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A refined dynamic theory for viscoelastic cylindrical shells and cylindrical laminated composites, Part 2: An application
&nnaZofSbzmdandVi&mtion(1989)
130(l), 69-77
A REFINED DYNAMIC THM)RY FOR VISCOEXASTIC CYLINDRICAL SHEXJS AND tXLlNDR;[C!ALUMlNATED COMI’OSl’IlB, PART 2: AN APPLICATION G.
DepmtmentofE&ineeringScience,
A.
BIRLIK
Mi&&&.st
Techniaar University,
Ankam,
!b%q
AND
Y. MENGI Depmtmentof Civil E&ineering, (Received7Mmh1988,
Cukurova University,
andinmisedfonn19
Adana, !lWz.q
September1988)
In this study, the general approximate theory developedin Part 1 for shells is assessed for axially symmetric elastic waves propagating in a closed circular cylindrical shell (hollow rod). The spectra predicted by zeroth and second order approximate theories are determined for various values of shell thicknesses and the Poisson ratios and they are compared with those of exact theory. It is found that the agreement between the two is good. Approximate and exact cut-off frequencies match almost exactly. The approximate theory is valid for thin as well as thick shells. These results, which are obtained without using correction factors, give an indication of the power of the general theories proposed in Part 1. 1. INTRODUCTION The dynamic approximate theories proposed in Part 1 are general. They are developed for both cylindrical shell and laminated composites, and they contain circular and flat shapes as a special case. The material may be elastic or viscoelastic. The order of the theories is arbitrary and the high frequency response of shell or composite structures can threebe taken into account by increasing the order. The theories accommodate
dimensional behavior of the aforementioned structures. The novelty of the general approximate theories lies in the fact that they take into account properly the lateral boundary or interface conditions of shells or composites, without causing any discrepancy between assumed deformation shapes and the above-mentioned conditions. We expect that this property of the theories will improve their dynamic predictions. With the object of the assessment of the general approximate theories, in this study the approximate theory proposed in Part 1 for shells is applied to a special problem involving axisymmetric motion of elastic closed circular shells “CC? (hollow rods). This special problem is chosen because exact data for this problem is available (see, e.g., references [ 1,2]). Some approximate theories exist already in the literature for axisymmetric motions of CCS (see, e.g., references [3-61, but these theories have some limitations. Their order is fixed. They contain some inconsistencies between assumed displacement shapes and lateral boundary conditions. To compensate for the error caused by these limitations, some correction factors are used in these theories. It may be noted that the determination of the correction factors involves lengthy computations and depends on the availability of the exact data. 69 0022-460X/89/070069+09
$03.00/O
@ 1989 Academic
Press Limited
70
G. A. BIRLIK AND Y. MENGI
The analysis presented in this study is carried out for two different orders of the approximate theory: zeroth order theory, which is a two-mode theory, and second order theory, which is a six-mode theory. The spectra of axisymmetric waves predicted by the approximate theory are established and compared with the exact spectra for various values of shell thicknesses and Poisson ratios. It is found that the agreement between exact and approximate spectra is very good. This perfect match is achieved without using correction factors and in spite of our using lower order theories. Further, it is observed that the shapes of the uncoupled displacement distributions predicted by the approximate theory at the cut-off frequencies agree closely with those of the exact data. We believe that the findings stated in the previous paragraph give an indication of the power of the general approximate theories proposed in Part 1. For complete assessment one should, of course, apply the theories to other problems involving various motions (axisymmetric, flexural, etc.) of shells and composites. This will be done in future work.
2. FORMULATION
OF THE EXAMPLE PROBLEM USING APPROXIMATE THEORY
With the object of appraising the theories proposed in Part 1, in this study a dynamic problem for which exact data is available in the literature is chosen. This example problem involves axially symmetric waves propagating in a hollow rod, which can be treated also as a closed circular cylindrical shell (CCS). The circular shell has the inner and outer radii a and b, respectively, and it has the thickness Zh (see Figure 1). The shell material is assumed to be linear elastic and has the Lam6 coefficients A and p. CCS is referred to the orthogonal curvilinear (y, , y,, y3) co-ordinate system described in Part 1. We note that the y,-y, surface coincides with the mid-surface of CCS, y1 is in the axial direction, and y2 measures the radial distance from the midsurface (see Figure 1). The inner and outer surfaces of CCS are free of traction.
IIQme L Geomettical desuiption of cimdar cylindxicalshell.
