# Analytical solution for stationary waves

We consider the stationary wave propagating with a constant velocity without changing of its form. In the latter case the solution satisfies the following conditions

where: *E,* is the running wave variable, *E, =x — vt,v* is the phase velocity.

We introduce the following non-dimensional deformation of the wave profile;

Substituting relations (4.29), (4.30) into equation (4.22) one gets

Integration with respect to the variable *E,* yields the following equations of the harmonic oscillator with squared nonlinearity

where: *a = E* (1 - v^{2}/v„) / (£//^{2}) , b = £2/ (*2E?,l*^{2}), *c* is the constant of integration.

A few researchers did not take into account the constant c while investigating the propagation of nonlinear waves in solid media [158]. The obtained results were erroneous from the physical standpoint. It is clear that the displacements *и* cannot increase unboundedly. It means that the average deformation with regard to the wave period should be equal zero:

Condition (4.33) allows to find constant *c.* In order to find the exact analytical solution, we multiply equation (4.32) by *df **jd%.* and after integration one gets

Relation (4.34) can be interpreted as the energy conservation principle of the anharmonic oscillator. The integration constant W_{0} presents the initial total energy of the system. Wo > 0; *W _{k}* = (1/2

*)(df/d^)*is the kinetic energy;

^{2}*W*+ (fo/3)/

_{p}= (a/2) f^{2}^{3}+ / is the potential energy; Wo =

*W*

_{k}+ W_{p}.Equation (4.34) can be recast to the following form:

A solution exists if the kinetic energy satisfies the following inequality

The cubic term W_{0} - *cf - (a/2)f ^{2}*

**- (fo/З)/**

^{3}, depending on the values of the coefficients, is responsible either for one or three real roots. Since the case of one real root corresponds to the unbounded solution it is not further considered, the solution is bounded and periodic if the function

*W*= W

_{k}_{0}-

*cf*

*—*

*(a/2)f*

^{2}**- (**

*b/3)f*

^{3}takes values on the interval between its real roots.

Let us present the polynom *W _{0} — *

*cf—(a/2)f*

^{2}*-*(

*b/3)f*

^{3}through its roots

*f,*/2, /3:

where

**Figure 4.4 **Kinetic energy of the oscillator with square nonlinearity: (a) *b <* 0, (b) *b* > 0 [reprinted with permission from Elsevier],

Equation (4.35) is recast to the following form

Assume the following order of the roots: *fa > fa > fa-* Form of the graph of the kinetic energy function depends on the soft (£2 < 0, *b* < 0) or hard (£2 > 0, *b* > 0) nonlinearity of the composite material (Fig. 4.4). In what follows we briefly consider both cases.

**Soft nonlinearity. **If *b* < 0, then the periodic solution exists for *fa* > / > *fa.* Let us carry out the following change of the variables

Equation (4.39) takes the following form:

Integration of equation (4.42) yields

We carry out inversion of the elliptic integral occurring on the right hand side of equation (4.43) [2]:

Let us introduce the following notations: *F = f _{2}* — /3,

*к = 2>J—b(f*—/3)/6 =

*J-2bF*/ (3j

^{2}), where F is the amplitude, F > 0, ic is the constant of propagation (analog of frequency).

Solution (4.44) has the period *L = 4K (s)/к.* Substituting (4.44) into condition (4.33) yields

where *К* (s) and *E* (.?) stand for complete elliptic integrals of the first and second kind, respectively [2].

Finally, we have

and solution (4.44) is recast to the following form

Modulus *s* of the elliptic function stands for solution to the following transcendental equation

The propagation constant *к* is coupled with wave length *L* in the following way
Velocity of the wave propagation is found from equation (4.38), and it reads

**Hard nonlinearity. **If *b >* 0, then the periodic solution exists for *f > f > fi- *Instead of relations (4.40), (4.41), we introduce the following dependencies

and we introduce the notation

Carrying out the transformations in a way analogous to the previous, we get

where modulus 5 is defined by the following transcendental equation

The relations (4.49), (4.50) are the same. Observe that the solutions (4.47), (4.48) in the case of soft nonlinearity (£2 < 0) and the solution (4.53), (4.54) in the case of hard nonlinearity (£2 > 0) get over to each other while changing the sign of coefficient £2.

# Analysis of solution and numerical results

The value of parameter .? estimates intensity of the nonlinear effects as well as allows to define how strong the nonlinear regime of the wave propagation differs from the linear one. Fig. 4.5 presents the form of nonlinear waves of deformation for different values of s. The computations have been carried out for the composite material with soft nonlinearity based on equation (4.47).

