Solutions to Time-Dependent Pure-End-Condition Problems of Elasticity Pressure-Step Wave Propagation...

8/3/2019 Solutions to Time-Dependent Pure-End-Condition Problems of Elasticity Pressure-Step Wave Propagation and End-R

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Solutions to Time-Dependent Pure-End-Condition Problems of Elasticity

Pressure-Step Wave Propagation and End-Resonance Effects

Irwin S. Goldberg; Robert T. Folk

SIAM Journal on Applied Mathematics, Vol. 53, No. 5. (Oct., 1993), pp. 1264-1292.

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004SlAM J . APPL. MATH 0 1993 Society for Industrial and Applied Mathemat~csVol. 53 , N o . 5, pp. 1264-1292. October 1993

SOLUTIONS TO TIME-DEPENDENT PURE-END-CONDITION PROBLEMS OF ELASTICITY: PRESSURE-STEP WAVE PROPAGATION

AND END-RESONANCE EFFECTS* IRWIN S . GOLDBERG? A N D R O B ER T T . F O LK S

Abstract. A double-transform method is developed for finding exact solutions to axisymmetric nonm ixedtime-dependent problems in elasticity in which the applied stresses are specified on the end of semi-infinitecircularly cylindrical bars with stress-free lateral surfaces. Th e response of the ba r to th e sudd en applicatio n ofpressure to the end is determined. In another application, the method is used to calculate the reflection of acon tinu ous train of waves off a free end of a semi-infinite cylindrical bar. F or the reflection problem , a n endresonance is found when the incident waves have an angu lar frequency in the neighborhood of w = 1.5 1v d / a ,for a rod with Poisson's ratio of 113, where a is the radius of the bar a nd Vdis th e dilatation wave velocity. T heneighborhood of the end-resonance frequency determined for the free-end reflection problem is shown to cor-respond to a range of frequencies of oscillation a t which large deviatio ns exist between the solu tion to t he pure-end-condition problem and the solution to the comparative mixed-end-condition problem describing the pressure-step response.

Key words. elasticity, time dep enden t, pure-end condition. nonm ixed-end condition. wave propagation,end resonanceA M S subject classifications. 73D10, 73D15. 73D30, 73D25, 73C02, 73C05. 73CIO. 35A22, 35615. 73V35

1 . Introduction. Boundary value problems for determining the response of semi-infinite elastic bars to applied surface stresses are classified according to the type of endconditions. The end conditions are called pure if either two components of stress or twocomponents of displacement are specified as boundary conditions at the end of the bar.If one end condition is given in terms of particle displacements and a second end conditionis given in terms of stresses, then these boundary conditions are referred to as mixed-end conditions. A general method to solve mixed-end-condition problems was first givenby Folk et al. [9], [lo]. Goldberg, in an unpublished dissertation [14], used these solutionsfor mixed-boundary-condition problems to obtain integral equations for pure-end-con-dition problems. He solved the integral equations for examples involving end conditionsin bars and slabs.

We present Goldberg's method in 4 2 by applying it to a specific example thatillustrates how the method can be used to solve any linear axisymmetric dynamic pure-end-condition problem for a homogeneous and isotropic semi-infinite circularly cylin-drical elastic bar with a stress-free lateral surface. The solution of the example solved in4 2, in which the response of a cylindrical bar to a suddenly applied pressure step at theend is determined, will be compared to earlier experimental and theoretical work on thisproblem [ lo] , [12]. In 43 similar methods are used to calculate the reflection of a con-tinuous train of waves off the free end of a semi-infinite circularly cylindrical bar. A freeend means that the normal and tangential stress components are specified to be zero onthe end, and so these are classified as pure-end conditions.

2. Response.of a circular bar to a pressure step. As noted above, the procedure forsolving axisymmetric pure-end-condition problems for cylindrical bars is developed interms of an example in which the semi-infinite bar is subjected to a sudden step-function


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DYNAMIC PURE-END-CONDITION PROBLEMS OF ELASTICITY 1265pressure pulse on its end. This example is referred to as apressure-step pure-end-conditionproblem. In this section, the Laplace transform in time of the solution is obtained, andthe procedure of using saddle-point methods for inverting the transform is presented inAppendix B.

With the assumption of radial symmetry with no torsion, using cylindrical coor-dinates having Z along the bar and R at right angles to the axis of the bar, the basic linearequations of motion can be written as

where

and1 aur acizQ = -z(az a ) .

and where U rand U , are the radial and axial displacement components, p is the density,and X and p are the Lam6 elastic constants. To express the above equations in dimen-sionless form, we define dimensionless quantities as 21, = C r r / a ,u , = CT, /a , r = R l a ,z = Z / a , and t = V d T / a ,where a is equal to the radius of the bar and Vdis equal to thedilatational velocity, which is given as Vd=m.he dimensionless forms of( 1 ) and ( 2 )become

d22tr d A 2 d R- + - - ,d t 2 d r C d z

where C = ( A + 2 p ) l p . Since A and R are dimensionless quantities, the dimensionlessexpressions for A and Q can be obtained by substituting lowercase dimensionless coor-dinate and displacement variables for corresponding uppercase dimensional variables in( 3 ) and (4) .

The stress components are determined by the stress-strain equations1 d ( r u r ) d u ,

7:- = ( C- 2 ) ---- + C - ,r dr d z


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1266 IRWIN S. GOLDBERG AND ROBERT T. FOLK( i ) The stress-free boundary conditions at the lateral surface of the bar are given as

and the pure-end conditions corresponding to a suddenly applied pressure loading atz = 0 are(9) T,,(z, r , t ) l z = O= PoS(t) and rrz(z,r , t ) lz=o= 0;

(ii) The quiescent initial conditions at t = 0 are

Here Pois the applied pressure (made dimensionless by dividing the actual applied pressureby P),which is specified to be uniform across the end of the bar, and S ( t ) is the unitstep function.To obtain the solution to the above pure-end-condition problem, first double-trans-form techniques are used to obtain the solution to two mixed-end-condition problems.For the first of these mixed problems, referred to as mixedproblem I, the end conditionsare chosen to be

The end conditions for the second mixed problem, referred to as mi xed problem 11, arechosen to be(12) uZ(z,r , t) z = O = uoz(r, ) and r , t) z = O~ ~ ( 2 , = 0.The two mixed problems are solved with the functions uorand uozstill unspecified.

A set of integral equations, which we call a correspondence relation, is then developedfrom which these unspecified functions are determined by the stipulation that the solutionto these two mixed problems be identical. Under that stipulation, the solution to eitherof these determined mixed-end-condition problems would be equal to the solution tothe pure-end-condition problem that satisfies the end conditions given by ( 9 ) . Amongthe various transforms that are used to obtain solutions to these mixed problems is theunilateral Laplace transform in time

where 7 is a dimensionless complex frequency variable defined b 7 = wa/Vd, with whaving units equal to the angular (Neper) frequency, and i =d.

The general procedure for using double-transform methods to solve dynamic end-condition problems is given in [2 ] , [9] , [l o ] , [14], [16], [22 ], [25] , [31]. An outlineof the application of this method to mixed problem I is shown in Appendix A.

