- Email: [email protected]
A class of controllable wave propagations in finite elasticity
Inr.
J Non-l,~near
Mechnnia.
Vol.
II.
pp
325-330
Pergamon PWSS 1976
PrInted III Great Britain
416
A CLASS OF CONTROLLABLE WAVE PROPAGATIONS IN FINITE ELASTICITY M. SHAHINPOOK~ and I. G. TADJBAKHSH~ College of Engineering, Pahlavi University, Shiraz, Iran (Received 18 Mnrch 1976) Abstract-Governing
equations of axisymmetric finite dynamic deformations of an incompressible. isotropic and elastic cylindrical shell made of Neo-Hookean materials are derived. The non-linear partial differential equations are simplified for the cases where all deformation variations along the thickness of the tube may be neglected. The simplified non-linear equations are then solved exactly to arrive at traveling wave solutions along the axis. These wave solutions are called controllable because they can be maintained by prescribable surface stresses, bounded amplitude and frequency of excitations alone.
Ericksen [I J in 1954 made the first systematic endeavor to find all deformations that any homogeneous isotropic, incompressible elastic body may sustain under suitable surface tractions alone. Rivlin [2] had observed in 1948 that some specific problems in elasticity may be solved exactly without any detailed knowledge of the form of the strain-energy fun~ion. In Ericksen’s work, El], concerning controllable static defo~ations in any homogeneous, incompressible isotropic elastic materials two special cases remained “too stubborn” [3] to yield to analysis. Ericksen [4] further showed that the only such deformations possible in every compressible homogeneous elastic materials were homogeneous. A further family of controllable deformations for the incompressible case was obtained subsequently by Klingbeil and Shield [S], Singh and Pipkin [ti], and extensively analysed by Marris and Shiau [7]. A further attempt on the last class of such deformations was initiated by Kafadar ES]. Carrol [9] extended the problem to incompressible simple materials and later Petroski and Carlson [lo], [ll], Hays, Laws and Osborn [12], and Laws [ 133, extended the problem to include thermal effects. Dynamically speaking, Truesdell [14] gave the first systematic solutions of counterpart dyn~i~ai problems for incompr~sib~e homogeneous isotropic elastic materials. Such dyn~ically controllable deformations were named “quasi-equilibrated motions” [14]. Truesdell’s theorems on “quasi-equilibrated motions” were further employed by Fosdick [15] for dynamic motions of incompressible isotropic simple materials, and by Shahinpoor and Nowinski [16], Shahinpoor [17-223 for some specific problems in finite dynamic elasticity. According to Truesdell [14] and Truesdell and No11 [23] the controllable dynamicaliy possible qu~i-equilibrated motions that can be produced in every isotropic, incompressible elastic material by the application of surface tractions alone, do not allow any coupling of the wave propagations type between the coordinates involved. We intend to show in the present analysis that for the isotropic, incompressible and perfectly elastic Neo-Hookean tube a certain kind of coupling of the wave propagation type between the radial and axial coordinates may exist, for large dynamic defo~ation of the tube. In this endeavor we employ the exact governing equations of axisymmetric non-linear motions of a circular cylindrical tube made of Neo-Hookean materials, derived originally by Rivlin [24]. The thickness of the tube is assumed to be small enough so that the variation of the deformation vector across the thickness of the tube may be neglected, and further the initial undeformed radius of the tube is replaced by an average mean radius upon which the large radial defo~ation is superposed. Upon this ~sumption the non-linear partial differential t Associate Professor of Mechanical and Aerospace Engineering. $ Professor and Chairman, Department of Mechanical Engineering. 325
M. SHAHINPOORand I. G.
326
TADJRAKHSH
equations of motion are simplified while still retaining their non-linear character. The simplified equations are then solved explicitly for obtaining the finite dynamic deformation field and the frequency of propagating waves under prescribed surface tractions. I.
