A pressurized cylindrical shell containing a longitudinal crack is considered. The shear modulus of the sheet may be evaluated from the measured Young's moduli and the Poisson's ratios rather than being an independent material constant. Two examples, one for a mildly orthotropic (titanium) and the other for a strongly orthotropic (graphite) material approximately satisfying the condition of special orthotropy are given. The results show that the stress intensity factors are rather strongly dependent on the degree of orthotropy

Erdogan, F.

Ratwani, M.

Yueceoglu, U.

English

NASA Technical Reports Server

IFSM- 73-30 /2-
LEHIGH UNIVERSITY .
ON THE EFFECT OF ORTHOTROPY IN A
CRACKED CYLINDRICAL SHELL
BY
F, ERDOGAN
M, RATWANI
U, YUCEOGLU
TECHNICAL REPORT NASA TR-73-2
(NASA-CR-13229 5) ON THE EFFECT OF N73-28887
OTTHOTROPY IN A CRACKED CYLINDRICAL SHELL
(Lehigh Univ.) 14 p HC $3.00 CSCL 20K
Unclas
G3/32 10463
FEBRUARY 1973
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
GRANT NGR-39-007-011
https://ntrs.nasa.gov/search.jsp?R=19730020155 2020-03-22T23:41:08+00:00Z
ON THE EFFECT OF ORTHOTROPY IN A
CRACKED CYLINDRICAL SHELL*
F. Erdogan, M. Ratwani, and U. YUceoglu
Lehigh University, Bethlehem, Pa.
Abstract. A pressurized cylindrical shell containing a
longitudinal crack is considered. It is assumed that the
material has a special orthotropy, namely that the shear modu-
lus of the sheet may be evaluated from the measured Young's
moduli and the Poisson's ratios rather than being an independ-
ent material constant. Two examples, one for a mildly ortho-
tropic (titanium) and the other for a strongly orthotropic
(graphite) material approximately satisfying the condition of
special orthotropy are given. The results show that the stress
intensity factors are rather strongly dependent on the degree
of orthotropy.
SOME REMARKS ON THE FORMULATION OF THE PROBLEM
A detailed treatment of the linear bending theory of aniso-
tropic shallow shells may be found in [1-3]. Assuming that
through a proper superposition the problem has been reduced to
a perturbation problem with the crack surface tractions as the
only external loads, in an eight order theory the differential
equations for an orthotropic shallow cylindrical shell may be
expressed as
2 2
D1V1 4W(x'X 2) 2 F(X',x 2 ) = 0
ax1
4 2hE2a 2
V2 4F(x1,x 2) h+ R w(x 1 ,x2 ) = 0 , (l.a,b)
@x1
*This work was supported by The National Aeronautics and Space
Administration under The Grant NGR 39-007-011.
-1-
where a,R,h are the dimensions of the crack and the shell (see
the insert in Figure 1),
x1 = X/a , x2  = Y/a , (2)
F is a stress function and w is the displacement component nor-
mal to the surface. The operators V14 and V24 are defined by
4 4  G 4 E 44  [ 12 a 2 a
1 4 + 2[v 2 + 2(1-v1 2 ) E 2 2 E 4 '
1  1  2 2
4 34 E2 _ 4  E2  a4V2  + 2 2G 22 + El x2 4  (3.a,b)
ax1  ax ax 1 ax
The notation for the orthotropic elastic constants are defined
by the following stress strain relations:
EII l 1 (I- v 1c 2 2 ) ' E22 - E2 (c 2 2 - v2 1 1 ) '
S1 V1  2  3
12 H2. '12 ' E l  E2  1 E h/[12(1-v2)] '
(4.a-e)
where x1 and x2 are the principal directions of orthotropy and
are taken respectively along the axial and circumferential
directions. The stress and moment resultants are related to
F and w through the usual expressions (see [3] or [4]).
In solving the problem, for example by expressing F and w
in terms of appropriate Fourier integrals, (1) may be reduced
to a system of two fourth order ordinary differential equations.
The characteristic function of this system will then be an 8th
