A. M. Samsonov, G. V. Dreiden, A. V. Porubov, I. V. Semenova and E. V. Sokurinskaya- Theory and...

8/3/2019 A. M. Samsonov, G. V. Dreiden, A. V. Porubov, I. V. Semenova and E. V. Sokurinskaya- Theory and observation of

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Theory and observation of strain solitons in solids

A. M. Samsonov1, G. V. Dreiden2, A. V. Porubov1, I. V. Semenova2

and E. V. Sokurinskaya11 Solid State Theory Sector2 Laboratory for High Temperature Plasma Physics

e-mail: [email protected]

1. Introduction

Even recently one of the most surprising things in the physics of condensed matter implied that there was no

individual mention of successful observations of a solitary strain wave in solids. Indeed, the descriptions of long

wave propagation in liquids and in solids are quite similar, providing a good forecast of soliton existence in solids.

Moreover, even by the time of the pioneering shallow water soliton experiments made by J. Scott Russell 1 [2], thetheory of elasticity had been developed sufficiently, and the nonlinearity of elastic materials and the stress-strain

relation was widely discussed, see, e.g., a review [1].

When a powerful strain wave propagates in a nonlinearly elastic and bounded solid, the curvating of the wave

front can increase rapidly right to the point of appearance of irreversible deformations. This phenomenon can be

balanced with wave dispersion inside a wave guide having small, finite, but not an infinitesimal cross section size

that is in close correspondence with an appropriate balance in a shallow water wave theory. However, the velocity

of a longitudinal (density) wave in solids is much greater then in liquids, and this was the essence of a formerexplanation of unsuccessful soliton observations in experiments in solids.

The solitary strain wave was not observed in solids till the last decade. Furthermore, the availability of powerful

and highly resolutive modern methods of wave detection in solids allowed one to believe that the problem is one

of the theory too. Solitary wave solutions have been found and thoroughly studied in multiple (and very useful)

numerical simulations of nonlinear elasticity problems; nevertheless, the difference between the virtualand physical

realities remained remarkable and, perhaps, suppressed further progress in the area.

The mathematical problem of nonlinear elastic guided wave propagation can be reduced to coupled nonlinear

PDEs. In order to initiate a physical experiment one has to study their reductions to a single PDE, to select types

of self-similar (or travelling) waves as the simplest detectable signals, to estimate the infl

uence of inhomogeneity,impurity and dissipation processes, as well as the results of a numerical simulation as the cheapest experiment,

being very keen on possible artefacts.

As a result of a theoretical analysis the appropriate type of material should be chosen, as well as the suitable

detection, generation and observation methods. After that an experimental setup will be determined. Our experi-

ence in this area has led us to the conclusion that the more complex a nonlinear phenomenon, the more simple

has to be the observation approach.

Nonlinearities of elastic solids have various sources; however, we will consider the intrinsic, or material

nonlinearity of elastic solids and the nonlinearity of deformation to be of primary importance for nonlinear

elastic wave propagation.

The subdivision of elastic materials nonlinearity is carrying out, as a rule, into the geometrical and physical

nonlinearities. The former is governed by the finite deformation tensor (the Green tensor) :

C= [U+ (U)T+U (U)T]/2 (1)

written in terms of a vector gradient U and its transpose (U)T, both defined with respect to the displacementvector U = {Uk}. The third term in (1) reflects small but finite, although not infinitesimal deformations (as itis in linear elasticity), and strains resulting from them.

The physical nonlinearity is based on a representation of the internal energy U, and the potential energy ofan adiabatic deformation of a material as a partial sum of power series written with respect to Ik(C), k= 1, 2, 3,and with coefficients depending on the elastic moduli (, ) of second order, (l,m, n) of third order, (i) of 4th

1It is a great pleasure to refer now to the second new successful experiment on solitons in natural conditions on shallow water surface,

which was made in the same place of the EdinburghGlasgow canal.

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order etc., which introduces a model proposed by Murnaghan (1954) for a hyperelastic medium2

= ( + 2)I21 /2 2I2 + (l+ 2m)I31 /3 2mI1I2 + nI3. (2)

It has been shown that both nonlinearities must be taken into consideration, even for metals under finite, but

not infinitesimal deformation.

