Combined Anelasticity and Nonlinearity : the Longitudinal Resonances of a Bar H

COMBINED ANELASTICITY AND NONLINEARITY : THE LONGITUDINAL RESONANCES OF A BAR H. Jeong, D. Beshers

To cite this version:

H. Jeong, D. Beshers. COMBINED ANELASTICITY AND NONLINEARITY : THE LONGITUDI- NAL RESONANCES OF A BAR. Journal de Physique Colloques, 1987, 48 (C8), pp.C8-317-C8-322. 10.1051/jphyscol:1987846. jpa-00227150

HAL Id: jpa-00227150 https://hal.archives-ouvertes.fr/jpa-00227150 Submitted on 1 Jan 1987

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. JOURNAL DE PHYSIQUE Colloque C8, supplément au n°12, Tome 48, décembre 1987 C8-317

COMBINED ANELASTICITY AND NONLINEARITY : THE LONGITUDINAL RESONANCE OF A BAR

H.S. JEONG and D.N. BESHERS

Henry Krumb School of Mines, Columbia University, New York, NY 10027, U.S.A.

Résumé - Nous avons ajoute un terme quadratique peu important I l'équation du modèle du solide anélastique standard et nous avons étudie son effet dans l'approximation du premier ordre. La résonance fondamen tale ne change pas, en première approximation, mais uneharmonique de deuxième ordre apparaît. En fait, la. deuxième harmonique résulte de deux ondes spatiales à même fréquence. Comme l'amortissement linéaire tend vers zéro, la forme mathématique de la solution change, les deux ondes dégénèrent en une seule qui n'est pas exactement sinusoïdale.

Abstract - The addition of a small quadratic term to the standard anelastic solid model has been investigated to first order approximation. The fundamental resonance does not change, to first order, but a second har monic wave appears. The second harmonic actually consists of two spatial waves with one frequency. As the linear damping goes to zero, the mathe matical form of the solution changes, the two waves degenerating into one that is not entirely sinusoidal.

I - INTRODUCTION

The longitudinal resonance of a bar has often been exploited for the measurement of internal friction. The analysis of the experiments has usually been con ducted in an elementary fashion, and often that suffices. However, when the internal friction is nonlinear, that is to say dependent on the amplitude of oscillation, then there is a need for more careful consideration of the problem, a need universally neglected for lack of a suitable theory. A first step towards such a theory, an approximate treatment for the case of a small nonli near effect co-existing with a linear damping, is presented in this paper.

II - THE MODEL

Because nonlinear internal friction usually occurs in materials that also exhi bit linear damping, it is important to treat the linear and nonlinear aspects together. The basic model for linear damping is the standard anelastic solid (SAS) /1/. The longitudinal resonance of a bar made from a SAS has been treated already /2/. We consider here a bar of homogeneous material obeying a consti^ tutive equation based on the standard anelastic solid, but augmented by one nonlinear term: * ~ 2 a + ba - ae + ce + de , (1 ) where a is the longitudinal stress, e the corresponding strain, and a, b, c, d

are material constants; in the usual notation, a = 1/(JU+6J), c = t a> and b - J c. The coefficient d is assumed subsequently to be small. For a long, thin bar the equations of stress equilibrium reduce to

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987846 JOURNAL DE PHYSIQUE

where p denotes the density of the material and u = u(x,t) the displacement of a cross-section of the bar. Strain and displacement are related by

We eliminate stress in favor of u by differentiating (1) with respect to x, then using (2), and replacing E with u by (3), obtaining the equation of motion to be solved:

putt + bputtt = au + cu + 2duxuxx. (4) XX xxt The bar is considered to be driven at one end, x = 0, by a periodic stress of amplitude 20 and circular frequency w and to be free at the other end, x = L. The boundaryOconditions are then, choosing the phase appropriately,

o(~,t)= ooeiWt + 0 e-iwt atx=O (5) 0

We seek a steady-state solution of this problem, expecting to find resonant modes, and to examine the behavior in the vicinity of resonance. By steady- state we mean that every term in the solution must repeat in time with the fundamental frequency; that is we assume that harmonics of the fundamental will occur, but not subharmonics. This assumption is consistent with experimental results in this laboratory.

The nonlinear term necessitates care in the use of complex exponential functions, which are otherwise the simplest representation with which to work. The real part of any complex number Z is given by

but R, as an operator, does not commute with multiplication. We shall therefore in effect require that R operate first by writing the trial solution in the form m u(x,t) = z [un(x)einwt + u *(x) e-inwt]. (8) n=O

In this way, all our expressions are real, although sometimes we will not bother to write out the conjugate part.

