Lie Symmetry Analysis and Approximate Solutions for Non-linear Radial Oscillations of an Incompressible Mooney Rivlin Cylindrical Tube

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1 Ž Journal of Mathematical Analysis and Applications 45, doi:116jmaa6748, available online at on Lie Symmetry Analysis and Approimate Solutions for Non-linear Radial Oscillations of an Incompressible MooneyRivlin Cylindrical Tube D P Mason 1 and N Roussos Centre for Differential Equations, Continuum Mechanics, and Applications and Department of Computational and Applied Mathematics, Uniersity of the Witwatersrand, Priate Bag 3, Wits 5, Johannesburg, South Africa Submitted by William F Ames Received December 8, 1999 The non-linear, second-order differential equation derived by Knowles Ž196, Quart Appl Math 18, 7177 which governs the aisymmetric radial oscillations of an infinitely long, hyperelastic cylindrical tube of MooneyRivlin material is considered It is shown that if the boundary conditions are time dependent, then the Knowles equation has no Lie point symmetries, while if the boundary conditions are constant it has one Lie point symmetry corresponding to time translational invariance The derivation by Knowles Ž 196, J Appl Mech 9, 8386 of bounds on the period of the oscillation for the heaviside step loading boundary condition is etended to obtain limiting oscillations that ehibit periods that bound the eact period above and below The Knowles equation for a MooneyRivlin material is epanded in powers of a dimensionless parameter,, defined in terms of the thickness of the tube wall To zero order in an ErmakovPinney equation is obtained which has three Lie point symmetries It is shown that the differential equation which is correct to first order in also has three Lie point symmetries which disappear at second order in For time independent boundary conditions, the three Lie point symmetries of the order equation are derived eplicitly and the associated first integrals are obtained The general solution is derived in terms of the three first integrals and it is illustrated for free oscillations and the heaviside step loading boundary condition The non-autonomous first order in equation is transformed to an autonomous ErmakovPinney equation and a non-linear superposition principle for the solution to first order in is derived and applied to a blast loaded applied pressure that decays linearly with time The solutions to first order in are compared with numerical solutions of the Knowles equation for a thick-walled cylinder and are found to be more accurate than the zero order solutions described by the ErmakovPinney equation Academic Press Key Words: finite elasticity; Lie point symmetries; symmetry breaking; limiting oscillations; first integrals; non-linear superposition; ErmakovPinney equation 1 To whom all correspondence should be addressed -47X $35 Copyright by Academic Press All rights of reproduction in any form reserved 346

2 ANALYSIS OF MOONEYRIVLIN TUBE INTRODUCTION The first investigation of a dynamic problem in a bounded medium in finite elasticity was undertaken by Knowles 1, 11 He considered the non-linear radial oscillations of an infinitely long, circular cylindrical tube of incompressible hyperelastic material and reduced the equation of motion to a second order ordinary differential equation For a MooneyRivlin material, the Knowles equation of motion is ln 1 ln 1 ž 1 / 1 Ž 1 ln PŽ t, Ž ž / where the overhead dot denotes differentiation with respect to time, r Ž t 1 t, 1, Ž 1 1 Ž Ž C1 C P1Ž t PŽ t, PŽ t, * * 1 1 Ž 1 C1 and C are the MooneyRivlin constants which are assumed positive, the constant * is the density of the homogeneous and elastically isotropic incompressible tube, 1 and are the inner and outer radii of the undeformed tube which deform continuously to give the inner and outer radii r Ž t and r Ž t of the deformed tube, and P Ž t P Ž 1 1 t is the net radial applied pressure, where P Ž t and P Ž 1 t are the applied pressures on the inner and outer surfaces of the tube The dimensionless parameter,, which is defined in terms of the thickness of the wall of the tube, is small for a thin-walled tube Since the left-hand side of Eq Ž 11 is of order, Pt Ž is of order one This paper is concerned with Eq Ž 11 and its epansion in powers of In the limiting case of a thin-walled tube, the Knowles equation for a MooneyRivlin material Ž 11 reduces to the ErmakovPinney equation 3, 19 Ž PŽ t 3 Ž 13

3 348 MASON AND ROUSSOS This was utilized by Nowinski and Wang 17 who considered a vanishing net applied load at the boundary to analyse free oscillations The general solution of the ErmakovPinney equation may be obtained from a non-linear superposition principle 3, 19 Thus Shahinpoor and Nowinski were able to derive eact solutions for free oscillations, heaviside step loading, blast loading, a harmonically varying load, and a periodic step pulse load at the boundary Rogers and Baker 1 etended their results to a larger class of strain-energy functions using a generalisation by Burt and Reid 1 of the non-linear superposition principle of Ermakov and Pinney Their work in this area has been reviewed by Rogers and Ames The coefficients in the Knowles equation Ž 11 can be epanded in powers of the thickness parameter and differential equations correct to successive orders of can be derived We will investigate the relationship between the order in to which the differential equation is epanded and the Lie point symmetry structure of the equation The approimation to zero order in, which is the ErmakovPinney equation Ž 13, and the differential equation correct to first order in each admit three Lie point symmetries For the epansion correct to second and higher orders in as well as for the eact equation Ž 11, symmetry breaking occurs, and the autonomous equations where the net applied pressure at the boundaries is time independent possess only one Lie point symmetry corresponding to time translational invariance, while the non-autonomous equations where the net applied pressure is time dependent possess no Lie point symmetries By adopting an approach similar to the one used by Knowles 11 to derive upper and lower bounds on the period we derive limiting oscillations for Eq Ž 11, subject to a heaviside step loaded boundary condition, which have periods which bound the eact period above and below for both net inward and net outward applied pressures For the non-linear radial oscillations correct to first order in we obtain the first-order correction of the results derived by Shahinpoor and Nowinski for the zero-order equation Ž 13 The three Lie point symmetries of the autonomous differential equation correct to first order in are obtained eplicitly from which three associated first integrals are derived The general solution for constant Pt Ž may thus be epressed in terms of the three first integrals When the boundary conditions are time dependent, the differential equation correct to first order in is transformed to an autonomous equation by insisting that the transformed equation possess the Lie point symmetry corresponding to time translational invariance in the transformed time variable The transformed equation is found to be an autonomous ErmakovPinney equation By starting

