Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

45, 2, pp. 259-275, Warsaw 2007

NONLINEAR TRAVELING WAVES IN A THIN LAYER
COMPOSED OF THE MOONEY-RIVLIN MATERIAL

Maciej Major

Izabela Major

Department of Civil Engineering, Technical University of Częstochowa

e-mail: admin@major.strefa.pl

In this paper the problem of studies nonlinear traveling waves in the
Mooney-Rivlin elastic layer is studied. By averaging the equations of
motions over the width of the layer we obtain a system of partial dif-
ferential equations in one dimensional space and time. A technique of
phase planes is used to study the waves processes. Based on the pha-
se trajectory method, we can make an interpretation of conditions of
propagation of nonlinear travelingwaves and can establish the existence
conditions under which the phase plane contains physically acceptable
solutions.

Key words: discontinuous surface, traveling waves, hyperelastic mate-
rials, phase plane

1. Introduction

The considered layer has two kinematically independent degrees of freedom
which are represented by two independent functions describingmotion in the
layer. The effect of finite lateral dimensions and inertia of the elastic layer
are considered by describing the layer as a one-dimensional elastic structu-
re with one scalar variable representing transverse symmetric motion. In the
simplest description, one scalar variable can be used to describe effects of fini-
te transverse dimensions in the elastic layer that undergoes longitudinal and
symmetrical transverse motion only.
Following this introduction, general equations describingmotion of an in-

compressible, nonlinear elasticmedium, symmetric lateralmotionof the elastic
layer and aprocedure of averaging of equations ofmotion are presented in Sec-
tion 2. The traveling waves are described in Section 3.We obtained a solution
for the traveling wave propagating with the speed V in the direction of the
coordinate X1 depending on one parameter only.



260 M. Major, I. Major

In Section 4, we use phase plane methods to classify different solutions
for traveling waves that are possible. Some of the solutions to the differential
equations do not correspond to physically acceptable waves propagating in
the layer, and so additional restrictions must be imposed from the physical
problem. We explore such restrictions in Section 5. We are able to establish
conditions for the existence of physically acceptable solutions as represented
by individual paths in the phase plane. Finally, in Section 6 we present a
numerical analysis for traveling waves in the layer composed of the Mooney-
Rivlin material.

2. Symmetric motion of the layer

Motion of a continuum is represented by a set of functions (Truesdell and
Toupin, 1960)

xi =xi(Xα, t) i,α=1,2,3 (2.1)

We assume that the traveling wave is propagating in the half-infinite elastic
layer which occupies thematerial region X1 > 0 (Fig.1) in the direction of the
axis X1. At the frontal area of the layer X1 =0, the boundary conditions for
deformations are given (FuandScott, 1989).Weassume thatmotiondescribed
by equation (2.2) undergos without imposing additional contact forces at the
lateral planes of the layer X2 = ±h (Coleman and Newman, 1990; Wright,
1981).

Fig. 1. Motion of the layer; (a) main motion in the longitudinal direction,
(b) secondarymotion in the transverse direction

Motion of the considered traveling wave is assumed as

x1 =X1+u1(X1, t) x3 =X3
(2.2)

x2 =X2+f(X2)ε2(X1, t)



Nonlinear traveling waves in a thin layer... 261

wheref(X2) is a functionoff ∈C1 (〈−h;h〉→R),which is odd for symmetric
motion in the transverse direction, however u1,ε2 ∈C3 (〈0;∞)×〈0;∞)→R).
Aftermultiplying by ε2(X1, t), the function f(X2) describesmotion in the

transverse direction of the layer.
We assume the simplest form of the function f(X2)=X2, then for (2.2)2

we have
x2 =X2+X2ε2(X1, t) (2.3)

The strain ε1, the gradient of the transversal strain κ and speeds of the
particle of the medium ν1 and ν2 in both directions of the layer are equal,
respectively

ε1 =u1,1 κ= ε2,1 (2.4)

ν1 = ẋ1 = u̇1(X1, t) ν2 = ẋ2 =X2ε̇2(X1, t) (2.5)

