6 Construction of the Solutions of Non-stationary …
95
u(r, τ ) = u
(0)
(r ) − 2P 0
∞
k=1
Y (r, χ k ){g(z k )e
z k τ
+ 2Re[g(α k + iω k )e
(α k +iω k )τ
]},
(6.26)
where
u
(0)
(r ) = −P 0
(1 − 2ν 0 )r 0
(1 − a/b)[(1 − 2ν 0 )r
2
0 + 1]
(
r
r 0
−
r 0
r
),
(6.27)
Y (r, χ) =
1
χ dX (χ )
dχ
[J 1 (r 0 χ)N 1 (r χ) − N 1 (r 0 χ)J 1 (r χ)],
(6.28)
X (χ ) = N 1 (r 0 χ)[J 1 (χ ) − wχ J 0 (χ )] − J 1 (r 0 χ)[N 1 (χ ) − wχ N 0 (χ )], (6.29)
g(s) =
(s + b)
2
2s 2 + (4b − 3a)s + 2b(b − a)
,
(6.30)
where J 0 , J 1 , N 0 , N 1 are Bessel functions of the first and second kinds of zero and
first indices; χ k (k = 1, 2, 3, . . .) are the real roots of the equation
X (χ ) = 0
(6.31)
while z k and α k + iω k (ω k > 0) are real and one of two complex conjugate roots of
the Eq. (6.22).
6.5 Notes and Comments
With a constant Poisson’s ratio in the case of a two-parametrical exponential kernel
(6.21), the solution of a non-stationary dynamic problem of linear viscoelasticity
will not be hard to obtain if a solution of the corresponding problem of the elasticity
theory is already known. In the above-stated example, the solution was presented in
a rather simple form and remains true within the whole range of time changing.
Acknowledgements The reported study was funded by Russian Foundation for Basic Research,
according to the research projects Nos. 18-08-00471 a, 19-38-70005 mol_a_mos.
References
Amendola, G., Fabrizio, M., Golden, J. M. (2012). Thermodynamics of materials with memory, 574
p. Heidelberg: Springer.