4 ANALYSIS OF MOONEYRIVLIN TUBE 349 from this ErmakovPinney equation, a non-linear superposition principle correct to first order in is derived An outline of the paper is as follows In Section the Lie point symmetries of the Knowles equation Ž 11 for a thick-walled tube are investigated In Section 3 the heaviside step loading boundary condition is considered and limiting oscillations are obtained which correspond to the upper and lower bounds on the period for both net inward and net outward applied pressures In Section 4 the Knowles equation is epanded in powers of the dimensionless parameter It is shown that the differential equations correct to zero or first order in each have three Lie point symmetries and that the reduction in the number of symmetries from three to one or zero, depending on whether the net applied pressure P is constant or time dependent, occurs at the second order in approimation For the special case in which P is constant the three Lie point symmetries are calculated correct to first order in In Section 5 the autonomous case of constant P is considered The first integral associated with each of the three Lie point symmetries is derived correct to first order in and the general solution for constant P is found in terms of the three first integrals The general solution is used to obtain eplicit solutions correct to first order in for free oscillations and for the heaviside step loading boundary condition In Section 6 the non-autonomous case of time dependent boundary conditions is considered The differential equation correct to first order in is transformed to an autonomous ErmakovPinney equation and a non-linear superposition principle, correct to first order in, is derived for the equation and used to solve to first order in the problem of blast loading with linear decay in time The new results correct to first order in are compared graphically with the zero order solutions obtained by Shahinpoor and Nowinski and with the numerical solution of the eact equation Ž 11 Finally, concluding remarks are made in Section 7 LIE POINT SYMMETRIES In this section we investigate the Lie point symmetries of Eq Ž 11 Etensive treatments of Lie group theory applied to differential equations have been given by Olver 18, Bluman and Kumei, Stephani 3, Mahomed and Leach 16, and Ibragimov and Anderson 7 In general, the second order differential equation FŽ t,, Ž 1

5 35 MASON AND ROUSSOS is said to admit a Lie point symmetry generated by X Ž t, Ž t,, Ž t if X Ž FŽ t,, whenever FŽ t,,, Ž 3 where 7 1 X X Ž t,,, Ž 4 ẍ is the second prolongation of the generator X, 1 X X 1Ž t,, Ž 5 ẋ is the first prolongation of the generator X, 1Ž t,, DŽ DŽ, Ž t,,, DŽ 1 DŽ, Ž 6 and D Ž 7 t ẋ For arbitrarily given FŽ t,,, the determining equation Ž 3 is t t t t F F ž 3 / F t t F Ž 8 t t ẋ ž / For Eq Ž 11, the epanded determining equation Ž 8 yields an equation which can be separated by equating the coefficients of like powers of, 1 3 :, ln 1 Ž

6 ANALYSIS OF MOONEYRIVLIN TUBE : 3 1 ln 1, Ž 1 t 1 1 : t 3 t t 1 ln 1 : lnž 1 1 Ž ln 1 1 ln 1 PŽ t ln 1, t 5 ln 1 1 ln 1 Ž 11 ln 1 1 Ž Ž 1 PŽ t 1 lnž 1 / lnž 1 1 Ž 1 5 ln 1 1 ln 1 PŽ t P Ž t t lnž 1 ln 1 / ž / Ž 1 The aim is to solve Eqs Ž 9 Ž 1 for Ž t, and Ž t,

7 35 MASON AND ROUSSOS Consider first Eq Ž 9 for Ž t, It is a first order, homogeneous, linear differential equation for with integrating factor ŽŽlnŽ1 1 1 and can be integrated once with respect to to obtain ž / 1 1 f1ž t ln 1, Ž 13 1 Ž 1 1 where f t is an unknown function of t and the factor is introduced Ž Ž 1 in the denominator because the numerator in 13 is O as The results derived here will therefore coincide with known results for the ErmakovPinney equation Ž 13 derived from the Knowles equation for a MooneyRivlin material in the thin-shell limit as The integration of Ž 13 with respect to gives Ž t, f1ž t HŽ fž t, Ž 14 Ž where f t is an unknown function of t and 1 H ž / HŽ ; ln 1 d Ž 15 1 Consider net Eq Ž 1 for Ž t, It may be integrated once with respect to to give 1 kž t, Ž 16 3 t ž 1 ln 1 / ž / where kt Ž is an unknown function of t On substituting Ž 14 for Ž t,, Eq Ž 16 becomes 1 f 1Ž t HŽ f3ž t, Ž 17 3 ž 1 ln 1 / ž / where f Ž t kt Ž f Ž t Equation Ž 17 3 is a first order differential Ž 1 equation for t, with integrating factor HŽ, where the prime denotes differentiation with respect to Hence, Eq Ž 17 may be integrated to give 1 Ž t, f1ž t H Ž f3ž t HŽ f4ž t, Ž 18 HŽ Ž where f t is an unknown function of t 4 1

8 ANALYSIS OF MOONEYRIVLIN TUBE 353 Ž Ž Ž Ž Ž The substitution of 14 for t, and 18 for t, into 11 yields, after simplification, where 1 f t f t y f Ž 3Ž 1Ž 1Ž t yž PŽ t f1ž t y3ž 3 3 f1ž t y4ž, Ž 19 y 1, 1Ž Ž y H, 1 Ž Ž Ž 1 y3ž, Ž 1 lnž 1 / ž ž // 1 4 ž 1 ln 1 / ž ž // 4 1 ln Ž 1 1 Ž 1 y Ž Ž In order to evaluate the unknown functions in Ž 19 we determine the Wronskian determinant Wy, y, y, y 8, 9 on some interval of tif Ž y Ž, y Ž, y Ž, and y Ž are defined by Ž to Ž , it can be verified using Mathematica that their Wronskian determinant is not identically zero The epansion of Wy, y, y, y is given in Appendi A Therefore, for a given radial oscillation there will be an interval of t Ž for which W y, y, y, y Ž Thus, y Ž, y Ž, y Ž, and y Ž are linearly independent and we can equate their respective coefficients to zero: f Ž t f Ž t, f Ž t, PŽ t f Ž t, f Ž t Ž Hence f Ž t even for free oscillations for which Pt Ž, and where c f3ž t f Ž t c 3, Ž 6 is a constant Thus, Ž 14 and Ž 18 reduce to 1 1 Ž t, fž t, Ž t, f Ž t c3 HŽ f4ž t HŽ Ž 7

9 354 MASON AND ROUSSOS In order to determine f Ž t, c, and f Ž t, we substitute Ž 7 for Ž t, 3 4 and Ž t, into Ž 1 and simplify to obtain the determining equation f Ž t z Ž f Ž t z Ž c z Ž f Ž t z Ž f Ž t z Ž where Ž Ž f Ž t c PŽ t z Ž f Ž t PŽ t z Ž f Ž t PŽ t 4c PŽ t z Ž, Ž z1ž HŽ, Ž zž KŽ HŽ GŽ, Ž 3 z3ž KŽ HŽ GŽ, Ž 31 z4ž 1, Ž 3 z5ž GŽ, Ž HŽ z6ž, Ž lnž 1 1 ln 1 / ž / ln 1 z Ž, Ž 35 ž / ž ž // Ž Ž 36 lnž 1 / and where ž / lnž 1 KŽ ; ln lnž 1 / 1 Ž 1 Ž ž 1 ln 1 / ž /