Fig. 2. Propagation of the traveling wave in the layer

For assumedmotion (2.3), the deformation gradient and the left Cauchy-
Green tensor have the form

F= [xiα] =







1+ε1 0 0
X2κ 1+ε2 0
0 0 1







(2.6)

B=







(1+ε1)
2 (1+ε1)X2κ 0

(1+ε1)X2κ (X2κ)
2+(1+ε2)

2 0
0 0 1







For an incompressible material, there is identity

detF=1 (2.7)



262 M. Major, I. Major

then for the considered material

(1+ε1)(1+ε2)= 1 (2.8)

We assume that the layer is made of the Mooney-Rivlin material charac-
terized by the strain-energy function

W =µ[C1(I1−3)+C2(I2−3)] (2.9)

where C1 and C2 are constitutive constants. The invariants I1 and I2 of the
deformation B are

I1 = I2 =(1+ε1)
2+(1+ε2)

2+(X2κ)
2+1 (2.10)

According to Wesołowski (1972a,b) or Dai (2001), the Cauchy tensor has the
form

T=−qI+2µ(C1B−C2B−1) (2.11)
where q is an arbitrary hydrostatic pressure.
The nominal stress tensor (the Piola-Kirchhoff tensor TR)may be expres-

sed by the Cauchy tensor T

TR =TF
−⊤ (2.12)

and its non-zero components are given by

TR11 =−(1+ε2)[q+2µC2(X2κ)2]+2µ(1+ε1)[C1−C2(1+ε2)4]
TR12 =2µC2X2κ[(1+ε1)

2+(1+ε2)
2+(X2κ)

2]+qX2κ

TR21 =2µX2κ(C1+C2) (2.13)

TR22 =−(1+ε1)[q+2µC2(X2κ)2]+2µ(1+e2)[C1−C2(1+ε1)4]
TR33 =−q+2µ(C1−C2)

For deformation gradient (2.6)1, the equations of motion

TRiα,α = ρRui,tt (2.14)

are reduced to a system of equations for the plane strain deformation

TR11,1+TR12,2 = ρRu1,tt

TR21,1+TR22,2 = ρRX2ε2,tt (2.15)

TR33,3 =0

The boundary conditions at the top and bottom surfaces of the layer have the
form

TR12(X1,±h,X3)=TR22(X1,±h,X3)= 0 (2.16)



Nonlinear traveling waves in a thin layer... 263

Fig. 3. A cross-section of the layer A (perpendicular to axis X1)

We employ a procedure, which was described by Wright (1981), consisting in
averaging equations of motion (2.15)1,2 along the cross-section of the layer A
(see Fig.3).

We assume that in the averaging procedure, boundary conditions (2.16)
at the lateral surface X2 =±h are satisfied. We multiply second equation of
motion (2.15)2 by X2 and average both resulting equations and first equation
(2.15)1 over the width of the layer, thus obtaining

1

2h

h
∫

−h

∂TR11
∂X1

dX2+
1

2h

h
∫

−h

∂TR12
∂X2

dX2 =
1

2h

h
∫

−h

ρRü1 dX2

(2.17)

1

2h

h
∫

−h

∂TR21
∂X1

X2 dX2+
1

2h

h
∫

−h

∂TR22
∂X2

X2 dX2 =
1

2h

h
∫

−h

ρRε̈2X
2
2 dX2

In each of equations (2.17), the second integral admits explicit integration of
the form

h
∫

−h

∂TR12
∂X2

dX2 =TR12
∣

∣

∣

h

−h
=0

(2.18)
h
∫

−h

∂TR22
∂X2

X2 dX2 =X2TR22
∣

∣

∣

h

−h
−
h
∫

−h

TR22 dX2 =−
h
∫

−h

TR22 dX2

Taking into account boundary conditions (2.16), one obtains an averaged equ-
ation of motion



264 M. Major, I. Major

∂

∂X1

( 1

2h

h
∫

−h

TR11 dX2
)

