Thin–Walled Structures 199 (2024) 111858
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Thin-Walled Structures
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Full length article
A three-dimensional progressive failure analysis of filament-wound
composite pressure vessels with void defects
Lei Ge a, b, Jikang Zhao a, Hefeng Li a, Jingxuan Dong a, Hongbo Geng c, Lei Zu d, Song Lin e,
Xiaolong Jia a, b, *, Xiaoping Yang a, b
a
State Key Laboratory of Organic-Inorganic Composites, College of Materials Science and Engineering, Beijing University of Chemical Technology, Beijing, 100029, China
Key Laboratory of Carbon Fiber and Functional Polymer, Ministry of Education, Beijing University of Chemical Technology, Beijing, 100029, China
Inner Mongolia Aerospace Hong Gang Machinery Corporation Limited, Inner Mongolia, 010076, China
d
School of Mechanical Engineering, Hefei University of Technology, Hefei, 230000, China
e
Department of Material Engineering, North China Institute of Aerospace Engineering, Hebei, 065000, China
b
c
A R T I C L E I N F O
A B S T R A C T
Keywords:
Composite pressure vessel
Void defects
Damage behavior
Burst pressure
Composite pressure vessels are promising solution for storing compressed gas with the high strength/stiffness to
weight ratio, but the accurate modeling of mechanical behavior under high pressure remains a huge challenge
considering manufacturing-induced void defects. The void generating method is developed to construct the finite
element model with voids explicitly constructed. A three-dimensional(3D) progressive damage model complied
with the user-defined material subroutine (UMAT) is performed to predict the void defect introduced damage
behavior and burst pressure of composite pressure vessels. It indicates that the predicted burst strength agrees
well with experimental value, and with the pressure increasing, the damage of composite layers initially appears
in the matrix of middle cylinder section and region near the equator. The matrix damage is dominant at the
outset, then achieves a balance with the fiber damage, and finally comes in second until the burst failure of the
composite pressure vessels. The void defects will result in the damage initiation, evolution and final failure of
composite layers, and the clustered voids will deteriorate the local damage status, but will not change the general
damage trend. The present damage failure analysis scheme can provide theoretical guidance for reliability
evaluation of composite pressure vessels with void defects.
1. Introduction
Hydrogen energy is regarded as the most promising new energy with
the advantage of zero release. A high focus has been placed on the
hydrogen storage technology with the development of hydrogen energy.
Now, the use of filament-wound thin-walled composite pressure vessel
to store hydrogen represents the most mature technology, and has been
applied in various fields of aerospace, aviation and automotive, etc. [1].
The internal pressure generates high stresses, which lead to the massive
use of composite materials and therefore the high cost. The design
optimization performed by an appreciate numerical method can
improve the structural efficiency of composite pressure vessels and
achieve balance of economic efficiency and performance. The composite
pressure vessels are intertwined by yarns with different winding angles,
and then thickened by the composite lay-up. The manufacturing process
results in high heterogeneous characteristics of wound composites, and
the reliability evaluation under high pressure remains a huge challenge
due to their structural complexity. Furthermore, void defects inevitably
produce in the manufacturing process, and highly influence the strength
and damage mechanism of composites [2,3], adding another layer of
difficulty in accurately capture the mechanical behavior of composite
pressure vessels.
Current numerical simulations of mechanical behavior are related to
different structural and damage models for composite pressure vessels.
The composite pressure vessel commonly consists of the cylindrical
section and dome zone of which the thickness and winding angle
constantly change due to the radius reduction. Consequently, variations
of winding angle and material properties pose threats to the finite
element modeling of pressure vessels [4,5]. A commercial plugin for
ABAQUS, known as Wound Composite Modeler (WCM), is regarded as
* Corresponding author at: State Key Laboratory of Organic-Inorganic Composites, College of Materials Science and Engineering, Beijing University of Chemical
Technology, Beijing, 100029, China.
E-mail address: jiaxl@mail.buct.edu.cn (X. Jia).
https://doi.org/10.1016/j.tws.2024.111858
Received 21 January 2024; Received in revised form 16 March 2024; Accepted 2 April 2024
Available online 3 April 2024
0263-8231/© 2024 Elsevier Ltd. All rights reserved.