I11 - THE SOLUTION

Substitution of (8) in (4) gives a very lengthy equation. Because exp(inwt) and exp(-inwt) are linearly independent functions, we require that the coefficient of each be zero, which has the effect of converting our partial differential equation into an infinite set of coupled ordinary differential equations. Each of these ordlnary equations may be viewed as consisting of two parts. One part contains four terms which correspond to the four terms of the standard linear solid at the frequency nw and the other part is an infinite series which has the coefficient d of the nonlinear term in (1). These ordinary equations are therefore inhomogeneous. Now the solution to an inhomogeneous equation is the gem- era1 solution of the corresponding homogeneous equation plus any particular solution of the inhomogeneous equation. We can obtain the solutions to the homogeneous equations immediately, and then approximate the infinite series in terms of these known solutions, after which particular solutions can be found. This strategy is facilitated by the fact that the terms of the infinite series, while nonlinear, become products of trigonometric functions of x and therefore expressible as linear combinations of trigonometric functions of other periodic- i ties: the equations become linearized.

The homogeneous equations

The reduced equation for any n has the form

where U' indicates dU/dx, and another equation which is the complex conjugate of (9). Equation (9) has the well-known form

O" + K 2~n0= 0 "n (10) where the complex propagation constant K = kn + ian is determined by

The general solution of (10) is

unO = C sinK x + DncosK x (1 2) where the constants C and D are to be determined eventually by the boundary conditions. For the Boundary conditions given by (5) and (6). the solution is and u(x,t) = ulo(x)eiwt + C.C. (14) where C.C. stands for complex conjugate. This is the solution given previously /2/ for the resonance of a bar made of a standard linear solid.

The Inhomogeneous equations

Returning to the inhomogeneous equation, we start the process of approximating to the nonlinear terms by assuming that

and that all other Un are small of order d, at least, and proceed to sort the terms by the powers of d, in the end neglecting all terms containing d to a power higher than 1. In the process of approximation we are also helped by assuming that 6J/J is small so that the strain amplitude at resonance is orders of magnitude less than unity. The only equations that survive this process are those for n = 0,1,2. For n = 1, the nonlinear term vanishes in this approxima.- tion. Therefore the present treatment will not, in fact, extend to the case of nonlinear damping. Rather, the approximation is one in which the fundamental is unaffected to first order, but small harmonic generation occurs. Our approxirnw tions have then reduced the infinite set of equations envisioned in (8) to three only. We focus our attention on the equations for n = 2, putting aside the equation for n = 0 which will not affect the results. Thereby, we discuss here a system with only two states, a closed system not an open one. We have then to solve the equations for n = 2

0 and the complex conjugate of (16). With U, given by (131, and denoting the JOURNAL DE PHYSIQUE

coefficient of cosK1!L-x) in (13) by A, the particular solution of (16) is by the method of variation of parameters

where 2 2 D = [(2K1) - K2 1 ( a + ic2w). The boundary conditions, (5) and (6), imply that the stress corresponding to U2 shall be zero at both ends of the bar, forcing the other terms in U to be proportional to the coefficient of the particular solution. Let P senote the coefficient of sin2K1 (L-x) in (17). Then the general solution of (16) is

U2 = C sinK (L-x) + C2cosK2(L-x) + Psin2K1(L-x) 12 (19) where Cl = (,-K2/2K1)P, and C2 = P(K~/~K,)[CO~~K~L- cosK2Ll/sinK2L.

Equation (19) shows that our solution is more complicated than we might have expected. While we have only one periodicity in time, the spatial variation involves two closely spaced periodicities, K and 2K1. The variation of the elastic field of the second harmonic along tge bar is thus an interference phenomenon. Use of equations (2) and (3) shows that the stress and strain of this field are not quite proportional to each other, because the propagation constants of the two waves differs slightly. The difference between K and 2K1 arises from the frequency dependence of the anelastic strain for the S~S. Further (17) and (19) state that the amplitude of U2 is proportional to the square of the amplitude of U1, and is therefore more sharply peaked at resonance than U1.

The degenerate Case

The degenerate case when the denominator of P vanishes (K2 2K ) requires 5 1 special consideration. We restrict our attention to the neighborhood of the fundamental resonance, klL = n, so w is not significantly variable. Using (111, we still find three cases for which K = 2K1: (i)when b = c = o, or T = o; (ii)when b = c = m, or T = -; and (fii) c = ba, or 6J = o, for any TO. Case (iii) corresponds to the Pemoval of the anelastic defects from the matgrial, as by purification or annealing. The damping is zero for all three cases, but the modulus defect vanishes only for the latter two.