10 ANALYSIS OF MOONEYRIVLIN TUBE 355 and lnž 1 GŽ ; ln 1 1 ln 1 ž / ž / ž / ž / Ž ln 1 ž / Ž 1 Ž 38 8 ž 1 ln 1 / ž / The Wronskian determinant Wz, z,, z with the z Ž 1 8 i, i 1,,8, given by Ž 9 to Ž 36 was evaluated using Mathematica but is too long to reproduce here To check that the Wronskian determinant is not identically zero, the term 3 47, Ž ž 1 ln 1 / ž / which is linearly independent of all other terms in the Wronskian determinant, was considered The sum of its coefficients was evaluated using Mathematica to be Therefore, the Wronskian determinant is not identically zero for a given radial oscillation and we conclude that f Ž t, f Ž t, c, and f Ž t P Ž t Hence f Ž 4 3 t c where c is an arbitrary constant and also c P Ž t Ž 4 Therefore, from Ž 7, ½ c, P Ž t, Ž t,, P Ž t, Ž t, Ž 41 Ž and we conclude from that the autonomous Knowles equation for a MooneyRivlin material possesses one Lie point symmetry, X, Ž 4 t

11 356 MASON AND ROUSSOS which corresponds to time translational invariance whereas the timedependent equation admits no Lie point symmetries 3 LIMITING OSCILLATIONS We have seen that it is not possible to solve the Knowles equation for a MooneyRivlin material through Lie point symmetry methods when Pt Ž When Pt Ž, a first integral can be derived 11 because the Knowles equation admits the symmetry X t and the problem can be reduced to quadrature Sometimes, however, the integral cannot be evaluated in closed form in terms of elementary functions and approimate methods for evaluating the integral have to be devised In this section we will consider the heaviside step loading boundary condition for which the problem has been reduced to quadrature by Knowles 11 Knowles derived upper and lower bounds on the period of the oscillation for both net inward and net outward applied pressures We will etend Knowles results to obtain the corresponding displacement fields or limiting oscillations for each of the upper and lower bounds on the period We will compare these limiting oscillations with the numerical solution and with the non-linear superposition solution of the Ermakov Pinney equation Ž 13 for the thin-walled tube approimation obtained by Shahinpoor and Nowinski Consider the Knowles equation for a MooneyRivlin material, Ž 11, with the net applied pressure Pt Ž given by the heaviside function Ž P1Ž t PŽ t, t PŽ t Ž 31 ½ P, t, * 1 where P is a constant, and subject to the initial conditions Ž 1, Ž Ž 3 The greatest or least value of during an oscillation is determined from the condition Knowles 11 showed that when either 1or a where for all values of P satisfying P lnž 1, Ž 33 a is given uniquely by 1 a Ž 34 P 1 e Ž 1

12 ANALYSIS OF MOONEYRIVLIN TUBE 357 For a net inward applied pressure, P, the inequality Ž 33 is always satisfied and a 1 For a net outward applied pressure, P, we consider only net applied pressures P for which the inequality Ž 33 is satisfied and 1 a It may be shown 11 that the period T of the oscillation is given by Ž Ž a lnž 1 z T sgnž a 1 ' H 1 1 Ž z 1ln Ž 1 z 1 a dz, Ž 35 where z and sgnž a 1 is the sign of the factor a 1 By a similar argument, the time t taken for the inner radius of the cylinder to reach a particular value of for the first time by epanding or contracting from rest at 1 is given by 1 1 z Ž lnž 1 z t sgnž a 1 ' H 1 1 Ž z 1ln Ž 1 z Ž 1 a 4 dz Ž 36 With the aid of the inequalities 11, r s 1 r r s ln, 1 r 1 s 1 s r s, Ž 37 r lnž 1 r r, 1 r r, Ž 38 we will briefly rederive Knowles results for the upper and lower bounds on the periods and then derive the limiting displacement fields corresponding to these bounds on the periods 31 Lower Bound on the Period and Associated Oscillation for Net Outward Applied Pressure Since P it follows that a 1 Thus, Eq Ž 35 for the period is 1 1 a ln 1 z T H Ž dz Ž 39 1 ' 1 Ž z 1ln Ž 1 z 1 a Ž 4 In order to obtain a lower bound on the period given by Ž 39 we use the Ž Ž inequalities 37 and 38 Since z a and z we have

13 358 MASON AND ROUSSOS respectively whence z a lnž 1 z Ž 1 a 4, 1 a z lnž 1 z, 1 z Ž Ž a a z T H dz Ž 311 ' 1 Ž z Ž z 1Ž a z Further, since z 1 it follows that zž z 1Ž 1, whence 1 Ž a a dz H L 1 1 ' 1 T T, Ž 31 Ž 1 Ž z 1Ž a z where T is the lower bound on the period Using the substitution 17 L z 1 u, Ž 313 a z we have a dz du H Ž H 1 Ž z 1Ž a z 1 u and therefore Ž Ž 1 a 1 TL, Ž 315 ' 1 which is a better estimate than Knowles 11 lower bound on the period for a net outward applied pressure, a TL, Ž ' Ž 1 owing to its being a greater lower bound We may now derive the displacement field associated with the lower bound on the period Ž 315 From Eq Ž 36, 1 1 z lnž 1 z t H dz, Ž ' 1 Ž z 1ln Ž 1 z Ž 1 a 4

14 ANALYSIS OF MOONEYRIVLIN TUBE 359 which may be approimated in the same way as was the period to yield the following oscillation associated with the lower bound on the period: Ž dz Ž Ž z 1Ž a z 1 a z 1 ' H 1 t 1 Ž By making the transformation 313 we obtain 1 z dz u du z H ž ž / 1 z 1 a z 1 u a z / H Ž Ž 1 Ž 318 tan Ž 319 Ž and hence, epressed in the original variable,, where z, 318 becomes Ž 1 1 a 1 t tan Ž 3 1 ' Ž 1 a Ž Equation 3 may be solved for the limiting oscillation ' Ž 1 ' Ž 1 Ž 1 1 Ž Ž a Ž a t cos t a sin t, 31 1 which is in the form of a non-linear superposition and where a is given by Ž 34 The displacement field Ž 31 of the approimate solution has period T given by Ž 315 L which is smaller than the eact period, but the maimum and minimum values, a and 1, respectively, of the dimensionless inner radius t Ž are the same as for the eact solution 3 Upper Bound on the Period and Associated Oscillation for Net Outward Applied Pressure As with the lower bound on the period for net outward applied pressure, P and therefore a 1 The equation for the period is again given by Ž 39, but, in order to obtain an upper bound on the period, we use the inequalities z a lnž 1 z Ž 1 a 4, lnž 1 z z, 1 z Ž 3 since z a and z Thus a a 1 z T H dz Ž 33 ' 1 Ž z 1Ž a z 1