L. Ge et al.
Thin-Walled Structures 199 (2024) 111858
an efficient way to the design and mechanical analyses of composite
vessels. Due to the rotating symmetry and thin-walled structural fea­
tures, the axisymmetric [6–10] and shell [11–16] elements are suitable
for modeling the composite layers, and the three-dimensional (3D) solid
element are commonly selected for the liner of vessels [6,15]. Shell el­
ements show advantages over the computational efficiency whereas
solid elements [16–22] are more appropriate for the accurate calcula­
tion and capturing the damage behavior across composite layers in spite
of their more complex modeling procedures. The efficiency drawbacks
of solid elements can be well relieved by reducing simulation structural
models (i.e. 1/4 [6],1/12 [18] and 1/72 [17] of the whole vessel)
coupling with periodical boundary conditions or symmetric constraints
on the relative deformation of opposite sides. Another focus is the
damage model which is equally critical to describe the damage failure
mechanism of layers of winding composites. Numerous damage failure
models for composites are built to simulate the constituent failures such
as the fiber rupture, matrix cracking and in-plane shear damage,
including the frequently-used progressive damage model with Hashin
criterion [7,8,13], Puck failure criteria with sudden material degrada­
tion model [11], ABAQUS built-in failure model [12,15],
Hashin–Rottem failure criterion based on progressive damage model
[16], and the linking of principle stress with failure criterion of
Maximum stress criterion, Hoffman, Tsai–Hill and Tsai–Wu criteria [18,
19]. Besides, some special damage models are also developed such as a
specific continuum damage mechanics modeled by the concept of fixed
damage directions [6,20], intra-laminar damage law model [9,10],
discrete ply model [14], maximum stress criterion with
temperature-dependent material strengths [17], and so on. Based on
above finite element models and damage models, the mechanical
behavior of filament-wound composite pressure vessels has been fully
investigated including the burst pressure [6–8,11,13,15,16,18], optimal
design [9,12], thermal effect on burst behavior [10], non-linear buck­
ling analysis [14], safety evaluation due to temperature rise [17], the
structural integrity under low-velocity car-to-car collision [19], tensile
or compression damage behavior of wound composite laminates [20]
and the design optimization by suitable fibers [21]. On the whole, most
of above literatures are based on the simplified 2D shell model or
axisymmetric model without considering out-plane mechanical prop­
erties of composites, which cannot effectively capture damage charac­
teristics in the lay-up direction and therefore lead to the poor predicting
accuracy [22]. Additionally, finite element analysis method can also be
achieved by coupling the ANSYS Mechanical with the ANSYS Composite
PrepPost (ACP) module to predict mechanical behavior of composite
pressure vessels under the internal pressure [23–25]. On the whole, the
more accurate 3D solid element model coupled with progressive damage
method has been rarely adopted to simulate the damage and failure
behavior of wound composite vessels under internal pressure.
In the winding process, the fiber bands are wound on the mandrel,
and there are still a lot of impregnant residing or process induced
unsoaked region. The manufacturing defects will be inevitably gener­
ated during the curing step, which has been attracting special attention
due to the detrimental threat to the strength and damage of composites
[26]. Some literatures have been conducted to figure out the defect
existence [27–29], defect grown [30] and defect effects [31–38] on
properties of wound composite vessels. Lin et al. [27] found
micro-cracks and defects by SEM images of the dome section under
hydraulic and atmospheric fatigue cycling tests of filament wound CFRP
vessels. Behera et al. [28] discussed the insight of various defects of
filament wound composite pressure vessel arising from the parameter
variation during the manufacturing process through the transmission
ultrasonic technique. Guillon et al. [29] carried out a global tomography
at the end of the tank prototyping, and examined the existence of
numerous small size voids near the nozzle of thermoplastic composite
overwrap pressure vessel. Wang et al. [30] developed a comprehensive
prediction model to figure out the relationships between the final void
size and time-dependent manufacturing characteristics in the filament
winding process. The effects of void defects on the mechanical behavior
of composite pressure vessels can be achieved by the experimental
[31–33], analytical [34,35] and the most promising numerical [36–38]
methods. Özaslan et al. [31] conducted a comprehensive experimental
study to understand how manufacturing defects affect bursting behavior
of pressure vessels with various stress ratios. Also, Arikan [32] con­
ducted a static internal pressure test to reveal the failure mechanism
with an inclined surface crack in the filament wound pipes.
