Solution of boundary-value problems of the theory of thermopiezoelasticity
for a half-plane
M. Chumburidze
Akaki Tsreteli State University. Associated professor
Preface. We has considered the basic nonstationary system of differential equations of mechanical thermopiezoelectricity
for isotropic homogeneous media taking account Toupin’s and Midlin’s assumptions.
Introduction. Basic nonstationary system of differential equations of mechanical
thermopiezoelectricity for isotropic homogeneous media taking account Toupin’s and Midlin’s
assumptions has the form[1]
d 2v
c 44 v x, t   (c12  c 44 ) graddivv x, t   d 44 q  x, t   (d12  d 44 ) graddivq x, t   gradv 6  x, t    2
dt
d 44 v x, t   (d12  d 44 ) graddivv x, t   b44  b77 q x, t   (b12  b44  b77 ) graddivq x, t  
(1)
 gradv 5  x, t   gradv 6  x, t   aq  x, t 
1
v5  x, t   divq x, t   0
e0
1 v6


v6  x, t  
  divv   divq
 t
t
t
where, v  v1 , v2   vk
T
q  q1 , q2   v2k
T
2 x1
2 x1
-diplacement vector(single-column matrix),
, k  1,2 -polarization vector, v5-elecric potential of the domain, v6-temperature
variation;x=(x1,x2)-point of two-dimensional Euclidean space E2, t-time, ∆- two-dimensional Laplacian
operator; c44,c12,d44,d12,b44,b12,b77,ϛ,a,….elastic, elecri and heat constants[2].
___
Object and methods of research.Further it will be assumed that v k  x, t , k  1,6 is represented by the
Laplace-Mellin integral (general dynamical case)
 i
1
vk x, t  
et u k x, d (2)

2i  i
where     i ,   0, i   1 in view of (2) the system (1) is reduced to the form (with respect of
U ( x, )  u, p, u5 , u6   u k
6 x1
):
c 44 u  (c12  c 44 ) graddivu  d 44 p  (d 12  d 44 ) graddivp  gradu 6   2 u  0
d 44 u  (d 12  d 44 ) graddivu  b44  b77 p  (b12  b44  b77 ) graddivp 
 gradu 5  gradu 6  ap  0
u 5 
(2)
1
divp  0
e0

u 6 divu  divp  0

For the components of force-stress and electric-stress we have[2]:
 u j u k 
 p

  d 21divp jk  d 44  j  p k 
 jk x,   c 21divu  u 6  jk  c 44 


 x

 x k x j 
 k x j 
 u j u k 
 p


p 
  b21divp jk  b44  j  p k   b77  p k  j 
 jk x,   d 21divu  u 6  jk  d 44 


 x

 x

 x k x j 
 k x j 
 j x k 
Lemma 1. Regular solution of the system (2) U ( x, )  u, p, u5 , u6   u k 6 x1 admits in the domain of
u 6  x, t  


regularity representation U ( x, )  u 1  u 2  , p 1  p 2 , u5 , u6  uk 6 x1
where
1
1
3
3
2
 u 2  


2 u
2  p
2












  ki 

  0,    k i  u   0,    k i 3  p 2    0,
u
i 1
i

1
i

1
 6 
 5 


 u 1 
 u 2  


rot  1   0, div  2    0
p 
p 
and the constants k i2 , i  1,5 are determined by the identities
 d12  2d 44       (b12  2b44 )     1e  a  c12  2c 44 
 0