= ρRü1

(2.19)

∂

∂X1

( 1

2h

h
∫

−h

X2TR21 dX2
)

− 1
2h

h
∫

−h

TR22 dX2 = ρRε̈2
h2

3

Equations (2.15) are the strict equations, however equations (2.19) are a con-
sequence of the applied average procedure.Formotion (2.15), the cross-section
of the layer remains plane and the normal to the surface of cross-sections over-
lap the axis X1 (Fig.2). The analogical assumptionwasmade in the paper by
Braun and Kosiński (1999).
In further analysis, we take the advantage of averaged equation (2.19)1 and

equation (2.15)2.
Substituting the components of Pioli-Kirchhoff stress tensor (2.13) into

(2.15)2 and integrating them with respect to X2, we obtain the equation of
motion in the direction of the axis X2

q=
µX22
(1+ε1)

[

κ,1(C1+C2)−2κ2C2(1+ε1)−
1

2
ν−2o ε2,tt

]

+
q1(X1, t)

1+ε1
(2.20)

where νo =
√

µ/ρR is the speed of infinitesimal shear waves and q1 =(X1, t)
is an arbitrary function.
Using boundary conditions (2.16) TR22

∣

∣

X2=±h
=0,we determine the func-

tion q1 = (X1, t). We substitute expression (2.20) into (2.13)4. The obtained
equation depends on (X2)

2, then both boundary conditions are satisfied. For
theMooney-Rivlin material, we obtain

q1 =−2µC2(1+ε1)(hκ)2+2µ(1+ε2)[C1−C2(1+ε1)4]+
(2.21)

−µh2
[

κ,1(C1+C2)−2κ2C2(1+ε1)−
1

2
ν−2o ε2,tt

]

Finally (2.20) has the form

q=
µ(X22 −h2)
1+ε1

[

−2κ2C2(1+ε1)+κ,1(C1+C2)−
1

2
ν−2o ε2,tt

]

+

(2.22)

+
2µ(1+ε2)

1+ε1
{C1−C2(1+ε1)2[(1+ε1)2+(hκ)2]}

Averaged equation of motion (2.19)1 has the form

( 1

2h

h
∫

−h

TR11 dX2
)

,1
= ρRü1 (2.23)



Nonlinear traveling waves in a thin layer... 265

Including (2.22), the left-hand side of equation (2.23) for the Mooney-Rivlin
material is

( 1

2h

h
∫

−h

TR11 dX2
)

,1
=2µ

{

C1[(1+ε1)− (1+ε2)3]+

+C2(1+ε2)
[

(1+ε1)
2− (1+ε2)2+

2

3
(hκ)2

]

+ (2.24)

−(1+ε2)2
h2

3

[

2C2κ
2(1+ε1)+

1

2
ν−2o ε2,tt−κ,1(C1+C2)

]}

,1

After differentiation and transformation, we obtain from (2.8)

κ= ε2,1 =−ε1,1(1+ε1)−2
(2.25)

κ,1 =2ε
2
1,1(1+ε1)

−3−ε1,11(1+ε1)−2

Including (2.8) and (2.25) in (2.24), we finally obtain an equation which con-
tains the function ε1(X1, t) only

{

C1[(1+ε1)− (1+ε1)−3]−
h2

6
ν−2o ε2,tt(1+ε1)

−2+

+C2
[

(1+ε1)− (1+ε1)−3+
2

3
(hε1,1)

2(1+ε1)
−5
]

+ (2.26)

−
h2

3
[2C2ε

2
1,1(1+ε1)

−5− (C1+C2)(2ε21,1(1+ε1)−5−ε1,11(1+ε1)−4)]
}

,1
=

=
1

2
ν−2o u1,tt

The above equation (2.26) is the governing equation describing nonlinear dy-
namics of the layers.