GOVERNING
EQUATIONS
We shall be brief in expressing the governing equations. Let X” be the material coordinates of a point and xi the spatial coordinates of the same point. with Z” and zm the rectangular fixed Cartesian coordinates corresponding to X” and s’. respectively. The pertinent equations for an isotropic, incompressible. elastic body possessing a strain energy function x and in absence of body forces are [22] : Equations of motion t~+rtjtij+r;ktjk=
p[(g)
+ rj,(i$j)(F)],
(1.1)
Constitutice equation (1.2) where t’j is the Cauchy stress tensor. and +i = SjQi,
(1.3) (1.4)
+ I,-
J
II,_,6j-I,_,c,~“+c;“c,~‘k.
c,:li,
=
G
II,-,
SX” 2x0’
G,, GO;‘= S;, rg
=
f[I,‘-I-c,‘“c;1q.
(1.6)
?_? 2-Y
_ dZ’?Z’
@
=
(1.5)
f@[gir.k
gij = ,-
(1.7)
7,
C’S’ c..YJ
gijgjk
= #i
(1.8)
f $?kr,j-gi,.r]-
(1.9)
and Einstein summation notation is used throughout the sequel. We employ a semi-inverse method and assume that a point of the tube which was originally at X” = R, 0, Z in the undeformed state B. at time t = 0, moves to xi = r,O.: at time t > 0 defined by y2 z!z(1= @
.x1 = r = R+u(R . Z , t).5
53 = z = Z + w(R, Z, t).
(1.10)
Clearly (1.10) completely describes a general axisymmetric dynamic deformation of the tube. Application to Neo-Hookean material Let us now assume that our material is rubber of the Neo-Hookean energy is
type whose strain
c = )cr(l,- I - 3).
(1.11)
where c( is a material constant. Considering that the only non-vanishing Christoffel symbols in this case are 1 r:2 = 1-:, = -,
T;, = -R,
we have the following exact governing equations of axisymmetric cylindrical rubber tube?
tThese equations were first derived exactly Rivlin’s equations.
by Rivlin
(1.12)
R
[23].
One should
motions of a circular
put a = E/3 and replace
p to -p
to get
321
A class of controllable wave propagations in finite elasticity
(I + :)[(I
+ %)(I
+ g)
- -$g]
= 1;
(incompressibility).
(1.15)
Let as assume that the th~~k~~s of the tube is smaB eziough so that the varj~t~o~s of u, w and p across the thickness of the tube may be disregarded, ~o~s~ue~tf~, equations (1.133-(1.15) reduce to
a% _$$+oLu=o, 5F El d2W
PW
sp @.l?)
Ps-q$=z
(I + ;)(I
+ gj
where R, is the mean u~def~~~ fxq-(1.3) reduces to
(1.16)
1 1,
(incomprasibifity),
radius af the tube. The corresponding
(lJ8) Stress system (1.19)
(1.22) t= = g21 = t’3 = $31 = 0 . Let us TIOW&mge
the vari&des
f1.23)
from Z and t into q such that
(1.24) q = Z+ut, where u is the unknown speed of propagation of waves along the length of the tube in such a way that it is the direction of Z, i.e. if 2 > 0 then D 1 ct and if 2 < 0, v < 0. In other words we are considering the poss~b~~ty of ~~~tjat~~ wave pro~~at~o~ from the middfe of the tube (Z = 0) towards both ends of the tube. Otrr governing equations (1.16)~ff.18) together with the stresses (1.19)-(1.23) reduce to -d*u (1.25) d$ +j@$E)u=O. CP
fpg-oa)__=
bg - _,
dq
fn.26) (1.27)
(1.29)
328
From
M. SHAHINPOOR and 1. G. TADJBARHSH
(1.23) (1.32)
where (1.33) and A and I$ are integration Further from (1.26)
constants.
