted. The strains can be expressed in terms of displacement through a straindisplacement relationship that is e exx Syy ezz gyz gxz 7xy T B U e 8.76 4 r. 3.5 3 Gj 2.5- 2 1.5 CD Ẹ 1 0.5 2 OT g --- 0 amp m --- 5000 10000 15000 -I 20000 25000 30000 -9- 35000 1.6 0.2 0.4 0.6 0.8 1 1.2 1.4 Elastic compressive strain x 10 5 ơ 1.8 0 . 0 Figure 8.14 Stress-strain curves for different magnetic field intensities in Terfenol-D. Modeling of Smart Sensors and Actuators 207 where B is the strain-displacement matrix and its evaluation is given in Chapter 7. For coupled analysis we need to take the magnetic field as the independent degree of freedom. In such cases we can write the magnetic field in the three coordinate directions as H HX Hy Hz NH H e 8.77 where H e is the nodal magnetic field vector and NH is the shape function associated with the magnetic field degree of freedom. The strain energy in a structure with magnetostrictive patches over a volume V is given by Ve 1 j e T s dV V Substituting for ơ from Equation 8.74 converts the above equation into terms of strains and magnetic field vectors. In this equation the strains are expressed in terms of displacement using Equation 8.76 and the magnetic field in terms of the nodal magnetic field vector using Equation 8.77 . The resulting expression for the strain energy will become Ve 2 U eT Kuu U e - 1 U eT KuH H e 8.78 where Kuu j B