Blanc-Vannet [33] confirmed the reduction of burst pressure due to the
impact damage to composite cylinders by impact tests. Farukoglu and
Korkut [34] introduced the void-inducted errors by an analytical
method to investigate effects of the void-introduced defects on stresses
and pressure of tubes. Kang et al. [35] proposed the analytical model of
composite riser with void defect based on the Flügge model and Galerkin
method, and the strong impact of void defects on the buckling resistance
has been fully discussed. Moskvichev et al. [36] assessed the fracture of
composite overwrapped pressure vessel with a constructed
semi-elliptical crack defect on the internal surface of liner. Ellul and
Camilleri [37] investigated the failure progression and response of
filament wound pressure vessels, which was highly dependent on the
developed two types of manufacturing defects. Wen et al. [38] intro­
duced random wrinkle defects by elements to study the mechanical re­
sponses, and a strong correlation existed between the defects and
circumferential displacement distortion of composite overwrapped
pressure vessels. To sum up, the existence of manufacturing defects has
been studied and verified, but how the defects influence the damage and
failure of composite pressure vessels has not been revealed.
In this paper, a 3D progressive damage method coupled with the 3D
finite element model is adopted to investigate the damage behavior
across layers of composite pressure vessels with void defects. A void
generation method is developed in order to introduce void defects in the
3D solid model, and therefore reveal the damage evolution and failure
mechanism of wound composite pressure vessels considering void de­
fects. The influences of void position are discussed by explicitly con­
structing three different void-models with real void data from
experiments.
2. 3D model and material system of composite pressure vessel
2.1. The geometry and finite element model
To compare with the experimental data of the literature [7], the
finite element model of the composite pressure vessel will be established
with the cylinder section and elliptical domes according to following
specific geometry sizes, as shown in Fig. 1: the diameter and length of
the cylinder section are 380 mm and 540 mm, respectively, and the total
length is 800 mm. The liner mainly plays the role of sealing hydrogen
gas of wound pressure vessel. The composite layers are the main
load-bearing structure, and ensure the service safety of the pressure
vessel. The 3D solid model is advantageous over other shell or axisym­
metric models, but will cost more computing time. Due to the circum­
ferential symmetry of pressure vessels, to improve the computing
efficiency, the 1/36 periodical symmetric model is adopted and con­
structed layer by layer, as shown in Fig. 1.
Based on the geometry characteristics, the solid composite layers are
modeled by the WCM plug-in of ABAQUS software 2017. Each layer is
automatically generated from the new constructed surface with the layer
winding angle and layer thickness. The winding angle along the cylin­
drical section remains constant, but the helical winding angle over the
dome is described using the following Eq. (1) [39]. With regards to the
dome structure, the winding angle of fiber band varies in a layer from
the initial value in the cylindrical section to 90◦ (i.e. perpendicular to the
vessel axis) at the layer end point. Then, the fiber band turns around the
pole and goes back to the cylindrical section. This winding process of
layers repeats until the whole vessel is completed. The netting theory is
well-suited and applied commonly to calculate the variation of winding
2
L. Ge et al.
Thin-Walled Structures 199 (2024) 111858
Fig. 1. 3D structural model of composite pressure vessel.
angle and layer thickness.
(r )
0
θ(r) = sin− 1
r
cylinder section; BW is the helical band width.
The filament winding can be divided into two types including the
hoop winding (i.e. fibers wound in the hoop style can provide the
circumferential strength) and helical winding (i.e. fibers wound in the
helical style with a defined angle can resist both axial and circumfer­
ential loads). The winding initial angles and composite layers of the
literature [7] are adopted to model wound composite pressure vessels so
as to compare the finite element solution with the experimental one, as
shown in Fig. 3a. The composite pressure vessel consists of three parts
including the boss, liner and composite layers. The different parts of the
3D solid model are assembled together by assuming a perfect junction,
and the tie connection is imposed on the contact surfaces of the liner,
composite layer and boss to avoid the possible relative slippage. The
boss, liner and composites layers are all meshed with 3D hexahedron
elements (C3D8R). To facilitate the application of periodical boundary
conditions, the symmetric nodes should be generated in opposite sides,
and then periodical elements can be meshed, as shown in Fig. 3b.
(1)
where, r denotes the radial distance from the center axis to the current
point in the layer; r0 is the radial distance from the center axis to the
turnaround point.