When the damping goes to zero, the solution does not diverge as suggested by the appearance of the denominator of (171, but in the limit as K .-2K goes to zero the functional form changes. A new solution must be sought $or this case; it is

2 3 2 3 where B1 = dA k /a = dA (k /2) /a, and A is the amplitude of the 1 fundamental. Owe obtainedo(20f by using the0method of variation of parameters, but the same result is found by putting 2K1 = K2 + n, where n is small, obtaining the first order approximation in 0, and taking the limit as n + 0. TO keep A finite in this limit, the driving stress must go to zero. The qualita- tive cRange in the solution as the damping goes to zero, represented by the factor of x in the second term of (20), is noteworthy. The true nature of the solution is obscured when the undamped wave functions are used in the nonlinear calculation. IV - DISCUSSION

To get a physical picture of these results, consider first the ideal elastic body with no damping and no nonlinearity. There are no energy losses in such a model and so at steady~statethere can be no energy input either. Such a state might be created by a driving force operating for a certain length of time to build up some prescribed amplitude, after which the driving force would be removed, and the state just created would last forever after. Although seew ingly convenient for calculation, and the usual starting point, this situation is quite unphysical.

The reality is that losses are always present, but the mechanism of the loss may take more than one form. The SAS is a linear model with the energy loss occur- ring everywhere in the vibrating body. The mechanical energy is converted to heat, each volume element receiving its due proportion, but the detailed mechar nism of dissipation is not specified. Steadycstate occurs when the input of energy from the driving stress equals the sum of the distributed losses. If, instead of anelasticity, nonlinearity is introduced, in the general case the effect is to provide for the continual conversion of energy from the fundamental mode to modes of higher frequency. We expect that these other modes will also be subject to the nonlinearity and so their energy will be converted to still other modes, both higher and lower in frequency, and so on until at steady-state there will be an infinite (in a continuum) number of modes excited, with net conversion from the fundamental to the other modes giving an energy outflow that will be just equal to the input by a driving force at the fundamental frequency. Note that, although the creation of harmonics draws energy from the fundamental mode, that energy is still in organized form in the harmonics, not yet lost to heat. This is quite in contrast to the anelastic losses, which go straight to the thermal bath.

The model presented here falls short of describing the general case. With only two modes treated, and the second one small, transfers of energy to higher modes have been neglected as higher order processes. The solution (20) to the undamped case represents an apparent loss-less steady-state. We believe that this is an artifact of the approximation, and that the general case will eventu? ally prove to be as suggested in the previous paragraph.

When we combine the two mechanisms, as in this problem, a11 modes created by nonlinear processes have anelasticity in their own right as shown by equations (10) to (12). The effect is to create small shifts in the resonant frequencies of the modes, shifts associated with the modulus defect, so that the exact resc nant frequencies are not related by integers /2/, while the harmonic frequencies are. Thus resonance of the fundamental mode corresponds to kl = r/L, with a frequency wl, while resonance of the second harmonic corresponds to k(w2) = 2r/L, with a frequency w2, such that w2 + 2wl. Therefore the second harmonic is imperfectly excited. The excitation in this case comes not from the driving stress but from the term on the right hand side of (16) that renders the equar tion inhomogeneous.

The excitation of harmonics is therefore distributed throughout the bar, at the frequency wl, generating a wave of frequency 2wl and propagation constant 2K 1' To satisfy the boundary conditions the latter generates another wave at the same frequency but with the propagation constant K2 = k2 + ia2 corresponding to 2wl, as given by (11).

As we have defined it, k is neither equal to 2kl nor is it equal when w = dl to 2+/L, differing from bot8 by terms of the order of 6J/J = (c/ab) - 1. However, the width of the resonance is also of order 6J/JU, so the mismatch in frequency is not significant in the present approximation. That is, if (19) is expanded for small n [as suggested below (20)], the result differs from (20) only by JOURNAL DE PHYSIQUE

small terms. Since we are assuming d small, these terms belong to the higher orders that lie outside this approximation.

For the approximation used here, the amplitude of the second harmonic is small and the energy losses from it, by either internal friction or mode conversion, are second order. Therefore, at steady-state, the apparent internal friction is unchanged to first order. Of course, before steady-state was achieved, energy had to be added to create the harmonic wave; once created, then the losses from the harmonic are second order.

Further work is necessary to treat the experimentally observed cases of amplitude-dependent damping. The assumption of small d will have to be dropped, although the assumption of small strain amplitude will remain eminently valid.

In conclusion, we have shown that the inclusion of anelasticity lifts a degener- acy between 2K1 and K2, revealing that the harmonic wave has an unexpectedly complicated spatlal form. We expect that the use of the SAS wave functions will be essential in the further development of the nonlinear theory.

References /1/ A. S. Nowick and B. S. Berry, Anelastic Relaxation in Crystalline Solids, Academic Press,New York, 1972. /2/ D. N. Beshers in Techniques of Metals Research, Vol. 7, part 2, Ed. R. Bunp shah, John Wiley and Sons, New York, 1976, pp531-707.