We now proceed to present the equations of the approximate theory for the axisymmetric motion of CCS under consideration. In the analysis, we will use two different orders of the approximate theory, namely, zeroth (m = 0) and second (m = 2) orders. Further, we will assume that the distribution functions are Legendre polynomials. Accordingly, the constants appearing in the approximate theory will be taken from Tables l-3 of Part 1. First we observe that the radius of curvature R of the mid-surface appearing in the general approximate theory is constant and is given by R=(a+b)/2
(1)
for our problem. Further, we note that since the problem concerns the axisymmetric motion of CCS we have for the displacements k=%(YI,Y2,t),
u2 = UZ(Yl, Y2, t),
113= 4
(2)
VISCOELASTIC
SHELLS
AND
COMPOSITES,
71
2
where r, as indicated in Part 1, denotes time. To avoid complicating the presentation, repetitions of the meanings of the symbols and the definitions which are given in Part 1 will be kept at minimum level, and only some definitions will be repeated for the sake of clarity. In view of equations (2), the definitions given in Part 1 and the fact that the constants 15~are all zero, which is the consequence of the mean radius R being constant for CCS, one has u;=~;=+-&),
2, =O, Ui
s;=o.
(3)
Note that since the problem is axisymmetric all of the variables appearing in the approximate theory are functions of yl and t. Next, the field equations of the zeroth and second order approximate theories can be written down. In view of equations (3), the constitutive equations of the approximate theory, equations (10) of Part 1, reduce to 7;,=(2~+h)a,u;+A(s;-ii~)+A~/R, ~;*=(~cL+A)(S~~--~)+A~~U;+A~/R, 7;3=(2~+A)(~/R)+Aal~;+A(S;-ii~), T;*=/.La*u;+~(s;-ii;),
773= r2”3= 0.
(4)
Here the 7;‘s are generalized stresses of the approximate theory defined in Part 1. In equations (4), n = 0 for zeroth order theory and n takes the values 0,l and 2 for second order theory. The displacement face variables S; are defined by s” = (1/2h)K, ’ 1 (1/2h)$,
forn=0,2 forn=l
S;=ut-u;
$=ut+u;
I’
(5)
where UT and uf denote, respectively, the displacements on the outer and inner surfaces of CCS and h is half of the shell thickness: that is, 2h=b-a.
(6)
18 view of the approximate theory presented in Part 1, we note that the variables # and ul appearing in equations (4) are related to the generalized displacements ul by
where the coefficients c,k and &k are, respectively, given in Tables 1 and 2 of Part 1, and m = 0 for zeroth order and m = 2 for second order theory. We now turn our attention to writing down the approximate equations of motion for the CCS under consideration. To this end, we first note that, in view of equations (2), the last of equations (4) and of the fact that Eii= 0, we have sy3= fi3+1)3=
#&=O,
*n 7ij=o,
R:=O.
(8)
Equations (3), (8) and the last of equations (4) reduce the equations of motion of the approximate theory, equations (5) of Part 1, to a,~;, + R; - F;, + fi,/ R = pii;,
G.
72
A. BIRLIK
AND
Y. MENGl
where n = 0 for zeroth order and n = 0, 1,2 for second order theory. Here, R1 is the stress traction face variable defined by R;c7zi-
RV =
(Vr)K, ’ 1 (1/2h)R+,
7yi for
Rt=Tli+T,
n=0,2
forn=l
I’
(10)
where 72:.and r;i designate the stress tractions on the outer and inner surfaces of CCS. The variables Fz and e are related to the generalized stresses T; by
(11)
k=O
Here, the coefficients c,,k and Enkare again given in Tables 1 and 2 of Part 1, and m is the integer representing the order of the theory. We now write down the constitutive equations for the face variables R:, equations (14) of Part 1, for our problem. In view of equations (3) and (8), they take the form
R;=? R;=? h
5
h
i
,kyktd;+p
k=O,Z,...
(Zp+A)y*u:+ha,S:+~ES:+
k=l,,,...
2 p R; = ; k=;*,,,. (2/J + h bk”2 *+A@;+;
ES;+
In these equations E = R’/(R*-
F = -Rh/(R*-
h2),
h2),
(13)
the constants (yiyi,y+, y-) are given in Table 3 of Part 1 and
p = 0,p’= - 1
for zeroth order theory,
p = 2, p’ = 1
for second order theory. (14)
Note that, for zeroth order theory, p’= -1; consequently, the summations involving p’ (the first terms) in the first and third of equations (12) drop out for this order of the theory. As lateral boundary conditions, we assume that the outer and inner surfaces of CCS are free of stress tractions, which leads to Tli = 0.
(15)
By using the definition R’ = 72:.F TTi these boundary conditions can be expressed in terms of the stress traction face variables R: appearing in the approximate theory. They are R;=O.
