Study of Composite Fiber Reinforcement of Cracked Thin-Walled Pressure Vessels Utilizing Multi-Scaling Technique Based on Extended Finite Element Method
One of the most important challenges of storing fluids in thin walled pressure vessels under internal pressure is preventing crack propagation. In low temperature, steel shows brittle crack propagation characteristic which is highly dangerous. In this paper, a new numerical model is presented in ord...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2018
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| Назва видання: | Проблемы прочности |
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| Цитувати: | Study of Composite Fiber Reinforcement of Cracked Thin-Walled Pressure Vessels Utilizing Multi-Scaling Technique Based on Extended Finite Element Method / S.H. Mirmohammad, M. Safarabadi, M. Karimpour, M.R.M. Aliha, F. Berto // Проблемы прочности. — 2018. — № 6. — С. 118-130. — Бібліогр.: 24 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1740162025-02-23T17:43:17Z Study of Composite Fiber Reinforcement of Cracked Thin-Walled Pressure Vessels Utilizing Multi-Scaling Technique Based on Extended Finite Element Method Оценка трещиностойкости тонкостенного сосуда давления с трещиной и накладкой из армированного волокнами композита с помощью мультимасштабирования и расширенного метода конечных элементов Mirmohammad, S.H. Safarabadi, M. Karimpour, M. Aliha, M.R.M. Berto, F. Научно-технический раздел One of the most important challenges of storing fluids in thin walled pressure vessels under internal pressure is preventing crack propagation. In low temperature, steel shows brittle crack propagation characteristic which is highly dangerous. In this paper, a new numerical model is presented in order to investigate the reinforcement of a cracked thin walled pressure vessel by composite patch. Одна из наиболее важных проблем хранения жидкостей в тонкостенных сосудах высокого давления предотвращение распространения трещин. При низких температурах имеет место хрупкое распространение трещин, которое является опасным. Представлена новая численная модель расчета усиления тонкостенного сосуда давления с трещиной композитной накладкой. Однією з найбільш важливих проблем зберігання рідини в тонкостінних посудинах високого тиску є запобігання розповсюдженню тріщин. При низьких температурах має місце крихке розповсюдження тріщин, що є небезпечним. Представлено нову числову модель розрахунку підсилення тонкостінної посудини тиску з тріщиною композитною накладкою. 2018 Article Study of Composite Fiber Reinforcement of Cracked Thin-Walled Pressure Vessels Utilizing Multi-Scaling Technique Based on Extended Finite Element Method / S.H. Mirmohammad, M. Safarabadi, M. Karimpour, M.R.M. Aliha, F. Berto // Проблемы прочности. — 2018. — № 6. — С. 118-130. — Бібліогр.: 24 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/174016 539.4 en Проблемы прочности application/pdf Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| topic |
Научно-технический раздел Научно-технический раздел |
| spellingShingle |
Научно-технический раздел Научно-технический раздел Mirmohammad, S.H. Safarabadi, M. Karimpour, M. Aliha, M.R.M. Berto, F. Study of Composite Fiber Reinforcement of Cracked Thin-Walled Pressure Vessels Utilizing Multi-Scaling Technique Based on Extended Finite Element Method Проблемы прочности |
| description |
One of the most important challenges of storing fluids in thin walled pressure vessels under internal pressure is preventing crack propagation. In low temperature, steel shows brittle crack propagation characteristic which is highly dangerous. In this paper, a new numerical model is presented in order to investigate the reinforcement of a cracked thin walled pressure vessel by composite patch. |
| format |
Article |
| author |
Mirmohammad, S.H. Safarabadi, M. Karimpour, M. Aliha, M.R.M. Berto, F. |
| author_facet |
Mirmohammad, S.H. Safarabadi, M. Karimpour, M. Aliha, M.R.M. Berto, F. |
| author_sort |
Mirmohammad, S.H. |
| title |
Study of Composite Fiber Reinforcement of Cracked Thin-Walled Pressure Vessels Utilizing Multi-Scaling Technique Based on Extended Finite Element Method |