degree polynomial the coefficients of which will be functions
of the transform variable. For the problem to be analytically
-2-
tractable it is essential that the roots of the characteristic
equation be obtainable in closed form. Apparently, for the
anisotropic shells in general and for the orthotropic shells
in particular this is not possible. In order to express the
roots in closed form the operators V14 and V24 must be properly
factorized. From (3) it is clear that these operators can
indeed be factorized and may be expressed in the following form
4 2 + E2/E1  2 2 (5)V 1 3 1 +V2 2 )  V2
ax  ax2
provided the elastic constants satisfy the following conditions:
[v 2 + 2(1-v 1v 2) E-- 12] E1 /E 2 = 1
1
(2G 122 - 2 ) / E 1/E 2 = 1 (6.a,b)
By direct substitution it may easily be shown that the conditions
(6) are satisfied if
(EIE 2 )G (7)
12 2[l + (v1 2 ) ]
Considering also the relation (4.d), this means that the sheet
material has only three independent constants. Such a material
is said to be specially orthotropic. The results given in this
note will then be valid only for those materials in which the
measured value of G1 2 and that calculated from (7) in terms of
measured Ei and vi, (i=1,2) are in reasonably good agreement.
Changing the variables once more as
-3-
x I = x , (E 1 /E 2 )4 x 2  y ,(8)
(5) becomes
2 2 24 4 = 4
V1 2  2 2) (9)ax ay
With (9), (1) becomes identical to the differential equations
for isotropic shells in which D = Eh3/[12(1-v2)] and E are
replaced by D1 and E2, respectively. The problem may then be
solved by following, for example, the same procedure as that
described in [4] or [5] and will not be repeated here.
STRESS STATE AROUND THE CRACK TIP AND STRESS INTENSITY
FACTORS
Following very closely the technique described in [4], for
a pressurized shell the membrane and bending components of the
stresses in the close neighborhood of the crack tip may be
obtained as
m(rO,) = A 1 (5cos cos ) + 0(r 2 )
yy m 427 4 2 2
k 2 0
x (r,) =m P- c (3cos- + cos-) + 0(r ) ,
a (r,) = -- (sin - sin-2 ) + 0(r2) , (10.a-c)
xy m 2-4 2
Uyb(re,z) = Ab kp - [ - 5 8vo s2
yy 2V-(vo -8vc + c2 cos
2Vc cos ] + O(r )
2-4-
-4-
xx b (r,,z) = Ab [k(8 + 8v c 2 + 3v2 )cos-S [(8+c2 1 2 c +2
- 2 (1-v 2 c)cos5 ] + (r ) ,
4c
k a2h3G v
Sxyb(r,O,z) =A ap z 1 2  c sin 5xy b h 12D 1  c2 2
- (8 + -c)sin ] + 0(r 2 ) , (ll.a-c)
C
where (r,O) are the polar coordinates at the crack tip, z is the
coordinate perpendicular to and measured from the neutral surface
of the shell, and
PoR 2
k p = a , c = (E 1 /E 2) = c -p h1 1
h3G E h3
Vc = c 2 - ( + 3D 2 )  D2 = 2  )  (12)
c 3D2 2 12(1-v1v2 ) (12)
Here po is the internal pressure. Thus, kp is the corresponding
stress intensity factor in the flat plate, and Am and Ab repre-
sent stress intensity factor ratios defined by
Am = ks/k A kb/kp (13.a,b)
m p b b p
where ks and ks are the membrane and bending components of them b
stress intensity factor in the shell.
NUMERICAL EXAMPLES
The elastic constants of the orthotropic shells which are
considered as examples are shown in Table 1. The Table also
shows the "average shear modulus" calculated from
-5-
G E1 E2) (14)
av 2[1 + (v 1 v 2 ) ]
Table 1. Elastic Constants of the Materials
Titanium Graphite
El  (psi) 1.507x107 1.5x106
E2 (psi) 2.08x107 40xl06
V1 0.1966 0.0075
V2  0.2714 0.2000
G 6.78x106 4.0x10612
Gav 7.15x106 3.73x106
Table 1 indicates that for the two materials under consid-
eration G1 2 and Gav are sufficiently close to justify the
assumption of special orthotropy. The calculated stress inten-