The Murnaghan approximation for is not unique, see [3]; indeed, the BlatzKo potential was proposed for a

rubber-like (compressible) material, the Mooney potentialfor incompressible media, etc.; each of these resultingin a coupled nonlinear p.d.e.

Several points can b e outlined here. Firstly, despite the common experience with erasers and childs catapults,

the physical nonlinearity should be taken into account not only for rubber-like materials, but even for metals,

see, e.g., [4]. For copper and iron, brass and lead the next order elasticity values are much larger then the Lameconstants, and therefore may provide a relative increase in potential energy of up to 20% for small second order

terms in tensor C, as shown in [5]. One can conclude that the potential energy density must remain fully nonlinear,unless a correctness of any particular linear approximation stated. In other words the condition |Ui/xj| 1does not necessarily lead to a relatively small addition in strains, (unonlin ulin)/ulin 1, which is typical of thelinear static theory of elasticity, and the reason for this difference consists, at least, in possible large intervals of

space and time.

Nevertheless, aiming to consider the simplest wave problem in a solid nonlinear wave guide, we will derivethe main equation from the first principles in an object-oriented form useful for further application to physical

experiments, see, e.g., [6] for details.

2. Theory of nonlinear waves in an elastic rod

Let us consider the nonlinear wave propagation problem for an isotropic homogeneous nonlinearly elastic rod.

Introducing the time variable t and the cylindrical space coordinates (x, r, ), where x is axis along the rod, [0, 2], 0 r R(x) R0; we write the displacement vector as: U(ui) U(U,V,W), and assume eachcross section to be circular after deformation.

Consider a nonlinearly elastic Murnaghans material, having the elastic moduli of 2nd order (E, ) and of 3rdorder (l,m, n). The Lagrangian density written in standard form:

L = (U/t)2 [Ik(C)] (3)will lead to the coupled p.d.e. after appropriate variation. To reduce them to a single equation we assume

that: (i) torsions will be neglected: U = 0; (ii) strains are small, finite, but not infinitesimal: i, k [1, 3],ui/xk 1; (iii) long waves are considered only, having the length: R0. Two more assumptions shouldbe made in order to reduce the coupled equations to the only one: (iv) each cross section remains planarafter deformation: Ur = 0; then ui = ui(x, t) and either: (v) the A. E. Love relationship is valid between themain component of longitudinal strain Ux and transversal displacement V: V = rUx, or: (vi) the refinedrelationship [7]: U = U(x, t) + r2Uxx/2 and V = rUx r32Uxxx/[2(3 2)], to avoid possible lack ofaccuracy in the asymptotics of stress tensor components on the lateral surface of the wave guide. The last leads

to different coefficients in the resulting equation, with its differential operators form remaining the same.

Taking the assumptions (i)(v) into consideration, we obtain from (3) the Lagrangian density for a longlongitudinal wave in a rod made of nonlinearly elastic material:

L = R2/4 2U2t 2EU2x 2U3x /3 + 2R2U2xt 2R2U2xx (4)where is the density per unit volume, E is the Young modulus, is the shear modulus, while the syntheticnonlinearity coefficient = (E, ; l,m, n) depends upon the elastic moduli of 2nd and 3d orders.

The action functional A defined as:

A =t1t0

W

L()dW

dt (5)

2Well before Murnaghan, a general power series expansion for an elastic energy was proposed by Landau and Rumer in 1937, (seealso textbook on mechanics by Landau and Livshitz).

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enables us to apply the Hamilton principle and to obtain from (4) the dimensional Double Dispersive Equation(DDE) in the form of a quasi-hyperbolic nonlinear p.d.e. of 4th order:

Utt x

1

R2ER2U

x

=

1

2R2

x

R2U2x +

2R4Uxtt

2R4Uxxx

(6)

as it was derived in 1982 (see [810]). When the wave guide is assumed to be homogeneous and isotropic, the

DDE can be reduced to the simplest form, written with respect to the strain component u Ux:utt c20uxx = (1/2)

c2u

2 + 2R2utt c21uxx

xx

(7)

where c20 = E/, c21 = /, c2 = /, = E/[2(1 + )] and R, are all constants.