| title_short |
Study of Composite Fiber Reinforcement of Cracked Thin-Walled Pressure Vessels Utilizing Multi-Scaling Technique Based on Extended Finite Element Method |
| title_full |
Study of Composite Fiber Reinforcement of Cracked Thin-Walled Pressure Vessels Utilizing Multi-Scaling Technique Based on Extended Finite Element Method |
| title_fullStr |
Study of Composite Fiber Reinforcement of Cracked Thin-Walled Pressure Vessels Utilizing Multi-Scaling Technique Based on Extended Finite Element Method |
| title_full_unstemmed |
Study of Composite Fiber Reinforcement of Cracked Thin-Walled Pressure Vessels Utilizing Multi-Scaling Technique Based on Extended Finite Element Method |
| title_sort |
study of composite fiber reinforcement of cracked thin-walled pressure vessels utilizing multi-scaling technique based on extended finite element method |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| publishDate |
2018 |
| topic_facet |
Научно-технический раздел |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/174016 |
| citation_txt |
Study of Composite Fiber Reinforcement of Cracked Thin-Walled Pressure Vessels Utilizing Multi-Scaling Technique Based on Extended Finite Element Method / S.H. Mirmohammad, M. Safarabadi, M. Karimpour, M.R.M. Aliha, F. Berto // Проблемы прочности. — 2018. — № 6. — С. 118-130. — Бібліогр.: 24 назв. — англ. |
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Проблемы прочности |
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| fulltext |
UDC 539.4
Study of Composite Fiber Reinforcement of Cracked Thin-Walled Pressure
Vessels Utilizing Multi-Scaling Technique Based on Extended Finite Element
Method
S. H. Mirmohammad,
a
M. Safarabadi,
a
M. Karimpour,
a
M. R. M. Aliha,
b,1
and F. Berto
c
a Mechanical Engineering Department, College of Engineering of the University of Tehran, North
Kargar, Tehran, Iran
b Welding and Joining Research Center, School of Industrial Engineering, Iran University of Science
and Technology (IUST), Narmak, Tehran, Iran
c Department of Engineering Design and Materials, Norwegian University of Science and Technology,
Trondheim, Norway
1 mrm_aliha@iust.ac.ir
ÓÄÊ 539.4
Îöåíêà òðåùèíîñòîéêîñòè òîíêîñòåííîãî ñîñóäà äàâëåíèÿ ñ òðåùèíîé è
íàêëàäêîé èç àðìèðîâàííîãî âîëîêíàìè êîìïîçèòà ñ ïîìîùüþ ìóëüòè-
ìàñøòàáèðîâàíèÿ è ðàñøèðåííîãî ìåòîäà êîíå÷íûõ ýëåìåíòîâ
Ñ. Õ. Ìèðìîõàììàä
à
, Ì. Ñàôàðàáàäè
à
, Ì. Êàðèìïóð
à
, Ì. Ð. Ì. Àëèõà
á
, Ô. Áåðòî
â
à Ìåõàíè÷åñêèé ôàêóëüòåò, Èíæåíåðíûé êîëëåäæ Òåãåðàíñêîãî óíèâåðñèòåòà, Òåãåðàí, Èðàí
á Íàó÷íî-èññëåäîâàòåëüñêèé öåíòð ñâàðêè è ñîåäèíåíèÿ, Èðàíñêèé óíèâåðñèòåò íàóêè è
òåõíîëîãèè, Íàðìàê, Òåãåðàí, Èðàí
â Êàôåäðà èíæåíåðíîãî ïðîåêòèðîâàíèÿ è ìàòåðèàëîâ, Íîðâåæñêèé óíèâåðñèòåò åñòåñòâåííûõ
è òåõíè÷åñêèõ íàóê, Òðîíõåéì, Íîðâåãèÿ
Îäíà èç íàèáîëåå âàæíûõ ïðîáëåì õðàíåíèÿ æèäêîñòåé â òîíêîñòåííûõ ñîñóäàõ âûñîêîãî
äàâëåíèÿ – ïðåäîòâðàùåíèå ðàñïðîñòðàíåíèÿ òðåùèí. Ïðè íèçêèõ òåìïåðàòóðàõ èìååò
ìåñòî õðóïêîå ðàñïðîñòðàíåíèå òðåùèí, êîòîðîå ÿâëÿåòñÿ îïàñíûì. Ïðåäñòàâëåíà íîâàÿ
÷èñëåííàÿ ìîäåëü ðàñ÷åòà óñèëåíèÿ òîíêîñòåííîãî ñîñóäà äàâëåíèÿ ñ òðåùèíîé êîìïîçèòíîé
íàêëàäêîé. Äëÿ ìîäåëèðîâàíèÿ ðîñòà õðóïêèõ òðåùèí ïî òîëùèíå òîíêîñòåííîãî ñîñóäà
äàâëåíèÿ èñïîëüçóåòñÿ ðàñøèðåííûé ìåòîä êîíå÷íûõ ýëåìåíòîâ â ñî÷åòàíèè ñ ìåòîäîì
ìóëüòèìàñøòàáèðîâàíèÿ. Ïîñðåäñòâîì ðàñøèðåííîãî ìåòîäà êîíå÷íûõ ýëåìåíòîâ ðàññ÷è-
òûâàåòñÿ êðèòè÷åñêàÿ ýíåðãèÿ, ò.å. ìàêñèìàëüíàÿ ýíåðãèÿ äåôîðìàöèè, êîòîðóþ ñîñóä
äàâëåíèÿ ìîæåò íàêîïèòü äî íà÷àëà õðóïêîãî ðàñïðîñòðàíåíèÿ òðåùèíû. Äëÿ óâåëè÷åíèÿ
êðèòè÷åñêîé ýíåðãèè èñïîëüçîâàëèñü ñâÿçóþùèå ýëåìåíòû è ñîñòàâíûå ó÷àñòêè ñ ðàçëè÷íîé
ïîñëåäîâàòåëüíîñòüþ óêëàäêè, êîòîðûå ðàíåå èññëåäîâàëèñü ýêñïåðèìåíòàëüíî è àíàëèòè-
÷åñêè, à îïòèìàëüíàÿ ïîñëåäîâàòåëüíîñòü óêëàäêè îöåíèâàëàñü ñ ïîìîùüþ ðàñøèðåííîãî
ìåòîäà êîíå÷íûõ ýëåìåíòîâ. Êðîìå òîãî, èñïîëüçîâàëàñü òðàäèöèîííàÿ ìåòîäèêà îïòèìè-
çàöèè àðìèðîâàíèÿ êîìïîçèòíûìè íàêëàäêàìè, îñíîâàííàÿ íà îïòèìàëüíîì ñîîòíîøåíèè
óâåëè÷åííîé êðèòè÷åñêîé ýíåðãèè è òîëùèíû íàêëàäêè. Ïîêàçàíî, ÷òî ïðè ïîñòîÿííîé
òîëùèíå íàêëàäêè è èçìåíåíèè óãëà óêëàäêè âîëîêîí ìàêñèìàëüíàÿ ýíåðãîåìêîñòü óâåëè÷è-
âàåòñÿ íà 7…11%. Ïðè óâåëè÷åíèè òîëùèíû àðìàòóðû íàáëþäàåòñÿ çíà÷èòåëüíûé ðîñò