For mixed problems I and 11, the Laplace-transformed analytic results for the stresses,the particle displacements, and the experimentally measurable sum E of the strain quan-tities (where E = coo+ c, with coo = ur/ r and c,, = du,/dz) [ 121 are (see Appendix A)


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DYNAMIC PURE-END-CONDITION PROBLEMS OF ELASTICITY 1267where

with( 15h)and( 15i)where the complex wavenumber y is defined in Appendix A in terms of the sine andcosine transforms of the variable z . The sum in ( 14) is over all the complex roots yn(n)with Im (7,) > 0 and all the real roots with dyn/dn > 0 of the Pochhammer-Chreefrequency equation(16) G(yn, 7) = 0, where G( yn , 7) is given by ( l5 g) [6 ], [l o] , [30 ]. (Roots at k = 0 are not included in the sum; see Appendix A.) For mixed problem I, an=- a(y,, 77) is given by (17a)

where t is a dummy variable replacing r .Solutions to mixed problem I1 have the identical form to that of mixed problem I

given by ( 14) and ( 15) with the normal-mode coefficient a ' replaced by a", where


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1268 IRWIN S. GOLDBERG AND ROBERT T. FOLKFor convenience, the roots of the Pochhammer-Chree frequency equation ( 16) will

be relabelled with a double subscript M and L, and the related mode contributions willbe referred to as the (M, L) modes. The subscript M orders the roots so that the magnitudeof Im (y,,,) for 7 = 0 increases with M. The subscript L denotes the quadrant of thecomplex y-plane in which the root lies when 7 = 0. Since, for solutions with z > 0, rootswith Im ( y ) < 0 are not included in the sum in ( 14 ), roots labeled with L = 3 or 4 areexcluded. There are two roots, which we label y , , , and y ~ , ~ ,hat are real for all real 7.We distinguish these two roots by the conditions d y l , l / d 7> 0 and d ~ ~ , ~ / d ~0 when7 is real and positive.

The position of all the roots y ~ , ~ =re analytic functions of 7 in the vicinity of 70, and thus a set of functions ~ ~ ~ ~ ( 7 )s defined by analytic continuation [9]. At lowreal frequencies, only the roots y , , , and y are real, and all other roots are complex. Athigher real frequencies, some higher-order modes (with M > 1) become real. As explainedby Folk et al., for real positive 7, only those real roots with d y Id 7 > 0 are included inthe normal-mode expansion given by ( 14) [9 ], [l o] , [20 ]. Therefore, for 7 > 0, the realroot y , , , is included, while y,, , is not included in the solution given by (14). As theanalytic continuation of the y,,, functions is extended to larger real values of 7 wheremodes with M > 1 become real, choices of branches are required for the multivaluedsolutions of the Pochhammer-Chree equation. Each of these choices is directed by thecriteria that, when 7 is real and positive, only those real roots with d y Id 7 > 0 are to beincluded in the sum-over-modes solutions.

Starting at 7 = 0, the roots of the frequency equation were evaluated at differentreal values of 7 with an iterative use of the Newton-Raphson method (generalized toapply to complex functions). The iterative procedure was initiated at 7 = 0 with theapproximation [35 ](18a) Y M , L I ~ = ~ 0.5( -1)L+1Log, [4n(M - I ) ] + in(M - 1) for M > 1, L = 1, 2and(18b)All complex (nonreal) roots are symmetrically placed with respect to the imaginary axis;that is, for each complex mode for any M and real 7, we have yM,2(7)= -Yh ,1(7) ,where * denotes complex conjugate. Also, it follows from the symmetry properties ofexpressions ( 15g) and ( 16) that Laplace-transformed solutions can be calculated for7 < 0 with the substitution y(- 7) = -y*(q)for each mode included in the sum in ( 14).Thus, contributions at +7 and -7 can be combined to provide real-valued solutions.Sketches of dispersion curves showing the positions of the complex y,,, roots as a functionof 7 are presented in [9 ] , [21], [29], [34] .

The argument principle [7 ] was used to determine the number of zeros of ( 16)contained within selected closed contours in the complex y-plane. Thereby, all appropriateroots of the Pochhammer-Chree frequency equation were verified to have been accountedfor and included as part of the solution [35]. (Specifically, to avoid difficulties fromsingularities at k = 0, the number of zeros of the expression (G l k ) = 0 was determined;see Appendix A.)

Each mode of mixed problem I has the same rand z dependence as the corresponding


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DYNAMIC PURE-END-CONDITION PROBLEMS OF ELASTICITY 1269( 14) eads to a countably infinite set of integral equations (referred to as a correspondencerelation) from which the unspecified functions uk, and uk, may be determined.

The first step canied out below in solving the correspondence relation is to reducethis set of integral equations to an infinite set of algebraic equations by expanding theLaplace transform of the unspecified end conditions uk, and uk, in modified series oforthogonal Bessel functions. The coefficients in these expansions are the unknowns inthe algebraic equations. To avoid poor convergence due to a Gibbs-type phenomena atr = 1 (uh, and uk, do not satisfy the same homogeneous boundary conditions as theBessel functions), the following expansions are used instead of simple expansions ofu;, and tik, in a series of Bessel functions:

and

where Jo and J , are Bessel functions of the first kind and order 0 and 1, respectively.The unspecified complex coefficients c,, 4, and do that are to be determined depend on7, while the constants el and g, are determined by the successive zeros of

Integrating ( 19) and (20) leads to the following expansions for ud, and ub, that are tobe substituted into ( l7a) and ( l7b):

and

The additional expansion terms coand d- ,,which are introduced as integration constants,help to improve the rate of convergence of the series [14] , [ 311.

Substituting expression (22) into ( 17a), the normal-mode coefficients for mixedproblem I are obtained as


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1270 IRWIN S. GOLDBERG AND ROBERT T. FOLKSimilarly, substituting expression (23) into ( l7b) , normal-mode coefficients for mixedproblem I1 are obtained as

For each value of 7, an infinite set of algebraic equations for the complex expansioncoefficients cj's and 4 's is obtained by equating (2 4) to (25) for each value for M and Lincluded in the sum-over-mode solution. Those expansion coefficients must be determinedfor a range of values for 7.

It is necessary to truncate the correspondence relation to a finite set of 2N - 1complex equations for 2N - 1 complex expansion coefficients cj and d, ( 0 < j < N - 2and d- ) included in the above sums. We retained the correspondence equations for themodes YM,Lwith 1 < M < N for L = 1 and 2 < M < N for L = 2. By substituting theexpansion coefficients c, or 4 , which are solutions of the correspondence relation, intoeither (24 ) or (25), the coefficients CYM,Lare determined and can be substituted into ( 14)to obtain the Laplace-transformed solution to the pure-end-condition problem.The correspondence relation consists of a diagonally dominant system of equations.Therefore, we expect that the truncation of the correspondence relation to a finite numberof modes with a finite number of terms in the Bessel function expansions is justified andthat these truncated expansions are expected to give a close approximation to any of thenormal-mode coefficients with a selected value of M if the number of terms, N, kept inthe series is sufficiently larger than M. Comparisons were made of the normal-modecoefficients ~ M , Lobtained by solving the correspondence relation with different numbersof modes included. Such comparisons clearly show stable convergence of each of thelower normal-mode coefficients obtained as the number of modes in the correspondencerelation is increased.