(16)
The formulation of our problem is now complete. The axisymmetric motion of CCS is governed by equations (4), (9) and (12) with R: = 0.!l?ky constitute 10 equations for the us, ST, S:) for zeroth order theory and 22 equations for unknowns ( $,,42,T;s, 7:2, d, the UnknOWllS (7111 , 7;~) 7;~) 71;) U: , U; , ST, S:) (n = 0, 1,2) for the second order theory. Through the substitution of equations (4) into (9), the number of unknowns can be decreased. After this reduction, the unknown variables become u;, u,“, s:, s:,
with
n = 0 for zeroth order,
n = 0, 1,2 for second order,
(17)
VISCOELASTIC
SHELLS
AND
COMPOSITES,
73
2
the number of variables thus being, respectively, 6 and 10 for zeroth and second order theories. As stated previously, the variables of the approximate theory are functions only of the axial distance JQ and the time t. 3. APPROXIMATE DISPERSION RELATIONS FOR AXISYMMETRIC WAVES The object of this study is to provide an assessment of the theories proposed in Part 1 by comparing the dispersion of axisymmetric waves predicted by the approximate theory with that by the exact theory. For this purpose we consider harmonic axisymmetric waves propagating in the axial y, direction of CCS. To obtain the approximate dispersion relation (frequency equation) we let the trial solution for the unknown variables of the approximate theory be of the form Ae
i(ky,-or)
,
(18)
where k is the wavenumber, w is the angular frequency, A is a constant describing the amplitude and i is the imaginary unit number. When we substitute the assumed solution, equation (18), into the reduced governing equations of the approximate theory expressed in terms of the variables appearing in equations (17), we find the following set of algebraic equations: C aGAj = 0
(i = 1,. . . , k).
(19)
j=l
Here, k = 6 for zeroth order theory and k = 10 for second order theory; Aj describes the amplitude vector for the variables in equations (17); the coefficient matrix elements aij are functions of w and k. To save space, the av’s are presented here only for the zeroth order theory: Go k a14= x G,
a
a22 =
2/L?& -pk* - -+pw*,
2P
a41 = h
as3 = Aik,
a62=
az6=--GOP R h’
az4= &ik, 2h
R2
w$++A)
h
PY+
yo,
a43 = h,
a,,=ik, PY-
a3,= -
h’
a35= cLiS
cl, = pik,
A
as5 = x J% Yov
%6=
44 = hik,
other ati = 0.
!4Uzefrequency equation can be obtained by requiring that equation (19) possess a non-trivial solution. It is det(aU)=O,
(21)
which governs the spectral lines relating the frequency o to the wavenumber k. We note that zeroth and second order theories accommodate two and six dispersion lines on the (w-k) plane, respectively; i.e., they constitute two- and six-mode theories, respectively.
74
G. A. BIRLIK
AND
Y. MENGI
4. NUMERICAL RESULTS Numerical results have been obtained for various values of shell thicknesses, namely a* = b/a = (1*1,2,4), covering the range between thin and thick tubes, and two different values of the Poisson ratio, namely, u = O-25 and 0.29. The roots of the frequency equation, equation (21), which governs the dispersion lines, were computed by using the secant method. The results are shown in Figures 2-4. In the figures, 5 and 0 designate the non-dimensional wavenumber and frequency, defined by R = w/w;,
e= ak/S.
(22)
In this equation wf = &/a is the first axial shear cut-off frequency determined from exact theory, where v2 = m is the shear wave velocity. The values of the nondimensional parameter S are tabulated in reference [5] for various values of a”. Equation (22) shows that the exact non-dimensional value of the first axial shear cut-off frequency is unity. Figures 2(a)-(c) are devoted to the comparison between exact [5] and approximate spectra for v =0*25 and a* = (1*1,2,4). Figures 3(a)-(c) give the same comparison for Y = O-29. For v = O-25, the predictions of both zeroth and second order theorries are included in the figures. As the general trend of the predictions of zeroth order theory is approximately the same for both Y = O-25and v = 0929, in the figures belonging to v = 0.29 only the dispersion curves of second order theory are indicated. Figure 4 shows the displacement distributions over the thickness of CCS at the cut-off frequencies determined from second order approximate theory for Y = 0.25 and a* = 4. These distributions are governed by the expression
(23) which can be obtained by using the displacement expansion given in equation (13) of Part 1. In equation (23) the 4;‘s are Legendre polynomials. The distributions in Figure 4 were established by solving equation (19) for the amplitudes and substituting them into equation (23).