For the helical layers in the dome section, the layer thickness in­
creases gradually and then sharply near the polar openings, as described
in Eq. (2) [39]. And, the calculation diagram of geometrical variables of
the angle and thickness is depicted in Fig. 2.
t(r) =
ttl ⋅cos(θtl )
rtl
⋅
(
)4
cos(θr )
r + 2⋅BW⋅ rrtltl−− rr0
(2)
where, ttl and θtl are the thickness and winding angle in the tangent line,
respectively; rtl stands for the radius in the tangent line of the dome or
2.2. The finite element model with void defects
To introduce void defects in the 3D finite element model, the
following method is developed: Firstly, the elements of composite
pressure vessel are meshed completely, as depicted in Fig. 3b. Secondly,
a Python script is written to randomly choose elements in the specified
region by ABAQUS software 2017. The following steps are required:
first, define the specified region using the element set. Next, implement
the strategy using Python’s random.sample function from the random
module to obtain random non-repeating elements from a sequence.
Finally, determine the random positions of the voids using the new void
set. The chosen elements are marked in brown, and should satisfy the
given void volume fraction of about 5 % in this literature [7], as depicted
in Fig. 4. The void defects with the same void volume fraction are
established in different positions, which are labeled as the MP-1(i.e.
random voids, Fig. 4a), MP-2 (i.e. clustered voids in the middle,
Fig. 4b) and MP-3 (i.e. clustered voids around the side, Fig. 4c). Finally,
the material properties of chosen elements are weakened and assigned to
simulate void defects and guarantee the satisfactory convergence. Spe­
cifically, based on the chosen void element set by the Python script, their
mechanical parameters are highly reduced by 99.99 % in the ABAQUS
software according to some other works [40,41]. In this case, these pore
defects will not bear the loading in the simulation steps.
Fig. 2. The calculation diagram of geometrical variables of the angle
and thickness.
3
L. Ge et al.
Thin-Walled Structures 199 (2024) 111858
Fig. 3. Finite element model of composite pressure vessels [7]: winding layout (a) and element meshing (b).
Fig. 4. Wound composites with voids in different positions [7]: MP-1 (a), MP-2(b) and MP-3 (c), location distribution of void(d), and finite element models(e).
2.3. The property of constitutes
and stress–strain curves are shown in Fig. 5. The reinforced composite
layers are wound by carbon fiber-epoxy bands, and the material model
of composite bands is regarded as anisotropic. Assuming the composite
bands are free from void defects, and the mechanical properties of the
comparative literature are adopted, as listed in Table 2.
The metal boss and liner of pressure vessel are made of the aluminum
alloy and nylon, respectively, and both materials are assumed to be
isotropic. The corresponding material parameters are shown in Table 1,
3. Progressive damage analyses of composite pressure vessel
Table 1
Properties of constituents: aluminum alloy 6061-T6 and nylon PA6 [7].
Constitutes
E (GPa)
Υ
σs (MPa)
σb (MPa)
δ
ρ (g.cm− 3)
6061-T6
PA6
74.12
1.88
0.28
0.4
280
47
368
55
12 %
–
–
1.06
3.1. Boundary conditions
The composite pressure vessels show periodical characteristics in the
circumferential direction, and therefore, periodical boundary conditions
4
L. Ge et al.
Thin-Walled Structures 199 (2024) 111858
convenient to be applied in the finite element method.
uleft − uright =
νleft − νright =
π Rj
γ
18 tr
(8)
π Rj
(9)
18
εt
wleft − wright = 0
(10)
where, Rj is the radius of the nodes on the periodic surfaces perpen­
dicular to the circumference; μ, υ, ω are the radial, circumferential and
longitudinal displacements, respectively.
3.2. Progressive damage model
The wound composites exhibit the inhomogeneity, and the failure of
heterogeneous composites occurs when the damage accumulates to a
certain extent. The composite layers of wound pressure vessel can be
regarded as transversely isotropic, and a 3D Hashin’s criterion is
adopted to describe damage initiation of composite layers in different
failure modes, such as longitudinal breakage (1-direcion), transverse
cracking (2, 3-direction) and shear failure (12, 13, 23-direction). The
damage criterions of different failure modes are listed as follows:
Tensile failure in 1-direcion (σ1 ≥ 0):
( )2
( )2
( )2
σ1
σ12
σ 31
f1t =
+
α
+
α
≥1
(11)
X1t
S12
S31
Fig. 5. The stress–strain curves of aluminum alloy and nylon PA6 [7].
Table 2
Mechanical properties of composite bands [7].