-B-&_lr2-~)
p(q) =
where B is the constant
of integration
(1.34)
and from the first of (1.27) and (1.32) (1.35)
From (1.35) a new result emerges. This result is the observation that the absolute value of A/R,,, cannot be greater than unity for if it is then for some kq+4 the product (A/R) cos(kq + 4) may just be equal to ( - 1) for which dridr7 becomes unbounded. Indeed the direct integration of (1.35) also will reveal the same observation. Therefore in order for the large amplitude oscillations under discussion to exist A ~R,
I
<1
(1.36)
Consequently one may infer from (1.36) that if R, + O+ there exist no motions of the wave propagation type initiated at the center Z = 0. Further. the amplitude of the radial oscillations cannot exceed the value of the original undeformed mean radius of the tube. Based on (1.32), (1.34) and (1.35) the stresses reduce to -2
t
l1 = B++(pC*-a)
1+
+cos(kq+&+
sin'(krl + +)I
E[ 1 -A’/?
(1.37)
1
Ritz2
=[I
+$coS(kq+~]
‘*{B+(i”-lljl
+&co~(kl’+@]l-~+z.
(1.38)
-2
1+
tr3t3’ = -ctAksin(kq+4)
(1.39)
$cos(kv+oj) m
1
(1.40) 2.
WAVE
CONTROLLABLE
Suppose that a large amplitude, symmetric middle of the tube in such a way that
PROPAGATIONS
and harmonic
u(R,Z,t)lz=o
= qcostor.
excitation
is induced
at the
(2.1)
where q is the intensity of excitation and UJ is its frequency. We would like to investigate the possibility for this excitation to propagate along the tube in both positive and negative directions of the Z-axis from the middle point Z = 0. In other words we would like to see if it is possible to maintain and control this finite wave propagations under prescribed surface conditions alone. Applying the condition (2.1) to equation (1.32) it is immediately found that (2.2)
u =
qcos
c i ~Z+tut 1’
(2.3)
A class of controllable wave propagations in finite elasticity
329
Based on (2.3) (24) (2.5) From the first of (2.2) and equation (1.33)
(2.6) and thus
In order forthe speed ofpropagation of waves to be real and thus ensuring the uniqueness and stability of motion
t2.a) which imposes limitation on the frequency of induced excitation. Note that in obtaining (2.8) we have taken o! as positive. This is of course true and is due to Baker-Ericksen ~n~ua~~ty [241, conce~~~~ the restrictions imposed on the strain energy.? The stresses, now, can be succinctly writlen as
As can be seen the system of surface tractions (2.9)-(2.13) are the only means necessary for sustaining and controlling the propagation of axisyrnmetric waves in a thin circular cylindrical tube made of Weo-Hookean materials_ Ofcourse this analysis is rather ~rjm~tive. Howezer, it opens a broad realm of ~o~tro~~ab~~ finite waves which is of utmost importance in engineering application. Afstt, the boundedness of the amplitude of such waves as well as the boundedness of excitation frequencies ofsuch waves indicate that they bear an intrinsic connection with dynamic stability of finite deformations. REFERENCES
2. R. S. Rivlin. Large elastic deformations of isotropic materials-VI. Further results in the theory of torsion, shear and flexure. Phil. Trans. R. Sot., London A242, 173-195 (1949). 3. C. Truesdell. Problems of Non-Lineur Elasticiry. Gordon & Breach (1965). 4. 3. L. Erieksen. Deformations possible in every isotropic compressible perfectly elastic materiaL J. marh. P&IX 34, IX-130 ff955).
t This was also experimentally confirmed by Rivlin and Saunders [25].