5. DISCUSSION AND CONCLUSIONS Discussion of the results presented in the previous section and the conclusions which can be drawn from them is now in order. The following observations can be made from study of the results presented in Figures 2-4. (a) As seen from Figures 2 and 3, the agreement between exact and approximate dispersion lines is fairly good for zeroth order theory and excellent for second order theory. The increase in the order of the approximate theory improves its prediction. The cut-off frequencies predicted by the exact and approximate theories are almost the same for second order theory, and are close and have some deviations only when u* > 2 for zeroth order theory (see Figures 1 and 2). The phase velocity at the origin of the w-k plane whirh.is also equal to the group velocity at that point, agrees with the exact value, e Y p, for both zeroth and second order theories. Here, E is Young’s modulus. It is known that this match is important for pulse propagation. (b) The approximate theory gives good predictions for various values of the Poisson ratio.
VISCOELASTIC
SHELLS
AND
COMPOSITES,
1.6.
0
0.2
0.4
0.6
0.0
1.6.
0
0.2
0.4
0.6
0.6
0
0.2
0.4
0.6
0.6
E
Figure 2. Spectra for Y=0*25 and (a) (I* = 1.1, (b) a* = 2, and (c) a* =4. -, zeroth order; . . .BBB. . . . , approximate second order.
Exact; -
-, approximate
(c) Second order theory gives good predictions for both thin and thick shells. Some deviations begin to appear when a* > 2 for zeroth order theory. (d) The frequency range of second order theory, which is a six-mode theory, is larger than that of zeroth order theory, which is a two-mode theory. If the analysis does not involve very high frequencies, in view of findings established above the use of zeroth order theory may be suggested, since its equations are simpler than those of second order theory. (e) The parts (a), (b), (c) and (d) of Figure 4 describe, respectively, the predictions of second order theory for the displacement distributions at the axial, first radial, lint axial shear and second radial cut-off frequencies. As seen from the figure, these distributions
76
G. A. BIRLIK
0
AND
Y. MENGl
0.4
0.2
0.6
0.8
(c
(b)
- ““--e
......~......... ~ .........o...
‘.
L .,..... /. /
_’
,......
..
,_ ..”
.’
,...
”
:’
0
0.2
0.4
0.6
0
0.8
0.2
0.4
0.6
0.8
c ~gure
3. Spectra
rorv = 0.29 and (a)a*= 1.1,(b) a*=2, and
(C) a* = 4. Key BS Rgure 2.
are uncoupled: i.e., the first and third distributions are purely longitudinal, and the second and fourth distributions are purely radial. This is in complete agreement with the findings established from the exact analysis. Further, the distribution shapes in Figure 8 agree very well with those of exact theory. In view of these preceding observations we may make the following conclusions. The good prediction of the approximate theory is obtained without using correction factors. This accomplishment is a natural consequence of the fact that the general approximate theory developed in Part 1 is capable of satisfying the lateral boundary conditions of the shell correctly. It is known that a comparison of the spectral properties of an approximate theory with those of the exact theory can be used as a criterion for the value of tab approxinutte
VISCOELASTIC SHELLS (a)
77
ANDCOMPOSITES, +(b)
Outer face lo0.5
O-
Inner face2
”1
for
u2
Figure 4. Displacement distributions at the cut-off frequencies predicted by second Y = 0.25 and a* = 4.(a) f = 0, 0 =O;(b) 5 = 0, f2 = 0.6608; (c) 5 = 0,R = 1.0042; (d)
order approximate theory 5 = 0, R = 1.8805.
theory. Therefore, in view of the preceding discussion, we expect that the zeroth and second order approximate theories discussed in this study describe the general axisymmetric dynamic behavior of CCS very well. The dynamic approximate theories proposed in Part 1 are general. They are developed for cylindrical shells and laminated shells (not necessarily circular). They may involve various types of motions (axial, flexural, etc.). The results obtained for the special problem considered in this study provide a partial assessment of the general theories proposed in Part 1.
REFERENCES 1. D. C. GAZIS 1959 &znnaZ of rheAwz&imZ Society ofAmerica 31568573. Three-dimensional investigation of the propagation of waves in hollow circular cylinders, I: Analytical foundation. 2. D. C. GAZIS 1959 &znnaZof theAwmtim2 Society ofAmerim31,573578. Three-dimensional investigation of the propagation of waves in hollow circular cylinders, II: Numerical results. 3. G. HERRMANN and I. MIRSKY 1956 eblanalof AppZied Me&cm& 23, 5fZM%& Threedimensional and shell-theory analysis of axially symmetric motions of cylinders. 4. I. MIRSKY and G. HERRMANN 1958 ebzndofApplidM&anics25,97-102. Axially symmetric motions of thick cylindrical shells. 5. H. D. MCNIVEN, A. H. SEAE and J. L. SACKMAN 1966 eblonaloftheAcoustica1 Sodetyof Ammku40, 784792. Axially symmetric waves in hollow, elastic rods, Part I. 6. H. D. MCNIVEN, J. L. SACKMAN and A. Ii. SEAE 1966 JmmZofrheAwz~tkdSixGtyof Anzekm40,10731U76. Axially symmetric waves in hollow, elastic rods, Part II.