Due to the faster exponential decay in z of the modes with larger values of M,truncation of the infinite set of integral equations to include only a finite set of the lowestvalues of M is equivalent to saying that, for a given value of q , the two complementarymixed-end-condition problems have equivalent Laplace-transformed solutions for allvalues of z greater than some finite value of 2 .

Numerical results were calculated using the Vax System computer in which double-precision and extended double-precision arithmetic were used. The correspondence re-lation was solved using the Gauss elimination method [19]. All numerical results arepresented in this paper for the special case where Po = 1.0 and where Poisson's ratio isequal to 113, which corresponds to setting C = 4 in all expressions. Also, all Laplace-transformed results presented in the tables and figures are evaluated for real values of 7that are required for the saddle-point approximations described in Appendix B.


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DYN AM IC PURE-END-CONDITION PROBLEMS OF ELASTICITY 1271reference, the dimensionless frequency qC2= 1.85 is referred to as the second-mode cutoffrequency.) The root y2 , , remains real when q 2 nc2 = 1.85, while y2, , is real when1.85 5 n 5 1.91 and when a > 2.16, and y2,2 is purely imaginary when 1.92 5 7 < 2.16.Additional modes with M 2 3 become real at higher frequencies.

For Po = 1 O, the Laplace transforms of the specified pure-end conditions given by(9 ) are

As demonstrated in Tables 1A and 1B for various values of q, the Laplace-transformedsolutions for 7:, and 7 i Z ,evaluated as sum-over-mode solutions at z = 0, closely matchthe Laplace transforms of the pure-end conditions as specified by (9 ) with Po = 1.0.Representative results shown on these tables demonstrate the closest match of the solutionsto the specified boundary conditions, which occurs for small values of 7 , as well as thepoorest match, which occurs when the frequency is in the neighborhood of q = 1.51(later identified as the end-resonance frequency), where the normal-mode convergenceis slowest. Results of Table 1A are obtained with the sum of 39 modes, and results ofTable 1B are obtained with the sum of 99 modes. As demonstrated when comparingTables 1A and 1B (illustrated for = 1.51 where convergence is slowest), the matchbetween the solution and the boundary conditions at z = 0 improves as the number ofmodes included in the sum is increased from 39 modes to 99 modes.

Additionally, the Laplace-transformed results for ub, and uh,, evaluated by ( 14)and ( 15) as sum-over-mode solutions at z = 0, closely agree with the series results givenby (2 2) and (23) after substitution of the c, and d, coefficients, which are obtained fromthe solution to the correspondence relation. Those results were observed to agree to atleast three decimal places for all evaluated values of q (unpublished results).

When 7 0, and so the factors exp ( iyz) decay with increasing z .For larger real 7 , some of the other modes (with M > 1 ) have Im ( y )= 0, but at thesehigher frequencies the contributions from these normal-mode coefficients C Y ~ , ~ ( ~ )resmall for the problem with the step-function loading considered. However, Fox andCurtis experimentally observed small oscillations due to the M = 2 modes of the solution181, [121.To complete the solution, the Laplace transforms of the results given by ( 14) mustyet be inverted. That is, the time-dependent solution for each of the stresses, particledisplacements, and strains appearing in ( 14) involves both a sum over the modes(M , L) and an integral over 77

For solutions at large z , two simplifications occur. First, the sum over y,,, convergesvery rapidly because the factors exp (iyz) are small for the higher modes. Folk et al. [ lo ]


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IRWIN S. GOLDBERG AND ROBERT T. FOLK

TABLEIAC'alc~rlated L~zplace-tratrsformedend stresses jbr the unit a,npI~tzidepre.~.~ure-step prire-end-condition problem evalziated at z= 0,hr various valueso f 1 and r. Trutzsformed resiilts are rnult~plit~dy ? f i r t7asl.cornparison ofthost, valztes with fh e Laplace transform of t h ~trre-end conditions given by(9) ( w ~ t ho = 1.0).The se representrrtive calculated re s~ tlts re obtained with39 modes incllided in the summation .

1 r Real I mag Real Imag0.20 0.00.20 0.20.20 0.40.20 0.60.20 0.80.20 1.o1 .OO 0.01.00 0.21 .OO 0.41.00 0.61 .00 0.8I .OO 1.01.51 0.01.51 0.21.51 0.41.51 0.61.51 0.81.51 I .02.00 0.02.00 0.22.00 0.42.00 0.62.00 0.82.00 1 .0Specified by

( 9 )

TABLE1BCalculated Lapluce-tra nsforzed solutions as described itl the legendto Table 1A; however. the results in th is table are obtained with 99 modes

included in the summation showing iwil~rovedagreement (compared toTable 1A)between th e two solutions and the boundary conditions at z =0 j i~r7 = 1.5 1, where convergence is slowest.

1 r Real Imag Real Imag


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DY NA MIC PURE-END -COND ITION PROBLEMS O F ELASTICITY 1273is presented in Appendix B, where the stress wave E ( z , r, t ) on the surface of the bar( r = 1.0) is determined [ 2] , [5 ], [ 9] - [l l] , [14].Fox and Curtis conjectured that, for large z,the time-dependent solution to thepure-end-condition pressure-step problem considered in this section can be closely ap-proximated by a solution to a specific mixed-end-condition problem I with end conditionsgiven by ( 1 I ) with uo, = 0 [1 2] . (For further reference, this specific mixed-end-conditionproblem is referred to as a compurutive mixed-end-condition problem .) To test the con-jecture of Fox and Curtis for the dominant first mode, at various values of q, comparisonsare given between the values of a, , ,obtained for the solution to the pressure-step pure-end-condition problem and the values of this a, , ,coefficient obtained for the solution tothe comparative mixed-end-condition problem. The ~ M , Lmode coefficients for this com-parative mixed-end-condition problem, referred to as a e ; ' , can be obtained fromexpression ( 17a) with all terms containing integrals set equal to zero, with the result

(see [ lo ] and Appendix A). As shown in Table 2, the first-mode coefficients a, , ,(7 ) forthe comparative pure and mixed problems are found to agree closely for values of q


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1274 IRWIN S. GOLDBERG AND ROBERT T. FOLK

Comparison of the a , , ,coeficien t obtaine d from the solzition to the prc.rsure-step11llr.e-end-conditionproblem with the sirnilar a',?"' coefficient ( 27 ) obtained Jiom thesohrtion to lhe comparative mixed -end-co ndition problem for varioiis real val~ res f 'q.'4 verj)close agreement between those res~rlts re shown for small values of q(? 1.40.Using the saddle-point method developedin Appendix B, the frequencies, amplitudes, and the time of occurrence of the oscillationsdue to the modes ( 1 , l ) , (2, l ) , and (2, 2) that were determined for the pure-end-condition problem are given in Tables 3A and 3B. Oscillations for higher modes werenot calculated because their amplitudes are very small.

