ON THE TOLERANCE WAVE - TYPE SOLUTIONS TO THE HYPERBOLIC HEAT CONDUCTION
IN MICROPERIODIC COMPOSITES
Lena Łacińska, Ewaryst Wierzbicki
Institute of Mathematics, Czestochowa University of Technology, Poland, lena@imi.pcz.pl
Abstract. In the note wave-type solutions to the hyperbolic heat conduction in the micrope- riodic composites are investigated. The considerations are related to the tolerance averaged model of the hyperbolic heat transfer problems in microperiodic composites. The Lapunov exponent notation is applied. The open form of these solutions are formulated in one shape function case.
Introduction
Many physical problems can be described by the Cauchy problem for a system of ordinary differential equations with constant coefficients, which in the consis- tent matrix form can be written in the form
, (0 )
T T
x & = Ax x + = x
o(1)
for known real matrices A and x
oof n n × and 1 n × dimensions, respectively, and unknown 1 n × matrix x = x(t), t > 0. The solution to this equation can be written in the form
T o At
T
e x
x = (2)
where
∑
∞=
=
0
!
n n n At
n t
e A (3)
is the Lapunov exponential of matrix At. It is the well known fact that
t B A Bt
At
e e
e =
( + )(4)
provided that AB = BA . Indeed if x
T= e
−Bty
Tthen and under (2) and
T o Bt T
o
e x
y =
−one can obtain
T o At Bt
T
e e y
y = (5)
and
y B Ae e
y & = (
Bt −Bt+ ) (6)
Hence under assumption AB = BA and (6)
T o t B A
T
e y
y =
( + )(7)
and then conditions (5), (7) yield (4).
In this note we are to apply mentioned above Lapunov exponents notation properties to the discussion of the wave-type solutions in the hyperbolic heat con- duction in the microperiodic solid.
1. Formulation of the problem
The starting point of consideration will be tolerance averaged model of the hy- perbolic heat conduction in the one dimensional microperiodic solid which occupy the interval [0,L] in the reference configuration. Denoting by c mean heat, by τ relaxation time and by k conductivity constant and assuming that the total tempera- ture together with temperature gradient in every constituent for a certain shape functions sequence h = [h
1,…, h
n] can be approximated with a sufficient accuracy by decompositions
( , ) x t u x t ( , ) h x ( )
T( , ), x t
x( , ) x t u x t
x( , ) h x
x( )
T( , ) x t
θ = + υ θ = + υ (8)
respectively. Here and in the subsequent considerations symbol ( ) ⋅
x, ( ) ⋅
xxdenote spatial derivatives. In above decompositions u = u(x) and υ = υ (x) = [ υ
1,…, υ
n] are new basic unknowns named as the averaged temperature and the vector of temperature amplitudes. Temperature introducing a certain finite which will be rewritten here in the form, cf. [1]
[ ] 0
{ } [ ] 0
T
xx x
T T
x
c u c u k u k
k k u
τ τ
υ γ υ γυ υ
〈 〉 + 〈 〉 − 〈 〉 − =
+ + + =
&& &
&& & (9)
where under the averaged operator over the unit cell [– λ /2, λ /2]
] 2 / , 2 / [ , ) 1 (
) (
2 /
2 /
λ λ λ
λ λ
−
∈
=
〉
〈 ∫
−
L x
dy y f x
f (10)
matrix coefficients are given by
{ } [ ]
1 1 1 1 1 1
1 1
1 1 1
1 1
, ,
, , ,
n n
n n n n n n
n
x x x x
n
x x
n n n
x x x x
ch h ch h c h h c h h
ch h ch h c h h c h h
kh h kh h
k k kh kh