ýíåðãèè äåôîðìàöèè (äî 40%). Äëÿ èñêëþ÷åíèÿ ïîâðåæäåíèÿ íàêëàäîê ïðè ðàñïðîñòðàíåíèè
òðåùèí èñïîëüçîâàëñÿ êðèòåðèé ïîâðåæäåíèÿ Õàøèíà.
Êëþ÷åâûå ñëîâà: òîíêîñòåííûé ñîñóä âûñîêîãî äàâëåíèÿ, ðàñøèðåííûé ìåòîä êî-
íå÷íûõ ýëåìåíòîâ, íàêëàäêà èç âîëîêíèñòîãî êîìïîçèòà, ìåòîä ìóëüòèìàñøòàáèðî-
âàíèÿ, ìåõàíèêà ðàçðóøåíèÿ, êðèòåðèé ïîâðåæäåíèÿ Õàøèíà.
© S. H. MIRMOHAMMAD, M. SAFARABADI, M. KARIMPOUR, M. R. M. ALIHA, F. BERTO, 2018
118 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 6
1. Background. Reinforcement of thin-walled pressure vessels has always been an
important subject in the industry since it can reduce the possibility of fracture. Whether the
property of the material is ductile or brittle, there is a high possibility that it shows brittle
behavior under certain circumstances such as cold temperature of the fluid inside the
pressure vessel. More over different shapes of the crack tend to become similar to
nail-shaped crack due to the lower levels of energy.
1.1. Fracture Mechanics Based on Extended Finite Element Method (XFEM)
Approach. There are several approaches to analyze crack stability and propagation. Early
theories were based on local formulations, however, soon it was realized that they are
unreliable due to the dependency to mesh or size. Three fundamentals approaches for
discussion of defects and failures are continuum-based plasticity, damage mechanics and
the crack based approach of fracture mechanics. Theory of plasticity and damage
mechanics are used to analyze problems where displacement field remains continuous
everywhere, and use material properties to calculate the status of the material after
experiencing an amount of load. On the other hand, fracture mechanics is specifically
formulated to deal with discontinuities where displacement field is discontinuous. The
theory of plasticity is based on softening plasticity models and is capable of predicting
initiation and propagation of the crack. The theory of damage mechanics is similar to the
theory of plasticity. In the theory of damage mechanics both stiffness and strength of the
material decrease if the material experience damage, however, in the theory of plasticity
stiffness remains unaltered but the strength is updated due to the softening/hardening
behavior. Last but not least, the theory of linear elastic fracture mechanics (LEFM) adopts
the laws of thermodynamic to formulate an energy or stress intensity based criterion to
analyze an existing crack. Moreover, the most important facet of LEFM is the capability to
derive the singular stress field by means of the analytical solution at the crack tip. Due to
the limitations of analytical methods in handling complex geometries and boundary
conditions, several numerical techniques were developed for solving fracture mechanics
problems. Among all of these techniques, fast developments of finite element method has a
great impact on the application of LEFM which helps to implement this theory for models
with complex geometries and engineering applications [1, 2]. Robinovitch and Frosting
conducted a research on the fiber reinforced plastic strips in order to predict interfacial
delamination. They carried out an analytical stress analysis and managed to introduce
energetic fracture criterion by which the basis of stable or unstable crack growth was
derived. Moreover, they proposed that this criterion is more reliable to predict catastrophic
failure that stress based failure criteria [3]. Colombi [4] examined the delamination failure
of metallic beams reinforced by externally bonded fibres reinforced polymers (FRP), and
illustrated a simplified criterion which is based on the evaluation of energy release rate
(ERR) using both analytical and numerical models. Consequently, an alternative approach
for the stress-based method was proposed to use ERR based criterion to evaluate the load
carrying capacity of the adhesive joints. Finally, the concluded that using this fracture
criterion is more convenient because it does not need burdensome evaluation of stress field
especially in a discontinuity, hence, it is more applicable to industrial cases [4]. Bruno et al.
