PSI - Issue 3

M. Colussi et al. / Procedia Structural Integrity 3 (2017) 153–161 M. Colussi et al. / Structural Integrity Procedia 00 (2017) 000–000

155

3

H

s

kij d H





T     

ij

ijkl kl

k

(2)

B d 

H

i

ikl kl

ik k

ijkl s , ikl d ,

where ij  are the components of the strain tensor and H

T ik  are respectively the magnetic field elastic

compliance, the magnetoelastic constants and the magnetic permittivity. Valid symmetry conditions are:

H H

H H

s

jikl      ijlk kji T s s

s

ijkl

klij

kij d d

(3)

T ij   ji

The relation between the strain tensor and the displacement vector u i is:

1 2





, j i u u 

ij  

(4)

, i j

The magnetic field intensity, named φ the potential, is written as:

H

 

(5)

,

i

i

For Terfenol-D, the constitutive relations can be written as:

0 0 0 0 0 0

d d d

11                     22 33 23 31 12 2 2 2      

11                     22 33 23 31 12      

         

         

11 s s s s s s 12 13 0 0 0 s s s H H H 13 33 H H H H H H 13 12 11 13

0 0 0 0 0 0 0 0 0

        

 

31

1            2 3 H H H   

31

33





(6)

0

0

d

H

0 0

s

15

44

  

15 0 0 0 0 0

d

H

0 0 0 0

0

s

44

0 0 0 0 0

H

s

66

     

11           22

    

0  T

0 0 0 0

0

0 0

B B

d

1     2       3 H H H

1      

    

15

11

33

T 

0 0 0

0 0

0  T

d





(7)

2    

23         31 12  

  



15

11

3    B d d d  31 31

0 0 0

0 0

33

33

where:

,

,

23                         32 31 13 12 21 23 32 31 13 12 21 11 s s 1111 1122 13 , , , , H H H H H s s s

  

H H

H

H H

,

s s

s

s s 

1133  

2222 12 4 , H s

2233 33

3333





2323     4 H s

1212 11 s s    4 2 H H H s 12

H

H

s

s

44

3131 66

15 311   2 2 , , d d d d d d d d  332 33 131 223 31

333