15 36 MASON AND ROUSSOS Further, since z 1 it follows that 1 z 1 and therefore 1 až 1 a dz T T U H Ž 34 1 ' 1 Ž z 1Ž a z But this integral was evaluated in Ž 314 The upper bound on the period is thus 1 až 1 TU, Ž 35 ' which was first derived by Knowles 11 The equation of the limiting oscillation with period TU is obtained from Ž 36 by performing the same approimations as were made to derive the period Ž 34 This gives 1 až 1 z dz t H Ž 36 1 ' 1 Ž z 1Ž a z Ž and by using 319 and transforming back to the original variable z we obtain 1 až t tan Ž 37 ž / ' a Solving Ž 37 for we obtain the limiting oscillation 1 ' ' Ž 1 1 Ž až 1 až 1 t cos t a sin t, 38 which is again in the form of a non-linear superposition It has period T given by Ž 35 U which is greater than the eact period but the same maimum and minimum values for the dimensionless inner radius, a and 1 respectively, as the eact solution 33 Lower Bound on the Period and Associated Oscillation for Net Inward Applied Pressure We suppose now that P, that is, that there is a net inward applied pressure Then, a 1, whence a t Ž 1 By following a similar procedure to that for net outward applied pressure we obtain the lower bound on the period, a 1 dz a TL H, Ž 39 1 ' a Ž 1 zž z a '

16 ANALYSIS OF MOONEYRIVLIN TUBE 361 and the corresponding limiting oscillation, which can be integrated to give a 1 dz t H, Ž 33 1 ' z Ž 1 zž z a 1 ' ' Ž Ž t cos t a sin t 331 a a Equation Ž 331 is in the form of a non-linear superposition and a 1 defines the minimum value of the dimensionless inner radius during the oscillation which is equal to the eact minimum value given by Ž 34 Clearly, the maimum value of the dimensionless inner radius is unity, in agreement with the initial conditions Ž 3 34 Upper Bound on the Period and Associated Oscillation for Net Inward Applied Pressure Finally, for P and a 1, the upper bound on the period is found to be 1 1 Ž a 1 dz Ž a H 1 TU Ž 33 ' a Ž 1 zž z a ' and the limiting oscillation with period T U is given by which leads to 1 Ž a 1 dz t H, Ž ' z Ž 1 zž z a 1 ' ' Ž 1 1 Ž Ž a Ž a t cos t a sin t 334 As before, a 1 and 1 are the minimum and maimum values of the dimensionless inner radius, respectively, of the eact oscillation and the oscillation, Ž 334, corresponding to the upper bound on the period For comparison, the displacement field in the thin-shell approimation, obtained by Shahinpoor and Nowinski by solving the ErmakovPinney equation Ž 13 for the heaviside step loading boundary condition using a

17 36 MASON AND ROUSSOS non-linear superposition principle, is where 1 ' ' Ž Ž t cos t a sin t, 335 a a 1 1 a lim Ž P 1 e Ž 1 P Ž 1 The period of the oscillation is a T Ž 337 ' The results Ž 335, Ž 336, and Ž 337 apply for both net outward and net inward applied pressures, P The limiting oscillation Ž 331 corresponding to the lower bound on the period for net inward applied pressure has the same form as Ž 335 The only difference is a given by Ž 34 in Ž 331 and a given by Ž 336 in Ž 335 The results of this section are illustrated in Fig 1 where the approimate displacement fields for a net outward applied pressure, Ž 31 and Ž 38, as well as for a net inward applied pressure, Ž 331 and Ž 334, the corresponding thin-shell displacement field Ž 335 and the numerical solution of the Knowles Equation Ž 11 with the heaviside step loading boundary condition, obtained by using the Mathematica NDSole routine, are plotted on the same system of aes We see that for P, the maimum displacement of the thin-shell solution of the ErmakovPinney equation is a poor approimation of the numerical solution, unlike the maimum displacements of the two limiting oscillations, even for the comparatively small value of 6 used in Fig 1 In addition, the period of the thin-shell oscillation is even less than the lower bound of the period, and the graph of the thin-shell oscillation lags behind the numerical solution even more than the limiting oscillation corresponding to the lower bound of the period For P, we see that the minimum value of the inner radius in the thin-shell approimation is a poor approimation of the numerical solution, unlike the minimum values given by the two limiting oscillations However, the graph of the thin-shell approimation lags only slightly behind that of the numerical solution and the period of the thin-shell approimation provides a better approimation to the period of the eact solution than do the two limiting oscillations

18 ANALYSIS OF MOONEYRIVLIN TUBE 363 FIG 1 The nondimensional inner radius, t, Ž for heaviside step loading given by Ž 31 plotted against t for P 5 and 6: Ž a oscillations Ž 31 for P and Ž 331 for P corresponding to the lower bounds on the period, Ž b a numerical solution of Ž 11, Ž c oscillations Ž 38 for P and Ž 334 for P corresponding to the upper bounds on the period, and Ž d thin-shell approimation Ž 335 '

19 364 MASON AND ROUSSOS 4 SYMMETRY BREAKING Nowinski and Wang 17 and Shahinpoor and Nowinski have shown that if the Knowles equation for a MooneyRivlin material is epanded in powers of the thickness parameter then for a thin-walled tube it reduces to the ErmakovPinney equation Ž 13 to zero order in But, it is well established 3, 19, that the ErmakovPinney equation possesses three Lie point symmetries for both the autonomous and non-autonomous cases In this section we consider an epansion in powers of of the functions in the symmetry-determining equation Ž 8 in order to investigate at which order in the reduction in the number of Lie point symmetries of the eact Knowles equation for a thick-walled MooneyRivlin tube occurs From Ž 7 the Lie point symmetries are given by 1 1 X f t f Ž Ž t c3 HŽ ; f4ž t, Ž 41 t HŽ ; Ž Ž where H ;, defined by 15, may be epanded in a Taylor epansion with respect to to give HŽ ; O Ž Ž The functions f Ž t and f Ž 4 t and the constant c3 are obtained from the determining equation Ž 8 We epand the coefficients z Ž to z Ž 1 8 in Ž 4 8 in Taylor epansions with respect to correct to order because it is only to this order that the epansions of the eight functions become linearly independent These epansions correct to order 4, with the Taylor epansions of KŽ ; and GŽ ; given by Ž 37 and Ž 38 and on which z Ž to z Ž 1 8 depend, are given in Appendi B The epansions are written in such a way that the functions which are linearly dependent and to which order in is immediately apparent 41 Zero Order in Ž The Knowles equation 11 for a MooneyRivlin material when epanded in powers of is, to zero order in, the ErmakovPinney equation Ž PŽ t Ž 43 3