3. Traveling waves

The phase ξ is defined by
ξ=X1−Vt (3.1)

where V is the speed of propagation of the traveling wave with a constant
profile displaced along the axis X1. For the traveling wave with any profile,
we express motion as a function of one parameter ξ only

u1(X1, t)=u1(ξ) ε2(X1, t)= ε2(ξ) (3.2)



266 M. Major, I. Major

Fig. 4. Propagation of the traveling wave with speed V

According to equations (3.1) and (3.2), the derivatives with respect to t
and X1 are equal

u1,t =−Vu1,ξ u1,tt =V 2u1,ξξ u1,1 =
∂u1
∂ξ
=u1,ξ

(3.3)

u1,11 =
∂2u1
∂ξ2
=u1,ξξ ε2,tt =V

2ε2,ξξ =V
2
[

−
ε1
1+ε1

]

,ξξ

Including (3.3) and integrating with respect to ξ, equation (2.26) for the tra-
veling wave has the form

(C1+C2)[(1+ε1)− (1+ε1)−3]+
h2

6
ν(1+ε1)

−2
[ ε1
1+ε1

]

,ξξ
+

(3.4)

+
h2

3
(C1+C2)[2ε

2
1ξ(1+ε1)

−5−ε1ξξ(1+ε1)−4] =
1

2
νε1+d1

where ν =V 2/ν2o and d1 is integration constant.
Multiplying (3.4) by ε1,ξ, we integrate once more to obtain

1

2
(C1+C2)[(1+ε1)

2+(1+ε1)
−2]+

h2

12
νε21,ξ(1+ε1)

−4+

(3.5)

−
h2

6
(C1+C2)ε

2
1,ξ(1+ε1)

−4 =
1

4
νε21+d1ε1+d2

where d2 is another integration constant.
This equation gives a solution for the traveling wave propagating with the

speed V in the X1 direction and depends on one parameter ξ (3.1) only.
If the constant ν = ρV 2/µ and the constants of integration d1 and d2 are

known, we could find the solution.



Nonlinear traveling waves in a thin layer... 267

4. Phase plane analysis of propagation of traveling waves in the
layer

By constructing phase portraits of the solution in the (ε1,ε1,ξ) plane, we can
madean interpretationof the conditionsofpropagationof anonlinear traveling
wave (Dai, 2001; Major andMajor, 2006).
First, we introduce dimensionless variables

b1 =
2C1
ν−2C1

b2 =
2C2
ν−2C1

D1 =
2d1
ν−2C1

D2 =
2d2
ν−2C1

(4.1)

Multiplying the equations ofmotion in form(3.5) by 4/(ν−2C1) and including
(4.1), we obtain an approximate form

(b1+ b2)[(1+ε1)
2+(1+ε1)

−2]+
h2

3
ε21,ξ(1+ε1)

−4(1−b2)=
(4.2)

= ε21(1+ b1)+2D1ε1+2D2

Now we introduce the following transformation

ζ =

√
3

h
ξ (4.3)

Apart from a scaling factor, ζ is just the current configuration coordinate X1
in terms of the phase ξ, and (4.2) takes the form

ε21,ζ =F(ε1,D2) (4.4)

where

F(ε1,D2)=
1

(1− b2)(1+ε1)2
·

(4.5)

·{(1+ε1)6[2D1ε1+2D2+ε21(1+ b1)]− (b1+ b2)[(1+ε1)8+(1+ε1)4]}

Wehavewritten D2 explicitly as an argument of F because different curves in
the phase plane correspond to different values of D2.More precisely, the para-
meters b1, b2 and D1 uniquely determine a portrait, and then D2 determines
the curves in that portrait.
We introduce a denotation

y= ε1,ζ =
dε1
dζ
=
√

F(ε1,D2) (4.6)



268 M. Major, I. Major

whose first derivative with respect to ζ is

y,ζ = ε1,ζζ =
dy

dζ
=
F ′(ε1,D2)

2
√

F(ε1,D2)
ε1,ζ =

1

2
F ′(ε1,D2) (4.7)

then

tanβ=
dy

dε1
=
d

dε1

√

F(ε1,D2)=
1
2
F ′(ε1,D2)
√

F(ε1,D2)
=
F ′

2y
(4.8)

where derivatives of F(ε1,D2) with respect to ε1 are denoted by prime. This
system shows immediately that equilibria in the phase plane satisfy y = 0,
F ′(ε1,D2)= 0.
This indicates a specific character of a nonlinear system, which have one

or several equilibrium positions, and depends on the function F(ε1,D2) (Dai,
2001).