If the refined relationships (vi) betweenU and Vmentioned above are assumed, then both components Prr, Prxof the PiolaKirchhoff stress tensor P are equal to zero, and the coefficients of the DDE are different:

utt c20uxx = (1/2)c2u

2 + R2

( 1)utt+ c20uxxxx

. (8)

Introduction of a small parameter follows from an assumption that the strain wave, having a typical amplitudeA and length , should be elastic: A

1 and a long one: R0/

1. The most interesting case consists in the

balance of nonlinear terms with the dispersive terms, which leads to the relationship: = A = (R0/)2 1.Neglecting all next-order terms in , we obtain the dimensionless form for both versions of the DDE as:

utt uxx = u2 + autt buxx

xx

(9)

where a, b are constant coefficients depending on the elasticity of the wave guide.Direct integration of the DDE (8) in order to find the travelling wave solution, depending upon the phase

variable z = xVt, leads to the exact solitary wave solution u and its wave length , given by:

u =3E

V2

c20 1

cosh2

xVt

A cosh2 xVt

2

= 2R2

[c20 (1 )V

2

] / [V2

c20]. (10)

Then the solitary wave exists if and only if its velocity belongs to the interval

for 1/2 > > 0 c0/

1 >V> c0 and for > 0 V2/c0 > 1 (11)

being either compressive or tensile, depending on the sign of = 3E+ f(l,m, n). Therefore an allowed zoneof velocities exists, in which an original weak shock pulse provides for the soliton generation.

Evidently for > 0 a subsonic (V< c0) solitary wave of compression does not exist for any elastic material,whereas a transonic soliton can exist for most materials if its velocity is expected to be from (11); therefore, e.g.,the solitary compression wave can be generated by the initial pulse (weak shock) with a transonic velocity.

The standard reduction to the nonlinear evolution equation (NEE) by means of substitution = tk+1, =k(xVt), results in the KdV equation: u + (u2) /2 + cu = O(), and the main problem is how to satisfythe conditions prescribed for the initial problem: u(t= 0) = u0(x). Moreover, the corresponding KdV equationleads to different and incorrect estimations for both the velocity V and the wave length , namely c0 < V< ,(semi-infinite interval!) and

1/A.

A detailed analysis of the allowed zone of velocities of waves, which will be considered as suitable sources

for soliton generation, will be published. Our attempts to generate a solitary wave in a rod suggest that correct

theoretical estimation of wave length and velocity is of crucial importance for successful experiments on soliton

generation in solids.

The soliton propagation in a rod with varying cross section was studied, too, based on a detailed asymptotical

analysis of the perturbed problem, see [6, 10] and resulted in the first successful experiments in soliton propagationalong such a geometrically inhomogeneous rod.

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3. Soliton generation and observation

Numerical simulation of generation and propagation of various initial pulses in the framework of the DDE model

(i.e., of a non-stationary problem) shows the dependence of both the mass and the type of tail behind a solitarywave on the mass (duration) of initial pulse. It is found that a sharp pulse provides the solitary wave with adecaying oscillating tail behind, while a massive one leads to a solitary wave train followed by oscillating wave

trains having an almost sinusoidal envelope wave form.

In order to avoid any plasticfl

ow in the specimen and to keep the elastic type of deformations, the parametersof the pulse should satisfy the following condition for the finite strain tensor component:1 + 2Cxx 1 < Y0,

where Y0 is the yield point of a material. The amplitude of the initial pulse should be chosen in order to balance

nonlinear and dispersive features of a wave guide. The parameters of a tentative initial pulse (pressure at the frontof shock and the pulse duration) and of the corresponding solitary wave in the rod were summarized in [11].Following the analytical results obtained we have chosen transparent polystyrene (PS) SD-3 with = 0.35, = 6 1010 N/m2 and c0 = 1.8 103 m/s as an appropriate material to manufacture a solid nonlinearly elasticwave guide.

The apparatus used for generation and observation of strain solitons consisted of a channel designed to produce

a strain wave in the solid from a weak shock wave in a water cell, a synchronizer, a holographic interferometer

and a control unit for measuring the laser pulse energy, see [12].