Our result describing the head of the pulse is similar in form to that obtained bySinclair and Miklowitz in which double-transform methods were used for a pressure-pulse problem involving a semi-infinite slab [311. However, only transformed solutionscorresponding to q -+ 0 are given in their report; therefore, their results did not describethe behavior of the wave beyond the head of the pulse.

3. Reflections off the free end of a bar. In this section, the procedure used previouslyin 5 2 is used to solve for the reflection of an infinite train of harmonic waves off the free


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D Y N A M I C P U R E - E N D - C O N D I T I O N P R O B L E M S O F E L A S T I C I T Y 1275T A B L E3ADimensionless frequencies ( q ), arrival times (equa l to 100 d - y l d ~ hen z = 100bar ra dii), and amplitudes of oscillations (A ,(M , L) ) or the pure-end-condition pressure-step problem. These results are calculated using the second-order saddle-point approx-imation given by (B 5 )o fAppendix B fo r E ( z , r, t)I,= l ,o.

M o d e ( I , 1 ) Mode (2, 1 ) Mode (2 , 2 )Aniva l Amval Amva l

7 t i m e A $ ( ] , ) t ime A,(2, 1 ) t ime A,(2, 2 )

T A B L E BTh e arrlval times, dimensionless.fiequencies (q,,), and co m pl e,~ ave-num bers (y,,) describing locations of third-order saddle pointsfor modes( I , I ) , ( 2 , I ) , a nd ( 2 , 2 ) . T h e t er ms A,, and x can be substituted into (B6),for the evaluation ofthird-orde r sadd le-po~nt ontributions to pure-end-con-dition solutions q f E( z, r, t ) ,= ,,o at z = 100 bar radii.Mode Ilrm Y sm Arrival tim e A,, X

and( 2 9 ) r , t ) l z = o= r r z ( z , r , t ) l z = o= 0 .2 2 ( ~ ,A body force at a distance zo from the end of the bar is assumed to be "sendingout" harmonic waves of angular frequency 170 for -co < t < co . That is, we include a


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1276 I R W I N S. G O L D B E R G A N D R O B E R T T . F O LKTABLEA

Complex residual stress amplitude of T, , evaluated at z = 0, r = 0.0.Incident wave + Incident wave +

In c~ de ntwave first refl. mo de 39 refl. mod es'lo Real Imag Real Imag Real Imag

TABLEBComplex residual stress amplitude of T,, evaluated at z = 0, r = 1.0

Incident wave + Incident wave +Inclden t wave first refl. mo de 39 refl. mod es

'lo Real Imag Real Imag Real Imag

by (30) , added to ( 2a ) (o r to ( 2 ) ) ; the remaining governing equations ( 1 ) - ( 7 ) are leftunchanged.

Since the domain of the time variable in this example is -co < t < co, the (two-sided) Fourier transform


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D Y N A M I C P U R E - E N D - C O N D I T I O N P R O B L E M S O F E L A ST IC IT Y 1277TABLEC Com plex residual stress am plitude of 7, ; evaluated at z= 0 , r = 0.5.

Incident wave + Incident wave +Inciden t wave first refl. mo de 39 refl. mod es

'lo Real Imag Real Imag Real Imag

Co mp ies resrdual stress amplrtude of.^, evaluated a( z = 0 , r = 0.5.Incident wave + Incident wave +

Incident wave first refl. mo de 39 refl. mod es'lo Real Imag Real Imag Real Imag

as the combination of separate Laplace transforms for negative and positive times [27].In this manner, we may justify the inclusion of modes with the same values of Y M , L usedin 2 for the pressure-step propagation problem.

As in 5 2, a correspondence relation will be obtained by equating the solutions oftwo mixed-end-condition problems, each with an unspecified end condition. For one ofthese mixed-end-condition problems (mixed-reflection problem I ) the end conditionsare chosen to be


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1278 IRWIN S. GOLDBERG AND ROBERT T. FOLKwhile, for the other mixed problem (mixed-reflection problem 11), the chosen end con-ditions are given by ( 12). The functions uo, and uor,which are initially unspecified, areto be determined by the correspondence relation.

The solutions to these two mixed-end-condition problems are obtained by the samemethod used in Appendix A and [ lo ] . Because of the body-force term (30), the solutionmust be written separately for z < zo and for z > zo. The solution for 0 S z < zo is

where the an functions are given by expression ( 15), and y, are the roots of the Poch-hammer-Chree frequency equation in the upper-complex plane, as described in termsof the (M, L) subscripts in 5 2.

The coefficient V M , L is

which is determined by the body-force term ( 30) and is independent of the end conditions.Note that this sum-over-mode solution is separated into two parts: the first part, withcoefficients p n , representing waves reflected from the free end at z = 0, and the secondpart, with coefficients Y , , representing waves generated by the body force and travellingtoward the free end.

The normal-mode coefficients PMSLfor mixed-reflection problem I, with end con-ditions given by expression (32) are given by

where azhl is defined as aLXL ,iven by (2 4), with the term iPo(C- 2)/7h2omitted. Inthe derivation of (35), the series expansion of the form given by expression (22) is usedfor the unspecified end displacement u&. For mixed-reflection problem 11, the normal-mode expansion coefficients are given by

where ak,L is defined by (25 ). In the derivation of (36), the series expansion of the formgiven by expression (23) is used for the unspecified end displacement, u& . Clearly, thevalues of the coefficients cl and dl in (2 2) and (23) are not the same for the present


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DYNAMIC PURE-END-CONDITION PROBLEMS OF ELASTICITY 1279Because each of the inhomogeneous terms in these linear equations contains the

factor [6(q - 70)- 6(17 + no)], the coefficients cj(q) and d,(n) also contain this samefactor. The presence of the Dirac-delta function in each additive term of the Fourier-transformed solution greatly simplifies the evaluation of the integral for the inversion ofthe Fourier transforms [23] . Combining contributions at +go and - no in the inversionintegrals, real time-dependent solutions for the stresses and displacements are obtainedfrom the evaluation of an integral expression of the form

with Re { .) denoting the real part of a complex function.The problem is especially simple to analyze for no smaller than the second-mode

cutoff frequency and when the location of the body force is far from the free end, i.e.,for zo large. The results reported here are for zo equal to 50 bar diameters. For theseconditions, a close approximation results when only the first-mode (nondecaying) term,involving the v,,, coefficient, is included for the incident wave.

The coefficients c, and d, were solved for different choices of the number of termsretained in the truncation and for a range of values of the "driving" frequency 70. Thesecoefficients are substituted into either ( 35 ) or ( 36) to obtain the values of /3M,L,whichcan be substituted into (33) to obtain time-transformed solutions. As in tj2, all numericalresults are calculated for Poisson's ratio equal to 1/ 3.