130(l), 69-77
A REFINED DYNAMIC THM)RY FOR VISCOEXASTIC CYLINDRICAL SHEXJS AND tXLlNDR;[C!ALUMlNATED COMI’OSl’IlB, PART 2: AN APPLICATION G.
DepmtmentofE&ineeringScience,
A.
BIRLIK
Mi&&&.st
Techniaar University,
Ankam,
!b%q
AND
Y. MENGI Depmtmentof Civil E&ineering, (Received7Mmh1988,
Cukurova University,
andinmisedfonn19
Adana, !lWz.q
September1988)
In this study, the general approximate theory developedin Part 1 for shells is assessed for axially symmetric elastic waves propagating in a closed circular cylindrical shell (hollow rod). The spectra predicted by zeroth and second order approximate theories are determined for various values of shell thicknesses and the Poisson ratios and they are compared with those of exact theory. It is found that the agreement between the two is good. Approximate and exact cut-off frequencies match almost exactly. The approximate theory is valid for thin as well as thick shells. These results, which are obtained without using correction factors, give an indication of the power of the general theories proposed in Part 1. 1. INTRODUCTION The dynamic approximate theories proposed in Part 1 are general. They are developed for both cylindrical shell and laminated composites, and they contain circular and flat shapes as a special case. The material may be elastic or viscoelastic. The order of the theories is arbitrary and the high frequency response of shell or composite structures can threebe taken into account by increasing the order. The theories accommodate
dimensional behavior of the aforementioned structures. The novelty of the general approximate theories lies in the fact that they take into account properly the lateral boundary or interface conditions of shells or composites, without causing any discrepancy between assumed deformation shapes and the above-mentioned conditions. We expect that this property of the theories will improve their dynamic predictions. With the object of the assessment of the general approximate theories, in this study the approximate theory proposed in Part 1 for shells is applied to a special problem involving axisymmetric motion of elastic closed circular shells “CC? (hollow rods). This special problem is chosen because exact data for this problem is available (see, e.g., references [ 1,2]). Some approximate theories exist already in the literature for axisymmetric motions of CCS (see, e.g., references [3-61, but these theories have some limitations. Their order is fixed. They contain some inconsistencies between assumed displacement shapes and lateral boundary conditions. To compensate for the error caused by these limitations, some correction factors are used in these theories. It may be noted that the determination of the correction factors involves lengthy computations and depends on the availability of the exact data. 69 0022-460X/89/070069+09
$03.00/O
@ 1989 Academic
Press Limited
70
G. A. BIRLIK AND Y. MENGI
The analysis presented in this study is carried out for two different orders of the approximate theory: zeroth order theory, which is a two-mode theory, and second order theory, which is a six-mode theory. The spectra of axisymmetric waves predicted by the approximate theory are established and compared with the exact spectra for various values of shell thicknesses and Poisson ratios. It is found that the agreement between exact and approximate spectra is very good. This perfect match is achieved without using correction factors and in spite of our using lower order theories. Further, it is observed that the shapes of the uncoupled displacement distributions predicted by the approximate theory at the cut-off frequencies agree closely with those of the exact data. We believe that the findings stated in the previous paragraph give an indication of the power of the general approximate theories proposed in Part 1. For complete assessment one should, of course, apply the theories to other problems involving various motions (axisymmetric, flexural, etc.) of shells and composites. This will be done in future work.
2. FORMULATION
OF THE EXAMPLE PROBLEM USING APPROXIMATE THEORY
With the object of appraising the theories proposed in Part 1, in this study a dynamic problem for which exact data is available in the literature is chosen. This example problem involves axially symmetric waves propagating in a hollow rod, which can be treated also as a closed circular cylindrical shell (CCS). The circular shell has the inner and outer radii a and b, respectively, and it has the thickness Zh (see Figure 1). The shell material is assumed to be linear elastic and has the Lam6 coefficients A and p. CCS is referred to the orthogonal curvilinear (y, , y,, y3) co-ordinate system described in Part 1. We note that the y,-y, surface coincides with the mid-surface of CCS, y1 is in the axial direction, and y2 measures the radial distance from the midsurface (see Figure 1). The inner and outer surfaces of CCS are free of traction.
IIQme L Geomettical desuiption of cimdar cylindxicalshell.