E1(GPa)
E2/
E3(GPa)
G12/
G31(GPa)
G23(GPa)
ν12/ν31
ν23
160
Xt (MPa)
7.42
Xc (MPa)
3.71
Yt/Zt (MPa)
1250
74
0.28
τ12/τ13
(MPa)
50
0.3
2500
4.79
Yc/Zc
(MPa)
180
Compressive failure in 1-direcion (σ 1 < 0):
( )2
σ1
f1c =
≥1
X1c
τ23
(MPa)
90
Tensile and shear failure in 2 and 3-direction (σ2 + σ3 ≥ 0):
( 2
) ( )2 ( )2
(
)
σ23 − σ 2 σ3
σ2 + σ3 2
σ 12
σ 31
t
f2(3)
=
+
+
+
≥1
X2t
S12
S31
(S23 )2
(PBCs) are necessary to use the 1/36 model for numerical simulations.
The opposite faces in the circumferential direction should satisfy the
displacement and force continuities [42,43] in the following equation:
ui = εik xk + ui ∗
(3)
(4)
uj−i
(5)
=ε
j−
ik xk
+ u∗i
( j+
)
j−
j−
= εik Δxjk
uj+
i − ui = εik xk − xk
≥1
(14)
When the material damage initiates, the composite material will
gradually fail with the further increasing load. A damage evolvement
model is necessary to describe the degeneration of the material prop­
erties. Therefore, a progressive damage model is adopted to describe the
stiffness degradation of composite components [44]. The damage law of
each damage mode is defined with the following equivalent
displacements:
(
)
I
I0
δI1
eq δeq − δeq
); I ∈ {1t, 1c, 2t, 2c, 3t, 3c, mt, mc}
dI = (
(15)
I0
δIeq δI1
eq − δeq
(6)
where, the superscript “j+” and “j− ” represent the opposite and negative
directions, respectively; xk is the node coordinate, and the difference
j
Δxk is constant for paired boundaries. The right side of Eq. (6) is
invariant with a specified εik .
A cylindrical-coordinate system is defined and PBCs in the circum­
ferential direction can be expressed as follows:
i
i
Uleft
− Uright
= εh
(13)
Compressive and shear failure in 2 and 3-direction (σ2 + σ 3 < 0):
[(
]
( 2
) ( )2
)2
)
(
σ − σ2 σ3
1
X2c
σ2 + σ3 2
σ12
c
f2(3)
= c
− 1 (σ2 + σ3 ) +
+ 23
+
X2
2S23
2S23
S12
(S23 )2
( )2
σ 31
+
S31
where, ui represents the displacement field; εik is the average strain
tensor; ui* is the periodic part of displacement components on boundary
surfaces.
Due to the unknown ui ∗ in Eq. (3), it is hard to be directly imposed on
symmetry boundaries. A transformed form is given in Eq. (4)–(6):
j+
∗
uj+
i = εik xk + ui
(12)
(7)
I1
Here, δI0
eq and δeq denote equivalent displacements of the initiation
and full damage of each damage mode I, respectively; δIeq is the equiva­
lent displacement of each mode I, as described in the literature [44].
The equivalent displacements of the initiation and full damage are
described by Eqs. (16)–(18):
where, Uileft and Uiright represent the displacements of corresponding
nodes on a pair of periodic surfaces in circumferential directions,
respectively.
Under the six strain loadings (i.e. εr, εt, εz, γzr, γ zt, γ tr) on the opposite
surfaces, the periodical boundary condition of the opposite surfaces
perpendicular to the circumference can be realized by the following Eqs.
(8)–(10). Then, the nodal displacement constraint equations are
δIeq
δI0
eq = √̅̅̅
fI
5
(16)
L. Ge et al.
δI1
eq =
Thin-Walled Structures 199 (2024) 111858
2GI
(17)
σ I0
eq
σ Ieq
σ I0
eq = √̅̅̅
v
ḊI =
here, fI is the damage initiation value of each damage mode I; GI is the
fraction energy density; σI0
eq denotes the equivalent initiation stress of
each damage mode I.
The voids are embedded in the composite layers of the finite element
model, and the chosen elements simulating void defects own the same
damage constitutive model as the composites. Differently, the material
properties of chosen elements are greatly weakened to simulate void
defects. The damage of constitutes can be described by Murakami–Ohno
damage model [45]. When the material damage initiates, the stiffness
matrix will be updated accordingly with the changing damage degree.