20 ANALYSIS OF MOONEYRIVLIN TUBE 365 In order to investigate the symmetries of Eq Ž 43, we consider the determining equation Ž 8 to zero order in : 1 3 f t f t 4 c f Ž Ž Ž t 1 f Ž t Ž Ž 3 f Ž t c PŽ t f Ž t PŽ t 3 4 f Ž t PŽ t 4c PŽ t Ž 44 We split Eq Ž 44 according to powers of, starting with the most negative powers By equating to zero the coefficients of 4 and 3 we find, respectively, that f Ž 4 t and c3 The remaining non-zero terms in Ž 44 are all proportional to By equating the coefficient of to zero we obtain f Ž t 4Ž PŽ t f Ž t P Ž t f Ž t Ž 45 Equation Ž 45 is a third order linear ordinary differential equation for f Ž t and its solution therefore contains three constants of integration The Lie point symmetry generators are given by Ž 41 with HŽ ; given by Ž 4 to zero order in : 1 X f t f Ž Ž t Ž 46 t Equation Ž 43 therefore admits three Lie point symmetries obtained by setting to zero in turn all constants ecept one in f Ž t 4 First Order in Ž The Knowles equation 11 for a MooneyRivlin material epanded to first order in is 1 PŽ t Ž PŽ t 1 Ž

21 366 MASON AND ROUSSOS Ž The determining equation 8 correct to first order in is c f 3 4Ž t 3 ž 3 5/ ž 4 6 / 1 1 f t f Ž Ž t f4ž t Ž f Ž t c3 PŽ t 4 f t P t f t P 4Ž Ž Ž Ž Ž t 4c3PŽ t Ž 48 4 We separate Ž 48 according to powers of, starting with the most negative powers By equating to zero the coefficients of 6 and 5 it follows that f Ž 4 t and c3 By equating to zero the coefficients of the remaining powers, 1 and, we obtain 1 : f Ž t 4 PŽ t f Ž t P Ž t f Ž t, Ž 49 Ž 1 : f t 4 1 P t f t P Ž Ž Ž Ž t fž t Ž 41 Ž Ž Equation 49 is consistent with 41 because terms of order are neglected Equation Ž 41 is a third order, linear ordinary differential equation for f Ž t and its solution will contain three constants of integration There are thus three Lie point symmetries of the first order in correction to Ž 43 with the Lie point symmetry generators given by Ž 41 with HŽ ; given by Ž 4 correct to order : If and we define 1 X f Ž t f Ž t Ž 411 t 1 PŽ t Ž 41 Ž t 1 PŽ t, Ž 413

22 ANALYSIS OF MOONEYRIVLIN TUBE 367 then Ž 41 can be written as f Ž t 4 Ž t f Ž t 4 Ž t Ž t f Ž t Ž 414 Consider now the special case in which Pt Ž P where P is a constant satisfying Ž 41 and define Ž 1 P Ž 415 We will obtain eplicit epressions for the three Lie point symmetries If we let ḟž t hž t, Ž 416 then Ž 414 becomes Thus where A and A 1 hž t 4 hž t Ž 417 hž t A1cosŽ t A sinž t, Ž 418 are constants and therefore A1 A fž t sinž t cosž t A 3, Ž 419 where A3 is a constant By setting one of the three constants A 1, A, A3 equal to zero in turn, three Lie point symmetries are obtained from Ž 411, X1, t Ž X sinž t cosž t, t ž / Ž X3 cosž t sinž t Ž 4 t Three first integrals of the autonomous differential equation Ž 47 with Pt Ž a constant will be derived in Section 5 from the three Lie point symmetries Ž 4 to Ž 4 Finally, in this subsection, we rewrite Ž 41 in a useful alternative form as a second order ordinary differential equation containing an arbitrary constant If Ž 41 is multiplied by f Ž t then it can be rewritten as ž / d 1 d fž t fž t f Ž t 1 PŽ t f Ž t dt dt Ž 43

23 368 MASON AND ROUSSOS and therefore ž / 1 fž t fž t f Ž t 1 PŽ t f Ž t C, Ž 44 where C is a constant of integration If we let f Ž t g Ž t, then Ž 44 becomes C gž t 1 PŽ t gž t, Ž 45 3 g Ž t which is an ErmakovPinney equation Therefore, the second order differ- Ž ential equation 47 admits the Lie point symmetry with generator X g Ž t gž t gž t Ž 46 t where gt Ž satisfies the second order differential equation Ž 45 In Section 6 we will derive, from this result, a non-linear superposition principle for Eq Ž 47 for the general, non-autonomous case in which Pt Ž can be a function of time The results derived in this subsection correct to order reduce to the results for the ErmakovPinney equation Ž 43 by letting Second Order in The Knowles equation Ž 11 for a MooneyRivlin material epanded correct to order is PŽ t PŽ t 1 PŽ t Ž Ž Now, in the determining equation 8 epanded correct to order, the highest negative power of is 8 and it occurs only in the epansion of z Ž But the coefficient of z Ž in Ž 8 is f Ž t and therefore f Ž t By equating to zero the coefficients of 7 and 5 in the remaining terms of the determining equation correct to order it can be verified

24 ANALYSIS OF MOONEYRIVLIN TUBE 369 that 7 7 : f Ž t c3, Ž : f Ž t 3 c3 Ž 49 6 But since the determinant of the coefficients of f Ž t and c3 in the homogeneous system of equations, Ž 48 and Ž 49, is non-zero it follows that f Ž t and c Thus f Ž 3 t c where c is a constant and the determining equation reduces to Ž 4 as with the Knowles equation for a thick-walled tube Thus if Pt Ž then c and from Ž 41 there are no Lie point symmetries, while if Pt Ž then c is arbitrary and from Ž 41 we regain the Lie point symmetry Ž 4 which corresponds to time translational invariance 44 Third Order in In the epansion of Ž 8 to third order in, the largest negative power 1 of is and it occurs only in z Ž The coefficient of z Ž in Ž is Ž Ž 9 f4 t and therefore f4 t By considering the coefficients of and 7 which occur only in z Ž and z Ž 3, it can be verified that c3 and f Ž t Thus the determining equation reduces again to Ž 4 and there are no Lie point symmetries if Pt Ž and there is the trivial Lie point symmetry, X t, if Pt Ž 45 Fourth and Higher Orders in If terms of order 4 or higher are retained in the determining equation then z Ž to z Ž are linearly independent Thus the coefficients of z Ž to z Ž in Ž 8 vanish and the determining equation reduces to Ž 4 8 We therefore see that the reduction in the number of symmetries from three to one or zero occurs at the second order in approimation It was not necessary for the functions z Ž to z Ž 1 8 to be linearly independent for this to occur and indeed z Ž and z Ž 1 6 are linearly dependent up to order 3 5 FIRST INTEGRALS FOR CONSTANT NET APPLIED PRESSURE TO ORDER Throughout this section we will consider the special case for which Pt Ž P where P is a given constant which satisfies the inequality Ž 41 The three Lie point symmetry generators are thus given by Ž 4 to