The accuracy of the results is tested by using them to evaluate stresses T, , and 7 , atthe end of the bar, which are specified to be zero by the end conditions (2 9). In Tables4A-4D, the calculated complex residual stress amplitudes are given for T,, and 7 , at thefree end of the bar (z = 0 ) . These complex stress amplitudes are evaluated by integratingthe Fourier transform of the stresses, as given by expression (3 3), with respect to 7 from17 = 0 to 17 = co. (The integrations are performed to remove the 6(7 - vO)and the6(7 + no) singularities, thereby facilitating the presentation of data in the tables.) Theresults are given for a range of frequencies 70 of the incident wave. The representativeresults presented in these tables are evaluated for the case where the body force generatingthe incident wave is located at zo = 100 bar radii from the free end. For comparison,three sets of columns in these tables show the real and imaginary parts of (i ) he complexresidual stress amplitudes resulting from the incident wave alone; (ii)the complex residualstress amplitudes resulting from the contributions of the incident wave plus the firstreflected mode; and (iii) the complex residual stress amplitudes resulting from the con-tributions from the incident wave plus 39 reflected modes. Note that these complexresidual stress amplitudes in which the reflected-mode contributions are included aremore than a magnitude smaller than the complex residual stress amplitudes due to theincident wave alone.

It is interesting to note that, for no < vc2, while only one mode is present in theincident wave, many decaying modes are present in the reflected wave near the end ofthe bar. For any mode with M > 1, the amplitude of the contribution from any of thesedecaying (M, 1 ) modes is equal to the amplitude of the contribution from the related(M , 2) modes. Therefore, the response due to these higher modes are exponentially


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1280 IRWIN S. GOLDBERG AND ROBERT T. FOLKThough the amplitudes of the incident and reflected first-mode waves are equal,there can be a phase shift between them. This phase-shift difference for the stress T,, atz = 0 is shown in Fig. 1 for a range of frequencies. If the phase shift was exactly an oddinteger multiple of T radians, then the reflected first-mode wave would cancel the incident

one exactly, and so the end condition 7,: = 0 would be satisfied with only the first-modeterms. Note that the phase shift is either close to T or -T radians except for a small rangeof frequencies in the neighborhood of q0 = 1.5 1, that is, in the neighborhood of w0 =1.51Vd / a .In the neighborhood of q0 = 1.5 1, the incident and reflected first-mode waves forrZ zdo not even approximately cancel on the stress-free end. Therefore, the amplitudesof some of the higher modes must be large to cancel the residual first-mode contributionson the end.An mth-mode axialreflection coeficient can be defined as the ratio of the magnitudeof the mth-mode (with M = m, L = 1 ) contribution to the magnitude of the first-mode(with M = 1, L = 1 ) contribution of the reflected-wave calculation of u, evaluated atz = 0 [ 3 4 ] . The second- and third-mode axial reflection coefficients are displayed asfunctions of'qo in Figs. 2(a ) and 2(b) , respectively. These results demonstrate thelarge contributions of higher modes (with M > 1 ) of the axial displacement in thevicinity of z = 0 when qo is in the neighborhood of the resonance peaks of those curvesat q0 = 1.51.The large contributions from the higher modes of u, in the range of frequenciesnear q0 = 1.5 1 are standing waves whose effects are confined to the region near the endof the bar because the factors exp (iy,z) decrease with z. This means that, within thatnarrow range of frequencies, the end of the bar oscillates with a much larger amplitudethan at other frequencies. It is therefore reasonable to identify these relatively large os-cillations near the end of the bar when q0 is close to 1.51 as an end-resonance effect. Theend-resonance behavior described in this paper can be shown to be independent of zo,

-1800.0 0.5 1 .O 1.5Dimensionless F r e quen c y q,


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DYNAMIC PURE-END-CONDITION PROBLEMS OF ELASTICITY

15-

it .a,0 -

10-- t .a,CK -Q) -u0 - 3 5 -D -'L(U

0 0 0 5 1 0 1 5D i m e n s i o n l e s s F r e q u e n c y q,

0 00c0 1 50 5D i m e n s i o n l e s s F r e q u e n c y rl,

F I G . 2. Second- and third-mode axial rejection coefficients: (a) Second-mode axial rcifection coefficientevalztated at r = 0.5, showing md-resonant behavior at 70 = 1.51; (b ) Third-mode axial rclflection coeflcientevahmted at r = 0.5, showing a smaller resonant peak at = 1.51 than thal obtained3)r the second-modecoeficient. Th e results of ( a ) and (b ) , shown,for r = 0.5, are representative o ft h e similar end-resonance peaksevalrrated at various values o f r from r = 0 to r = 1.0; that I S , ur unpublished results indicate that only theheight of'th e resonance peak varies with r , while the end-resonance fkq ue nc y rema ins invariant at qo = 1.51.Th e curves end at the second-mode cuto fffieq uen cy at 70 = 1.85.

provided that zo is sufficiently large such that the contributions from each of the decayingmodes generated at the body force decay to negligible values before reaching the free endat z = 0.


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1282 IRWIN S. GOLDBERG AND ROBERT T. FOLKAt 170 = 1.5 1 , the complex wavenumbers of the dominant decaying ( 2 , 1 ) and( 2 , 2 ) modes are y = k 1.23 + i1.32. These slowest decaying modes decay in Z with a

factor exp ( - 1 . 3 2 Z / a ) , from which an estimate can be made for the distance from theend over which end-resonance effects predominate. In particular, at end-resonance fre-quencies, all decaying mode contributions decay at least by a factor of 5 X to withinfour bar radii from the end of the bar.

End-resonance effects have also been shown to exist for semi-infinite plates fromthe theoretical calculations of Torvik [ 3 3 ] ,using variational principles, and by Gregoryand Gladwell [17 1, using their method of projection for determining the reflected wavefield. End-resonance behavior has also been predicted from previous solutions to theapproximate equations of elasticity in slabs [ 1 3 ]and bars [ 21 1 , [ 2 4 ]and has been observedand measured in bars in experiments conducted by Oliver [ 2 8 ]and Zemanek [ 3 4 ] .

Zemanek's experimentally measured end-resonance frequency at 170 = 1.5 13 was inclose agreement with his theoretically calculated end-resonance frequency at 170 = 1.503(for a bar with Poisson's ratio equal to 0 .3 3 1 7 ) . In those calculations, the normal-modecoefficients were determined from a system of equations obtained by setting the truncatednormal-mode sum for the stresses at the end of the bar to zero at selected nodal circles.However, the solution obtained by Zemanek was not shown to satisfy the end conditionsover almost all points at the end. The results given in this paper include quantitativedetails of the solution for the response of the bar, including the solution at and near theend under end-resonance conditions. Our calculated end-resonance frequency is in closeagreement (within 1 percent) of the calculated and measured end-resonance frequencygiven by Zemanek [ 341.

From the period of vibration of 8 . 4 ~econds measured at resonance by Oliver [ 2 4 ] ,[ 2 8 ] ,we estimate the end-resonance frequency to be at 170 = 1.63 for a steel bar with anestimated Poisson ratio of 0.29 . For comparison with Oliver's experimental result, wecalculated the end-resonant frequency for a bar with Poisson's ratio equal to 0.290(C = 3.381 ) to be at 170 = 1.59, which is within 2.5 percent of Oliver's result. In contrast,McNiven's solution of the approximate equations [ 2 4 ]results in a calculated end reso-nance at 170 = 1.4 1 , which deviates by 13 percent from the experimental result given byOliver.