[5] investigated edge debonding in beams strengthened with externally bonded composite
laminated plates and evaluated strain energy release rate (ERR) by means of interface
displacement jumps which is a very efficient numerical procedure. Results obtained by this
procedure showed agreement with refined 2D-continuum FE investigation with high
accuracy. Moreover, the analysis carried out by this approach does not suffer of
non-convergence problem in the energy release rate mode components revealed by FE
computations [5]. Greco et al. [6] evaluated both total and mixed mode energy release rate
with an analytical solution by using stress and strain discontinuities across a crack tip for
the analysis of typical edge debounding problems in concrete strengthened with externally
bonded composite, and compared it with a thorough FE investigation. Their model consists
ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 6 119
Study of Composite Fiber Reinforcement ...
of two mathematical layers which adopt first-order shear deformation laminate theory.
Their analytical solutions for evaluating energy release rates utilizes an interface fracture
analysis based on interfacial concentrated forces. This analytical approach contributes to
less computation costs and avoids the complexity of detailed mesh required to capture the
singularities. Therefore, the analytical solution avoids complexity of FE procedure [6]. All
of the aforementioned carried out studies which adopted FEM to evaluate stress and strain
in order to calculate parameters such as stress intensity factor, the energy release rate of
J integral to analyze cracks. Additionally, various techniques were used to produce
singularity in the stress field at a crack tip, and facilitate arbitrary crack propagation which
led to the introduction of the technique of Singular finite elements which was the
introduction of elements with singularity in their formulation [7]. This technique was prior
to the development of XFEM and suffered from some shortages. Since it has to be used in a
finite element mesh, where crack face have to match element boundaries, its application
had limitations unless combined with at least a local adaptive finite element scheme.
However, this drawback was dealt with by introduction of XFEM which was introduced
and developed by Belytschko and Black [8] they developed an algorithm based on partition
of unity (PUFEM) which utilizes approximation functions which can be guessed from
analytical solutions to represent the crack tip discontinuity. However, in case of severely
curved cracks, minimal remeshing was needed far away from crack tip that is costly
efficient. Moreover, stress intensity factors which was calculated by this method agrees
with analytical solutions with high accuracey for some 2D problems [8]. Furthermore,
Moës et al. [9] improved this method to make it more convenient to model long cracks.
They used crack tip asymptotic functions to reproduce the discontinuity of the crack tip and
Haar functions to represent the discontinuous field across the crack faces. The
aforementioned method, not only is advantageous for nonlinear materials; but also utilizes
mapping of the discontinuous crack tip functions for curved cracks. Last but not least, the
accuracy of this method is quite independent of mesh size for a large range of sizes and
overcome the need for transitions and very small mesh size near the crack tip which was
proved by comparing the stress intensity factors that was calculated by this method and the
analytical solutions for some 2D problems [9]. Additionally, Areias and Belytschko [10]
transitioned this methods for 3D brittle quasi-static problems. They carried out this study
utilizing tetrahedral elements and a viscosity-regularized continuum damage constitutive
model in order to improve the stability of equilibrium equations. Moreover, the evolving
discontinuity surface is discretize through a C 0 surface formed in order to separate each
cracked elements in two. Although, there are accuracy defects in tetrahedral elements, they
found that cracking results are accurate even with coarse meshes. They proposed an
explanation for this adequacy which was decoupling of surface geometrical definition and
dissipation mechanism in XFEM algorithm. Moreover, the novelty of this paper was taking
advantage of continuum damage after the discontinuity is introduced [10]. It needs to be
mentioned that formulation of the XFEM method is based on partition of unity finite
element method (PUFEM) which was proposed earlier by Melenk and Babuðka [11],
PUFEM is a method that contributes to solution of the differential equations by using local
approximations that reflects the rough behavior of the solution. Interestingly, the meshless
aspect of this method is helpful for problems which have to deal with frequent remeshing.
However, one of the potential drawbacks of this aspect is difficulty in integrating according
to the shape functions which are not tied to the mesh. Fortunately, this issue has been dealt
with in closed form integration method of previous studies [11]. Belytschko et al. [12]
presented a technique to model and analyze arbitrary cracks in which both discontinuities in
the function and its derivatives were considered. Additionally, they took advantage of level
set method to update the position of the discontinuity. In this technique the surface of
discontinuity is described by signed distance function. Hence, the discontinuity
approximation involves only nodal data. However, the major drawback of this method is
120 ISSN 0556-171X. Ïðîáëåìè ì³öíîñò³, 2018, ¹ 6
S. H. Mirmohammad, M. Safarabadi, M. Karimpour, et al.
the quadrature of the weak form such that utilizing quadrature elements requires certain
modifications [12]. A variational (weak) form of any equation involves multiplying an
arbitrary vector with a free index and integrating the result over the domain which is
analogous to integrating work over a closed domain which equals to zero. In solving partial
differentials equations this technique is used to reduce derivatives which contributes to
reduce computational costs [13]. Areias and Belytschko [14] proposed a new formulation
for the analysis of arbitrary crack propagation in shells. Moreover, they used a modified
formulation for the 4-noded cracked shell elements to avoid locking. They compared the
results of intensity factors from this method to previous methods which showed good
agreement [14]. Budarapua et al. [15] proposed a multi-scale numerical method for quasi
static crack growth. Phantom node method is used to model the crack in the continuum
region and a molecular statics model is used near the crack tip [15]. Furthermore, several
research have been conducted recently which investigated the applications of this method.