sity factor ratios based on this assumption are shown in Tables
2 and 3 and Figures 1-4. In the examples El is taken in the
axial direction. The tables also show the related (orthotropic)
shell parameter o0 calculated from
A0  = [12(1-v 1 2 )E2/E1 L]  a/(Rh) . (15)
Since Xo is dependent on the elastic constants, in comparing
the orthotropic results for different material orientation and
the results for the isotropic material, a/(hR) is used as the
independent variable.
-6-
Table 2. The Stress Intensity Factor Ratios
for a Pressurized Titanium Cylinder
(E1 /E 2 ) = 0.724
x a/(Rh) Am Ab
1 0.5025 1.1962 0.1453
2 1.0050 1.6053 0.2645
3 1.5075 2.0795 0.3091
4 2.0100 2.5638 0.2853
5 2.5126 3.0419 0.1983
6 3.0151 3.5094 0.0590
7 3.5176 3.9648 -0.1255
8 4.0201 4.4081 -0.3493
(E1/E 2 ) = 1.380
1 0.5903 1.1962 0.2010
2 1.1807 1.6053 0.3643
3 1.7711 2.0794 0.4265
4 2.3615 2.5638 0.3927
5 2.9518 3.0420 0.2734
6 3.5422 3.5094 0.0808
7 4.1326 3.9649 -0.1743
8 4.7229 4.4082 -0.4835
-7-
Table 3. The Stress Intensity Factor Ratios
for a Pressurized Graphite Cylinder
(E1/E 2 ) = 0.0375
x a/(Rh)2 Am Ab
1 0.2365 1.1928 0.0217
2 0.4730 1.5906 0.0464
3 0.7095 2.0493 0.0645
4 0.9460 2.5179 0.0726
5 1.1826 2.9818 0.0697
6 1.4191 3.4373 0.0564
7 1.6556 3.8833 0.0338
8 1.8922 4.3196 0.0027
(E1/E 2 ) = 26.66
1 1.2214 1.1928 0.5787
2 2.4428 1.5906 1.2393
3 3.6642 2.0493 1.7220
4 4.8856 2.5179 1.9368
5 6.1070 2.9818 1.8593
6 7.3284 3.4373 1.5049
7 8.5498 3.8833 0.9007
8 9.7712 4.3196 0.0726
-8-
For the purpose of comparison, Figures 1-4 also show the
results for an isotropic shell (with a Poisson's ratio of 1/3)
which are given in [5]. It is seen that in the specially ortho-
tropic shells the stress intensity factors are strongly dependent
on the modulus ratio E1/E2 , and Am and Ab generally increase with
decreasing E1/E2 . However, this does not necessarily mean a
reduction in the resistance of the shell to cleavage as E1/E 2
decreases (i.e., as the shell becomes stiffer in circumferential
direction). For this, one also has to consider the fracture
strength of the shell in the plane parallel to E1 as a function
of E1/E2 . Intuitively, it is expected that this strength too
would increase as E1/E2 decreases.
REFERENCES
1. S. A. Ambartsumyan, "Theory of Anisotropic Shells", NASA
Tech. Transl. F-118 (1964).
2. W. FlUgge and D. A. Conrad, "Singular Solutions in the
Theory of Shallow Shells", Tech. Report No. 101, Stanford
University (1956).
3. K. Apeland, "Analysis of Anisotropic Shallow Shells", Acta
Technica Scandinavica, No. 22 (1963).
4. U. YUceoglu and F. Erdogan, "A Cylindrical Shell with an
Axial Crack under Skew-Symmetric Loading", To appear in
Int. J. Solids Structures (1973).
5. F. Erdogan and J. J. Kibler, "Cylindrical and Spherical
Shells with Cracks", Int. J. Fracture Mechanics, Vol. 5,
pp. 229-237 (1969).
-9-
5A
m E/E= 0.724
4 1.0
1.381
3
h
2 Eo 7-2a1 R
E, x
1 Y
0 1 2 3 4 5
Figure 1. Membrane component of the stress intensity
factor ratio for a pressurized Titanium cylinder.
Ab
0Q4
a/RE
0 1 2 3 5
E/E = 11 2
0.724
- 0.4 1.381
-0.8
Figure 2. Bending component of the stress intensity
factor ratio for a pressurized Titanium cylinder.
5A
4
3
E /E= 2 6.66
1.0
0.0375
0 5 a //I 10o
Figure 3. Membrane component of the stress intensity
factor ratio for a pressurized Graphite cylinder.
2A bb
E/ 6.66
1.0
0.0375
0 ,
o 5 a //- 10
-1
Figure 4. Bending component of the stress intensityfactor ratio for a pressurized Graphite cylinder.


On the effect of orthotropy in a cracked cylindrical shell

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