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Finite amplitude oscillations of a hyperelastic spherical cavity
FINITE A~~LIT~~E ~SGr~~A~~~S OF A HYPERELASTIC SPHERICAL CAVITY R. BALAKRISHNAN and N. SHAHINPOOR College of Eagineerine, Pablavi University, Shiraz, kaa
Abstract--We present numerical results for the finite oscillations ofa hyperelastic spherical cavity by emptoying the goveruiu,g equations far finite amplitude oscillations of hyperefastic spherical shdls and simplifying it for a spheric& cavity in au ir&tite medium and then applying a fourth-order Rtmge-Kutta numericai technique to the resutting non-linear first-order ~~ffere~~a~equation. The results are plotted for ~~on~~-~~~~~ type materials for free and forced osciflations under Iieaviside type step loading. The results for Neo-Rookean materials are also discussed. Dependence of the amplitudes and frequencies of oscillations on different parameters of the probiem is also discussed in length.
The rapid dev~lopmeut of the theory of large elastic deformations during the past two decades has been stimulated by the existence and widespread use of vulcanized rubber. Contributions were, however, made to the finite theory of elasticity by Figner, Brillouin, Riviiin, Ericksen, Truesdell El], Toupin, NoIl, Nowinski and others. The reader is referred to Knowles f2,3], Guo and Solecki f4], Knowles and Jakub [S], Wang. f6]? Nowinski and Schuffz [7], Nowinski [g], ~ha~inpoor and Nowinski f9] and ~ha~upoor f14 1l] for an extensive review on exact solutions in finite h~ereIasticity. It is the purpose of the present work to study the finite amplitude oscillations of a spherical cavity in an infinite medium. It is assumed that the cavity is made of homogeneous, isotropic, hyperelastic Mooney-Rivlin and Neo-Hookean materials.? The governing equation is the one originaffy derived by Guo and Sole&i [4] for an arbitrary wall thickness, and then simplified to a spherical cavity in an infinite medium. The numerical solutions to the governing equations are obtained using a fourth-order Runge-Kutta method. The results are presented graphically for the cases of free and forced oscillations.
Let us consider a spherical shell made of an elastic, homogeneous, isotrupic and incompressible material. Let X” be the material coordinates of a typical particle in the elastic medium in its undeformed state, and let X’ = R, 0, 0, and inner and outer radii RI and R2 respectively. It is assumed that the shell remains sphericai in shape through the deformations also, It has spatiai coordinates xi = r, 0, 4, in its deformed state. Since the motion ofthe shell is sphe~c~ly symmetric, the motion can be represented by the function
We assume that the motions of the shell are finite, of arbitrary magnitude and induced by time dependent hydrostatic surface pressures Pr(t) and P2(g) on the inner and outer surfaces of the shell, respectivefy. For an incom~r~sible material of mass density p, the associated governing equation of motion, was originalIy derived by Guo and Sole&i for the case of a i The strain energy density for the Mooney-Rivlin
materials is
tt’(ll,it)-~ir,-3!~~~~,-3) where I,, I, are the strain invariants and a, ~3are cotnstants. If/? = 0 then the X.r.M~ f3D-P If1
materialbecomes ~~*~ao~eau.
