Electronic Supplementary Information
Three-Dimensional Nonlinear Finite Element Analysis of Thick Imperfect Cross-Ply
(Very) Long Cylindrical Shells (Plane Strain Rings)
S1. The Method of Virtual Work and Fully Nonlinear Finite Element-Based
Discretized Incremental Equations of Motion
The second Piola-Kirchhoff stress tensor and the Green-Lagrange strain tensor are
conjugate quantities, because their properties are invariant under rigid body motions. The
equilibrium of a deformable body at time t+t can be expressed using the principle of
virtual displacements with tensor notation, using these two conjugate quantities in the
total Lagrangian formulation, as follows:

0
in which
t  t
t  t
0
S ij  t  0t ij 0 dV 
t  t
(S1)
R
V
R,
t  t
0
S ij and
t  t
0
 ij denote the external virtual work, Cartesian components
of the second Piola-Kirchhoff stress tensor and the total Green-Lagrange strain tensor,
respectively, defined at time t + ∆t, referred to the initial configuration. The latter two
quantities can further be expressed as follows:
t  t
0
S ij  0t S ij  0 S ij ,
(S2)
and

t  t
0 ij
where
t
0
S ij and
0
 0t  ij  0  ij ,
0
 ij  0 eij  0ij ,
(S3a,b)
S ij denote components of the second Piola-Kirchhoff stress tensor
defined at time t, and the incremental components of the same during the subsequent time
step t, respectively, both referred to the initial configuration. The quantities 0 eij and 0 ij
in Eq. (S3b) represent the linear and the nonlinear incremental strains, respectively, that
are referred to the initial configuration. The linear strain vector
two parts, namely the pure linear part,
 e  is here resolved into
0 ij
 e  and the linearized part,  e . The
L
0 ij
N
0 ij
incremental constitutive relation, which relates the components of incremental stress and
incremental strain both referred to the initial configuration, is given by
0
Sij  0 Cijrs 0  rs ,
in which
0
(S4)
Cijrs is the incremental elastic stiffness (material property) tensor, referred to
the initial configuration. Substitution of Eqs. (S2), (S3) and (S4) into the left hand side of
the Eq. (S1) finally yields the equations needed for the finite element formulation. The
details are available in Chaudhuri and Kim [36].
The fact that the variation in the strain components is equivalent to the virtual strains
permits the right hand side of Eq. (S1) to represent the virtual work done, when the body
is subjected to a virtual displacement at time t+∆t. The corresponding virtual work is
given by
t +t
R=

t  t
where the
 0u sj
t+t
t +t
f js  0u sj
t +t
(S5)
ds,
s
f js is the surface force vectors applied on the surface S at time t+∆t, and
is the ith component of the incremental virtual displacement vector evaluated on
the loaded surface. When the hydrostatic pressure is applied, the loading-path is always
deformation-dependent. This requires that the load vector should be evaluated at the
current configuration. The external virtual work can, however, be approximated to
sufficient accuracy using the intensity of loading corresponding to time t+t, integrated
t  t (i -1)
over the surface area,
s
calculated at the (k-1)th iteration (see Section 5 of
Chaudhuri and Kim [9]) as follows:
t+t
R=

t+t
f js  0u sj
t+t
(S6)
ds.
t  t (i -1)
s
S2. Isoparametric Finite Element Discretization and Iterative Scheme
On equating the left hand side of Eq. (S1) to the right hand side of Eq. (S6), followed
by discretization using curved 16-node isoparametric quadratic (in the surface-parallel
coordinates) layer-elements and satisfaction of boundary conditions (see Chaudhuri and
Kim [9, 36] for details), the incremental equations of motion can be described as follows:
K L 0V K N 0V  f L   f N ,
(S7)
in which K L  and K N  represent the linear global stiffness matrix, and nonlinear
contribution to the global geometric stiffness matrix, respectively, while fL  and f N 
denote the applied load vector and nonlinear internal force vector, respectively, with
0V being the global nodal displacement vector. These are as given below:
. K L  
T
   B T  Q  B T  
hk
NL NS
m 1 k 1 S ( m ) hk 1
(k )
LL
K N   2   B
m 1 k 1
S
NL NS
 
(m)
 
(k )
LL
(k )
BT
(k )
dz dS ,
(S8a)
(k )
LL
T  Q  B T  
(k )
BT
(k )
(k )
LN
(k )
BT
(k )
dz dS
hk 1
T
  B T  Q  B T  
hk
m 1 k 1 S ( m ) hk 1
NL NS
(k )
T
hk
NL NS
(k )
BT
(k )
LN
(k )
BT
(k )
  B T  
T
hk
m 1 k 1 S ( m ) hk 1
(k )
NN
(k )
BT
t
0
Sˆ ( k )
(k )
LN
(k )
BT
 B T  
(k )
NN
(k )
BT
(k )
(k )
dz dS
dz dS ,
(S8b)
NL
fL   

m 1 S (m)
BLL S
(N )
TBT S

n(NS 1) 

 Pr (N S 1) dS,
 0 

 