Sharma et al investigated the stress intensity factor of a semi-elliptical part through
thickness axial/circumferential crack in a pipe bend under internal pressure loading using
XFEM method. They concluded that axial crack is more severe than the circumferential
crack under same loading and boundary conditions. At the end, they compared the results
of SIF evaluated form XFEM to the results from regular FEM and claimed that not only it
showed a good agreement; but also; using XFEM is more convenient [16]. Zhang et al. [17]
investigated ductile fracture of API X65 buried pipeline including crack initiation and
propagation using XFEM. At the very first step, they evaluated GIC and Maxps of the
materials using experimental data of three point bending test and numerical analysis. These
material properties have the most important effect on the fracture behavior of a material
such that the value of the Maxps is the critical maximum principal stress at which a crack
initiates. Also G cI is the critical value that indicates the resistance of the material against
the crack propagation. More G cI value; gentler decrease in the load bearing capacity. After
evaluating the closest values to the experimental data, they used them to investigate the
fracture of the same pipe-line which is buried in the soil which is subjected to land-sliding.
They used linear elastic Drucker–Prague model as the mechanical properties of the soil.
Last but no least they concluded that using XFEM method is more conservative than using
nonlinear stabilization algorithm in case of investigating load bearing capacity since it takes
the damage effects of material into the consideration and provides a fundamental support
for the integrity assessment and safety evaluation of buried pipes [17].
1.2. Fiber Composite Reinforcement. There has been plenty of studies both analytically
and experimentally to investigate the effects of the angles of fibers and stacking sequence
in a composite on the reinforcement of the pressure vessels. Also since crack propagation
results in a shear stress load on the cohesive element and the composite reinforcement,
considering damage analysis of the both cohesive element and composite reinforcement is
of importance. Hashin proposed four distinct failure modes-tensile and compressive fiber
and matrix modes for three-dimensional unidirectional fiber composites. The novelty of this
work is to predict distinct failure modes which is helpful for progressive damage analysis
which contributes to set up a FE procedure and predict the exact situation of the fiber and
matrix [18]. Al-Khalil et al. [19] investigated several filament wound glass fiber reinforced
epoxy cylinders with winding angles of �35, �55, and �75 and stated that the mode of
the failure depends on the winding angle. Liu et al. [20] performed an analysis based on
continuum damage mechanics, the progressive failure analysis using explicit finite element
method to predict failure properties and burst strength of aluminum-carbon fiber/epoxy
composite cylindrical laminate structures. The used viscous damping effect for convergence
of the problem to properly deal with strain softening phenomenon. Also, they concluded
that in case of progressive damage explicit method shows a better convergence in
comparison with the implicit method [20].
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2. Numerical Modeling.
2.1. Crack Modeling Based on XFEM. Abaqus software is used to develop a FE
model. One of the most recent and novel algorithms to analyze discontinuity is using
XFEM procedure. Using this method, both modeling of crack initiation which uses LEFM
and propagation which uses traction separation cohesive behavior is possible. Either of
these methods use an energy based law according to the thermodynamics principals which
says a system always tries to reach the lowest level of energy. Modeling the problem starts
with investigating a thin-walled pressure vessel modeled with shell elements under static
internal pressure. The pressure vessel diameter is 1m and the length of the cylindrical part
is 4 m. Since the XFEM crack can only be represented in solid elements, using multi-
scaling technique which can be implemented in the model as submodeling technique in
abaqus is needed. Hence, shell to solid submodeling is used in order to drive the nodes of
the solid elements using displacement of the shell elements.
Validation of this method and its implementation was done under the static pressure
by the analytical solution. As it is known form the analytical solution the hoop stress is
calculated by PD tm 2 , in which the P, t , and Dm are internal pressure, thickness, and
mean diameter, respectively. As it is shown in Fig. 1 the hoop stress under 10 MPa
internal pressure for a pressure vessel with mean diameter of 1 m and thickness of 10 mm
is 500 MPa.
In order to model the XFEM crack, damage for traction-seperation laws is used as the
appropriate theory and parameters of steel for this theory is as Table 1, where E and � are
elastic modulus and Poisson’s ratio, respectively. Also MPS and G are the maximum
principal stress in which the crack initiates and the critical energy release rate, respectively.