1
 k i2 
2
3
(b12  2b44 )c12  2c 44   d12  2d 44 
i 1

3
k
i 1
k 
2
2
i 1 i
3

 1  a c  2c    2  1  a    3 
 12
 e

44
  e0

 0

2
(b12  2b44 )c12  2c 44   d12  2d 44 
(b12  2b44 )  
 3
 1  a
 e



 0

2
k


i
2
(b12  2b44 )c12  2c 44   d12  2d 44 
i 1
2
3
 k i23 
i 1
2
 k i23 
i 1
ac 44   2 b44  b77 
2
c 44 b44  b77   d 44
a 2
2
c 44 b44  b77   d 44
Lemma2. Parameters k i2 , i  1,5 are complex analytic functions of  , and for large values  ,
k    o  i  1,5
Lemma3. if scalar functions u ji , p ji i  1,3, j  1,5,6; u 54 , p14 and vector functions u 2l  , p 2l  , l  1,2
satisfy the equations:
 u ji 
u 
  0,  54   0 ,
p 

 14 
 ji 
and are related by the equaties:
  k  p
2
i

2l 
  k  u 
2
l 3

  0, divu 2l   0
2l  
p 

 k i2 (c12  2c 44 )   2 u1i  k i2 (d 12  2d 44 ) p1i  u 6i  0

 (k i2 (b12  2b44 )  a ) p1i  k i2 (d 12  2d 44 )u1i  u 6i  0

u  1 p  0
 5i e 1i
0

ap

u
 14
54  0

2
2
2l 
2
2l 
 k l 3 c 44   u  k l 3 d 44 p  0
 k 2 b  b   a p 2l   k 2 d u 2l   0
l 3
44
77
l  3 44




then vector
U  gradu1  u 2 , gradp1  p 2 , u5 , u6
where






3
2
i 1
i 1
u 1   u1i , u 2    u 2i  ,
4
2
p 1   p1i
p 2    p 2i 
i 1
4
u 5   u 5i
i 1
i 1
3
u 6   u 6i
i 1
is a regular solution of the piezoelectricity equations (2)
Fairness of there lemmas is established by analogy of couple-stress thermoelasticity[2]
Statement of problem. Let D-half-plane: D  x  E 2 , x2  0 Consider the first boundary-value
problem: it is required to find in D the regular solution U ( x, )  u, p, u5 , u6   u k 6 x1 of system (2),


satisfying the boundary condition: on x2  0 are given displacement, polarization, electric potential and
temperatur u x1 ,0   1 x1 , px1 ,0   2 x1 , u 5 x1 ,0   5 x1 , u 6 x1 ,0   6 x1  e, i.e.
where  1  1 , 2   ( 2)  3 , 4  , 5 , 6 -given functions representable by Foutier’s integrals:
T

T

1
1
 k x1  
f r  e ix1 d , k  1,6  f k x1  
 r  e ix1 d


2 
2 
Solution of problem, according to Lemma 3, is sought in the form:
U  grad u11  u12  u13   u 21  u 22 , grad  p11  p12  p13  p14   p 21  p 22 , u 51  u 52  u 53  u 54 , u 61  u 62  u 63 
where,
u lj 
1
2
1
p1 j 
2
u 54 
1
2
1
p14 
2
u 2,n  
p
2,n 


lj
exp(  x 2  2  k 2j ) exp( ix1 )d



1j
exp(  x 2  2  k 2j ) exp( ix1 )d ,
54
exp( ix 2 ) exp( ix1 )d
j  1,3; l  1,5,6






14
exp( ix 2 ) exp( ix1 )d

1
2
1

2


2 n 
exp(  x 2  2  k n23 ) exp( ix1 )d
2 n 
exp(  x 2  2  k n23 ) exp( ix1 )d , n  1,2




where  lj , 1 j , j  1,3; l  1,5,6,  54 , 14 -unknown scalars, while  2 n  ,  2 n  are unknown vectors.
Result and discussion.Thus , we have 22 unknown scalars. According to Lemma 3 and the boundary
conditions, to determine these scalars we obtain complete algebraic system with 22 scalar equations:


 k i2 (c12  2c 44 )   2  1i  k i2 (d 12  2d 44 ) 1i   6i  0

 (k i2 (b12  2b44 )  a )  1i  k i2 (d 12  2d 44 ) 1i   6i  0

  1   0 i  1,3
 5i e 1i
0

a 14   54  0

2
2
2l 
2
2l 
 0;  k l23 b44  b77   a  2l   k l23 d 44 2l   0
 k l  3 c 44     k l 3 d 44 
 i 2l   r  2l   0 l  1,2
1
l 3 1

