Saxton, R

English

Brunel University Research Archive

                    TR/12/82       November 1982  Solitary and Travelling Waves  in a Rod  by  R. Saxton   Solitary and Travelling Waves in a Rod  We are concerned with the existence and partial  analysis of longitudinal t rave l l ing  waves  in  a  th in  rod ,  wi th  a  la te r  appl ica t ion  to  a  growth  condition in incompressible elasticity.  We consider only solutions               u(x,  t )  =   which have the form of  s teady t ravel l ing waves          ( ) ,ctz,z −φfor the equation of motion, which is derived most simply by means of  Hamilton's principle (cp. [4],[5],[6]) applied to the functional            (1) ( ) .dtIkT0∫ − Here, the total kinetic energy is given by    ( ) ,dt]))t,x(xu(2tβ)t,x(2tu[tK 21 ∫∞∞−+     (2)  where β(.) is a measure of the lateral deformation undergone during any elongation or contraction of the rod which is assumed to remain straight          a t  a l l  t imes.   u(x,  t )  i s  the  longi tudinal  displacement  a t  t ime t  of  a    material point at x measured from some chosen point along the rod,           which is now taken to be of infinite length.  We have taken the density            in the current and reference configurations to be unity. In terms of ,         xuby adopting the limit of incompressibility (Poisson's ratio 21ν→  in the   special case of small deformation) β  is given here by     ,1)u1(1)u(21xx −+=β       (3)  although we need not assume this.   To avoid material inversion or   - 2 -  f l a t t e n i n g  i n  g e n e r a l ,  i t  i s  o n l y  n e c e s s a r y  t h a t    I ,  t h e  .1β,ux −>total   strain  energy,   is   of   the  form        (4) ( ) dx)))t,x(xuβ(),t,x(xu(WtI ∫∞∞−= where W( .  ,  .) is the strain energy function.  On carrying out the variations, we obtain from (1) the Euler equation    ( ) ( ) 0,'βW'Wuβ'u'β'β'u x`xttx22xttt =+−+−   (5)  w h e r e  s u b s c r i p t s  d e n o t e  d i f f e r e n t i a t i o n  w i t h  r e s p e c t  t o  x  a n d  t  ,  a n d  superscr ip t s   '  ,  `   denote  d i f fe ren t ia t ion  wi th  respec t  to  dependent  v a r i a b l e s   and   β  r e s p e c t i v e l y .  xu We now consider the substitution in (5) of    ( ) ( ) ,,,, RIcctxzztxu ∈−== φ    (6) giving   ⎟⎟⎠⎞⎜⎜⎝⎛ φ⎟⎠⎞⎜⎝⎛ φβ−φ2332222dzdc.dzd'dzddzdc             ))dzdβ,dzd(W')dzd().cdzd("β).dzd(β'(dzd 2222 φφ+φφφ−     .0dzd'.dzd,dzd'W =⎟⎟⎠⎞⎜⎜⎝⎛ ⎟⎠⎞φβ⎟⎟⎠⎞⎜⎜⎝⎛ ⎟⎠⎞⎜⎝⎛ φβφ+     (7)  On further substituting  ( )dzd'cx,dzd ψψβ=φ=ψ  and                                     integrating (7) with respect to z,  we obtain    ( ) ( ) 0'`W'WdzdX'cc2 =ψβ++ψβ+ψ−     (8)  where we have omitted the constant of integration. - 3 -  At some point  the initial data for (8) is given by 0z   ( ) ( ) ( ) ( ) ,'czx,zdzd,z 1001000 ψψβ=ψ=ψψ=ψ  (9)  and on multiplying (8) by dzdψ  and integrating with respect to  z  ,                we have the energy equation ( )( ) ,E,WXzc 22122=ψβψ++ψ−   (10) where  E  is a constant depending only on the values of  10,ψψ  and  c .   