We now proceed to present the equations of the approximate theory for the axisymmetric motion of CCS under consideration. In the analysis, we will use two different orders of the approximate theory, namely, zeroth (m = 0) and second (m = 2) orders. Further, we will assume that the distribution functions are Legendre polynomials. Accordingly, the constants appearing in the approximate theory will be taken from Tables l-3 of Part 1. First we observe that the radius of curvature R of the mid-surface appearing in the general approximate theory is constant and is given by R=(a+b)/2
(1)
for our problem. Further, we note that since the problem concerns the axisymmetric motion of CCS we have for the displacements k=%(YI,Y2,t),
u2 = UZ(Yl, Y2, t),
113= 4
(2)
VISCOELASTIC
SHELLS
AND
COMPOSITES,
71
2
where r, as indicated in Part 1, denotes time. To avoid complicating the presentation, repetitions of the meanings of the symbols and the definitions which are given in Part 1 will be kept at minimum level, and only some definitions will be repeated for the sake of clarity. In view of equations (2), the definitions given in Part 1 and the fact that the constants 15~are all zero, which is the consequence of the mean radius R being constant for CCS, one has u;=~;=+-&),
2, =O, Ui
s;=o.
(3)
Note that since the problem is axisymmetric all of the variables appearing in the approximate theory are functions of yl and t. Next, the field equations of the zeroth and second order approximate theories can be written down. In view of equations (3), the constitutive equations of the approximate theory, equations (10) of Part 1, reduce to 7;,=(2~+h)a,u;+A(s;-ii~)+A~/R, ~;*=(~cL+A)(S~~--~)+A~~U;+A~/R, 7;3=(2~+A)(~/R)+Aal~;+A(S;-ii~), T;*=/.La*u;+~(s;-ii;),
773= r2”3= 0.
(4)
Here the 7;‘s are generalized stresses of the approximate theory defined in Part 1. In equations (4), n = 0 for zeroth order theory and n takes the values 0,l and 2 for second order theory. The displacement face variables S; are defined by s” = (1/2h)K, ’ 1 (1/2h)$,
forn=0,2 forn=l
S;=ut-u;
$=ut+u;
I’
(5)
where UT and uf denote, respectively, the displacements on the outer and inner surfaces of CCS and h is half of the shell thickness: that is, 2h=b-a.
(6)
18 view of the approximate theory presented in Part 1, we note that the variables # and ul appearing in equations (4) are related to the generalized displacements ul by
where the coefficients c,k and &k are, respectively, given in Tables 1 and 2 of Part 1, and m = 0 for zeroth order and m = 2 for second order theory. We now turn our attention to writing down the approximate equations of motion for the CCS under consideration. To this end, we first note that, in view of equations (2), the last of equations (4) and of the fact that Eii= 0, we have sy3= fi3+1)3=
#&=O,
*n 7ij=o,
R:=O.
(8)
Equations (3), (8) and the last of equations (4) reduce the equations of motion of the approximate theory, equations (5) of Part 1, to a,~;, + R; - F;, + fi,/ R = pii;,
G.
72
A. BIRLIK
AND
Y. MENGl
where n = 0 for zeroth order and n = 0, 1,2 for second order theory. Here, R1 is the stress traction face variable defined by R;c7zi-
RV =
(Vr)K, ’ 1 (1/2h)R+,
7yi for
Rt=Tli+T,
n=0,2
forn=l
I’
(10)
where 72:.and r;i designate the stress tractions on the outer and inner surfaces of CCS. The variables Fz and e are related to the generalized stresses T; by
(11)
k=O
Here, the coefficients c,,k and Enkare again given in Tables 1 and 2 of Part 1, and m is the integer representing the order of the theory. We now write down the constitutive equations for the face variables R:, equations (14) of Part 1, for our problem. In view of equations (3) and (8), they take the form
R;=? R;=? h
5
h
i
,kyktd;+p
k=O,Z,...
(Zp+A)y*u:+ha,S:+~ES:+
k=l,,,...
2 p R; = ; k=;*,,,. (2/J + h bk”2 *+A@;+;
ES;+
In these equations E = R’/(R*-
F = -Rh/(R*-
h2),
h2),
(13)
the constants (yiyi,y+, y-) are given in Table 3 of Part 1 and
p = 0,p’= - 1
for zeroth order theory,
p = 2, p’ = 1
for second order theory. (14)
Note that, for zeroth order theory, p’= -1; consequently, the summations involving p’ (the first terms) in the first and third of equations (12) drop out for this order of the theory. As lateral boundary conditions, we assume that the outer and inner surfaces of CCS are free of stress tractions, which leads to Tli = 0.
(15)
By using the definition R’ = 72:.F TTi these boundary conditions can be expressed in terms of the stress traction face variables R: appearing in the approximate theory. They are R;=O.
