Strain solitary waves 195
Hence the microstructure affects only dispersion in Eq.19. The solitary wave solution of Eq.19 is
20 v =
6ν E R
2
k
2
β 1 + 4γ
1 − 4γ −
1 − νV
2
c
2 ∗
cosh
− 2
k x − V t , where V is a free parameter while the wave number k is defined by
21 k
2
= 2ρ
V
2
− c
2 ∗
ν E R
2 1+4γ
1−4γ
−
1−νV
2
c
2 ∗
. Therefore the contribution of the microstructure results in the widening of the permitted
solitary wave velocities, 1
V
2
c
2 ∗
1 1 − ν
1 + 4γ 1 − 4γ
. Also the characteristic width of the solitary wave proportional to 1 k becomes larger relative
to the wave width in pure elastic case, γ = 0. We consider γ to be rather small due to the experimental data from Ref. [20]. Then the type of the solitary wave compressiontensile is
defined by the sign of the nonlinearity parameter β like in case without microstructure.
4. Nonlinear waves in a rod with Le Roux continuum microstructure
The procedure of obtaining the governing equations is similar to those used in previous section. The nonzero components of the tensor Ŵ
K L M
are Ŵ
x x x
= −
u
x x
, Ŵ
x xr
= Ŵ
r x x
= − u
xr
, Ŵ
xr x
= −w
xr
, Ŵ
xrr
= Ŵ
rr x
= −w
xr
, Ŵ
r xr
= − u
rr
, Ŵ
rrr
= −w
rr
. The b.c. 11, 12 are satisfied for the strain tensor components
P
rr
= λ +
2µ w
r
+ λ w
r + λ
u
x
+ λ +
2µ + m 2
u
2 r
+ 3λ + 6µ + 2l + 4m
2 w
2 r
+ λ +
2l w
r
w r
+ λ +
2l 2
w
2
r
2
+ λ + 2l u
x
w
r
+ 2l − 2m + n u
x
w r
+ λ +
2l 2
u
2 x
+ λ +
2µ + m 2
w
2 x
+ µ + m u
r
w
x
+ 2J
∗
2u
x t t
+ w
rt t
− 2a
1
u
x x x
− 2a
1
+ 2a
2
w
x xr
− 2a
1
+ a
2
1 r
r w
rr r
− a
1
1 r
r u
xr r
, 22
P
r x
= µ
u
r
+ w
x
+ λ + 2µ + m u
r
w
r
+ 2λ + 2m − n u
r
w r
+ λ +
2µ + m u
x
u
r
+ 2m − n
2 w
x
w r
+ µ + m w
x
w
r
+ µ +
m u
x
w
x
+ 2J
∗
u
rt t
− a
1
w
xrr
− 2a
1
+ 2a
2
u
x xr
− 2a
2
1 r
r u
rr r
. 23
Then the approximations for the components of the displacement vector have the form 24
u = U x, t + ν
r
2
2 1
1 − N U
x x
,
196 A.V. Porubov
w = −ν r U
x
− 4J
∗
2 − ν1 + ν1 − 2ν E 3 − 2νR
2
r
3
U
x t t
− ν
2
− 1 − 2ν1 − N G1 − ν − 2ν N
23 − 2ν1 − N r
3
U
x x x
− ν1 + ν
2 +
1 − 2ν1 + ν E
l1 − 2ν
2
+ 2m1 + ν − nν
r U
2 x
, 25
where G = 2a
1
µ R
2
, N = 2a
2
µ R
2
. Like in previous section the governing equation for longitudinal strain v = U
x
is the double dispersive equation 19 whose coefficients are defined now as
α
1
= c
2 ∗
, α
2
= β
2ρ , α
3
= ν
R
2
21 − N −
ν
2
R
2
2 +
2J
∗
ν 2 − ν, α
4
= ν
c
2 ∗
R
2
21 − N .
Solitary wave solution has the form 26
v = 6ν E R
2
k
2
β 1
1 − N −
1 1 − N
− ν + 4J
∗
2 − ν R
2
V
2
c
2 ∗
cosh
− 2
k x − V t , where V is a free parameter, and the wave number k is defined by
27 k
2
= 21 − N ρ
V
2
− c
2 ∗
ν E R
2
h c
2 ∗
− V
2
1 − ν1 − N + 4J
∗
1 − N 2 − νR
2
i . Physically reasonable case corresponds to rather small N , N 1. Then the influence of the
microstructure yields an alteration of the permitted solitary wave velocities interval, 1
V
2
c
2 ∗
1 1 − ν1 − N + 4J
∗
1 − N 2 − νR
2
. The widening or narrowing of the interval depends upon the relationship between N and the
parameter of microinertia J
∗
. Like in previous section the type of the solitary wave is governed by the sign of the nonlinearity parameter β. At the same time the characteristic width of the
solitary wave proportional to 1 k turns out smaller than the wave width in a pure macroelastic case, N = 0, J
∗
= 0.
5. Discussion