Now it is needed to specify the shape of the crack which is a nailed-shaped crack in
the thickness of the submodel.
The width of the crack is 1 mm and only the curved part of the crack geometry has
been inserted to the submodel. It needs to be mentioned that a part of the crack must be
outside of the thickness so the software can comprehend it as a crack also a mesh and a
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T a b l e 1
Material Properties of Steel [21]
E, GPa MPS, MPa � G , J/m2
210 440 0.3 107,142.8571
Fig. 1. Hoop stress contour in submodel.
material property should not be assigned to the crack. In order to assign the XFEM property
to the model step-special-crack-create-XFEM is used. Also a hard contact to the surfaces of
the crack is assigned to avoid the surfaces of the crack to go inside each other [22].
For improving the convergence of the solution a viscosity for the material in the
material property which can be 1-E5 Pa s� is assumed. Moreover, an automatic stabilization
can be used in the step module to improve the unstable response. As the crack starts to
grow, there would be an abrupt fall in the strain energy of the submodel after absorbing an
amount of strain energy which represents the toughness of the material.
2.2. Reinforcement Patch Modeling. In this study, it is tried to increase this critical
energy by cohesive elements and composite reinforcement. For this mean, a shell to which
a composite lay-up is assigned is modeled. The composite lay-up has the same length and
width as the submodel pressure vessel. In this paper, the carbon-epoxy (T300/5208)
composite has been used. The mechanical properties of this composite is as Table 2 that has
been extracted from [23].
Here, E1, E2, and �12 are the longitudinal and transversal moduli and Poisson’s
ratio, respectively, G12, G13, and G23 are regarding shear moduli, X T and X C are the
longitudinal tensile and compressive strength, respectively, Y T and Y C are transverse
tensile and compressive strength, respectively, and S L and S T are shear longitudinal and
transverse strength, respectively.
2.3. Cohesive Element Modeling. In order to model a cohesive element a solid
element with the thickness of 0.1 mm is modeled. Cohesive element has the same length
and width as the submodel pressure vessel. Also the cohesive element which has been used
to attach the composite lay-up to the submodel is Araldite 420 that has uncoupled
mechanical properties as Tables 3 and 4. In which Enn , Ess , and Ett are the main
diagonal entries of the stiffness matrix. Also, in order to investigate the damage of the
cohesive element under shear stress resulted from crack propagation, maximum damage
properties for the cohesive element have been used. The material properties has been
extracted from [24]. It is worth mentioning that that properties has been extracted form an
upcomng paper by Shahriyarifard et al. [24].
2.4. Assembly of the Model. To assemble the model to be a good representative of the
reality, the cohesive element and the submodel have been merged. Moreover, the composite
lay-up has been tied to the cohesive element which is a decent assumption during the
analysis.
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T a b l e 2
Elastic and Strength Properties of Carbon-Epoxy [23]
E1 ,
GPa
E2 ,
GPa
�12 G12 ,
GPa
G13 ,
GPa
G23 ,
GPa
X T ,
MPa
X C ,
MPa
Y T ,
MPa
YC ,
MPa
S L ,
MPa
S T ,
MPa
181 10.3 0.28 7.17 6.0 6.0 1500 1500 40 246 68 68
T a b l e 3
Elastic and Strength Properties of Cohesive Element Araldite 420 [24]
Enn,
GPa
Ess,
MPa
Ett ,
MPa
Nominal stress
normal-only mode
(MPa)
Nominal stress
first direction
(MPa)
Nominal stress
second direction
(MPa)
1.85 7.17 7.17 30.0 14.5 14.5
2.5. Mesh Study. Also mesh study has been conducted on both submodel and
composite lay-up to reach a mesh size that results a solution which is independent of the
mesh size. To reach this objective different mesh sizes for submodel were used and for each
mesh size, the critical energy as the objective of the mesh study process was calculated by
numerical methods.
As it was mentioned earlier in the literature review, XFEM algorithm is independent
of the mesh size after some practice so the mesh size with 85833 DOFs is considered.
Moreover, the shape of the evolved crack differs according to the integration points in the
elements and since full integration elements results more accurate solution, full integration
linear Hex elements which have 8 integration points are considered. The shape of the crack
with both reduced integration points and full integration points is shown in Fig. 2.
Additionally, a mesh study has been conducted on the composite lay-up. Hence, linear
quad elements with 13,668 DOFs were used.
Last but not least, the global model of pressure vessel which was modeled by shell
elements, must be partitioned to be possible to have a composite lay-up section at the exact
same place of the submodel.
3. Results and Discussion. In this paper, crack propagation in the thickness of a
pressure vessel was studied utilizing the combination of XFEM approach in fracture
mechanics and multi-scaling technique. Several stacking sequences and stacking angles
were used to reinforce the thin-walled pressure vessel. The nailed-shape crack has the
width of 1 mm and the depth of 0.5 mm. The submodel, cohesive element and the
composite patch all have the length and width of 50 mm which would be big enough to
avoid edge effects of submodel. In the parametric studies, several angle sequences were
examined which were extracted from previous studies [19]. Also, the thickness of the
cohesive element has been set as 0.1 mm both in the geometry and the section property
which is based on [22].