172
R. BALAKRISHNAN
and M. SHAHINPOOR
spherical shell made of an incompressible Mooney-Rivlin equation becomes
type materials. The corresponding
where
with qI(0) = R,, and a and p the Mooney-even material constants. Here the coefficient @is a measure of wall thickness of the shell. For the case of a spherical cavity in an infinite medium, we let p + cc as R, -+ co. Upon dividing the governing equation (2.1) by p and putting l/p = 0, we get
as the governing equation, which is exactly the same as the equations obtained by Knowles and Jakub [5], using a quasi-equilibrated approach. We now put the above equation into the following form 3x2
5 ! 4x
----G
2x x+--cc,
1 xz
1
t-c,
1
I----
i
1 2X3
1 PT =c’x 2s )
12.4)
where
*u
C 1 --pR:’
c
2=x’
2P
1
3 - ~RZ 1 _
c
and
PT = PI(f)-P*(t)
and we impose the following initial conditions x(0) = xg,
3i(O)= &)
(2.6)
where the dot represents the time derivative, and x0 and R0 are the initial displacement and velocity at the inner surface of the cavity. 3. NUMERlCAL
SOLUTIONS
TO THE
GOVERNING
EQUATION
The numerical solution to the governing equation (2.4) is obtained using the fourth-order Runge-Kutta algorithm for solving a system of Mfirst-order differential equation [ 121: Yj,i+ 1 = yji + h(kj, + 2kjz c 2kjs + kja)/6 k jr =fj(xi,Yli,Y2i,...,Y,i) y$ = yji++hkj,
k jz = fjtxi + $4 .YZ*.Y:iT. * ’ 2Yni) - = yji++hkj2 Yji kjs =fj(xi +$h jlit )?zir*. ., l?zJ $ = yji + hkj, k j4 =fi(xi + h, js:i, .Vzi,’ . . >J?ni)
which is the algorithm for solving a system of n first-order differential equations, where I. = 1,2,. _. , n number of first-order differential equations i = ith step of integration.
(3.1)
Finite amplitude oscillations of a hyperelastic spherical cavity First
we
173
rewrite the governing equation into a system of two first-order differential. Let Y, = x Y, = i
(3.2)
t=t
then
F
2
=dY,=& dt
dt2
The initial conditions are
y1.0= x0 Y2.0
=
(3.4)
lo.
Then this system of two first-order differential equations (3.3) with the initial conditions (3.4) are solved using the computer. The computer solutions are represented graphically for the cases of free oscillation and forced oscillations under Heaviside-type step loading. Case 1. Free oscillations PT=O,
t>O.
(3.5)
Figure 1 shows the numerical solution for the case of free oscillations. The curves are plotted for x0 = 1.0 with 1, as a parameter. It has been noted that the amplitude of oscillations during expansion is less than that during contraction. Also the time duration for contraction is less than that for expansion. The frequency of oscillations increases as we increase the initial velocity.
16
Material constants, 16 -
c
l-l.5
C2=15 *
C3=C1/24
14
-
16
-
14
_
I
I
0
02
0.4
PT=O
i
I
I
06
08
Time,
Fig. 1. Mooney-Rivlin
spherical cavity-free
I IO
I2
f, set
oscillations-~,
as a parameter.
174
R. BALAKRISHNAN and M.
Case 2. Forced oscillation-Heaviside
SHAHINPOOR
step load for
t < 0,
for
t > 0.
(3.6)
The numerical solutions for this case are given in Fig. 2. This shows the same trend as Fig. 1, but the amplitude of oscillations is slightly more than that in the case of free oscillations. This is due to the forcing function. 18
1 Material constants
16
-
c I-l.5 C2=15
* _ 14
C3=C1/24
-
1.6 -
14
I
I
I
I
I
I
0
02
0.4
06
06
10
12
Time, t, set
Fig. 2. Mooney-Rivlin
spherical cavity-forced oscillations-Heaviside parameter.
4. SPHERICAL
CAVITY
MADE
step loading--i,
OF NEO-HOOKEAN
as a
MATERIAL
If /I is taken to be zero in (2.3), there follows the governing equation for the so-called NeoHookean material. Thus, the governing equation takes the following form (3.7)
Material constants, c
l-100
c2=0 C3=Cl/24 PT-0
0
01
02
03
04
05
06
Time, f, set
Fig. 3. Neo-Hookian spherical cavity-free
oscillations--i
as a parameter.
Finite amplitude oscillations of a hyperefastic spherical cavity
Material
175
constants, Cl-100 c2=0
16
C3= Cl/24 f-r= PO =2
Time,
t,sec
Fig. 4. Neo-Hookian spherical cavity-Heaviside
step loading-&,
as a parameter.