(16)
The formulation of our problem is now complete. The axisymmetric motion of CCS is governed by equations (4), (9) and (12) with R: = 0.!l?ky constitute 10 equations for the us, ST, S:) for zeroth order theory and 22 equations for unknowns ( $,,42,T;s, 7:2, d, the UnknOWllS (7111 , 7;~) 7;~) 71;) U: , U; , ST, S:) (n = 0, 1,2) for the second order theory. Through the substitution of equations (4) into (9), the number of unknowns can be decreased. After this reduction, the unknown variables become u;, u,“, s:, s:,
with
n = 0 for zeroth order,
n = 0, 1,2 for second order,
(17)
VISCOELASTIC
SHELLS
AND
COMPOSITES,
73
2
the number of variables thus being, respectively, 6 and 10 for zeroth and second order theories. As stated previously, the variables of the approximate theory are functions only of the axial distance JQ and the time t. 3. APPROXIMATE DISPERSION RELATIONS FOR AXISYMMETRIC WAVES The object of this study is to provide an assessment of the theories proposed in Part 1 by comparing the dispersion of axisymmetric waves predicted by the approximate theory with that by the exact theory. For this purpose we consider harmonic axisymmetric waves propagating in the axial y, direction of CCS. To obtain the approximate dispersion relation (frequency equation) we let the trial solution for the unknown variables of the approximate theory be of the form Ae
i(ky,-or)
,
(18)
where k is the wavenumber, w is the angular frequency, A is a constant describing the amplitude and i is the imaginary unit number. When we substitute the assumed solution, equation (18), into the reduced governing equations of the approximate theory expressed in terms of the variables appearing in equations (17), we find the following set of algebraic equations: C aGAj = 0
(i = 1,. . . , k).
(19)
j=l
Here, k = 6 for zeroth order theory and k = 10 for second order theory; Aj describes the amplitude vector for the variables in equations (17); the coefficient matrix elements aij are functions of w and k. To save space, the av’s are presented here only for the zeroth order theory: Go k a14= x G,
a
a22 =
2/L?& -pk* - -+pw*,
2P
a41 = h
as3 = Aik,
a62=
az6=--GOP R h’
az4= &ik, 2h
R2
w$++A)
h
PY+
yo,
a43 = h,
a,,=ik, PY-
a3,= -
h’
a35= cLiS
cl, = pik,
A
as5 = x J% Yov
%6=
44 = hik,
other ati = 0.
!4Uzefrequency equation can be obtained by requiring that equation (19) possess a non-trivial solution. It is det(aU)=O,
(21)
which governs the spectral lines relating the frequency o to the wavenumber k. We note that zeroth and second order theories accommodate two and six dispersion lines on the (w-k) plane, respectively; i.e., they constitute two- and six-mode theories, respectively.
74
G. A. BIRLIK
AND
Y. MENGI
4. NUMERICAL RESULTS Numerical results have been obtained for various values of shell thicknesses, namely a* = b/a = (1*1,2,4), covering the range between thin and thick tubes, and two different values of the Poisson ratio, namely, u = O-25 and 0.29. The roots of the frequency equation, equation (21), which governs the dispersion lines, were computed by using the secant method. The results are shown in Figures 2-4. In the figures, 5 and 0 designate the non-dimensional wavenumber and frequency, defined by R = w/w;,
e= ak/S.
(22)
In this equation wf = &/a is the first axial shear cut-off frequency determined from exact theory, where v2 = m is the shear wave velocity. The values of the nondimensional parameter S are tabulated in reference [5] for various values of a”. Equation (22) shows that the exact non-dimensional value of the first axial shear cut-off frequency is unity. Figures 2(a)-(c) are devoted to the comparison between exact [5] and approximate spectra for v =0*25 and a* = (1*1,2,4). Figures 3(a)-(c) give the same comparison for Y = O-29. For v = O-25, the predictions of both zeroth and second order theorries are included in the figures. As the general trend of the predictions of zeroth order theory is approximately the same for both Y = O-25and v = 0929, in the figures belonging to v = 0.29 only the dispersion curves of second order theory are indicated. Figure 4 shows the displacement distributions over the thickness of CCS at the cut-off frequencies determined from second order approximate theory for Y = 0.25 and a* = 4. These distributions are governed by the expression
(23) which can be obtained by using the displacement expansion given in equation (13) of Part 1. In equation (23) the 4;‘s are Legendre polynomials. The distributions in Figure 4 were established by solving equation (19) for the amplitudes and substituting them into equation (23).
5. DISCUSSION AND CONCLUSIONS Discussion of the results presented in the previous section and the conclusions which can be drawn from them is now in order. The following observations can be made from study of the results presented in Figures 2-4. (a) As seen from Figures 2 and 3, the agreement between exact and approximate dispersion lines is fairly good for zeroth order theory and excellent for second order theory. The increase in the order of the approximate theory improves its prediction. The cut-off frequencies predicted by the exact and approximate theories are almost the same for second order theory, and are close and have some deviations only when u* > 2 for zeroth order theory (see Figures 1 and 2). The phase velocity at the origin of the w-k plane whirh.is also equal to the group velocity at that point, agrees with the exact value, e Y p, for both zeroth and second order theories. Here, E is Young’s modulus. It is known that this match is important for pulse propagation. (b) The approximate theory gives good predictions for various values of the Poisson ratio.