The material constitutive relation is described with the damage vari­
ables, as shown in Eq. (19):
bIJ =
4b2I b2J
(bI + bJ )2
, IJ ∈ {12, 23, 31}
(23)
According to the periodical meshing on paired surfaces, the PBCs can
be imposed by a Python script in ABAQUS software 2017, as shown in
Fig. 6. The proposed progressive damage model will be achieved by
writing the user subroutine “UMAT” with the Fortran Code and imple­
mented in the ABAQUS Software 2017. On the basis of the developed 3D
finite element model and material system in Section 2, the established
boundary conditions and damage model are applied to conduct nu­
merical analyses on the damage and failure behavior of composite
pressure vessels, and the flowchart of finite element method with UMAT
subroutine is shown in Fig. 7. The internal pressure loading is gradually
added to the inside of the liner until the final burst failure of the com­
posite pressure vessel.
here, the damage stiffness matrix C(D) is expressed by the following
equation:
⎧
⎫
⎪
⎪
⎪
⎪
⎪ b2 C
⎪
⎪
⎪
b
b
C
b
b
C
0
0
0
1 2 12
1 3 13
⎪
⎪
1 11
⎪
⎪
⎪
⎪
2
⎪
⎪
⎪
⎪
b
C
b
b
C
0
0
0
2 3 23
⎪
⎪
2 22
⎪
⎪
⎪
⎪
⎨
b23 C33
0
0
0 ⎬
C(D) =
(20)
⎪
sym
b12 C44
0
0 ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
b31 C55
0 ⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
⎪
b
⎪
23 C66 ⎪
⎩
⎭
bI = 1 − DI , I ∈ {1, 2, 3}
)
DI − DvI , I ∈ {1, 2, 3}
3.3. Numerical implementation
(19)
σ = C(D)ε
η
where, η is the viscosity coefficient representing the relaxation time; DvI
andDvm represent the regularized damage variables of mode I for com­
posite layers.
(18)
fI
1(
4. Results and discussion
4.1. The stress analysis of composite layers of wound pressure vessels
The on-axis stresses in the fiber longitudinal direction (σ11) and
transverse direction (σ 22) of pressure vessels are plotted under the in­
ternal pressure before the damage initiation. The stress values of first ten
composite layers (i.e. Layer 1–10) are extracted as a representative
including hoop and helix layers with the winding angle ranging from
◦
◦
20 to 89.5 . As shown in Fig. 8, in general, the stresses σ 11 of helical
layers are obviously higher than the hoop layers. In addition, for the
helical layers in the cylinder section, the higher winding angle means
the higher ultimate stresses. However, in the dome section, the opposite
trend has been found. From the dome section to the cylinder section,
there is a sharp stress fluctuation in the transition region, which may
cause a stress concentration. The stress variation of hoop layers is more
stable than helical layers in the ‘W’ shape, and only a small fluctuation is
observed. The stresses σ 11 of the random voids are about zero, but the
existences of the void defects may lead to the stress fluctuation in the
fiber-direction, as shown in Fig. 8. The voids show more effects on the
stress σ11 in the hoop layers than the helical layers.
The results of stress σ22 are shown in Fig. 9. On the whole, similar as
that in σ11, there is a sharp fluctuation in the helical layers, which may
(21)
(22)
where, Cij (i, j = 1, 2, 3) represents the component of the undamaged
stiffness tensor.
Severe convergence deficiencies may come with the material soft­
ening behavior and stiffness degradation in the implicit analysis pro­
grams. Duvaut and Lions regularization model is introduced to alleviate
the deficiencies in Eq. (23) [46]:
Fig. 6. Boundary conditions of the finite element model.
6
L. Ge et al.
Thin-Walled Structures 199 (2024) 111858
Fig. 7. The flowchart of finite element method with UMAT subroutine.
cause a stress concentration in the transition region. The stress decreases
with the increasing of winding angles of helical layers in the cylinder
section. However, the stresses σ22 of hoop and helical layers are almost
the same and vary with the distance along the axis direction in the ‘C’
shape. The hoop and helical layers in the cylindrical region are all
critical factors in tolerating the internal pressure of the vessel. Similar as
that of σ 11, the stresses σ 22 of the random voids are about zero, which has
verified the effectiveness of the simulation method of void defects in
Section 2.2. Also, the void defects have brought about the stress fluc­
tuation of stress σ 22 in the helical and hoop layers.
4.2. The defect induced damage behavior of composite pressure vessels
Based on the finite element models coupling with the damage model,
the progressive damage analysis is conducted to predict the pore defect
induced progressive damage behavior and burst pressure by the UMAT
subroutine in ABAQUS software 2017. The burst pressure of composite
pressure vessel is predicted to be 133 MPa, with an error of about 3 %
between the predicted value and experimental one in the literature [7].