4. Summary. A method was developed for obtaining analytic normal-mode ex-pansion solutions to time-dependent, nonmixed, end-condition problems in elasticity.The general method was applied to two specific problems involving (i ) he far-field prop-agating wave resulting from a pressure step applied to the end of a bar, and ( ii) thereflection of an incident wave off the stress-free end of a bar. For both of these problems,all transformed solutions were observed to closely match the specified end conditions atz = 0.

The solution to the end-reflection problem demonstrated an end-resonance effectin which, for a narrow range of frequencies in the vicinity of the end-resonance frequencyat 170 = 1.5 1 , large-amplitude vibrations occur at and near the stress-free end of the bar.These end-resonance effects were shown to be related to large exponentially decayingstanding-wave contributions from the decaying modes and a large variation in the phaseshift of the incident (nondecaying) mode upon reflection at frequencies in the vicinity


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DYNAMIC PURE-END-CONDITION PROBLEMS OF ELASTICITY 1283mixed-end condition of the type considered by Folk et al. [ lo ] was specified. To test thatconjecture, a comparison was made (for large z ) between the solutions obtained fromboth of these types of end-condition problems. That comparison showed a close agreementbetween those solutions for the main features of the results involving the shape andarrival time of the head of the pulse, the amplitudes and arrival times of lower-frequencyoscillations (with 17 < 1.40) following the head of the pulse, and the long time behaviorof the strain. These results were shown by Fox and Curtis to closely agree with experimentalmeasurement [12 1.

As determined from the saddle-point method (Appendix B ) , the arrival times forall oscillation components in the wave train of the pressure-step solution are independentof the type of end condition. However, the amplitudes for the higher-frequency oscillations(that is, for frequencies with 17 > 1.4) differ significantly between the solution obtainedfor the pure-end-condition problem and the solution obtained for the comparative mixed-end-condition problem. The greatest deviation between these solutions occurred for os-cillations at frequencies in the neighborhood of 17 = 1.5 1; these oscillations occurredduring the time interval when the saddle points were in the neighborhood of the end-resonance frequency (0 = 1.51 ), as determined from the end-reflection problem.

Typically, for a 2"-diameter aluminum bar (V dm 6.32 km/sec, Poisson's ratio0.33), the oscillation frequencies would be scaled as o(rad/sec) = 2.49 X l o50.

Numerical approximation methods provide a valuable tool for the analysis of time-dependent elasticity problems, particularly for the wide class of problems for which analyticsolutions are not available. Finite-difference and finite-element methods have been ex-tensively developed for this purpose. As new variations of those methods develop, testsmust be conducted to compare their performance with older, more established techniques.Also, as new classes of problems are considered with existing numerical methods, basicquestions such as the adequacy of the grid spacing and the time integration proceduremust be addressed if reliable numerical simulations are to be performed. Analytic solutionsof the type presented in this report can provide benchmark solutions that can be usedto evaluate the numerical approximation methods and answer the above questions. Theusefulness of these types of analytic results for benchmark solutions has been demonstratedin a recent paper by Goldberg et al. [I51, in which analytic sum-over-mode solutions ofa time-dependent, fluid-dynamic, entrance-flow problem was successfully used to bench-mark a finite-element code. Time-harmonic analytic solutions of the type presented forthe end-reflection problem would be particularly useful for testing timestep codes fornumerical time integration procedures. That is, deviations from the known periodicbehavior of the analytic solutions can be evaluated as the numerical time integrationcode steps through increasing numbers of time periods.

The methods applied in this report to solutions of pure-end-condition circular barproblems have been applied to time-dependent pure-end-condition problems involvingsemi-infinite slabs. These results can be found in the Ph.D. thesis of Goldberg [14], whereit was shown that. in the static limit (as 17 + 0), those solutions agree with the staticpure-end-condition solutions obtained by Herrmann [18].

Appendix A. U se of the double-transform method for solving mixed-end-conditionproblems. In this appendix, an outline of the double-transform method is presented for


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1284 I RW IN S . GOL DB ERG A N D ROB E RT T . F OLKThe Laplace transform in time given by ( 13) is used in addition to the Fourier sinetransform in z defined by

and the Fourier cosine transform in z defined by

where y is the dimensionless transform variable for z .For solutions to mixed problem I, we take sine transforms in z of ( 1), (3) , (5) , and(6 ) and cosine transforms in z of (2) , (4 ) , and ( 7) , followed by the Laplace transformsin time of all equations.With the above choice of transforms, a consistent set of transformed equations isobtained in which either the sine transform or the cosine transform (not both) of eachdependent variable is contained in the resulting equations. Also, in those transformedequations, the only end conditions necessary to be specified are those given by ( 1 1).The transformed equations can be combined, resulting in two uncoupled differentialequations

and

where

and

In the above results, we have used the notation(A51 h 2 = 9 2 - y 2 and k 2 = C V 2 - y 2 ,where i = G.The solutions to these uncoupled equations (which are finite at r = 0 )are

and


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DYNAMIC PURE-END-CONDITION PROBLEMS OF ELASTICITY 1285After substituting (A6) and (A7) into the doubly transformed equations for stressand displacement components, the coefficients A(y, 7) and B (y, 7) can be obtained bysetting 7:: and 7:: to zero at r = 1, with the result

and

where

Using results (A8) and (A9), doubly transformed solutions u", and US' are obtainedby substituting A" and Q"', given by (A6) and (A7), into the equations resulting fromtaking the appropriate double transforms of ( la ) and (2a) of 9 2. By substituting thoseresults into the equations resulting from taking the appropriate double transform of (5) -( 7 ) , doubly transformed solutions T:',, T::, and 7:; are obtained.The inverse sine transform of a doubly transformed function is given as

The inverse cosine transform is given as

The Cauchy residue theorem is used (with Jordan's lemma) to evaluate integralsin the forms contained in expressions (A1 1) and ( A 12). The integrals are equal to 2 ~ itimes the sums of the residues of the poles of the transformed solutions in the uppercomplex y-plane [9] , [l o] , [14], [20 ], [ 32] . These poles correspond to the roots of theequation G(y, 17) = 0 (referred to as the Pochhammer-Chree frequency equation [6],[30] ), where G(y, 7) is given by expression (A10). For each real value of 7, therecorresponds a countably infinite set of complex roots with Im ( y ) # 0, in addition to afinite set of real roots to the Pochhammer-Chree frequency equation. At first glance,they appear to be additional poles that may contribute in the evaluation of the inversionintegrals; however, it can be shown that the residues of all such isolated singularities