Fig. 5. The slope of a straight line tangent to the phase trajectory at the phase
plane (ε1,ζ,ε1) at the point B, (ε1c – denotes the center point, ε1s – saddle point)

Equation (4.8) describesa straight line tangent to the trajectory in function
of thephasecoordinates (ε1,y).Thephasepoints are called ordinaryor regular
points if the tangent is determinated, however if the tangent is indeterminate,
i.e.

dy

dε1
=
yζ
ε1,ζ
→ 0 (4.9)

the points are called singular points or equilibrium points.
Equilibria are solutions to the simultaneous system

y= ε1,ζ =0 ⇒ F(ε1,D2)= 0
yζ =0 ⇒ F ′(ε1,D2)= 0

(4.10)

Discharging necessary equilibriumcondition (4.10) after substituting (4.5), we
have

(1+ε1)
6[2D1ε1+2D2+ε

2
1(1+ b1)]− (b1+ b2)[(1+ε1)8+(1+ε1)4] = 0

(4.11)
6(1+ε1)

5[2D1ε1+2D2+ε
2
1(1+ b1)]+2(1+ε1)

6[D1+ε1(1+ b1)]+

−4(b1+ b2)[2(1+ε1)7+(1+ε1)3] = 0



Nonlinear traveling waves in a thin layer... 269

Eliminating D2 and simplifying, we obtain a polynomial equation

{

(1+ε1)
5[D1+ε1(1+b1)]+(b1+b2)[(1+ε1)

2−(1+ε1)6]
}

(1+ε1)
2 =0 (4.12)

The character of each equilibrium can be found by linearization of (4.6) and
(4.7). If ε1 = ε1e, y = 0 is the solution to (4.12), and according with (4.10)
we have

F(ε1e,D2)=F
′(ε1e,D2)= 0 (4.13)

then close to the equilibrium point

y=Y ε= ε1e+Λ (4.14)

where Y and Λ are small perturbations.
Substituting (4.14) into (4.6), we have

Y =(ε1e+Λ)ζ (4.15)

which entails that
Λζ =Y (4.16)

Similarly, substituting (4.13) into (4.7), we obtain

Yζ =
1

2
F ′[(ε1e+Λ),D2] =

(4.17)

=
1

2
F ′(ε1e,D2)+

1

2
F ′′(ε1e,D2)(ε1e+Λ−ε1e)

According to (4.13) 1
2
F ′(ε1e,D2)= 0, then

Yζ =
1

2
F ′′(ε1e,D2)Λ (4.18)

where D2 is a parameter value representing the equilibrium point.
It follows from the analysis described by Dai (2001) and Osiński (1980)

that if F ′′(ε1e,D2) < 0 the singular point (equilibrium point) at the phase
plane is a center. Such a point sets a stable state of equilibrium. However, if
F ′′(ε1e,D2)> 0, the singular point is a saddle and the state of equilibrium is
unstable. In the degenerate case in which F ′′(ε1e,D2) = 0, we obtain a cusp
point (see Fig.6).
We obtain the curve in the phase plane directly by taking square roots of

F(ε1,D2).
The foregoingdiscussion indicates connection between the location and the

nature of equilibria as well as the formof graphs of F(ε1,D2). The real curves
in the phase plane are described by equation (4.5) y=±

√

F(ε1,D2).



270 M. Major, I. Major

Fig. 6. Graphs of functions F (a) and phase trajectory y in the phase plane (b)

5. Discussion about physically acceptable solutions

The phase portrait method allows one to find solutions to differential system
(4.4).However, not all curves in thephaseplaneare interesting for thephysical
problem at hand. Our main task consists in characterizing such portraits,
whose values of b1, b2 and D1, represent physically meaningful behaviour.