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T a b l e 4
Damage Properties of Cohesive Element Araldite 420 [24]
Shear mode fracture
first direction (N/m)
Shear mode fracture energy
second direction (N/m)
Normal mode fracture energy
(N/m)
4700 4700 4000
Fig. 2. Shape of the evolved crack with (a) reduced integration points elements and (b) full
integration points elements.
a b
Before the beginning of the case studies, the maximum strain energy that the
submodel of the pressure vessel can absorb before the crack propagates entirely in the
thickness of the solid submodel of the pressure vessel was calculated. The diagram of the
strain energy in Fig. 3 shows that after absorbing a specific amount of strain energy the
material losses its load bearing capacity and the strain energy abruptly falls. In this paper,
the goal is to increase this critical energy using cohesive elements and composite patches
with different reinforcement angles and thickness.
It has been observed in Fig. 4 that by reinforcing the submodel the maximum
principal stress at the crack tip decreases which contributes to prevent the unstable crack
propagating in a brittle material.
3.1. Effects of Reinforcement Thickness. In this case study, a stacking angle which
would be 0/45/�45/90 is selected to evaluate the optimum ratio of the increase in
reinforcement to the thickness of the reinforcement which can increase by repeating the
angle pattern. Since each lamina has the thickness of 0.2 mm, then each pattern would have
the thickness of 0.8 mm. Figure 5 shows the results.
As the results show, increasing the thickness more than 2.4 mm is a waste of material
because it does not result in adequate increase in the reinforcement. Moreover, as the
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Fig. 3. Evolution of the strain energy as the crack propagates.
a b
Fig. 4. Maximum principal stress at the crack tip: non-reinforced (a) and reinforced (b)
submodels.
thickness of the reinforcement is increased the shape of the evolved crack differs and would
get wider as it is shown in Fig. 6.
3.2. Effects of Stacking Angle. In this case study, the effects of different stacking
angles which are popular angles or are extracted from [19] on the degradation of the
cohesive element have been investigated. It is crystal clear that each angle pattern that
provides more reinforcement, can distribute the load more effectively in the cohesive
element, hence the cohesive element would experience less degradation. Degradation of the
cohesive element can be calculated using SDEG output and the results for two stacking
angle is shown in Fig. 7. As it is observed in the results, 75/�75/75/�75 and 55/�55/55/�55
stacking angles result less cohesive element degradation than the other stacking angles.
3.3. Traditional Optimization. In this case study, it has been tried to reach the
optimum reinforcement thickness using traditional optimization method for fiber- reinforcing
problems. In each run, the thickness of the reinforcement has been increased and the critical
strain energy has been extracted from the FEM software. Figure 8 shows the percentage of
increase in the critical energy as the thickness of reinforcement is increased. Also, Fig. 9
shows the ratio of increase in energy resulted from each step to the thickness of each step.
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Fig. 5. Effect of increase in the thickness on the reinforcement.
a b
Fig. 6. Evolved crack shape with (a) 4 layers of reinforcement and (b) 20 layers of reinforcement.
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a b
Fig. 7. SDEG output of the cohesive element resulted from (a) 75/�75/75/�75 and (b) 0/0/90/90
stacking angles.
Fig. 8. Effect of the thickness of the reinforcement on the increase of the critical strain energy.
Fig. 9. Effect of thickness of the reinforcement on the ratio of critical energy increase to the
reinforcement thickness.
According to the diagrams, it can be concluded that:
Reinforcing with 0/45/�45/90: In case that the reinforcement is more important,
26.35% reinforcement which can be achieved by 2.4 mm reinforcement thickness is the
best. On the other hand, in case that mass of the system is more important, 19.33%
reinforcement which can be achieved by 1.6 mm reinforcement thickness is the best.
Reinforcing with 0/0/90/90: The best state of reinforcement is 20.4% which can be
achieved by 1.6mm reinforcement thickness.
Reinforcing with 0/90/0/90: In case that the reinforcement is more important, 27.51%
reinforcement which can be achieved by 2.4 mm reinforcement thickness is the best. On the
other hand, in case that mass of the system is more important, 18.9% reinforcement which
can be achieved by 1.6 mm reinforcement thickness is the best.
Reinforcing with 35/�35/35/�35: In case that the reinforcement is more important,
22.5% reinforcement which can be achieved by 2.4 mm reinforcement thickness is the best.
On the other hand, in case that mass of the system is more important, 15.4% reinforcement
which can be achieved by 1.6 mm reinforcement thickness is the best.
Reinforcing with 55/�55/55/�55: In case that the reinforcement is more important,
18.13% reinforcement which can be achieved by 1.6 mm reinforcement thickness is the
best. On the other hand, in case that mass of the system is more important, 10.6%
reinforcement which can be achieved by 0.8 mm reinforcement thickness is the best.
Reinforcing with 75/�75/75/�75: in case that the reinforcement is more important,
39.4% reinforcement which can be achieved by 3.2 mm reinforcement thickness is the best.