Besides, compared to the numerical result in this cited literature, the
prediction accuracy has been highly improved due to the 3D finite
element model with void defects and the progressive damage model. As
can be seen, the present 3D model has been demonstrated the reliability
to predict the burst strength of composite pressure vessels. The damage
results of the composite layers are shown in Fig. 10. The initial damage
of composite pressure vessel is the matrix damage which appears in the
middle of the cylinder section and the region near the equator under the
pressure of 44 MPa. With the internal pressure increasing to 54 MPa, the
matrix damage of the cylinder and dome region gradually evolves to a
greater degree, and expands to each other. Meanwhile, under the pres­
sure of 54 MPa, the fiber damage emerges in the dome and cylinder
section. When the internal pressure increases to 95 MPa, the matrix
damage further deteriorates and has almost reached the full damage
state compared to the damage state under the pressure of 120 MPa.
Numerous fibers fractures under the pressure of 95 MPa, and almost all
the fibers fractures under the pressure of 120 MPa. When the fiber
failure of composite layers further deteriorates to the critical failure
extent, the wound pressure vessel loses the load-bearing ability. It
Fig. 8. The stress in the fiber longitudinal direction of helical(a) and hoop
(b) layers.
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Thin-Walled Structures 199 (2024) 111858
Fig. 9. The stresses in the fiber transverse direction of helical(a) and hoop(b) layers.
Fig. 10. Damage evolution of composite pressure vessels under pressure loading.
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Thin-Walled Structures 199 (2024) 111858
concludes that the burst of composite pressure is mostly determined by
the fiber failure, even if the matrix failure produces earlier. Therefore,
the fiber damage is the dominating regulation factor on the mechanical
strength of composite layers, and the matrix damage shows lesser ef­
fects. The composite pressure vessels finally burst under the internal
pressure of 130 MPa, which agrees well the experimental data in the
literature [7]. To sum up, before the 54 MPa internal pressure, the
matrix damage is the main damage mode. In the pressure scope of 54
MPa to 95 MPa, the matrix damage and fiber damage play the almost
same important role. After the pressure reaches 95 MPa, the fiber
damage has dominated the damage process until the burst failure of
composite pressure vessels.
To deeply figure out influences of the local void defect and layer type
on the damage behavior of composite pressure vessels, the damage
process under the internal pressure is investigated detailedly including
the dome and cylinder section, as shown in Figs. 11 and 12. On the
whole, the cylinder section shows more higher matrix damage range and
damage rate than those of the dome section. And, more fiber damage
produces in the cylinder section rather than the dome section, which is
consistent with the failure mechanism of the experimental results [7].
For the fiber damage evolution in Fig. 11, the fiber damage initials firstly
around the void defects in the cylinder section and the dome section
near the equator. The fiber damage in the dome section tends to evolve
from the middle region to fiber direction and then the sides, but in the
cylinder section, the damage evolution process is that the fiber damage
firstly starts at the inside helical winding layers, then extends to the
outside helical winding layers and finally reaches the hoop winding
layers. As shown in Fig. 12, for the matrix damage, the void defects
induce the damage initiation around the edges of one side of the dome
region, then extend almost in parallel to the other side, and further to­
ward the fiber direction. This damage evolution mechanism is different
from that of the fiber damage in the dome section. As for the matrix
damage in the cylinder section, similarly, the damage initials around the
void defects of both edge regions, then rapidly extends to transverse
regions, and in turn evolve along the fiber direction. As can be seen, the
void defects will contribute to the damage initiation, evolution and final
failure of composite layers of pressure vessels. As a conclusion, the
present 3D finite element model with void defects is effective in pre­
dicting the damage failure mechanism and burst pressure of the com­
posite pressure vessels. The present damage failure analysis scheme can
provide theoretical guidance for reliability evaluation of composite
pressure vessels with void defects under the internal pressure.
4.3. Comparative analysis of defect models of composite pressure vessels
The void distribution may influence the strength and damage
behavior of the composite pressure vessels, and to compare with the
experimental data [7], the average void volume fracture of 5 % is
assigned for three different finite element models in Section 2.2 with the
abbreviation of MP-1(i.e. random voids), MP-2 (i.e. clustered voids in
the middle) and MP-3(i.e. clustered voids around the side). That is to say
on the basis of the same void volume fraction, three different void types
own randomly distributed or clustered void defects are generated to
investigate the effects of large void defects on the mechanical properties
of composite pressure vessels. Three type void models are simulated
under the same internal pressure, and almost no stresses are generated in
the void defects due to their weakened material properties. However,
the void defects have changed the stress distribution and deteriorate
concentrated stresses in the partial region.