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1286 IRWIN S. GOLDBERG AND ROBERT T. FOLKAfter inversion of the sine and cosine transforms, the time-transformed solution tomixed problem I may be expressed in the form given by ( 14) and ( 15) with the normal-mode coefficients given by ( l7a ) in 2.Solutions to mixed problem I1 can be obtained by interchanging the use of sine and

cosine transforms in z for the above procedure. That is, by taking cosine transforms inz of (1 ) , (3) , (5 ) , and (6 ), and taking sine transforms in z of (2 ) , (4 ), and ( 7 ) , aconsistent set of doubly transformed equations is obtained in which the only end con-ditions necessary to be specified are those given by ( 12).Appendix B. U s e of the saddle-point method for inversion of the Laplace transformto obtain time-dependent solutions. A brief discussion of the application of the saddle-point method for the evaluation of Bromwich inversion integrals of the type given by(26) is presented in this appendix [ 2] -[ 5] , [9] , [l o] , [ l l ] , [14]. In particular, themethod will be applied to obtain time-dependent solutions to the pressure-step pure-

end-condition problem. A more complete discussion of the application of these methodsto elastic wave propagation problems of the type presented in this paper has been presentedby Folk et al. [9 ] , [ l o ] .First, the positions of the saddle points, identified as 17 = v,, are determined for theintegrand of (26) by solving the equation

where V, is defined as the group velocity, which is a function of 17 Note that, for aconstant z, the positions of the saddle points 17 = q,, as given by (Bla ) and illustratedin Fig. 3, vary with time.Next, the path of integration in (26) is distorted to one along which the integrandis small except when passing through a saddle point along its path of steepest descent.The time-dependent position of the saddle points for mode ( 1, 1) is illustrated in Figs.3(a)-3(f), where the integration paths through the saddle points are shown to passthrough the indicated cross-hatched region. The value of the original integral is equal tothe sum of the contributions for each saddle point crossed by the distorted integral path(not all saddle points are used) plus 27ri times the residues of the isolated singularitiesthat lie between the original and new paths of integration.To demonstrate this method of integration, consider the evaluation of the inversionintegral in (26) of the Laplace-transformed result given in ( 14) for the experimentallymeasurable strain ~ ' ( z ,, v), evaluated at r = 1. Evaluated results are given for a Poisson'sratio of 1 3, and for unit amplitude input Po = 1 O. For this quantity, the integrand in(2 6) for mode (M , L) is


25/31

DYNAMIC PURE-END-CONDITION PROBLEMS OF ELASTICITY 1287The nature of the saddle points is different in different time intervals. For t < z , theintegral in ( 2 6 )is zero, since the path of integration can be distorted to 7 = im .That is,no signal travels faster than the dilatation wave [ 9 ] , l o ] .For z < t < z V d / V h ,where V h = \IYIP ( Y is Young's modulus) is called the bar

velocity, the path of integration can be distorted through a second-order saddle pointthat moves down in time along the imaginary 7-axis, as shown in Fig. 3 (a) (see figurelegend). Thus, the exponential term in (B2) decreases with z , approaching zero forlarge z . This very small first signal is called the precursor preceding the head of the pulse1 9 1 , l o ] .At t = z V d / V h ,the two second-order saddle points that were moving along theimaginary 7-axis collide at 7 = 0 forming a third-order saddle point for an instant, asillustrated in Fig. 3 ( b ) . For those saddle points in the vicinity of 7 = 0 , a special extendedsaddle-point method developed by Cerrillo (for third-order saddle points in the vicinityof a pole) must be used instead of the regular method [ 5 ] .The behavior of the head ofthe pulse can be determined from the first-mode contribution of the Laplace-transformedsolution in the vicinity of 7 = 0 . The first-mode Laplace-transformed contribution to thepure-end-condition solution and to the comparative mixed-end-condition solution haveidentical first-order terms for the Taylor-Laurent expansions about 7 = 0 , which can bewritten as

Therefore, the pure-end-condition problem and the comparative mixed-end-conditionproblem have identical first-order asymptotic solutions describing the head of the pulseof the form [ l o ]

where

a is Poisson's ratio (in this paper, a = 113; thus B = 1 . 8 6 9 ~ - ' ' ~ [ t 1 . 2 2 5 z ] ) ,and Ai({)is the Airy function [ I ] , [ 2 6 ] .Though there is a difference between the pure- and mixed-end contributions in higher-order asymptotic terms describing the head of the pulse, thesecond-order term can be shown to have a z -213dependence and can therefore be neglected(along with the other higher-order terms) for large values of z [ 9 ] , 1 11, [ 1 4 ] , 2 0 ] , 2 1 .Equation (B4a) describes the suddenly rising head of the pulse for t approaching z V d /V b ,followed by large oscillations about the value -0.2500 [ l o ] ,as experimentally verifiedby Fox and Curtis [ 12 1 .

After t = z V d /V h , he third-order saddle point at 7 = 0 splits into two second-ordersaddle points that move along the real axis away from the origin; see Fig. 3(c ) . At t =z V d / V R ,where V Ris the Rayleigh wave velocity, two new saddle points appear at infinity


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I R W I N S. G O L D B E R G A N D R O B E R T T . FO L K

FI G. 3. Position of the saddle points and steepest-descent integration paths a s a function of tim e for thefirst-mode contribution. AN steepest-descent integration paths are shown to go through the designated cross-hatched regions at the saddle points ( se e [ 9 ] - [ I ] , [ I41). A jirst-order pole is also shown at the origin: ( a ) or1 < t / z < Vd /V b , here Vd= dilatation velocity and Vb = bar velocity; ( b ) the collision of seco nd-order saddlepoints forming a third-order saddle point at the origin when t / z = V d / V b ; c ) f o r V d / V b< t / z < V d / V R , hereV R= velocity of Rayleigh surface wave; ( d )for Vd/VR< t / z < Vd/V,,,, where V,,,= minimum group velocityfor the jirst m od e. A second pair of saddle points are shown to form when t l z > V d / V R ;( e ) the collision ofsecond-order saddle points forming a third-order saddle point when t / z = Vd/V,,,;( f ) separation o ft he saddlepoints for t / z > V d /V m ;For a bar with Poisson's ratio of 1 / 3,first-mode resultsfrom the solution to ( 16 ) ndicatetha t Vd /Vb= d- yl , l / d1 71 n= 01.225, Vd /V R= 2.17, and Vd/Vm= 3.140.

constant residue of the pole plus oscillations calculated by the saddle points

where y, is equal to Y M , L evaluated at q = qr. The above sum is over the second-ordersaddle points for propagating modes (with ~ M , Leal) taken in pairs +q,, with q, > 0, to


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DYNAMIC PURE-END-CONDITION PROBLEMS OF ELASTICITY 1289

The phase 0 in ( B 5 )is given by

0 = i.-i + tan-' Im [F ' (TM ,L , , v ) 14 Re [ F ' ( YM . L ,1, v ) 1where Re [ 1 , Im [ 1 , and I I , respectively, denote the real part, the imaginary part,and the magnitude of a complex function and where the sign of the 7i/4 term is that ofd ' yM . L / d q 2 .For a specified value of z , the arrival time of the oscillations for mode(M , L) is equal to z d ~ ~ , ~ / d v ,valuated at the oscillation frequency vs.