(N )
T
T
f N      BLL( k ) TBT( k )  0t S ( k )   ( k ) dz dS ,
m1 k 1
hk
NL NS
S
(m)
(S9a)
(S9b)
hk 1
and
0VT

{0 U b(11) ,..., 0 U b(18) , 0Vb(11) ,..., 0Vb(81) , 0Wb(11) ,..., 0Wb(81) , 0 U t(11) ,..., 0 U t(81) , 0Vt1(1) ,..., 0Vt 8(1) , 0Wt1(1) ,..., 0Wt 8(1) ,
0
U b(1k ) ,..., 0 U b(8k ) , 0Vb(1k ) ,..., 0Vb(8k ) , 0Wb(1k ) ,..., 0Wb(8k ) , 0 U t(1k ) ,..., 0 U t(8k ) , 0Vt1( k ) ,..., 0Vt 8( k ) , 0Wt1( k ) ,..., 0Wt 8( k ) ,
0
U b(1N ) ,..., 0 U b(8N ) , 0Vb(1N ) ,..., 0Vb(8N ) , 0Wb(1N ) ,..., 0Wb(8N ) , 0 U t(1N ) ,..., 0 U t(8N ) , 0Vt1( N ) ,..., 0Vt 8( N ) , 0Wt1( N ) ,...., 0Wt 8( N ) }.
(S10)
Here NL and NS denote the number of elements for each layer and number of layers,
respectively. It may be noted that the total number of elements, N equals NLNS.
The differential operators,
matrix
B , B  and B , are as presented below. The
(k )
LL
(k )
LN
(k )
NN
B  referred to in Eqs. (S8a,b) and (B3a, b) is given as shown below:
(k )
LL












(k )
B

 LL  











while the matrix
B 
(k)
LN

x
0
0
0
1 
)
g(k
 
1
)
g(k

0
0

z
0

1
 (k )
 z g
1 
(k )
g 

z
0

x
1 
)
g(k
 

x
0












,












B  referred to in Eqs. (S8b) is given as follows:
(k )
LN
(S11)






















R11( k )

x
R12( k )
(k )
R21

g ( k ) 
(k )
R31
1
g ( k )
(k )
R32
(k )
R31
(k )
R22


 R11( k )
x
z
(k )
R22
1
g ( k )

x










,











 (k ) 
(k ) 
 R23

 R22




z
(k )
R33
 R32( k )  R33( k )


z g ( k )  g ( k )
(k )
R32
R11( k ) 
(k ) 
 R21
(k )
x
g  
R13( k )
 (k ) 
(k ) 
 R22

 R23




z
(k )
R31

(k ) 
 R21
(k )
z
g  

x

z
(k )
(k )
R32
R33

(k ) 


R
23
z
g ( k ) g ( k ) 


 R12( k )
x
z
(k )
R33
 R12( k )  R13( k )


x g ( k )  g ( k )
(k )
R23


 R13( k )
x
z
R (k )
 R13( k ) 
 (k )
 12( k )
x g   g 
(S12)
in which
R  B v ,
(k )
ij
(k )
NN
(k )
(S13)
with
R 
(k )
ij
R
(k )
11
(k )
R12
(k )
R13
(k )
(k )
R21
R22
(k )
R23
(k )
R31
(k)
R32
R33  ,
(k ) T
(S14)
v   u
(k )
t
0
(k )
t (k )
0
v
t
0
,
(k ) T
w
where the components of the vector v
 are known displacements at time t. The matrix,
(k )
B , referred to in Eqs. (S8b) and (S13), is given as follows:
(k )
NN
B 
(k )
NN
(S15)


x



0



0


0
0
1 
)
g(k
 

x
0
0
1 
)
g(k
 
0

x
0
1
)
g(k

0

0

z
0
1
)
g(k

0

z
0
0
1 
)
g(k
 
T
0 




0  .


 

z

(S16)
The layerwise linear distribution of displacement matrix, [TBT], referred to in Eqs. (S8)
and (S9) can be written as follows [36]





(k )
TBT ( z )  






1
z
hk
0
1
0
0
z
hk
0
0
z
hk
0
0
0
0
z
hk
0
0
0
z
hk
1
z
hk





.




(S17)
The quadratic global interpolation function matrix, [], referred to in Eqs. (S8) and (S9)
is given by
ψ
 0

 0
(r , s)  
 0
 0

 0
wherein
0
ψ
0
0
0
0
0
0
ψ
0
0
0
0
0
0
ψ
0
0
0
0
0
0
ψ
0
0
0
0
,
0
0

ψ
(S18)
{} = { 1 2 3 4 5 6 7 8}
(S19)
and {0} is 1x8 null matrix. k(r,s) , k = 1,...,8, are the shape functions as used for
displacements and coordinates.
t
t
Finally, the stress matrix, [ o Sij ] , and stress vector, { oSij } , referred to in Eqs.
(B2b) and (B3b), respectively, are given as follows:
[
t
o Sij
  0 0 
 S  0 ,
0  S 
 0t S *
] =  0