25 37 MASON AND ROUSSOS Ž 4 We will derive the first integral associated with each of the symmetry generators and obtain the general solution for constant net applied pressure in terms of the three first integrals The general solution will be used to obtain the particular solutions for free oscillations and for the heaviside step loading boundary condition All final solutions will be correct to first order in A function Jt, Ž, is a first integral associated with the symmetry generator X if 1 X 1 J and DJ, Ž 51 1 where X is the first prolongation of X defined by Ž 5 and the operator D is defined by Ž 7 Consider first the symmetry generator X t From Ž 51, J J J Ž 5 t ẋ and solving the differential equations of the characteristic curves of Ž 5 gives the three independent solutions c 1, c, J c 3, Ž 53 where c, c, and c are constants Hence the general solution of Ž is J JŽ u,, Ž 54 where u and Now, from the second relation in Ž 51, J J u Ž 55 u and from the differential equations of the characteristic curves of Ž 55 it follows that d Ž 56 du u But, and u and by using Ž 47 for, Ž 56 may be written as the first order differential equation for, d už P 1, du u u u u Ž 57 1 which is readily integrated to give, correct to order, 1 1 u 1 1 J ˆ, Ž 58 1 u u u

26 ANALYSIS OF MOONEYRIVLIN TUBE 371 where Jˆ is a constant Equation Ž 58 1 may be epressed concisely in terms of the original variables as where 1 J1 1 1, Ž 59 J Jˆ 1 1 Ž P Ž 51 The constant J1 is the required first integral corresponding to the symmetry generator X 1 Consider net the symmetry generator X defined by Ž 41 This gives, from Ž 51, the first order quasi-linear partial differential equation 1 J 1 J sinž t cosž t t ž / 1 J 1 cosž t sinž t Ž 511 The differential equations of the characteristic curves yield the invariants 1 cosž t u c 1, Ž c, sinž t sinž t Ž 51 where c and c are constants The reduced equation is given by Ž 56 1 Now by using in Ž 51 to eliminate it can be shown that u 513 sin t Ž Ž Also, using Ž 51 and Ž 47 to eliminate and, respectively, and by epressing in terms of u using Ž 51 it can be verified that 1 1 u 1 Ž 514 sinž t u u Ž Ž Ž The substitution into 56 of 513 and 514 yields the first order differential equation for, d 1 1 u 1, Ž 515 du u u

27 37 MASON AND ROUSSOS which is readily solved to give 1 J u 1, Ž 516 u u where J is a constant Substituting Ž 51 for u and, the first integral J may be rewritten, correct to order, as sin t 1 Ž ž / cosž t J 1 1 sinž t Ž 1 1 sin t 1 sinž t Ž 517 The first integral, J, associated with X given by Ž 4 3 3, is calculated in the same way as was J It can be verified that, correct to first order in, cos t 1 Ž ž / sinž t J3 1 1 cosž t Ž 1 1 cos t 1 cosž t Ž 518 The three first integrals, J 1, J, and J 3, are given correct to first order in by Ž 59, Ž 517, and Ž 518 Since Ž 47 is a second order ordinary differential equation, only two of the three first integrals are independent By using any two of the first integrals, the solution for can be obtained by eliminating However, the solution can also be written in terms of all three first integrals and in a concise way It follows directly from Ž 59, Ž 517, and Ž 518 that Ž t J J sinž t J cosž t Ž 519 ' Ž Equation 519 can be written alternatively as ' Ž t J J J cosž t, Ž 5 '

28 ANALYSIS OF MOONEYRIVLIN TUBE 373 where tan J J From Ž 5 3, the maimum and minimum displacements, ma and min, are given by ' ma J1 J J 3, min J1 J J 3, ' ' ' Ž 51 provided is such that Ž 1 J1 J J 3 5 Ž Also 1 1 ma min 1 3 J J J Ž 53 The solution Ž 519 may also be rewritten in terms of cosž t and sinž t as 1 J Ž t Ž J1 J3 cosž t sinž t ' J1 J J1 J J3 sin Ž t Ž 54 J J We will now use the general results derived in this section to obtain the solution for free oscillations and for heaviside step loading correct to order In both cases we will compare graphically the solution correct to order with the solution to zero order in obtained from the ErmakovPinney equation and with the numerical solution of the eact equation Ž 11 calculated using the standard Mathematica built-in numerical differentiation function, NDSole 51 Free Oscillations For free oscillations, P and the inequality Ž 41 is identically satisfied since we assume 1 The initial conditions are Ž,

29 374 MASON AND ROUSSOS Ž It is readily verified from Ž 59, Ž 517, and Ž 518 that ž / 1 J OŽ, Ž 55 ' J 1 OŽ, Ž J3 ž / OŽ, Ž 57 Ž as It follows directly from 54 that ½ ž / ž / 5 Ž t cos ' 1 t 1 sin ' 1 t 4 ' 4 4 ½ ' ž 4 / 1 sin 1 t, Ž 58 where terms of OŽ have been neglected Equation Ž 58 is the correction to first order in of the solution for free oscillations given by Shahinpoor and Nowinski From the relation between and given by Ž 53 ma min and by using Ž 55 to Ž 57 it can be verified that, for free oscillations, 1 ma min 1 OŽ, Ž 59 4 as For oscillations described by the ErmakovPinney equation, in which terms of OŽ are neglected, Ž 59 reduces to ma min 1, a result first derived by Knowles 1 It follows from Ž 59 that if terms of OŽ are included then 1 1 Ž 53 ma min In Fig, t Ž is plotted against t for,, and For these values of the parameters, the inequality Ž 5 is ' '

30 ANALYSIS OF MOONEYRIVLIN TUBE 375 FIG The nondimensional inner radius, t, Ž for free oscillations plotted against t for, ', and : Ž a solution correct to O Ž given by Ž 58, Ž b numerical solution of Ž 11 with Pt Ž, Ž c zero order approimation given by Ž 58 with '

31 376 MASON AND ROUSSOS satisfied Three solutions for t Ž are plotted for comparison in Fig, namely, the solution correct to first order in given by Ž 58, the solution to zero order in derived by Shahinpoor and Nowinski which can be obtained from Ž 58 by putting, and the numerical solution of the eact equation Ž 11 with Pt Ž We see that the approimate solution Ž 58 is a better approimation than the zero order in solution obtained from the ErmakovPinney equation Ž 43, especially for early times The maimum values of the dimensionless inner radius of the order approimation and the numerical solution are almost equal and the minimum value of the dimensionless inner radius of the order approimation bounds the minimum value attained by the numerical solution from below 5 Heaiside Step Loading For the heaviside step loading boundary condition given by Ž 31 where P is chosen to satisfy Ž 41, and subject to the initial conditions Ž 3, the first integrals, Ž 59, Ž 517, and Ž 518 take the form J1 Ž P ž 1 /, J, J3 Pž 1 / Ž 531 Thus, from Ž 54, to first order in, 1 ž 1 P / Ž t cos Ž t sin Ž t, Ž 53 ž 1 P / where is defined by Ž 415 From Ž 53 it can be seen that the amplitude depends on the ratio P We first investigate the range of values of P for which the approimate solution eists for given 1 From Ž 415, P 1, Ž 533 which places an upper bound on P when P but which is always satisfied when P Now since J, when P, 1 1 ž 1 / P 1 ma J1 J3, Ž 534 ' ž 1 P / 1 1 min 1 3 J J 1, Ž 535 '