Our analysis of equation (7) is in the phase plane.  (For a related discussion in another problem, c.p. Ball [l] ,  Calderer [2]).    We therefore use (8) and (9) to obtain the equivalent f irst  order system      ( ) ( )X,~mz~xdzd ψ=      (11)  where        )(X~xψ=        ,     (12)                      ,    (13) 2RIRI[,1:]~m →×∞−   ( ) ( ) ⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛β−−ψψβ=ψ ``w'w2cX'c1X,m      ,   (14) and           (15) ( ) ⎟⎟⎟⎠⎞⎜⎜⎜⎝⎛ψψβψ=ψ=1)00 ('c00X0zx )( We now presen t  a  r esu l t  on  loca l  ex i s t ence  o f  t r a jec to r ies   ~x :   Proposition 1  Assume  is locally Lipschitz continuous                     )x,(~m ψon i ts  domain of  defini t ion.   Then the ini t ia l  value problem (12)  -  (15)    has a unique maximally defined solution     where )I(C)I(C)z(~x11 ×∈ [z,z]I 00 α+α−=  for some  The solution  satisfies .RI∈α )z(~x     )(00 X)z(x 0ψ= . - 4 -   P r o o f   F o r   t o  b e  l o c a l l y  L i p s c h i t z ,  i t  i s  e n o u g h  f o r               ( X,m ψ )Β  ( . ) ,  W ’ ( . , . )  a n d  W `  t o  b e  l o c a l l y  L i p s c h i t z  c o n t i n u o u s  i n  e a c h  var iab le ,  and  for  β  ( . )  to  be  s ign  def in i te .   This  impl ies  we  requi re        t h e  e x i s t e n c e  o f  s o m e  )R,(,0 ∈Γ=Γ∞<Γ<  s u c h  t h a t  f o r  a n y    .,),(),`(),(',0,[,,][,1]),`(),,(212122112211etcXXRXWXWRRRRXWX−+−∈Γ≤−>ψ ψ ∈ ∈+− × − ∈ ψψψψand α < 0 such that .' 21α>β  We may then apply s tandard resul ts  in  ordinary different ia l  equat ions  (e .g .  Fr iedr ichs  [3] )  to  obta in  loca l  ex is tence  and  uniqueness  through .)(0X0ψ  Referr ing to  equat ion (10)  we def ine    ))(,(W2c)(V 22ψβψ+ψ−=ψ     (16) and denote    222dVdc)(' ψ+=ψσ      '.'`W'`'w2''W β+β+≡     (17)  We may character ize  the equi l ibr ium points  for  (11)  -  (14)  by the  following proposition. P ropos i t ion  2     Le t  c  be  g iven .   Then  the  equ i l ib r ium so lu t ions   ( ) ( ) RIz,z,0zX ∈ψ=ψ=  o f  ( 1 1 )  a r e  t h e  z e r o s  o f  t h e  f u n c t i o n   ( )ψσ−ψ2c  where ( ) ψ+ψ=ψσ ddvc 2   . Fu r the r ,  f o r  ψ  s a t i s fy i n g  )0,(),('c2 ψψσ<  i s  a  c e n t r e  a n d  )(v ψ  i s  a   r e l a t ive  min imum.   When  ψ  s a t i s f i e s  )0,(,)('c, 2 ψψσ<                 i s  a  sadd le  po in t  and  ( )ψv  i s  a  r e la t ive  maximum.  Proof  Fol lows from (14)  and i ts  l inear izat ion     ⎟⎟⎠⎞⎜⎜⎝⎛ ψψ⎟⎟⎠⎞⎜⎜⎝⎛−ψσ−ψβ−ψ X0)(c10)(,c1~)X,(m 2   In  a  neighbourhod of  ( ).0,ψ  - 5 -  Remark In addition to the above cases we may also have (ψσ= 'c2 ) which corresponds to a point of inflection of  V. We now assume that  for  each,  c   ,  there  exis ts  a  f ini te  number   n   of  Zeros      1iii }ni0,{ +ψ≤ψ≤≤ψ    (18)  to the function  . )('c2 ψσ=ψFurther we consider two possible hypotheses for σ  :-  (H1)   As  1~)(, ψψσ+∞→ψ  ; (H2)   As  B~)(, ψψψσ+∞→ψ ,  where B > 0.  (HI) and (H2) are compatible with the Lipschitz assumptions made earlier.  We use these conditions to investigate the region            0≥ψof the phase plane and obtain the next two results.  Proposition 3   Let  throughout,  the let 0≥ψ ),( 10 ψψ  be chosen so that    .ni0),(V),(E i10 ≤≤ψ≠ψψ  Then  a) when c2   ≤   1 and either (HI) or (H2) hold, the trajectory through ),( 10 ψψ  is periodic with period     ∫ −=1J0J21))V(ψ(E)dψ('c2τ ψβ     (19) w h e r e   a n d   a r e  t h e  i n t e r s e c t i o n s  o f  t h e  t r a j e c t o r y    10 JJ < )0,J(),0,J( 10with the axis X = 0. b)  When c2  > 1 and (H2) holds, we again have the result of a). c)  When c2  > 1 and (HI) holds, no periodic orbits exist through  any point   with )0,(ψ nψ>ψ  .   - 6 -   Proof  From (10) and (18) ,  and the Theorem of Poincaré-Bendixon. Corollary  When ),( 10 ψψ  are chosen so that   ( )i10 V),(E ψ=ψψ    for some of i= 0,1,…..,n t h e n  e i t h e r  A )   i s  a  c e n t r e ,  o r  w h e n  c),( 10 ψψ 2  <  1   and (HI) and (H2) hold or c2 > 1 and (H2) holds, then  B) i) for  we have 01 ≥ψ   0)z(xzlim,)z(zlim i =+∞→ψ=ψ+∞→ where  is   the   closest   saddlepoint   to  the   left    of  ( ,0) 0i ψ≤ψ 0ψ      ii)        for  01 ≤ψ 0)z(xzlim,)z(zlim j =+∞→ψ=ψ+∞→  where   is the closest saddle point to the right of  (0j ψ≥ψ 0ψ ,0).  Remark The case B) above is that which provides for the existence of  solitary waves.  Note that since centres and saddle points alternate (we    have excluded the case c2 = )),(' ψσ  given the same 0ψ  in B  i)  and ii)     above implies  .2ij +ψ=ψ  We conclude by looking at a case for ,01 ≤ψ<−  with )(ψβ  given by (3) .   Hypothesis (H3)    For  as [,1]p ∞−∈ +∞→q       .0,RIγ,~)q,p(w γq >δ∈δProposition 4  Under hypothesis (H3), a sufficient condition preventing flattening of the incompressible rod is  . 6γ >Proof  With  B  given by (3),  (10) becomes  2322 )1.(E)1()1(,(w2'8/c)1(2/c2122 ψ+=ψ+ψ+ψ+ψ+ψ+ψ− −  (20)  and so as   1−→ψ   -  7  -    2/32 )1()cE()31(~'8/c232 λ−ψ+δ−++ψand  the  resu l t  fo l lows  s ince  δ  >  0  and  i t  i s  imposs ib le  fo r       .1u x −=ψ=    ______________   References[1]     J .  Bal l ,  Discont inuous Equi l ibr ium Solut ions and Cavi ta t ion in Nonlinear Elasticity, Preprint.   [2]   C. Calderer.   The Dynamic Behaviour of Nonlinear Elastic Spher ica l  She l l s ,  J .  E las t i c i ty ,   to  appear .   [3]    K.O. Friedrichs, "Advanced Ordinary Differential Equations",     Gordon and Breach Science Publishers, Inc., New York (1965).  [4]    W. A. Green, "Dispersion Relations for Elastic Waves in a Bar",        in Progress in Solid Mechanics I,  Ed. Sneddon and Hill .   [5]   W. Jaunzemis, "Continuum Mechanics", New York MacMillan (1967). [6]    A. E. H. Love, Theory of Elasticity (4th ed.)  278, Cambridge University Press (1927).                                                                  

Solitary and travelling waves in a rod

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