As shown in Table 3A, the amplitudes, frequencies, and times of arrival of theseoscillations are evaluated using the second-order saddle-point method to calculate


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IRWIN S. GOLDBERG AND ROBERT T. FOLK

F I G . 4. Plot ~?fdimens~onlessealfieqzrency v versus t l z , here t I S the ditnensionless arrival fl m e at dlstance;from the m d ~ ' i l hressure-sfep loading. T he m odes ( 1 , 1 ), ( 2 , 1 ), a nd ( 2 , 2 ) are represenfed as indicated.The ratio t l z is deferm inedjrom fhe re lut~onship l z = d y l d ~ , vulua fedjbr each m ode as ajitnction o f ' s .R eg lo n D ( 2 , 2 ) denotes th e,fact [hat y2 ,2 s real (a n d negative) in the nurro\v,frequency range 1.85 < 11 < 1.91.A disconnected second region here y2.2 I S real ( an d posifive) is indicated hen 7 > 2.16. (72.2 is pure imag inaryin the intermediate,fiequency range hen 1.92 iv i2 .16 . )

At t = zVd/V m, where Vm is the m ini m um grou p velocity for th e first mode, thepairs of saddle points collide, forming a n insta ntane ous third-order sadd le point, as show nin Fig. 3 ( e ) . Th e extended third-order sadd le-point metho d mu st be used again becausethe nature of the saddle points is changing rapidly. However, there is no pole in thevicinity of these third-order saddle points, as show n in Fig. 3 ( b ) , an d so the result ofapplying the extended saddle point is not given by ( B 4 a ) . Instead, the response for tim esnear t = zV d/V m , ncluding the residue from the pole at 17 = 0 , is given as

where

a n d

a n d wh ere y,,, is ~ I \ I , Levalua ted at v,,, the position of the third-order saddle point whent = zV~/V, , (v,,, is defined a s th e value of 17 f or w hic h d 2 y / d V 2 0 ) . In ( B6 ) t he "p lu ssign" is used in the arg um ent of the Airy functio n when V,, is a min im um , an d the


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DYNAM IC PURE-END-CON DITION PROBLEMS O F ELASTICITY 1291( 1, 1 ), third-order saddle-point contributions are evaluated using (B6) with the constantresidue term omitted.

After t = ~l.'~/v,,,,he third-order saddle points break into pairs of second-ordersaddle points that move into the complex 7-plane,as illustrated in Fig. 3(f). Consequently,the oscillations then exponentially damp out, and the signal remains constant at thevalue given by the residue, the constant term, equal to -0.2500, in (B5) and (B6)[91, [l o] .All the features described in this appendix were reported in the experimental resultsof Fox and Curtis [121.

Acknowledgments. We thank Dr. Clyde Scandrett at the Naval Post GraduateSchool, Dr. Christopher J. Freitas at Southwest Research Institute, and Dr. Robert McLayat the University of Texas at Austin for helpful advice and suggestions.

REFERENCES[ 1 M. ABRAM OWITZ Handbook of Mathematical Functions, Dover Press, New York, 1972,ND I . STEGU N, pp. 446-452, 475-478. [2 ] J . D. ACHENBACH, ave Propagation in Elastic Solid s, H. A. Lauwerier and W. T. K oiter, eds., AppliedMathematics and Mechanics, Vol. 16, Elsevier, New York, 1984.[3] N. BLEISTEIN,athematical M odels for Wa ve Phenomenon , Academic Press, New York, 1984.[4 ] L . BRILLOUIN, ave Propagation and Group Velocity, Academic Press, New York, 1960.[ 5 ] M . V. CERRILLO,n E lementary Introduction to the Theory of the Saddle Point Method of Integration,Tech. Report 55:2a, Research Lab. of Electronics, MIT, Cambridge, MA, 1950.[6 ] C . CHREE, he equations of an isotropic elastic solid in polar and cylindrical coord inates, their solutionand application, Trans. Cambridge Phil. Soc., 14 ( 188 9), p. 351.[7] R. V. CHURCHILL, omp lex Variables and Applications, McGraw-Hill, New York, 1960.[ 8 ] C . W. CURTIS,Second mode vibrations of the Pochhammer-Chreefrequency equation, J. Appl. Phys., 25

( 195 4), p. 928.[9 ] R. T . FOLK,Time Dependent Boundary Value Problems in Elasticity, Ph.D. thesis, Lehigh University,

Bethlehem, PA, 1958.[ l o ] R . FOLK,G. FOX,C. A. SHOOK ,AN D C. W. CURTIS,Elastic strain produced by a sud den application ofpressure to one end of a cylindrical bar. I. Theory, J. Acoust. Soc. Amer., 30 ( 1958), pp. 552-558.[ 11 R. FOLKAND A. HERCZYNSKI,olutions of elastodynamic slab problems using a new orthogonality con-dition, J. Acoust. Soc. Amer., 80 ( 1986), pp. 1103-1 110.[ I21 C . Fo x A N D C. W. CURTIS,Elastic strain produced by sudden upplication of pressure to one end of acylindrical bar. 11. Experimental observations, J. Acoust. Soc. Amer., 30 ( 195 8), pp. 559-563.[ 131 D. C. GAZIS N D R. D. MIND LIN, xtensionul vibrations and waves in a circular disk and a semi-infiniteplate, Trans. ASM E Ser. E J. Appl. Mech., 27 (19 60 ), pp. 541-547.[I4 1 I . S. GOLDBERG,olutions to Dynamic Pure End Condition Problems in Elasticity, Ph.D. thesis, LehighUniversity, Bethlehem, PA, 1970.[ 151 I. S. GOLD BERG, . F. CAREY , . M CLA Y,AND L. PHINNEY, eriodic viscousflow:A bench-markproblem,Internat. J. Numer. M eth. Fluids, 11 (19 90 ), pp. 87-97.[16] I. S. GOLDBERGND R. T. FOLK,Solutions for steady and nonsteudy entranc ejlow in a semi-infinitecircular tube at very low Reynolds numbers, SIAM J. Appl. Math., 48 ( 1988 ), pp. 770-791.[ 171 R. D. GREGORY ND I. GLADWELL,he rejection of a symmetric Rayleigh-Lamb wave at the fi xe d orfree edge of a plate, J. Elasticity, 13 ( 198 3), pp. 185-206.[ 181 J. W. H E R R M A N N ,xact S olutions to Static and Dynam ic Problems of the Semi-Inf inite Slab with PureEnd Conditions, Ph.D. thesis, Lehigh University, Bethlehem, PA, 1967.[ 191 F. B. HILDEBRAND,ethods ofApplied M athematics, 2nd ed., Prentice-Hall, Englewood Cliffs, NJ , 1965.[ 20 ] 0 . E. JONESA N D F. R. NORWOOD, xially symmetric cross-sectional strain and stress distributions in


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Solutions to Time-Dependent Pure-End-Condition Problems of Elasticity Pressure-StepWave Propagation and End-Resonance Effects


SIAM Journal on Applied Mathematics, Vol. 53, No. 5. (Oct., 1993), pp. 1264-1292.

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16 Solutions for Steady and Nonsteady Entrance Flow in a Semi-Infinite Circular Tube at VeryLow Reynolds Numbers


SIAM Journal on Applied Mathematics, Vol. 48, No. 4. (Aug., 1988), pp. 770-791.

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