The numerical solutions to the above equation for the case of free os~lIations show the same trend as before. That is, the amplitude of oscillations is less during expansion than contraction, and also the time duration for contraction is less than that for expansion. But in the cases of forced oscillations the amplitude of oscillations is more during expansion during contraction. This is due to the forcing function, which tends to expand the cavity more and also since the Neo-Hookean materials are more elastic than Mooney-Rivlin materials. These results (Figs. 3 and 4) again prove the physical significance of the spherical cavity in an infinite medium. Acknowledgement-The authors are grateful for the financial support of the College of Graduate Studies of Pahlavi University. Thanks are also due to Miss M. Shaterpouri for typing the manuscript.
REFERENCES 1. C. Truesdeff and W. Noff, The non-linear field theories of mechanics, Encycfopediu of Physics, Vol.
2. 3. 4. 5. 6.
1 I l/3. Springer, Berlin (1965). J. K. Knowles, Large amplitude oscillations of a tube of incompressible elastic material, Q. appl. Math. 18, 71 (1960). J. K. Knowles, On a class of oscillations in the finite-deformation theory of elasticity, J. uppi. Me&. 29,283 (1962). Z. H. Guo and R. Solecki, Free and forced finite amplitude o~ffations of an elastic thick-walled hollow sphere of incompressible material, Arch. Me&. Stos. 15,425 ff 964). J. K. Knowles and M. T. Jakub, Finite dynamic deformation of an incompressible elastic medium containing a spherical cavity, Archs. ration. Mech. Analysis 14, 367 (1965). C. C. Wang, On the radial oscillations of a spherical thin shell in the finite elasticity theory, Archs. ration.
Meeh. Analysis 23,270 (1965).
7. J. L. Nowinski and A. B. Schultz, Note on a class of finite longitudinal oscillations of thick-walled cylinders, Proc. Ind. mtn. Congr. Theor. appt. Mech. 31 (1964). 8. J. L. Nowinski, On a dynamic problem in finite elastic shear, Int. J. Engng Sci. 4, SO1(1966). 9. M. Shahinpoorand J. L. Nowinski, Exact solution to the problem offorced large amplitude radiaf oscillations of a thin hyperefastic tube, Int. J. Non-linear Mech. 6, 193 (1971). 10. M. Shahinpoor, Combined radial-axial large amplitude hyperelastic cylindrical tubes, J. Math. Phys. Sci. VII, III (1973). 11. M. Shahinpoor, Exact solution to finite amplitude oscillation ofan anisotropic thin rubber tube, J. Acoust. Sac. Am. 56,477 (1974). 12. B. Carnahan, H. A. Luther and J. 0. Wilkes, Applied ~a~e~ica~ Methods. John Wiley, New York (1969).
176
R. BALAKRISHNAN and M. SHAHINFQOR
Resume: On donne les resultants numeriques pour les oscillation finies d'une cavite spherique hyperelastique en utilisant les eauations reaissant les oscillations d'amolitude finie de cdques spheriques hype&lastiques et en les simplifiant pour une cavitg sDh&iaue dans un milieu infini et en aooliauant ensuite une methode numbrique de Runge - Kutta du quatri&me ordre a l'equations differentielle non-lineaire du premier ordre en resultant. On trace les courbes de resultants pour des materiaux de type Mooney - Rivlin avec des oscillations libres et for&es sous des chargements en fonctions d'Heaviside. On s'inte'resse aussi aux resultats dans le cas de materiaux neo-Hookeens. On discute egalement longuement la dependance des amplitudes et des frequences des oscillations par rapport a differents paramhtres du probl&ne.
Zusammenfassung: Es werden numerische Ergebnisse fur die endlichen Schwingungen eines hyperelastischen kugelformigen Hohlraum anoeoeben. Die Bestimnunqsaleichunsen fur die Schwing&gin mit endlicher Amplitude eine; hyperelastischen Kugelschale werden fur einen Kuqelhohlraum in unendlich ausgedehntem Medium vereinfacht-und dann wird ein nutterisches Verfahren der vierten Ordnung nach Runge-Kutta auf die sich eraebenden nichtlinearen Differential-oleichunaen erster Ordning angewendet. Die.Eroebnisse werden fur Materialien der Mooney-Rivlinschen Art fur freie und erzwungene Schwingungen-unter Heavisideschen Sprungbelastungen aufgetragen. Die Ergebnisse fur Neo-Hookesche Stoffe werden ebenfalls Die Abhangigkeit der Schwingungsamplituden diskutiert. und-frequenzen von verschiedenen Parametern des Problems wird ausfuhrlich behandelt.