kh h kh h
τ τ
τ
τ τ
γ γ
= =
= =
K K
K K K K K K
K K
K
K K K K
K
(11)
We are to investigate wave-type solutions of the tolerance equations (9) of the form
) ( ) , ( ), (
) ,
( x t F x t x t G x t
u = − µ υ
T= − µ (12)
where F ( ) ⋅ and G ( ) ⋅ are new unknowns. Similar problem has been investigated in [1]. Under (12) tolerance model equations (9) take form
2 2
( ) " ' [ ] ' 0
" ([ ]
T) ' { } 0
c k F c F k G
G k G k G
τ τ
µ µ
γ µ γµ
〈 〉 − 〈 〉 − 〈 〉 − =
+ − + = (13)
where prime denotes the derivation of the wave-type generators F and G. Introduc- ing additional unknowns U ( ) ⋅ and V ( ) ⋅ by interrelations
( ) ' ( ) ( ) , ' ( )
U y = F y V y = G y (14)
and denoting unit n n × matrix by I the equations (13) can be rewritten in matrix
nform
2
2
'
0 0 0 [ ]
0 0 ' 0 0
0 0 ' 0 { } ([ ] )
n n
T
U U
c k c k
I G I G
k k
V V
τ
τ
µ µ
γ µ γµ
〈 〉 − 〈 〉 〈 〉
=
− − −
(15)
which, together with a certain initial conditions and under condition U = F ′ , is
equivalent to the Cauchy problem (1) for
2
1 1
2 2
0 0 0 0
0 [ ]
0 0 ,
1 1
0 { } ([ ] )
[ , , ]
n
T
T
c k
c k
A I
k k
x U G V
τ
τ τ
µ µ
γ γ γµ
µ
−µ
− 〈 〉
〈 〉 − 〈 〉
=
− − −
=
(16)
The problem we are to discuss the solution to the Cauchy problem (1) specified by (16).
2. Analysis
To discuss solutions to the Cauchy problem (1) specified by (16) we are to de- compose matrix A given in (16) into sum
2 1
1
2 1
nA H trA I
n
+= + + (17)
and hence
2
1 1
2 2
1 0 [ ]
2 1
0 1
2 1
1 1 1
0 { } ([ ] )
2 1
n n
T
n
c trA k
n
c k
H trA I I
n
k k trA I
n
τ
τ τ
µ µ
γ γ γµ
µ
−µ
− 〈 〉 −
〈 〉 − 〈 〉 +
= − +
− − − −
+
(18)
Since terms in decomposition (17) commute solution to the mentioned above Cauchy problem can be written in the form
0 Hy trAy
U
G e e x
V
=
(19)
It must be emphasized that the term e
trAyof the right hand side of (19) is a sca- lar one. At the same time H a certain traceless matrix coefficient, J
1= trH = 0.
Hence the Caqyley-Hamilton polynomial related to this matrix has the form
2 1 2 1 2 1
2 3
...
2det
n n n
z
++ J z
−+ J z
−+ + J z
n+ H (20)
where J
1, J
2,... J
2n, J
2n+1= det H are invariants of H. It is mean that term trHz in
2n(20) is equal to zero.
Now we are to restrict considerations to the special case in which the tolerance model of the hyperbolic heat conduction includes exclusively one shape function.
In this case matrix A as well as H are of 3 3 × dimension and hence polynomial (20) takes the form
3
2
det
z + J z + H (21)
It is mean that (21) has three roots of the form a, − 0.5a+bi, − 0.5a − bi and hence
/ 2
/ 2
0 0
0 (cos sin ) 0
0 0 (cos sin )
ay
Hy ay T
ay
e
e O e by i by O
e by i by
−
−
= +
−
(22)
for a certain orthogonal matrix O.
More detailed analysis and the physical reliability discussion of the solution (19) to the considered Cauchy problem will be explained elsewhere.
References
[1] Woźniak C., Woźniak M., 2D-dynamics of a stratified elastic subsoil layer, Arch. Appl. Mech.
1996, 66, 284-290.
[2] Moller C.,Van Loan C.,Nineteen dubious ways to compute the exponential of the matrix. Twen- ty-five years later, SIAM Review, Society for Industrial and Applied Mathematics 2003, 74, 1, 3000.