330
M. SHAH~NP~~R and I. G.
TADIBAKHSH
I. A. W. Marris and J. F. Shiau. Universal deformations in isotropic. incompressible. hyperelastic materials when deformation tensor has equal proper values, Archs rurion. Mech. Analysis 36, 135-148 (19701 8. C. B. Kafadar. On Ericksen’s problem, Archs ration. Mech. Analysis 47. t 5--27 C1972 t. deformation of incompressible simple materials. fnr. J. Enyng .Sc,i. 5. No. h. 9. M. M. Carrel. Controllable 515-522 (1967). states of rigid heat conductors. ZA.%fP 19. 371-3X3 (19681. 10. H. J. Petroski and D. E. Carlson. Controllable states of elastic heat conductors. Ar& rarion. iif(~l~. .Anul~srr 11. H. J. Petroski and D. E. Carlson, Controllable 31, 127-139 (1968). Q. ccppl. ~%tur/i. 12. M. Hays. N. Laws and R. B. Osborn, The paucity of universal motions in thermoelasticity, 27,41&420 (1969). states in thermoelasticity, Q. up&. Matlt. 28,432-436 (1970). 13. N. Laws, Controllable quamplurimorum de motu corporum elasticorum 14. C. Truesdell, Solution generalis et accurata problematum incomprimibi~ium in deformationibus valde magnis, Archs rarion. Me&. Anul~sis 11. 106-l 18 (1962). isotropic simple materials. .4rcl~ rurion. Me& 15. R. L. Fosdick, Dynamically possible motions of incompressible Analysis 29. 272-282 (1968). and J. L. Nowinski, An exact solution 10 problem of forced large amplitude oscillations of a 16. M. Shahinpoor thin hyperelastic tube, Int. J. Non-Linear Mech. 6, 193-207 (1971). Non-Iinear differential equations reducible to linear ones, lraninn J. Sri. Twit. 1, Nu. 2. 17. M. Shahinpoor, 99-112 (1971). Analysis of a class ofqu~i-equilibrated motions. Rend. Scirnz AtO4. 145-157 (1972). 18. M. Shahinpoor, Quasi-equilibrated motions of incompressible micropolar media, Proc. Zst irm. Cony. Ckii 19. M. Shahinpoor, Engng & Engng Mech. 1,30&317 (1972). 20. M. Shahinpoor. Large amplitude oscillations of a hollow sphertcal dieleclrlc. Inr. J. Non-Lirteur Me& 7. No. 4, 527-534 (1972). Combined radial.-axial large amplitude oscillations of hyperelastic cylindrical tubes. J. hlutk. 31. M. Shahinpoor, Phy. Sci. VII, 2, 11 I-128 (1973). 22. M. Shahinpoor. Exact solution to finite amplitude oscillation of an anisotropic thin rubber tube. J. UCOII.SI. Sot. Am. 56.2,477-480 (1974). 23. C. Truesdell and W. No% The non-linear field theories of mechanics. Et~c~&pac&c~ of Pil~.sic.\ (edited by Fliigge), Iii/3. Springer, Berlin (1965). 24. R. S. Rivlin, Large elastic deformations of isotropic materials, I-V, PM. Trans. R. Sot. A240 (1948): A195 (1949). 25. M. Baker and J. L. Ericksen, Inequalities restricting the form of the stress deformation relations for isotropic elastic solids and Reiner-Rivlin ifuids, J. Wash. Acad. Sci. 44. 33-35 (1954). 26, R. S. Rivlin and D. W. Saunders, Large elastic deformation of isotropic mateflals-VU. Experiments on the deformation of rubber, Phil. Trans. R. Sm., London A243.251-288 (1951).
On Gtablit les Equations des d;formations dynamiques finies a axe de sym6trie d'une coque cylindrique incompressible, glastique et isotrope faite d'un mat&iau &o Hook&en. On simplifie les equations aux d&-i&es partielles non lin
Zusammenfassung: Die Bestimnungsgleichungen fiir die achsensymmetrischen, endlichen. dynamischen Verformunged einer inkompressiblen, isotropischen und elastischen Zylinderschale aus neohookesthen Material werden hergeleitet. Die nicht~inearen partiellen Djfferentia~gleichungen werden vereinfacht fiir den Fall, dass alle Verformungsunterschiede iiber die Zylinderdicke vernachl%ssigt werden konnen. Die vereinfachten nichtlinearen Gleichungen werden dann genau gel&t und ergeben Ltisungen fiir sich entlang der Achse ausbreitende Wellen. Diese WellenlGsungen werden als kontrollierbar bezeichnet. weil sie nur durch Vorgabe von Oberflschenspannungen und begrenzte Erregungsamplitude und - frequenz aufrecht erhalten werden k&nen.