32 ANALYSIS OF MOONEYRIVLIN TUBE 377 and the approimate solution eists provided Ž 533 is satisfied When P, 1 1 ma 1 3 J J 1, Ž 536 ' 1 1 P 1 min J1 J3, Ž 537 ' ž 1 P / 1 and the approimate solution eists provided P 1 Ž 538 Ž Ž By combining 533 and 538 we see that the approimate solution correct to first order in eists for given 1 provided P 1 1 Ž 539 For P 1, min and as P 1, ma Graphs of ma and min plotted against P for, 5, 5, and 75 are presented in Fig 3 When the approimate solution to order gives an upper bound on the magnitude of the net inward applied pressure as well as on the magnitude of the net outward applied pressure However, since min is physically unattainable, the upper bound on the magnitude of the net inward applied pressure is not physical and is a consequence of the approimations made In Fig 4, t Ž is plotted against ' t for 3 and P 5 The graphs are of the solution correct to first order in given by Ž 53, the solution derived by Shahinpoor and Nowinski for zero order in obtained from Ž 53 by putting, and the numerical solution of the eact equation Ž 11 with Pt Ž given by Ž 31 The solution to order is a better approimation than the zero order solution for both net outward and inward applied pressures In addition, as with the free oscillations, the maimum value of the dimensionless inner radius, t, Ž attained by the numerical solution is bounded above and below by the maimum value of t Ž for the order and zero order approimations, respectively In the net section, a non-linear superposition principle is derived which provides a systematic way of solving Eq Ž 47 when the net applied force is time dependent

33 378 MASON AND ROUSSOS FIG 3 Maimum and minimum values of the nondimensional inner radius, t, Ž for heaviside step loading plotted against P for: Ž a Ž P 1, Ž b 5 Ž 7 P 78, Ž c 5 Ž 3 P 34, and Ž d 75 Ž 53 P 58 The values P and P correspond to net outward and net inward applied pressures, respectively

34 ANALYSIS OF MOONEYRIVLIN TUBE 379 FIG 4 The nondimensional inner radius, t, Ž for heaviside step loading plotted against t for P 5 and 3: Ž a solution correct to O Ž given by Ž 53 Ž, b numerical solution of Ž 11 with Pt Ž given by Ž 31, and Ž c zero order approimation given by Ž 53 with '

35 38 MASON AND ROUSSOS 6 TRANSFORMATION TO AN AUTONOMOUS EQUATION AND NON-LINEAR SUPERPOSITION PRINCIPLE FOR ARBITRARY Pt Ž TO ORDER We consider now the time dependent differential equation Ž 47 which admits the Lie point symmetry generator Ž 46 We proceed by transforming Eq Ž 47 into an autonomous equation by insisting that under the transformation Ž t, Ž t*, * the transformed equation admits the time translation generator t* We therefore insist that the symmetry generator Ž 46 transform to the generator X* t*; thus Therefore t* and equations X* XŽ t* XŽ * Ž 61 t* * t* * must satisfy the quasi-linear partial differential t* t* XŽ t* 1: g Ž t gž t gž t 1, Ž 6 t * * XŽ * : g Ž t gž t gž t, Ž 63 t where gt Ž satisfies Ž 45 Two independent solutions of the differential equations of the characteristic curves of Ž 63 are where c and c 1 1 * c 1, 1 c, Ž 64 gž t are constants A special solution for * is therefore 1 * 1 1 OŽ, 65 gž t gž t 4 Ž as Also, the differential equations of the characteristic curves of Ž 6 yield where c 3 is a constant t dt t* H c, Ž 66 3 g Ž t

36 ANALYSIS OF MOONEYRIVLIN TUBE 381 The transformation from Ž t, to Ž t*, *, correct to first order in, is thus given by Ž 65 and Ž 66 Under this transformation, Ž 47 becomes, neglecting terms of order, the autonomous equation d * 1 C* 1, Ž 67 3 dt* * where C is the arbitrary constant in Ž 45 Now now derive a non-linear superposition principle for the solution, t, Ž of Ž 47 using the method described by Rogers and Ames Since the constant C is arbitrary, we choose C Equations Ž 67 and Ž 45 thus reduce, respectively, to d g dt Ž Integration of 68 with respect to * gives d * 1 1, Ž 68 3 dt* * 1 PŽ t g Ž 69 1 d* 1 1 J, Ž 61 dt* * where J is a constant called the Lewis invariant 14, 15 When epressed in terms of and t, Ž 61 becomes J 1 gž t 1 gž t ž 1 / 1 g Ž t, Ž 611 where gt Ž satisfies the second order differential equation Ž 69 Let g Ž 1 t and g Ž t be two linearly independent solutions of Ž 69 and define J1 1 g Ž t 1 g Ž t ž 1 / 1 g1 Ž t, Ž 61 J ž 1 gž t 1 gž t / ž 1 / 1 g Ž t Ž 613

37 38 MASON AND ROUSSOS Equations Ž 61 and Ž 613 are two equations for and We solve for by eliminating This gives the non-linear superposition where 1 Ž t ag1ž t bgž t cg1ž t gž t, Ž J J 4 J J 1 W 1 a, b, c, Ž 615 W W W 1 and W is the Wronskian determinant of g Ž t and g Ž t Since g Ž 1 1 t and g Ž tare linearly independent solutions of Ž 69 it can be verified that W is a non-zero constant The constants a, b, and c are related by 1 ab c Ž 616 We need to impose initial conditions on g Ž t and g Ž 1 t such that the initial conditions Ž, Ž, where, are satisfied We take as initial conditions W g1ž, g 1Ž 1, Ž 617 g Ž, g Ž, Ž 618 Ž where g is not specified Then, if terms of order are neglected, W g Ž, 1 a 1, b 1 1, c g Ž Ž Ž 619 It can be verified by putting t in Ž 614 and by evaluating t Ž from Ž 614 that if terms of OŽ are neglected then the initial conditions, Ž and Ž, are satisfied by Ž 614 We can now state the