With some approximation we can assume that in the case of compression
or tension of a thin rubber layer the physically acceptable value ε1 is in the
interval from −0.5 to 0.5.
According to Theorem 1 from the paper by Dai (2001, p.104), in order

that there be a physically acceptable solution we must obtain for the func-
tion F(ε1,D2) a center point in the region of physically acceptable value ε1.
Supposing that this point exists for ε1 = ε1c (then F(ε1c,D2) = 0 and
F ′(ε1c,D2) = 0), we can find D1 and D2 as functions of ε1c, which deter-
mines this center

D1 =
(b1+ b2)[(1+ε1c)

6− (1+ε1c)2]
(1+ε1c)5

−ε1c(1+ b1)
(5.1)

D2 =
ε1c(b1+ b2)[(1+ε1c)

2− (1+ε1c)6]
(1+ε1c)5

+

+
(b1+ b2)[(1+ε1c)

6+(1+ε1c)
2]

2(1+ε1c)
4

+
1

2
ε21c(1+ b1)



Nonlinear traveling waves in a thin layer... 271

After substituting (5.1) into (4.12), we obtain

(ε1−ε1c)(1+ε1)2
{

(1+ε1)
5(1+ b1)−

(b1+ b2)(1+ε1)
2

(1+ε1c)3
·

(5.2)

·[(1+ε1)3(1+ε1c)3+3(1+ε1+ε1c)+ε21+ε1ε1c+ε21c]
}

=0

Other equilibrium points are then given by the roots of

(1+ε1)
5(1+ b1)−

(b1+ b2)(1+ε1)
2

(1+ε1c)3
·

(5.3)

·[(1+ε1)3(1+ε1c)3+3(1+ε1+ε1c)+ε21+ε1ε1c+ε21c] = 0

Finally, by computing F ′′(ε1c,D2)

F ′′(ε1c,D2)=
1

(1− b2)(1+ε1c)2
{

12(1+ε1c)
4[5D1ε1c+2D1(1+ε1c)+5D2]+

+2(1+ b1)(1+ε1c)
4[15ε21c+12ε1c(1+ε1c)+(1+ε1c)

2]+ (5.4)

−4(b1+ b2)[14(1+ε1c)6+3(1+ε1c)2]
}

and substituting (5.1) into (5.4), we find that ε1c will be a center if

1

2
F ′′(ε1c,D2)=

(1+ b1)(1+ε1c)
6− (b1+ b2)[3(1+ε1c)2+(1+ε1c)6]
(1− b2)(1+ε1c)2

< 0

(5.5)
In order to obtain physically acceptable solutions, we must have ν > 2C1. If
ν < 2C1, equation (5.5) is not satisfied.

It results from the paper byDai (2001), that there is a second point except
for the point of stable state of equilibrium. It is a point of unstable state of
equilibrium – the saddle point.

Since ν > 2C1, we see from (4.1) that b1 > 0 and b2 > 0. The difference in
the signs ofmarks of terms in expression (5.3) suggests that there is a positive
root, which we assume to be equal ε1 = ε1s.

Equation (5.2) take the form

(ε1−ε1c)(ε1−ε1s)(1+ε1)2
(1+ε1c)2

(1− b2)(1+ε1)2(1+ε1c)2
3(1+A1)+ε

2
1s+ε1sε1c+ε

2
1c

·

·
[

8E+3E2+3(ε21A1+ε
2
1cA2+ε

2
1sA)+6(1+F)+AF + (5.6)

+ε21sB+ε
2
1ε
2
1c+3(ε1ε1c+ε1ε1s+ε1cε1s)

]

=0



272 M. Major, I. Major

where we have eliminated b1 using the fact that ε1 = ε1s is a root

b1 =
(1− b2)(1+ε1s)3(1+ε1c)3

3(1+ε1s+ε1c)+ε
2
1s+ε1sε1c+ε

2
1c

−b2 (5.7)

and we used the following variables

A= ε1+ε1c A1 = ε1s+ε1c

A2 = ε1s+ε1 B= ε
2
1+ε1ε1c+ε

2
1c

E = ε1+ε1s+ε1c F = ε1ε1sε1c

(5.8)