J Non-l,~near
Mechnnia.
Vol.
II.
pp
325-330
Pergamon PWSS 1976
PrInted III Great Britain
416
A CLASS OF CONTROLLABLE WAVE PROPAGATIONS IN FINITE ELASTICITY M. SHAHINPOOK~ and I. G. TADJBAKHSH~ College of Engineering, Pahlavi University, Shiraz, Iran (Received 18 Mnrch 1976) Abstract-Governing
equations of axisymmetric finite dynamic deformations of an incompressible. isotropic and elastic cylindrical shell made of Neo-Hookean materials are derived. The non-linear partial differential equations are simplified for the cases where all deformation variations along the thickness of the tube may be neglected. The simplified non-linear equations are then solved exactly to arrive at traveling wave solutions along the axis. These wave solutions are called controllable because they can be maintained by prescribable surface stresses, bounded amplitude and frequency of excitations alone.
Ericksen [I J in 1954 made the first systematic endeavor to find all deformations that any homogeneous isotropic, incompressible elastic body may sustain under suitable surface tractions alone. Rivlin [2] had observed in 1948 that some specific problems in elasticity may be solved exactly without any detailed knowledge of the form of the strain-energy fun~ion. In Ericksen’s work, El], concerning controllable static defo~ations in any homogeneous, incompressible isotropic elastic materials two special cases remained “too stubborn” [3] to yield to analysis. Ericksen [4] further showed that the only such deformations possible in every compressible homogeneous elastic materials were homogeneous. A further family of controllable deformations for the incompressible case was obtained subsequently by Klingbeil and Shield [S], Singh and Pipkin [ti], and extensively analysed by Marris and Shiau [7]. A further attempt on the last class of such deformations was initiated by Kafadar ES]. Carrol [9] extended the problem to incompressible simple materials and later Petroski and Carlson [lo], [ll], Hays, Laws and Osborn [12], and Laws [ 133, extended the problem to include thermal effects. Dynamically speaking, Truesdell [14] gave the first systematic solutions of counterpart dyn~i~ai problems for incompr~sib~e homogeneous isotropic elastic materials. Such dyn~ically controllable deformations were named “quasi-equilibrated motions” [14]. Truesdell’s theorems on “quasi-equilibrated motions” were further employed by Fosdick [15] for dynamic motions of incompressible isotropic simple materials, and by Shahinpoor and Nowinski [16], Shahinpoor [17-223 for some specific problems in finite dynamic elasticity. According to Truesdell [14] and Truesdell and No11 [23] the controllable dynamicaliy possible qu~i-equilibrated motions that can be produced in every isotropic, incompressible elastic material by the application of surface tractions alone, do not allow any coupling of the wave propagations type between the coordinates involved. We intend to show in the present analysis that for the isotropic, incompressible and perfectly elastic Neo-Hookean tube a certain kind of coupling of the wave propagation type between the radial and axial coordinates may exist, for large dynamic defo~ation of the tube. In this endeavor we employ the exact governing equations of axisymmetric non-linear motions of a circular cylindrical tube made of Neo-Hookean materials, derived originally by Rivlin [24]. The thickness of the tube is assumed to be small enough so that the variation of the deformation vector across the thickness of the tube may be neglected, and further the initial undeformed radius of the tube is replaced by an average mean radius upon which the large radial defo~ation is superposed. Upon this ~sumption the non-linear partial differential t Associate Professor of Mechanical and Aerospace Engineering. $ Professor and Chairman, Department of Mechanical Engineering. 325
M. SHAHINPOORand I. G.
326
TADJRAKHSH
equations of motion are simplified while still retaining their non-linear character. The simplified equations are then solved explicitly for obtaining the finite dynamic deformation field and the frequency of propagating waves under prescribed surface tractions. I.