Solitons in a transparent solid rod were formed in our experiments from an initial shock wave initiated in aliquid surrounding the rod by means of the laser explosive evaporation of a metal covering a foil target surface

placed near the input rods cross section inside the water cell. The pulse duration was chosen to be 20 ns. The

power density of laser radiation acting on the target was measured by the energy control unit and was kept constant

and equal to 2.3 108 W/cm2 during the experiment. Observations were made in the perpendicular directionby means of a second pulsed ruby laser used for holographic interferometry, and two parallel cut-offs were made

along the specimen in order to make transparent the central part of the rod.

Recently, a study was made of the weak shock in water [13], generated by laser light evaporation of a metalfrom a foil. This wave was shown to look like a very narrow compression zone (0.1 m) followed by a large(1 mm) tension area. The parameters of this wave are quite similar to those necessary for strain soliton generation;moreover, the values of acoustical wave resistance of both water and PS are quite close, which allows input of a

wave in a specimen without considerable power loss at the interface.To detect a soliton and to study the soliton evolution we used a PS rod 1 cm thick, probed by a plane light

beam (50 mm in diameter) at different regions, and the holograms of focused images were recorded. The setupallowed us to change precisely the time delay between the pulse from the laser inducing the shock and the pulse

from the recording one.

First exposure was made in the absence of shock wave in the specimen, which allowed us to produce a

hologram in an unperturbed wave guide; the second one was synchronized with the prescribed stage of elastic

wave propagation. Carrying of fringes on interferograms made after the recovery of doubly exposed holograms

occurred due to rotation of the optical wedge between exposures.

Therefore the wave picture was visualized through the carrier fringe shift K, whose amplitude was usedfor direct calculations. The wave patterns were recorded at various distances from the input edge of the rod.

Figure 1 shows a hologram of the wave picture near the input tip of the rod. The initial shock (A) in the rod ismoving faster than the shock (D) in the surrounding water, where the Poisson conical waves (P) are observedalso, generated by the transversal displacement of the lateral surface.

A typical solitary wave propagation picture obtained in experiments is shown in Fig. 2, where the direction of

wave motion is shown by an arrow. The left and right edges of the frame are located, respectively, at a distance

of 75 mm and 125 mm from the input cross section of the rod [14].

The sharp kink on the right side of the photograph corresponds to the weak shock wave (A), which hasproduced the strain soliton. The fringe shift, representing a soliton, was extracted and plotted separately below

the observation area image for convenience only. The longitudinal strain soliton is, in fact, a density wave, but

not at the whole lateral surface elevation wave (as it could be on a water surface). The secondary shock waveis imprinted in the middle of the soliton. This shock is produced by reflections of the initial shock from the tip

of the rod and from the target foil consequently. The fringes in the surrounding water remain undisturbed, which

confirms the fact that the wave under study propagates in the rod. The inner area of the rod on each photograph

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10mm

A

waves in water

waves in rodinput tip of rod

D

P

Figure 1: Wave pattern near the input edge of the rod.

solition length,

10mm

10mm

wave motion direction

Kfringe shift, strain solution, C

A

Figure 2: Interferogram of the soliton in the rod of constant cross section.

is bounded by two black rectangles representing the (opaque) regions, whose lateral surface is not perpendicularto the probing laser light.

The recorded interferograms allowed us to trace the evolution of the wave pattern in the rod and to visualize

the process of soliton formation and the subsequent propagation of the soliton along the wave guide. We observed

that shock waves (which are residuals of the initial shock wave used for soliton generation) decay quickly, moving

along the rod; however, their velocities remain higher than both the sound (rods) velocity in polystyrene andthe soliton velocity (from one frame to another the soliton lags behind them). The soliton exhibits rather anextended water-basinshaped longitudinal wave, which is not followed by any tensile wave. Moving along the

rod, the soliton does not undergo any noticeable change in shape and amplitude; confirming thus the well known

major property of any nonlinear solitary wave, detected now in nonlinear elastic wave propagation.

The lack of attenuation is the second distinctive feature of the soliton in comparison with any linear pulse which

is expected to disappear in the PS at a 10 radii distance from the input through wave dispersion and dissipation

losses; even a shock wave attenuates dramatically moving along the PS rod.

The following quantitative estimations of the soliton parameters were made. The soliton length was measureddirectly from the interferogram, and the soliton amplitude A was calculated from the carrier fringe shift K,obtained by reconstruction of doubly exposed holograms and shown on the interferogram.