Substituting b1 (see (5.7)) into (5.5), we obtain the following expressions for
ε1c and ε1s, respectively

1

2
F ′′(ε1c,D2)=

ε1c−ε1s
(1− b2)(1+ε1c)2

·

·(1− b2)(1+ε1c)
5(6+4ε1c+ε

2
1c+2ε1cε1s+8ε1s+3ε

2
1s)

3(1+ε1s+ε1c)+ε
2
1s+ε1sε1c+ε

2
1c

(5.9)

1

2
F ′′(ε1s,D2)=

ε1s−ε1c
(1− b2)(1+ε1s)2

·

·(1− b2)(1+ε1s)
5(6+4ε1s+ε

2
1s+2ε1sε1c+8ε1c+3ε

2
1c)

3(1+ε1c+ε1s)+ε
2
1c+ε1cε1s+ε

2
1s

According to conclusions featured at condition (4.18), we can see that if

1

2
F ′′(ε1c,D2)< 0 or

1

2
F ′′(ε1s,D2)> 0 (5.10)

we obtain a center point or saddle point in the phase plane, respectively.

6. Numerical analysis

The numerical analysis is carried out for the function F(ε1,D2) based on
equation (4.4) obtained for theMooney-Rivlin material

F(ε1,D2)=
(6.1)

=
(1+ε1)

6[2D1ε1+2D2+ε
2
1(1+ b1)]− (b1+ b2)[(1+ε1)8+(1+ε1)4]
(1− b2)(1+ε1)2



Nonlinear traveling waves in a thin layer... 273

The constant D1 depends on ε1, and according to (4.12) we have

D1(ε1)=
(b1+ b2)[(1+ε1)

6− (1+ε1)2]
(1+ε1)5

−ε1(1+ b1) (6.2)

analogously, the constant D2 (which depends on ε1 too), according to (4.10)1
is equal

D2(ε1)=
ε1(b1+ b2)[(1+ε1)

2− (1+ε1)6]
(1+ε1)5

+

(6.3)

+
(b1+ b2)[(1+ε1)

6+(1+ε1)
2]

2(1+ε1)
4

+
1

2
ε21(1+ b1)

Then, for a chosen value of ε1 (in this paper ε1 =0.5) we can determine the
constants D1 and D2 from (6.1) and (6.2), respectively.
In the analysis, we assumed the rubber density ρ = 1190kg/m3 and the

shear modulus µ=1.432 ·105N/m2. The constants C1 and C2 are characte-
ristic for a kind of rubber described by Zahorski (1962) and take the values

C1 =4.299 ·104
N

m2
C2 =0.604 ·104

N

m2
(6.4)

the constants b1 and b2 are calculated according to (4.1)1,2.
In Fig.7, there are four graphs of the functions Fi(ε1) ≡ Fi(ε1,D2),

i = 1,2,3,4 for constant D2 calculated according to (6.3) and for ε1 = 0.5.
The functions y(ε1) denote respectively

yi(ε1)≡
√

Fi(ε1) yia(ε1)≡−
√

Fi(ε1) for i=1,2,3,4

Figure 7b shows phase trajectories in the coordinate system (ε1,ζ = y,ε1) for
the functions Fi(ε1), i = 1,2,3,4, found from Fig.7a for the Mooney-Rivlin
material.
The constants D1 and D2 for ε1 = 0.5 are calculated according to (6.2)

and (6.3). InFig.7, the constant D1 is −0.207 and the constant D2 calculated
from (6.3) is 0.452. The constants D2 =0.46,D2 =0.44 and D2 =0.37 have
been established arbitrarily, but here it is fixed at D2 =0.452.
The center point is obtained for ε1 ∼=0.062, and the graph contains phy-

sically acceptable solutions in the interval ε1 = 〈−0.435;0.5〉 (see Section 5).
We find that propagation of the traveling wave in a thin layer is possible

for compression and tension. The solution has a periodic character for closed
curves in the area limited by the solid line shown in Fig.7b, and can be a
solitarywave for solutions represented by a homoclinic orbit (see the solid line
in Fig.7b).