GOVERNING
EQUATIONS
We shall be brief in expressing the governing equations. Let X” be the material coordinates of a point and xi the spatial coordinates of the same point. with Z” and zm the rectangular fixed Cartesian coordinates corresponding to X” and s’. respectively. The pertinent equations for an isotropic, incompressible. elastic body possessing a strain energy function x and in absence of body forces are [22] : Equations of motion t~+rtjtij+r;ktjk=
p[(g)
+ rj,(i$j)(F)],
(1.1)
Constitutice equation (1.2) where t’j is the Cauchy stress tensor. and +i = SjQi,
(1.3) (1.4)
+ I,-
J
II,_,6j-I,_,c,~“+c;“c,~‘k.
c,:li,
=
G
II,-,
SX” 2x0’
G,, GO;‘= S;, rg
=
f[I,‘-I-c,‘“c;1q.
(1.6)
?_? 2-Y
_ dZ’?Z’
@
=
(1.5)
f@[gir.k
gij = ,-
(1.7)
7,
C’S’ c..YJ
gijgjk
= #i
(1.8)
f $?kr,j-gi,.r]-
(1.9)
and Einstein summation notation is used throughout the sequel. We employ a semi-inverse method and assume that a point of the tube which was originally at X” = R, 0, Z in the undeformed state B. at time t = 0, moves to xi = r,O.: at time t > 0 defined by y2 z!z(1= @
.x1 = r = R+u(R . Z , t).5
53 = z = Z + w(R, Z, t).
(1.10)
Clearly (1.10) completely describes a general axisymmetric dynamic deformation of the tube. Application to Neo-Hookean material Let us now assume that our material is rubber of the Neo-Hookean energy is
type whose strain
c = )cr(l,- I - 3).
(1.11)
where c( is a material constant. Considering that the only non-vanishing Christoffel symbols in this case are 1 r:2 = 1-:, = -,
T;, = -R,
we have the following exact governing equations of axisymmetric cylindrical rubber tube?
tThese equations were first derived exactly Rivlin’s equations.
by Rivlin
(1.12)
R
[23].
One should
motions of a circular
put a = E/3 and replace
p to -p
to get
321
A class of controllable wave propagations in finite elasticity
(I + :)[(I
+ %)(I
+ g)
- -$g]
= 1;
(incompressibility).
(1.15)
Let as assume that the th~~k~~s of the tube is smaB eziough so that the varj~t~o~s of u, w and p across the thickness of the tube may be disregarded, ~o~s~ue~tf~, equations (1.133-(1.15) reduce to
a% _$$+oLu=o, 5F El d2W
PW
sp @.l?)
Ps-q$=z
(I + ;)(I
+ gj
where R, is the mean u~def~~~ fxq-(1.3) reduces to
(1.16)
1 1,
(incomprasibifity),
radius af the tube. The corresponding
(lJ8) Stress system (1.19)
(1.22) t= = g21 = t’3 = $31 = 0 . Let us TIOW&mge
the vari&des
f1.23)
from Z and t into q such that
(1.24) q = Z+ut, where u is the unknown speed of propagation of waves along the length of the tube in such a way that it is the direction of Z, i.e. if 2 > 0 then D 1 ct and if 2 < 0, v < 0. In other words we are considering the poss~b~~ty of ~~~tjat~~ wave pro~~at~o~ from the middfe of the tube (Z = 0) towards both ends of the tube. Otrr governing equations (1.16)~ff.18) together with the stresses (1.19)-(1.23) reduce to -d*u (1.25) d$ +j@$E)u=O. CP
fpg-oa)__=
bg - _,
dq
fn.26) (1.27)
(1.29)
328
From
M. SHAHINPOOR and 1. G. TADJBARHSH
(1.23) (1.32)
where (1.33) and A and I$ are integration Further from (1.26)
constants.