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Fig. 10. Hashin damage criterion for fibers in tension.
Fig. 11. Hashin damage criterion for matrix in tension.
On the other hand, in case that mass of the system is more important, 10.29% reinforcement
which can be achieved by 0.8 mm reinforcement thickness is the best.
Last but not the least, using relative criterions for damage in fibers and matrix, none
of them experience damage based on the Hashin damage criteria which is shown in Figs. 10
and 11.
Conclusions. In this paper, reinforcement of thin-walled pressure vessels with
composite patches utilizing Multi-scaling technique was studied. Different angle patterns
that were introduced by means of experiments in previous studies were used and were
investigated using XFEM code. Various decisions can be made according to the results of
current FEM code.
To wrap it up, in case the reinforcement is more important, reinforcing the pressure
vessel with 75/�75/75/�75 angles with thickness of 3.2 mm is the best among all of the
other reinforcements. Moreover, in case where the mass of the system is more important
either 0/45/�45/90 scheme with 1.6 mm thickness or 55/�55/55/�55 scheme with thickness
of 0.8 mm can be used.
Ð å ç þ ì å
Îäí³ºþ ç íàéá³ëüø âàæëèâèõ ïðîáëåì çáåð³ãàííÿ ð³äèíè â òîíêîñò³ííèõ ïîñóäèíàõ
âèñîêîãî òèñêó º çàïîá³ãàííÿ ðîçïîâñþäæåííþ òð³ùèí. Ïðè íèçüêèõ òåìïåðàòóðàõ
ìຠì³ñöå êðèõêå ðîçïîâñþäæåííÿ òð³ùèí, ùî º íåáåçïå÷íèì. Ïðåäñòàâëåíî íîâó
÷èñëîâó ìîäåëü ðîçðàõóíêó ï³äñèëåííÿ òîíêîñò³ííî¿ ïîñóäèíè òèñêó ç òð³ùèíîþ
êîìïîçèòíîþ íàêëàäêîþ. Äëÿ ìîäåëþâàííÿ ðîñòó êðèõêèõ òð³ùèí ïî òîâùèí³ òîíêî-
ñò³ííî¿ ïîñóäèíè òèñêó âèêîðèñòîâóºòüñÿ ðîçøèðåíèé ìåòîä ñê³í÷åííèõ åëåìåíò³â ó
ïîºäíàíí³ ç ìåòîäîì ìóëüòèìàñøòàáóâàííÿ. Çà äîïîìîãîþ ðîçøèðåíîãî ìåòîäó
ñê³í÷åííèõ åëåìåíò³â ðîçðàõîâóºòüñÿ êðèòè÷íà åíåðã³ÿ, òîáòî ìàêñèìàëüíà åíåðã³ÿ
äåôîðìàö³¿, ÿêó ïîñóäèíà òèñêó ìîæå íàêîïè÷èòè äî ïî÷àòêó êðèõêîãî ðîçïîâñþä-
æåííÿ òð³ùèíè. Äëÿ çá³ëüøåííÿ êðèòè÷íî¿ åíåð㳿 âèêîðèñòîâóâàëèñü çâ’ÿçóâàëüí³
ä³ëÿíêè ç ð³çíîþ ïîñë³äîâí³ñòþ óêëàäàííÿ, ÿê³ ðàí³øå äîñë³äæóâàëèñü åêñïåðèìåí-
òàëüíî é àíàë³òè÷íî, à îïòèìàëüíà ïîñë³äîâí³ñòü óêëàäàííÿ îö³íþâàëàñü ðîçøèðåíèì
ìåòîäîì ñê³í÷åííèõ åëåìåíò³â. Îêð³ì òîãî, âèêîðèñòîâóâàëàñü òðàäèö³éíà ìåòîäèêà
îïòèì³çàö³¿ àðìóâàííÿ êîìïîçèòíèìè íàêëàäêàìè, ùî áàçóºòüñÿ íà îïòèìàëüíîìó
ñï³ââ³äíîøåíí³ çá³ëüøåíî¿ êðèòè÷íî¿ åíåð㳿 ³ òîâùèíè íàêëàäêè. Ïîêàçàíî, ùî ïðè
ïîñò³éí³é òîâùèí³ íàêëàäêè ³ çì³í³ êóòà óêëàäàííÿ âîëîêîí ìàêñèìàëüíà åíåðãî-
ºìí³ñòü çá³ëüøóºòüñÿ íà 7…10%. Ïðè çá³ëüøåíí³ òîâùèíè àðìàòóðè çíà÷íî çðîñòàº
åíåðã³ÿ äåôîðìàö³¿ (äî 40%). Äëÿ âèêëþ÷åííÿ ïîøêîäæåííÿ íàêëàäîê ïðè ðîç-
ïîâñþäæåíí³ òð³ùèíè âèêîðèñòîâóâàâñÿ êðèòåð³é ïîøêîäæåííÿ Õàøèíà.
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Received 05. 12. 2017
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