Abstract--We present numerical results for the finite oscillations ofa hyperelastic spherical cavity by emptoying the goveruiu,g equations far finite amplitude oscillations of hyperefastic spherical shdls and simplifying it for a spheric& cavity in au ir&tite medium and then applying a fourth-order Rtmge-Kutta numericai technique to the resutting non-linear first-order ~~ffere~~a~equation. The results are plotted for ~~on~~-~~~~~ type materials for free and forced osciflations under Iieaviside type step loading. The results for Neo-Rookean materials are also discussed. Dependence of the amplitudes and frequencies of oscillations on different parameters of the probiem is also discussed in length.
The rapid dev~lopmeut of the theory of large elastic deformations during the past two decades has been stimulated by the existence and widespread use of vulcanized rubber. Contributions were, however, made to the finite theory of elasticity by Figner, Brillouin, Riviiin, Ericksen, Truesdell El], Toupin, NoIl, Nowinski and others. The reader is referred to Knowles f2,3], Guo and Solecki f4], Knowles and Jakub [S], Wang. f6]? Nowinski and Schuffz [7], Nowinski [g], ~ha~inpoor and Nowinski f9] and ~ha~upoor f14 1l] for an extensive review on exact solutions in finite h~ereIasticity. It is the purpose of the present work to study the finite amplitude oscillations of a spherical cavity in an infinite medium. It is assumed that the cavity is made of homogeneous, isotropic, hyperelastic Mooney-Rivlin and Neo-Hookean materials.? The governing equation is the one originaffy derived by Guo and Sole&i [4] for an arbitrary wall thickness, and then simplified to a spherical cavity in an infinite medium. The numerical solutions to the governing equations are obtained using a fourth-order Runge-Kutta method. The results are presented graphically for the cases of free and forced oscillations.
Let us consider a spherical shell made of an elastic, homogeneous, isotrupic and incompressible material. Let X” be the material coordinates of a typical particle in the elastic medium in its undeformed state, and let X’ = R, 0, 0, and inner and outer radii RI and R2 respectively. It is assumed that the shell remains sphericai in shape through the deformations also, It has spatiai coordinates xi = r, 0, 4, in its deformed state. Since the motion ofthe shell is sphe~c~ly symmetric, the motion can be represented by the function
We assume that the motions of the shell are finite, of arbitrary magnitude and induced by time dependent hydrostatic surface pressures Pr(t) and P2(g) on the inner and outer surfaces of the shell, respectivefy. For an incom~r~sible material of mass density p, the associated governing equation of motion, was originalIy derived by Guo and Sole&i for the case of a i The strain energy density for the Mooney-Rivlin
materials is
tt’(ll,it)-~ir,-3!~~~~,-3) where I,, I, are the strain invariants and a, ~3are cotnstants. If/? = 0 then the X.r.M~ f3D-P If1
materialbecomes ~~*~ao~eau.
172
R. BALAKRISHNAN
and M. SHAHINPOOR
spherical shell made of an incompressible Mooney-Rivlin equation becomes
type materials. The corresponding
where
with qI(0) = R,, and a and p the Mooney-even material constants. Here the coefficient @is a measure of wall thickness of the shell. For the case of a spherical cavity in an infinite medium, we let p + cc as R, -+ co. Upon dividing the governing equation (2.1) by p and putting l/p = 0, we get
as the governing equation, which is exactly the same as the equations obtained by Knowles and Jakub [5], using a quasi-equilibrated approach. We now put the above equation into the following form 3x2
5 ! 4x
----G
2x x+--cc,
1 xz
1
t-c,
1
I----
i
1 2X3
1 PT =c’x 2s )
12.4)
where
*u
C 1 --pR:’
c
2=x’
2P
1
3 - ~RZ 1 _
c
and
PT = PI(f)-P*(t)
and we impose the following initial conditions x(0) = xg,
3i(O)= &)
(2.6)
where the dot represents the time derivative, and x0 and R0 are the initial displacement and velocity at the inner surface of the cavity. 3. NUMERlCAL
SOLUTIONS
TO THE
GOVERNING
EQUATION
The numerical solution to the governing equation (2.4) is obtained using the fourth-order Runge-Kutta algorithm for solving a system of Mfirst-order differential equation [ 121: Yj,i+ 1 = yji + h(kj, + 2kjz c 2kjs + kja)/6 k jr =fj(xi,Yli,Y2i,...,Y,i) y$ = yji++hkj,
k jz = fjtxi + $4 .YZ*.Y:iT. * ’ 2Yni) - = yji++hkj2 Yji kjs =fj(xi +$h jlit )?zir*. ., l?zJ $ = yji + hkj, k j4 =fi(xi + h, js:i, .Vzi,’ . . >J?ni)
which is the algorithm for solving a system of n first-order differential equations, where I. = 1,2,. _. , n number of first-order differential equations i = ith step of integration.