-B-&_lr2-~)
p(q) =
where B is the constant
of integration
(1.34)
and from the first of (1.27) and (1.32) (1.35)
From (1.35) a new result emerges. This result is the observation that the absolute value of A/R,,, cannot be greater than unity for if it is then for some kq+4 the product (A/R) cos(kq + 4) may just be equal to ( - 1) for which dridr7 becomes unbounded. Indeed the direct integration of (1.35) also will reveal the same observation. Therefore in order for the large amplitude oscillations under discussion to exist A ~R,
I
<1
(1.36)
Consequently one may infer from (1.36) that if R, + O+ there exist no motions of the wave propagation type initiated at the center Z = 0. Further. the amplitude of the radial oscillations cannot exceed the value of the original undeformed mean radius of the tube. Based on (1.32), (1.34) and (1.35) the stresses reduce to -2
t
l1 = B++(pC*-a)
1+
+cos(kq+&+
sin'(krl + +)I
E[ 1 -A’/?
(1.37)
1
Ritz2
=[I
+$coS(kq+~]
‘*{B+(i”-lljl
+&co~(kl’+@]l-~+z.
(1.38)
-2
1+
tr3t3’ = -ctAksin(kq+4)
(1.39)
$cos(kv+oj) m
1
(1.40) 2.
WAVE
CONTROLLABLE
Suppose that a large amplitude, symmetric middle of the tube in such a way that
PROPAGATIONS
and harmonic
u(R,Z,t)lz=o
= qcostor.
excitation
is induced
at the
(2.1)
where q is the intensity of excitation and UJ is its frequency. We would like to investigate the possibility for this excitation to propagate along the tube in both positive and negative directions of the Z-axis from the middle point Z = 0. In other words we would like to see if it is possible to maintain and control this finite wave propagations under prescribed surface conditions alone. Applying the condition (2.1) to equation (1.32) it is immediately found that (2.2)
u =
qcos
c i ~Z+tut 1’
(2.3)
A class of controllable wave propagations in finite elasticity
329
Based on (2.3) (24) (2.5) From the first of (2.2) and equation (1.33)
(2.6) and thus
In order forthe speed ofpropagation of waves to be real and thus ensuring the uniqueness and stability of motion
t2.a) which imposes limitation on the frequency of induced excitation. Note that in obtaining (2.8) we have taken o! as positive. This is of course true and is due to Baker-Ericksen ~n~ua~~ty [241, conce~~~~ the restrictions imposed on the strain energy.? The stresses, now, can be succinctly writlen as
As can be seen the system of surface tractions (2.9)-(2.13) are the only means necessary for sustaining and controlling the propagation of axisyrnmetric waves in a thin circular cylindrical tube made of Weo-Hookean materials_ Ofcourse this analysis is rather ~rjm~tive. Howezer, it opens a broad realm of ~o~tro~~ab~~ finite waves which is of utmost importance in engineering application. Afstt, the boundedness of the amplitude of such waves as well as the boundedness of excitation frequencies ofsuch waves indicate that they bear an intrinsic connection with dynamic stability of finite deformations. REFERENCES
2. R. S. Rivlin. Large elastic deformations of isotropic materials-VI. Further results in the theory of torsion, shear and flexure. Phil. Trans. R. Sot., London A242, 173-195 (1949). 3. C. Truesdell. Problems of Non-Lineur Elasticiry. Gordon & Breach (1965). 4. 3. L. Erieksen. Deformations possible in every isotropic compressible perfectly elastic materiaL J. marh. P&IX 34, IX-130 ff955).
t This was also experimentally confirmed by Rivlin and Saunders [25].