The value ofA was found by means of the known formula (see, e.g., [12]):

A = 0K2h [(n1 1) (1 2) + (n1 n0)] (12)

where n0 and n1 are the transparency indices of water and deformed material respectively, 0 is the light wavelength,2h < R0 is the rods thickness along the detecting laser light direction. We obtained the following parameters fora typical recorded soliton: A = 2.82 104; = 32.5 mm R0 = 5 mm, which is in good agreement with ourtheoretical estimations.

We studied also the soliton focusing in a nonlinearly elastic wave guide. For this reason a PS rod was chosen,

consisting of an interval (070 mm) with constant cross section followed by a linearly narrowing part, wherethe rod radius decreased from 5.0 mm to 2.75 mm. A soliton snapshot is shown in Fig. 3, where one can see

a decrease of the wave length. Details will be published elsewhere, we note only that, again, the soliton has no

negative wave behind and keeps its shape unchanged.

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6.9mm

8.9mm

amplificated solition,

Figure 3: Interferogram of the soliton in the narrowing rod.

Finally a theory has been established and successful experiments were done in order to generate and observe

strain solitons in 2D wave guides, e.g., in a PS plate (see [15] for details).

4. Conclusions

Thus, using a laser generator of weak shock waves and a holographic interferometry setup, we have made probably

the first generation, detection and records of a strain solitary wave (the soliton) inside a nonlinearly elastic solidwave guide (a rod).

Soliton detection may allow one to introduce a new one-pulse technology of nondestructive testing, to determine

the intrinsic physical properties of nonlinearly elastic materials and wave guide parameters, e.g., the 3d order elasticmoduli whose values could not be reliably measured for many structural materials up to now; to explain possible

damage of optical fibers due to the light stimulated backscattering, as mentioned in [16], etc. They can be of usefor study of inhomogeneities and cavities, in particular, when the ultrasonic technique cannot be applied.

However, it seems that the main and general result of the study is that a proof is given of energy transfer

by means of elastic solitons for quite a long distance in solids made of materials nontransparent for linear strain

waves and ultrasonic waves due to their dispersive and dissipative nature.

References

[1] Nature Soliton Wave Receives Crowd of Admirers 376 3 Aug 1995, 373

[2] J. F. Bell 1973 in Encyclopaedia of Physics YIa/1 (Springer: Berlin)

[3] A. I. Lurie 1990 Nonlinear Theory of Elasticity (Elsevier)[4] D. F. Parker 1984 J. Appl. Math. Phys. (ZAMP) 35 833

[5] P. Pleus and M. Sayir 1983 J. Appl. Math. Phys. (ZAMP) 34 192

[6] A. M. Samsonov 1994 Nonlinear Waves in Solids eds A. Jeffrey and J. Engelbrecht (Springer: WienNewYork) p 349

[7] A. V. Porubov and A. M. Samsonov 1993 Sov. Tech. Phys. Lett. 19 365

[8] A. M. Samsonov 1982 in Structural Optimization under Dynamical Loading, Workshop of Junior Scientistsed U. Lepik (Tartu University Press: Tartu, Estonia) p 71

[9] A. M. Samsonov 1984 Sov. Phys. Dokl. 29 586

[10] A. M. Samsonov 1988 Sov. Phys. Dokl. 33 298

[11] A. M. Samsonov and E. V. Sokurinskaya 1988 Sov. Phys. Tech. Phys. 33 989

[12] G. V. Dreiden, Yu. I. Ostrovsky, A. M. Samsonov, I. V. Semenova and E. V. Sokurinskaya 1988 Sov. Phys.Tech. Phys. 33 1237

[13] G. V. Dreiden, G. V. Ostrovskaya and Yu. I. Ostrovsky 1994 Thermophysics & Aeromechanics 1 59 (inRussian)

[14] G. V. Dreiden, A. V. Porubov, A. M. Samsonov, I. V. Semenova and E. V. Sokurinskaya 1995 Sov. Tech. Phys. Lett. 21 415

[15] G. V. Dreiden, A. V. Porubov, A. M. Samsonov and I. V. Semenova 1996 Tech. Phys. Lett. 22 891

[16] C. Montes 1987 Phys. Rev. A 36 2976

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