 0

t
0
*
t
0
(S20)
*
with
   S   S 
     S   S ,
   S   S 
 0t S11

t *
  0t S12
0S
 0t S13

t
0 12
t
0 22
t
0 23
t
0 13
t
0 23
t
0 33
(S21)
and
 S   S
t
0
T
ij
t
0 11
t
0
S22
t
0
S33
t
0
S23
t
0 13
S
t
0 12
S

(S22)
The Newton-Raphson iteration scheme in conjunction with Aitken acceleration is
used to obtain the limit point (hydrostatic) pressure. Beyond this load, the post-buckling
behavior is computed by an incremental displacement control scheme rather than the
usual incremental force control scheme, when a limit point appears on the equilibrium
path (see Sec. 5 of Chaudhuri and Kim [36] and also Appendix-C of Chaudhuri and Kim
[9]). The details pertaining to these iterative schemes for solution to nonlinear finite
element discretized equations under consideration are available elsewhere [9, 36], and
will not be repeated here in the interest of brevity.
S3. Effect of thickness induced transverse shear/normal deformation on localization
and delocalization
Figure S1 shows plots of the normalized pressure, p* = p/pcr,Donnell, versus the
normalized transverse displacement (deflection), w* 
w
at  = 0o, of imperfect
Ri  w0
[90/0/90] (plane strain) rings, with w** = w0/Ri = 0.005, for different Ri/h. Here, p and
pcr represent, respectively, uniform hydrostatic pressure and classical buckling pressure of
a long cylindrical shell. pcr,Donnell, is available in Table 1 of Chaudhuri and Kim [41]
These curves are analogous to isotherms of the phase transition literature (see e.g., Parthia
[S1]).
Chaudhuri and Kim [36] have earlier studied the effect of thickness shear on
localization; however, no occurrence of delocalization in the large displacement/strain
regime in the thickness range of 60  Ri/h  6 has been observed. Consequently, the
effect of thickness-shear on the appearance of delocalization is yet to be addressed in the
literature, which is the primary focus here. The appearance of a limit (localization) point
on the equilibrium path is a measure of the degree of the effect of interlaminar
shear/normal (primarily shear) deformation. This limit point is known as a saddle-node or
backward tangent bifurcation [S2], which represents the localization (onset of
deformation softening) hydrostatic pressure of an imperfect moderately thick to thick
cross-ply long cylindrical shell (plane strain ring). A saddle-node bifurcation is created
when stable (extended or periodic mode) and unstable (localized mode) orbits coalesce
and obliterate each other [S2]. Results presented in Figure S1 suggest that the thicknessshear is primarily responsible for the appearance of a limit (maximum or localization
pressure) point on the post-buckling equilibrium path, generally associated with a
periodic (modal or harmonic) buckling pattern for which a modal imperfection serves as a
perturbation. Localization of the buckling pattern results from "bifurcation" at or near this
limit point, and can be viewed as a symmetry breaking phenomenon.
S4. Effect of average (uniformly distributed) fiber misalignments on localization and
delocalization
Chaudhuri [37] has earlier investigated the effect of reduced transverse shear
*
modulus, GLT
, of a lamina weakened by the presence of uniformly distributed fiber
*
misalignments. A simple expression for the reduced transverse shear modulus, GLT
, of a
layer material is derived in terms of the average fiber misalignment angle, which is given
as follows [37]:
*
G LT
 G 

f
f
 2 
G LT .
(S23)
Figure 8 of Chaudhuri [37] displays the variations of the normalized localization
pressure, p* = p/pcr,Donnell, and normalized transverse displacement (deflection),
w* 
w
at  = 0o, of an imperfect (w** = w0/Ri = 0.005) [90/0/90] thick (Ri/h = 6)
Ri  w0
(very) long cylindrical shell (plane strain ring), with respect to normalized average
(uniformly distributed) fiber misalignment angle, f (For HS carbon fibers, this
corresponds to  ≈ 2o; see Chaudhuri [8]); however, no occurrence of delocalization in
the large displacement/strain regime has been reported. Consequently, the effect of
average (uniformly distributed) fiber misalignment on the appearance of delocalization is
yet to be addressed in the literature, which is the primary focus here. Figure S2 shows
plots of the normalized pressure, p*, versus the normalized transverse displacement
(deflection), w* 
w
at  = 0o, of an imperfect (w** = w0/Ri = 0.005) thick (Ri/h =
Ri  w0
6) [90/0/90] (very) long cylindrical shell, for f = 0 and 1.5.
Supplementary References
S1. Pathria, R. K. 1977 Statistical Mechanics. Pergamon, Oxford.
S2. Ott, E. 1993 Chaos in Dynamical Systems. Cambridge Univ. Press, Cambridge, UK.
Figure Legends
Figure S1. Computed equilibrium paths of a [90/0/90] plane strain ring (w** = 0.005) for
different radius-to-thickness ratios
Figure S2. Sensitivity of the post-buckling behavior of a thick [90/0/90] plane strain ring
(Ri/h = 6.0, w** = 0.005) to average fiber misalignment, /f
Fig. S1
Fig. S2