VISCOELASTIC
SHELLS
AND
COMPOSITES,
1.6.
0
0.2
0.4
0.6
0.0
1.6.
0
0.2
0.4
0.6
0.6
0
0.2
0.4
0.6
0.6
E
Figure 2. Spectra for Y=0*25 and (a) (I* = 1.1, (b) a* = 2, and (c) a* =4. -, zeroth order; . . .BBB. . . . , approximate second order.
Exact; -
-, approximate
(c) Second order theory gives good predictions for both thin and thick shells. Some deviations begin to appear when a* > 2 for zeroth order theory. (d) The frequency range of second order theory, which is a six-mode theory, is larger than that of zeroth order theory, which is a two-mode theory. If the analysis does not involve very high frequencies, in view of findings established above the use of zeroth order theory may be suggested, since its equations are simpler than those of second order theory. (e) The parts (a), (b), (c) and (d) of Figure 4 describe, respectively, the predictions of second order theory for the displacement distributions at the axial, first radial, lint axial shear and second radial cut-off frequencies. As seen from the figure, these distributions
76
G. A. BIRLIK
0
AND
Y. MENGl
0.4
0.2
0.6
0.8
(c
(b)
- ““--e
......~......... ~ .........o...
‘.
L .,..... /. /
_’
,......
..
,_ ..”
.’
,...
”
:’
0
0.2
0.4
0.6
0
0.8
0.2
0.4
0.6
0.8
c ~gure
3. Spectra
rorv = 0.29 and (a)a*= 1.1,(b) a*=2, and
(C) a* = 4. Key BS Rgure 2.
are uncoupled: i.e., the first and third distributions are purely longitudinal, and the second and fourth distributions are purely radial. This is in complete agreement with the findings established from the exact analysis. Further, the distribution shapes in Figure 8 agree very well with those of exact theory. In view of these preceding observations we may make the following conclusions. The good prediction of the approximate theory is obtained without using correction factors. This accomplishment is a natural consequence of the fact that the general approximate theory developed in Part 1 is capable of satisfying the lateral boundary conditions of the shell correctly. It is known that a comparison of the spectral properties of an approximate theory with those of the exact theory can be used as a criterion for the value of tab approxinutte
VISCOELASTIC SHELLS (a)
77
ANDCOMPOSITES, +(b)
Outer face lo0.5
O-
Inner face2
”1
for
u2
Figure 4. Displacement distributions at the cut-off frequencies predicted by second Y = 0.25 and a* = 4.(a) f = 0, 0 =O;(b) 5 = 0, f2 = 0.6608; (c) 5 = 0,R = 1.0042; (d)
order approximate theory 5 = 0, R = 1.8805.
theory. Therefore, in view of the preceding discussion, we expect that the zeroth and second order approximate theories discussed in this study describe the general axisymmetric dynamic behavior of CCS very well. The dynamic approximate theories proposed in Part 1 are general. They are developed for cylindrical shells and laminated shells (not necessarily circular). They may involve various types of motions (axial, flexural, etc.). The results obtained for the special problem considered in this study provide a partial assessment of the general theories proposed in Part 1.
REFERENCES 1. D. C. GAZIS 1959 &znnaZ of rheAwz&imZ Society ofAmerica 31568573. Three-dimensional investigation of the propagation of waves in hollow circular cylinders, I: Analytical foundation. 2. D. C. GAZIS 1959 &znnaZof theAwmtim2 Society ofAmerim31,573578. Three-dimensional investigation of the propagation of waves in hollow circular cylinders, II: Numerical results. 3. G. HERRMANN and I. MIRSKY 1956 eblanalof AppZied Me&cm& 23, 5fZM%& Threedimensional and shell-theory analysis of axially symmetric motions of cylinders. 4. I. MIRSKY and G. HERRMANN 1958 ebzndofApplidM&anics25,97-102. Axially symmetric motions of thick cylindrical shells. 5. H. D. MCNIVEN, A. H. SEAE and J. L. SACKMAN 1966 eblonaloftheAcoustica1 Sodetyof Ammku40, 784792. Axially symmetric waves in hollow, elastic rods, Part I. 6. H. D. MCNIVEN, J. L. SACKMAN and A. Ii. SEAE 1966 JmmZofrheAwz~tkdSixGtyof Anzekm40,10731U76. Axially symmetric waves in hollow, elastic rods, Part II.