The burst strengths of MP-2 and MP-3 are predicted to be 131 MPa
and 127 MPa, respectively. And, the burst strength of MP-2 is closer to
the burst pressure value 133 MPa of MP-1. That is to say the void type
Fig. 11. Fiber damage evolution of composite pressure vessels under pressure loading.
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Thin-Walled Structures 199 (2024) 111858
Fig. 12. Matrix damage evolution of composite pressure vessels under pressure loading.
with clustered void defects shows stronger influences than random void
defects on the burst pressure, and the clustered void defects in the edges
show the greatest adverse effects on the strength of composite pressure
vessels. Taking MP-1 as the reference, Fig. 13 shows the damage evo­
lution process of the same region of different models under the internal
pressure loading. The region around the clustered voids of MP-2 expe­
riences almost no damage, but more damage initiates around the small
defects in the neighbor layers (i.e. the region of Layer 18–21). And, the
damage status is similar to that of MP-1 in the same region. However,
more serious damage produces in the same layers of MP-3 compared to
MP-1(i.e. the region of Layer 35–38). As a conclusion, the clustered
voids will result in the damage deterioration of the basic extent brought
by small void defects, but will not change the general damage trend of
specific composite layers. This has explained why the predicted burst
pressure of MP-3 is the lowest of these three models.
5. Conclusions
In this paper, the WCM plugin coupled with a void generation
method is employed in the ABAQUS software to accurately model the 3D
composite pressure vessel with void defects. A 3D progressive damage
model compiled in UMAT subroutine is developed to predict the damage
evolution and failure modes of composite layers under the internal
pressure, and therefore determine the burst pressure considering void
defects which has been verified by the experimental data. To further
explore the effects of void characteristics on the damage behavior and
composite strength, three different finite element models have been
constructed basing on the experimental data and clearly discussed.
Based on the above research, the following conclusions have been
drawn.
(1) The 3D finite element model and progressive damage model are
effective in predicting the burst pressure of composite pressure
vessels with void defects, and the error of less than 3 % between
the predicted value and experimental data has demonstrated the
reliability of the present model.
(2) With the internal pressure increasing, the matrix damage of
composite layers initially appears in the middle of cylinder sec­
tion and the region near equator. Before the pressure of 54 MPa,
the matrix damage is the main damage mode. In the pressure
scope of 54–95 MPa, the matrix damage and fiber damage play
almost the same important role. After the pressure reaches 95
MPa, the fiber damage has dominated the damage process until
the final burst failure of composite pressure vessels.
Fig. 13. Damage evolution of different models under pressure loading.
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Thin-Walled Structures 199 (2024) 111858
L. Ge et al.
(3) The void defects will result in the deterioration of damage initi­
ation, evolution and final failure of composite layers, but the
clustered voids will not change the general damage evolution
trend of composite layers.
(4) The present damage failure analysis scheme can provide the
theoretical guidance for reliability evaluation of composite
pressure vessels with void defects under the internal pressure.
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CRediT authorship contribution statement
Lei Ge: Writing – original draft, Software, Methodology, Investiga­
tion, Funding acquisition. Jikang Zhao: Validation, Resources, Inves­
tigation. Hefeng Li: Resources, Conceptualization. Jingxuan Dong:
Software, Resources. Hongbo Geng: Resources, Data curation. Lei Zu:
Software, Resources, Methodology. Song Lin: Resources, Conceptuali­
zation. Xiaolong Jia: Writing – review & editing, Project administra­
tion, Funding acquisition. Xiaoping Yang: Project administration,
Methodology.
Declaration of competing interest
The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.
Data availability
No data was used for the research described in the article.
Acknowledgment
This work was financially supported by the National Natural Science
Foundation of China (Grant No. 52303071), China Postdoctoral Science
Foundation (Grant No. 2022M720373), Beijing Natural Science Foun­
dation (Grant No. 2192044), Fundamental Research Funds for the
Central Universities (Grant No. XK1802-2, BH2314), BUCT Youth Talent
Plan, 2020–2023 Open Project of State Key Laboratory of OrganicInorganic Composites (Grant No. Oic-202001008, Oic-202101008,
Oic-202201007, Oic-202301003), Consulting Research Project of Chi­
nese Academy of Engineering (No. 2020-XY-81).
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