3
2

)

i



 1( 2 j  f 1



1j
j 1
j 1

2
 3
)
   1 j r j    2( 2 j  f 2
 j 1
j 1

4
2
 i   1 j    1( 2 j )  f 3

j 1
j 1
 3
2
  r  i   ( 2 j )  f

1j j
14
2
4
 
j 1
j 1
4

(3)
  5 j  f 5
 j 1
3
  6 j  f 6
 j 1




where, r j   2  k 2j ; r j 3   2  k 2j 3
By analogy can be considered other boundary-value problems. For example, for the second problem on
x2  0 are given force-stresses electric stresses, charge of line and heat flow. Boundary conditions have
form:
 u
 p
u 
p 
1   21  x,   c 44  2  1   d 44  2  1 
 x1 x 2 
 x1 x 2 
u
p
2   22  x,   c 21divu  u 6  2c 44 2  d 21divp  2d 44 2
x 2
x 2
 u
 p
 p p
u 
p 
3   21  x,   d 44  2  1   b44  2  1   b77  1  2
 x1 x 2 
 x1 x 2 
 x 2 x1
u
p
4   22  x,   d 21divu  u 6  2d 44 2  b21divp  2b44 2
x 2
x 2
5  e0
6 



u 5
 p2
x 2
u 6
x 2

1
1
 r  e ix1 d , k  1,6   k x1  
where , k x1  

2 
2

   e
r
ix1
d

The system (3 )will be replaced by the last six equation:
3
2
2
3
2
2




c 44  2i  r j  1 j  i   22 j     12 j  r j 3   d 44  2i  r j  1 j  2 2  14    22 j     12 j  r j 3    1
j 1
j 1
j 1
j 1
j 1
j 1




3
2
3
3
2
c 21 (   1 j  i   12 j  )    r j  6 j  2c 44 ( r j  1 j   r j 3 22 j  ) 
j 1
j 1
j 1
3
2
3
j 1
j 1
j 1
2
j 1
j 1
2
3
j 1
j 1
2
 d 21 ( 2   1 j    12 j    r j 1 j   r j 3  22 j  )  2d 44 ( r j  1 j   2  14   r j 3  22 j  )   2
2
2
j 1
3
2
2


d 44  2i  r j  1 j  i   22 j     12 j  r j 3  
j 1
j 1
j 1


3
2
2
2


 2

 b44  2i  r j  1 j  2 2  14    22 j    12 j  r j 3   b77   12 j  r j 3    22 j     3
j 1
j 1
j 1
j 1


 j 1

3
2
3
3
j 1
j 1
j 1
j 1
2
d 21 (   1 j  i   12 j     r j  6 j )  2d 44 ( r j  1 j   r j 3 22 j  ) 
3
2
3
j 1
j 1
j 1
2
j 1
2
3
j 1
j 1
2
 b21 ( 2   1 j    12 j    r j  1 j   r j 3  22 j  )  2b44 ( r j  1 j   2  14   r j 3  22 j  )   4
3
3
j 1
j 1
2
e0 ( r j  5 j  i54 )   r j  1 j  i14   5
3
  r j 6 j   6
j 1
where, r j   2  k 2j ; r j 3   2  k 2j 3
2
j 1
References
1. W.Nowaski. Theory of Elasticity. Issue, 1975.p85
2. W. Kpraze, T.Gegelia, M. Bashaleisvili, T.Bucladze. Three-Dimensional Problems of the
mathematical Theory of Elasticity and Thermoelastisity. North-Holland publishing company.
Amsterdam, New York, Oxford, 1979,p.300