(3.1)
Finite amplitude oscillations of a hyperelastic spherical cavity First
we
173
rewrite the governing equation into a system of two first-order differential. Let Y, = x Y, = i
(3.2)
t=t
then
F
2
=dY,=& dt
dt2
The initial conditions are
y1.0= x0 Y2.0
=
(3.4)
lo.
Then this system of two first-order differential equations (3.3) with the initial conditions (3.4) are solved using the computer. The computer solutions are represented graphically for the cases of free oscillation and forced oscillations under Heaviside-type step loading. Case 1. Free oscillations PT=O,
t>O.
(3.5)
Figure 1 shows the numerical solution for the case of free oscillations. The curves are plotted for x0 = 1.0 with 1, as a parameter. It has been noted that the amplitude of oscillations during expansion is less than that during contraction. Also the time duration for contraction is less than that for expansion. The frequency of oscillations increases as we increase the initial velocity.
16
Material constants, 16 -
c
l-l.5
C2=15 *
C3=C1/24
14
-
16
-
14
_
I
I
0
02
0.4
PT=O
i
I
I
06
08
Time,
Fig. 1. Mooney-Rivlin
spherical cavity-free
I IO
I2
f, set
oscillations-~,
as a parameter.
174
R. BALAKRISHNAN and M.
Case 2. Forced oscillation-Heaviside
SHAHINPOOR
step load for
t < 0,
for
t > 0.
(3.6)
The numerical solutions for this case are given in Fig. 2. This shows the same trend as Fig. 1, but the amplitude of oscillations is slightly more than that in the case of free oscillations. This is due to the forcing function. 18
1 Material constants
16
-
c I-l.5 C2=15
* _ 14
C3=C1/24
-
1.6 -
14
I
I
I
I
I
I
0
02
0.4
06
06
10
12
Time, t, set
Fig. 2. Mooney-Rivlin
spherical cavity-forced oscillations-Heaviside parameter.
4. SPHERICAL
CAVITY
MADE
step loading--i,
OF NEO-HOOKEAN
as a
MATERIAL
If /I is taken to be zero in (2.3), there follows the governing equation for the so-called NeoHookean material. Thus, the governing equation takes the following form (3.7)
Material constants, c
l-100
c2=0 C3=Cl/24 PT-0
0
01
02
03
04
05
06
Time, f, set
Fig. 3. Neo-Hookian spherical cavity-free
oscillations--i
as a parameter.
Finite amplitude oscillations of a hyperefastic spherical cavity
Material
175
constants, Cl-100 c2=0
16
C3= Cl/24 f-r= PO =2
Time,
t,sec
Fig. 4. Neo-Hookian spherical cavity-Heaviside
step loading-&,
as a parameter.
The numerical solutions to the above equation for the case of free os~lIations show the same trend as before. That is, the amplitude of oscillations is less during expansion than contraction, and also the time duration for contraction is less than that for expansion. But in the cases of forced oscillations the amplitude of oscillations is more during expansion during contraction. This is due to the forcing function, which tends to expand the cavity more and also since the Neo-Hookean materials are more elastic than Mooney-Rivlin materials. These results (Figs. 3 and 4) again prove the physical significance of the spherical cavity in an infinite medium. Acknowledgement-The authors are grateful for the financial support of the College of Graduate Studies of Pahlavi University. Thanks are also due to Miss M. Shaterpouri for typing the manuscript.