330
M. SHAH~NP~~R and I. G.
TADIBAKHSH
I. A. W. Marris and J. F. Shiau. Universal deformations in isotropic. incompressible. hyperelastic materials when deformation tensor has equal proper values, Archs rurion. Mech. Analysis 36, 135-148 (19701 8. C. B. Kafadar. On Ericksen’s problem, Archs ration. Mech. Analysis 47. t 5--27 C1972 t. deformation of incompressible simple materials. fnr. J. Enyng .Sc,i. 5. No. h. 9. M. M. Carrel. Controllable 515-522 (1967). states of rigid heat conductors. ZA.%fP 19. 371-3X3 (19681. 10. H. J. Petroski and D. E. Carlson. Controllable states of elastic heat conductors. Ar& rarion. iif(~l~. .Anul~srr 11. H. J. Petroski and D. E. Carlson, Controllable 31, 127-139 (1968). Q. ccppl. ~%tur/i. 12. M. Hays. N. Laws and R. B. Osborn, The paucity of universal motions in thermoelasticity, 27,41&420 (1969). states in thermoelasticity, Q. up&. Matlt. 28,432-436 (1970). 13. N. Laws, Controllable quamplurimorum de motu corporum elasticorum 14. C. Truesdell, Solution generalis et accurata problematum incomprimibi~ium in deformationibus valde magnis, Archs rarion. Me&. Anul~sis 11. 106-l 18 (1962). isotropic simple materials. .4rcl~ rurion. Me& 15. R. L. Fosdick, Dynamically possible motions of incompressible Analysis 29. 272-282 (1968). and J. L. Nowinski, An exact solution 10 problem of forced large amplitude oscillations of a 16. M. Shahinpoor thin hyperelastic tube, Int. J. Non-Linear Mech. 6, 193-207 (1971). Non-Iinear differential equations reducible to linear ones, lraninn J. Sri. Twit. 1, Nu. 2. 17. M. Shahinpoor, 99-112 (1971). Analysis of a class ofqu~i-equilibrated motions. Rend. Scirnz AtO4. 145-157 (1972). 18. M. Shahinpoor, Quasi-equilibrated motions of incompressible micropolar media, Proc. Zst irm. Cony. Ckii 19. M. Shahinpoor, Engng & Engng Mech. 1,30&317 (1972). 20. M. Shahinpoor. Large amplitude oscillations of a hollow sphertcal dieleclrlc. Inr. J. Non-Lirteur Me& 7. No. 4, 527-534 (1972). Combined radial.-axial large amplitude oscillations of hyperelastic cylindrical tubes. J. hlutk. 31. M. Shahinpoor, Phy. Sci. VII, 2, 11 I-128 (1973). 22. M. Shahinpoor. Exact solution to finite amplitude oscillation of an anisotropic thin rubber tube. J. UCOII.SI. Sot. Am. 56.2,477-480 (1974). 23. C. Truesdell and W. No% The non-linear field theories of mechanics. Et~c~&pac&c~ of Pil~.sic.\ (edited by Fliigge), Iii/3. Springer, Berlin (1965). 24. R. S. Rivlin, Large elastic deformations of isotropic materials, I-V, PM. Trans. R. Sot. A240 (1948): A195 (1949). 25. M. Baker and J. L. Ericksen, Inequalities restricting the form of the stress deformation relations for isotropic elastic solids and Reiner-Rivlin ifuids, J. Wash. Acad. Sci. 44. 33-35 (1954). 26, R. S. Rivlin and D. W. Saunders, Large elastic deformation of isotropic mateflals-VU. Experiments on the deformation of rubber, Phil. Trans. R. Sm., London A243.251-288 (1951).
On Gtablit les Equations des d;formations dynamiques finies a axe de sym6trie d'une coque cylindrique incompressible, glastique et isotrope faite d'un mat&iau &o Hook&en. On simplifie les equations aux d&-i&es partielles non lin
Zusammenfassung: Die Bestimnungsgleichungen fiir die achsensymmetrischen, endlichen. dynamischen Verformunged einer inkompressiblen, isotropischen und elastischen Zylinderschale aus neohookesthen Material werden hergeleitet. Die nicht~inearen partiellen Djfferentia~gleichungen werden vereinfacht fiir den Fall, dass alle Verformungsunterschiede iiber die Zylinderdicke vernachl%ssigt werden konnen. Die vereinfachten nichtlinearen Gleichungen werden dann genau gel&t und ergeben Ltisungen fiir sich entlang der Achse ausbreitende Wellen. Diese WellenlGsungen werden als kontrollierbar bezeichnet. weil sie nur durch Vorgabe von Oberflschenspannungen und begrenzte Erregungsamplitude und - frequenz aufrecht erhalten werden k&nen.