38 ANALYSIS OF MOONEYRIVLIN TUBE 383 following non-linear superposition principle: Correct to terms of order, the solution of the differential equation 1 PŽ t Ž PŽ t 1, Ž subject to the initial conditions is where Ž, Ž, Ž 61 1 Ž t ag1ž t bgž t Ž 6 1 a 1, b 1 1, Ž 63 g Ž Ž and g Ž t and g Ž 1 t are linearly independent solutions of the second order differential equation d g dt subject to the initial conditions 1 PŽ t g, Ž 64 g1ž, g 1Ž 1, Ž 65 g Ž, g Ž Ž 66 The non-linear superposition principle is a generalisation, to first order in, of the non-linear superposition principle for the ErmakovPinney equation which was used by Shahinpoor and Nowinski to obtain eact solutions to problems with time dependent net applied surface pressures When the inequality Ž 41 is satisfied, Ž 64 is the time dependent harmonic oscillator equation When P is a constant, for eample in the cases of free oscillations or heaviside step loading, the non-linear superposition principle gives the same result as may be derived using the first

39 384 MASON AND ROUSSOS integrals obtained in Section 5 When P is not constant it gives new results Consider the solution of Eq Ž 47 subject to the boundary condition t Ž P1Ž t PŽ t M 1, t T*, PŽ t Ž 67 T* *1, t T*, where M and T* are constants, which describes the response of the tube to a blast loading with linear decay in time The maimum intensity of the blast is M and T* is the duration of the blast The derivation of the solution to first order in is similar to that given by Shahinpoor and Nowinski for zero order in and therefore only the main results are presented here Let t 1 M 1 Ž 68 T* and assume that Define 1 M Ž 69 T* 1 3 g už y, y, Ž 63 3 M Then the companion equation Ž 64 may be transformed to Bessel s equation of order 13, d u 1 du 1 1 u Ž 631 dy y dy 9 y The two independent solutions of Ž 631 which satisfy the initial conditions Ž 617 and Ž 618 are Ž Ž g1 AJ 1 13 BY 1 13, Ž Ž g AJ13 BY 13, Ž 63

40 ANALYSIS OF MOONEYRIVLIN TUBE 385 where J13 and Y13 are Bessel functions of order 13 of the first and second kind, respectively, and 1 A 1 M Y 1 3Ž M Y13Ž, Ž B 1 M J 1 3Ž M J13Ž, Ž A 1 M g Y g A ˆ Ž 13Ž Ž, Ž B 1 M g J g B ˆ Ž 13Ž Ž, Ž 636 where T* 3 3M 1 M, Ž M T* Ž The non-linear superposition solution 6 then gives ˆ 3 ˆ 3 ž ½ Ž Ž 5 / Ž t 1 AJ 1 13Ž BY 1 13Ž A J BY, 1 Ž 638 Ž where we denote by t the solution for t T* 1

41 386 MASON AND ROUSSOS We denote by Ž t the solution for t T* For t T*, Pt Ž and the solution Ž 58 for free oscillations applies: where ½ ž / t cos ' Ž 1 Ž t T* sin 1 Ž t T* ' ž 4/ ž ž 4/ / ' / ½ ž 1 ž / sin ' 1 Ž t T*, Ž Ž Ž T*, Ž Ž T* Ž Ž It can be verified using 638 that 1 1 AJ Ž BY Ž ½ Aˆ J Ž BY ˆ Ž, Ž ž / 1 AJ Ž B Y Ž AJ Ž B Y Ž ½ ½ Aˆ J Ž BY ˆ Ž Aˆ J Ž BY ˆ Ž, Ž 64

42 ANALYSIS OF MOONEYRIVLIN TUBE 387 where T* 3 1 Ž M In Fig 5, Ž t and Ž 1 t are plotted against t for 1,, T* 1, M 1, and We consider the case for which T* equals ' the period,, of free oscillations to zero order in ; thus At time t T* 1, the initial conditions for free oscillations are given by Ž 641 and Ž 64 The maimum and minimum displacements of the numerical solution of the Knowles equation Ž 11 subjected to a blast load with linear decay are bounded above and below by the maimum and minimum displacements of the approimation to order, Ž 638 for t T* and Ž 639 for t T*, and the zero order approimation obtained by setting in Ž 638 and Ž 639, respectively It is seen that the solutions correct to first order in are better approimations to the numerical solution of Eq Ž 11 than the corresponding zero order approimations, particularly for t T* 7 CONCLUDING REMARKS Figures 5 show that Eq Ž 47 is a better approimation of the Knowles equation Ž 11 for a MooneyRivlin material than is the ErmakovPinney equation Ž 13 However, there is the restriction Ž 5 on the value of for the first order solution to be valid which does not apply to the solution of the ErmakovPinney equation In addition, for the heaviside step loading boundary condition, a bound on the magnitude of the net inward applied pressure eists for the order solution which is a result of the approimations made and is not physical The general solutions, Ž 519, Ž 5, and Ž 54 of Eq Ž 47 are epressed in terms of the first integrals, J 1, J, and J 3, and are therefore invariant solutions They are invariant solutions because Eq Ž 47 can be transformed to the ErmakovPinney equation Ž 67 and all solutions of ErmakovPinney equations are invariant solutions 5, 6 The non-linear superposition principle derived in Section 6 is required when solving dynamic boundary value problems with non-constant net applied pressure Pt Ž The differential equation Ž 47 is then non-autonomous and cannot be solved by elementary means, as can the autonomous equation when P is a constant, because then a first integral corresponding to time translational invariance does not eist The approimate solutions derived for the displacement with the heaviside step loading boundary condition take the form of non-linear superpo-

43 388 MASON AND ROUSSOS FIG 5 The nondimensional inner radius, t, Ž for blast loading with linear decay plotted against t for 1,, T* 1, M 1,, and : Ž a solution correct to O Ž given by Ž 638 for t T* and Ž 639 for t T*, Ž b numerical solution of Ž 11 with Pt Ž given by Ž 67, and Ž c zero order approimation given by Ž 638 with for t T* and Ž 639 with for t T*

44 ANALYSIS OF MOONEYRIVLIN TUBE 389 sitions The maimum and minimum values of the radial displacement are given eactly by the approimate solutions and their periods are upper and lower bounds on the eact period They give better approimations than the thin-shell solution of the ErmakovPinney equation ecept for the approimation to the period for net inward applied pressure which is given more accurately by the thin-shell solution ACKNOWLEDGMENTS We thank Professor Fazal Mahomed of the University of the Witwatersrand, Johannesburg, South Africa, for his assistance and guidance with the Lie group theory in this paper We also thank the National Research Foundation, Pretoria, South Africa, for financial support APPENDIX A: THE WRONSKIAN DETERMINANT Wy, y, y, y W y, y, y, y ž / ln 1 9 Ž Ž 3 4 5Ž Ž 5 3 4ln 1 4Ž 6 Ž Ž ž / ln 1 43Ž 8 Ž ž / 3 ln 1