REFERENCES 1. C. Truesdeff and W. Noff, The non-linear field theories of mechanics, Encycfopediu of Physics, Vol.
2. 3. 4. 5. 6.
1 I l/3. Springer, Berlin (1965). J. K. Knowles, Large amplitude oscillations of a tube of incompressible elastic material, Q. appl. Math. 18, 71 (1960). J. K. Knowles, On a class of oscillations in the finite-deformation theory of elasticity, J. uppi. Me&. 29,283 (1962). Z. H. Guo and R. Solecki, Free and forced finite amplitude o~ffations of an elastic thick-walled hollow sphere of incompressible material, Arch. Me&. Stos. 15,425 ff 964). J. K. Knowles and M. T. Jakub, Finite dynamic deformation of an incompressible elastic medium containing a spherical cavity, Archs. ration. Mech. Analysis 14, 367 (1965). C. C. Wang, On the radial oscillations of a spherical thin shell in the finite elasticity theory, Archs. ration.
Meeh. Analysis 23,270 (1965).
7. J. L. Nowinski and A. B. Schultz, Note on a class of finite longitudinal oscillations of thick-walled cylinders, Proc. Ind. mtn. Congr. Theor. appt. Mech. 31 (1964). 8. J. L. Nowinski, On a dynamic problem in finite elastic shear, Int. J. Engng Sci. 4, SO1(1966). 9. M. Shahinpoorand J. L. Nowinski, Exact solution to the problem offorced large amplitude radiaf oscillations of a thin hyperefastic tube, Int. J. Non-linear Mech. 6, 193 (1971). 10. M. Shahinpoor, Combined radial-axial large amplitude hyperelastic cylindrical tubes, J. Math. Phys. Sci. VII, III (1973). 11. M. Shahinpoor, Exact solution to finite amplitude oscillation ofan anisotropic thin rubber tube, J. Acoust. Sac. Am. 56,477 (1974). 12. B. Carnahan, H. A. Luther and J. 0. Wilkes, Applied ~a~e~ica~ Methods. John Wiley, New York (1969).
176
R. BALAKRISHNAN and M. SHAHINFQOR
Resume: On donne les resultants numeriques pour les oscillation finies d'une cavite spherique hyperelastique en utilisant les eauations reaissant les oscillations d'amolitude finie de cdques spheriques hype&lastiques et en les simplifiant pour une cavitg sDh&iaue dans un milieu infini et en aooliauant ensuite une methode numbrique de Runge - Kutta du quatri&me ordre a l'equations differentielle non-lineaire du premier ordre en resultant. On trace les courbes de resultants pour des materiaux de type Mooney - Rivlin avec des oscillations libres et for&es sous des chargements en fonctions d'Heaviside. On s'inte'resse aussi aux resultats dans le cas de materiaux neo-Hookeens. On discute egalement longuement la dependance des amplitudes et des frequences des oscillations par rapport a differents paramhtres du probl&ne.
Zusammenfassung: Es werden numerische Ergebnisse fur die endlichen Schwingungen eines hyperelastischen kugelformigen Hohlraum anoeoeben. Die Bestimnunqsaleichunsen fur die Schwing&gin mit endlicher Amplitude eine; hyperelastischen Kugelschale werden fur einen Kuqelhohlraum in unendlich ausgedehntem Medium vereinfacht-und dann wird ein nutterisches Verfahren der vierten Ordnung nach Runge-Kutta auf die sich eraebenden nichtlinearen Differential-oleichunaen erster Ordning angewendet. Die.Eroebnisse werden fur Materialien der Mooney-Rivlinschen Art fur freie und erzwungene Schwingungen-unter Heavisideschen Sprungbelastungen aufgetragen. Die Ergebnisse fur Neo-Hookesche Stoffe werden ebenfalls Die Abhangigkeit der Schwingungsamplituden diskutiert. und-frequenzen von verschiedenen Parametern des Problems wird ausfuhrlich behandelt.