**Figure 4.5 **Forms of periodic nonlinear waves of deformation for *E _{2}* < 0 [reprinted with permission from Elsevier],

In the case of 5 = 0 the following limiting transitions take place: £ (0) = тг/2, *К* (0) = тг/2, £ (0) /А" (0) ~ 1 *-s ^{2}/*2, sn(z.O) = sin (z). The found solution describes the following harmonic wave

In the case of 5 = 1, the periodic nonlinear wave is transformed into the localized wave of the bell-form (soliton) - see Fig. 4.6. Having in mind that: £(1) = 1,

lim *К* (.?) = oo, sn(z, 1 )^{2} = 1 — sech(z)^{2}, one gets

JT—> 1

where the parameter Д stands for the soliton width.

Figure 4.6 Localized nonlinear waves of deformations (solitons): (a) soft nonlinearity (£2 < 0); (b) hard nonlinearity (£2 > 0) [reprinted with permission from Elsevier].

Analysis of solution (4.56) allows to achieve important conclusions regarding properties of the localized nonlinear waves propagation in composite materials. Since £ > 0, then in the case of soft nonlinearity (£2 < 0) we have / < 0. It means that in such material the localized waves of compression propagate. On contrary, in the case of hard nonlinearity (£2 > 0) we have / > 0, and consequently only localized waves of extension may exist.

Increase of the soliton amplitude £ yields decrease of its width Д and increase of its velocity v. Therefore, the “high” solitons have the small width and propagate faster than “low” solitons.

The velocity of solitons is higher than the velocity of waves propagating in linear homogeneous medium: v > vo- It is the ultrasonic regime. The analogous effect is observed while studying nonlinear waves of deformation in homogeneous solid [158, 398].

Further numerical examples is devoted to the case of the steel-aluminium composite. Properties of the components are the same: = *c^ =* 0.5. Owing to the obtained solution (4.23)—(4.25), we compute the effective elastic coefficients: £| = 160 GPa, £_{2} = -2385 GPa (soft nonlinearity) for £_{3} = 2.73 GPa.

**Figure 4.7 **Phase velocity of the nonlinear wave of deformation [reprinted with permission from Elsevier].

Fig. 4.7 presents the results for the phase velocity v of the nonlinear wave. For *s the super sonic (v < v*

_{0}) regime is realized, whereas for

*s >*so we deal with the ultrasonic regime (v > v

_{0}). Owing to relation (4.50) the threshold value is defined by the following condition

Solving numerically equation (4.57) yields so = 0.9803.... It should be noticed that the given value is defined only by the character of the obtained analytical solutions and does not depend on the properties of the composite material.

The obtained numerical results (Figs. 4.5, 4.7) yield conclusion that influence of nonlinearity on the form and velocity of the wave become important for *s > *0.6.. .0.8. In the case of smaller values of s, the wave form almost does not differ from harmonic one, whereas the regime of propagation is very close to the linear regime.

We consider how the value of parameter s depends on the wave amplitude *F* and on the ratio of the wave length *L* and the size of the material microstructure /. Note that the parameter *r = l/L* characterizes intensity of the effect of dispersion. The higher value t), the bigger is influence of the microstructure on the dissipation of the wave energy. The parametric dependencies of the modulus sonF and *t)* are shown in Fig. 4.8 (they have been found by solving numerically equation (4.48)).

The obtained data illustrate how phenomena of nonlinearity and dispersion compensate each other. Increase of the amplitude F(with constant *rj)* implies increase of ^ which yields increase of intensity of the nonlinear effects. In contrary, decrease of the wave length and increase of t) (with constant amplitude *F)* yields decrease of the parameter the solution becomes more close to its linear counterpart.

**Figure 4.8 **Modulus *s* characterizing intensity of the nonlinear effects [reprinted with permission from Elsevier],

Analysis of the results presented in Fig. 4.8 allows to estimate areas of applibility of various approximate theories, which are employed for modeling of elastic waves propagating in solids. For the majority of materials used for mechanical constructions, the area of elastic deformations is bounded by the value of F < 1СГ^{3}. Nonlinear effects are exhibited for s > 0.6 (in the latter case q < 0.13). Consequently, modeling of nonlinear waves can be carried out in the scope of the length wave linear approximation (for instance by employing of the higher order continuous or homogenize models).

Occurrence of dispersion plays a key role when the wave length is close to the size of the internal structure of material, i.e. for q > 0.3 (see Fig. 4.3). Then we get *s* < 0.28, which means that the nonlinear effects can be neglected. Therefore, the problem of propagation of short and strongly dispersive waves can be analyzed in the scope of linear theory. The solution can be obtained with a help of the Floquet-Bloch method [98].

The so far given statements are not valid if the model allows for large values of elastic deformations *(F* > 1СГ^{2}... 1СГ^{1}). In the latter case, the nonlinear and dispersive effects may appear simulteneously. They are exhibited by the rubber-type materials and elastomers, molecular and atomic chains, nano tubes, etc.