274 M. Major, I. Major

Fig. 7. Graphs for the rubber OKA-1made of theMooney-Rivlinmaterial
(µ=1.46kG/cm2, ρ=1190kg/m3) for the speed V =20.5m/s and constants
b1 =0.335, b2 =0.047 and D1 =−0.207 (according to (6.2) for ε1 =0.5);

(a) distribution of functions: for F1(ε1) the constant D2 is 0.452 (according to (6.3)
for ε1 =0.5), for F2(ε1) – D2 =0.46, for F3(ε1) – D2 =0.44 and for F4(ε1) –

D2 =0.37, respectively, (b) phase trajectory

References

1. Braun M., Kosiński S., 1999, Evolution behavior of transverse shocks in a
nonlinear elastic layer, International Series of Numerical Mathematics, 129,
119-128

2. ColemanB.D., NewmanD.C., 1990,Onwaves in slender elastic rods,Arch.
Rational. Mech. Anal., 109, 39-61

3. Dai H.-H., 2001, Nonlinear dispersive waves in a circular rod composed of a
Mooney-Rivlin material, nonlinear elasticity: theory and applications, London
Mathematical Society Lecture Note Series, 283, 392-432

4. FuY.B., ScottN.H., 1989,Accelerationwaves and shockwaves in transver-
sely isotropic elastic non-conductors, Int. J. Eng. Sci., 22, 11, 1379-1396

5. Major I., Major M., 2006, Phase plane analysis for nonlinear traveling wa-
ves in incompressible hyperelastic circular rod,Vibrations in Physical System,
XXII, 243-249



Nonlinear traveling waves in a thin layer... 275

6. Osiński Z., 1980,Teoria drgań, PWN,Warszawa

7. Truesdell C., Toupin R.A., 1960,The classical field theories,Handbuch der
Physik, III/1, Springer-Verlag, Berlin

8. Wesołowski Z., 1972a, Fala akustycznaw cylindrze odkształconymw sposób
skończony,Rozprawy Inżynierskie, 20, 4, 613-628

9. Wesołowski Z., 1972b,Wprowadzenie do nieliniowej teorii sprężystości,Wy-
dawnictwoUczelniane Politechniki Poznańskiej, Poznań

10. WrightT.W., 1981,Nonlinearwaves in rods,Proc. of the IUTAMSymposium
on Finite Elasticity, D.E. Carlson and R.T. Shields (edit.), 423-443,Martinus
Nijhoff The Hague

11. Zahorski S., 1962, Doświadczalne badania niektórych własności mechanicz-
nych gumy,Rozprawy Inżynierskie, 10, 1, 193-207

Nieliniowe fale biegnące w cienkiej warstwie wykonanej z materiału
Mooneya-Rivlina

Streszczenie

Referatdotyczypropagacjinieliniowej fali biegnącejwcienkiej sprężystejwarstwie
wykonanej z materiałuMooneya-Rivlina. Dla przybliżonego rozwiązania zagadnienia
propagacji fali biegnącej w warstwie hipersprężystej zastosowanometodę polegającą
na uśrednieniu równań ruchu w przekroju poprzecznym warstwy przy założeniu, że
uśrednione wielkości spełniają równania ruchu i warunki brzegowe.Otrzymanew ten
sposób równania zastosowanodo opisu procesów falowych dla rozpatrywanychwpra-
cy fal biegnących. Do analizy procesów falowych użyta została technika płaszczyzny
fazowej.W oparciu o metodę trajektorii fazowej zinterpretowanowarunki propagacji
nieliniowej fali oraz ustalono warunki istnienia fizycznie akceptowalnych rozwiązań.

Manuscript received May 22, 2006; accepted for print November 22, 2006