INTERNATIONAL JOURNAL OF PROFESSIONAL ENGINEERING STUDIES Volume 9 /Issue 3 / NOV 2017
IJPRES
MODELING AND ANALYSIS OF PRESSURE VESSEL USING FRC MATERIALS 1 Mr. Zafar Ullah Shareef, 2 Dr. Asheesh Kumar, 3 V. Chandra shekar goud
1 PG Scholar, Department of MECH, Aurora’s Scientific, Technological and Research Academy, Bandlaguda, Hyderabad, Telangana.
Email:[email protected] 2 Assistant Professor, Department of MECH, Aurora’s Scientific, Technological and Research Academy
Bandlaguda, Hyderabad, Telangana. Email:[email protected]
3 HOD, Department of MECH, Aurora’s Scientific, Technological and Research Academy Bandlaguda, Hyderabad, Telangana.
Email:[email protected]
Abstract
A pressure vessel is a container which contains gases
or liquids at high pressure when compared with
ambient pressure, In this project pressure vessel is
modeled using solid works 2016 version 24 design
software, and static structural analysis is carried out
force (fiber reinforced composite) materials in
ANSYS 14.5 software to analyze the best suitable
material for pressure vessel which can replace
stainless steel when compared with other FRC
materials that are E-glass Epoxy, S2 glass, Saffil and
Kevlar-49.The pressure vessels are used in a various
applications in both industrial and domestic sector.
They are used in this field as an industrial air
compressors and tanks. The high and uncontrolled
pressure differential is dangerous. Fatal accidents
have happened in the history of pressure vessel
development and operations. Pressure vessels can
theoretically be almost any shape, but shapes made of
sections of spheres, cylinders, and cones are usually
employed. A common design of pressure vessel is a
cylinder with end caps called heads. More
complicated shapes are difficult to analyze for safe
operations and are usually far more difficult to
construct. In this project the materials of the pressure
vessel changed and analyzed for five different
materials by keeping the dimensions of the pressure
vessel constant for all materials and the load applied
is also given same for all the materials. After the
results of simulation through ANSYS software it has
been observed that the Kevlar-49 material is showing
less maximum stress under load of 2MPa when
compared to all the other materials. Other useful
properties of Kevlar- 49 material are high tensile
strength relative to its weight, high heat resistance
and high impact resistance by observing all this
properties it has been found that Kevlar-49 material
can easily replace stainless steel as material for
pressure vessels.
INTRODUCTION
Pressure vessel is a container that has pressure
different than atmospheric pressure. Also, any
container which under certain condition pressurizes a
liquid or gas is called as a pressure vessel. The
pressure differential is dangerous, and fatal accidents
have occurred in the history of pressure vessel
development and operation. Consequently, pressure
vessel design, manufacture, and operation are
regulated by engineering authorities backed by
legislation. For these reasons, the definition of a
pressure vessel varies from country to country, but
involves parameters such as maximum safe operating
pressure and temperature, and are engineered with
a safety factor, corrosion allowance, minimum design
INTERNATIONAL JOURNAL OF PROFESSIONAL ENGINEERING STUDIES Volume 9 /Issue 3 / NOV 2017
IJPRES
temperature (for brittle Fracture), and
involve nondestructive testing, such as ultrasonic
testing, radiography, and pressure tests, usually
involving water, also known as a hydro test, but
could be pneumatically tested involving air or
another gas. The preferred test is hydrostatic testing
because it's a much safer method of testing as it
releases much less energy if fracture were to occur
(water does not rapidly increase its volume while
rapid depressurization occurs, unlike gases like air,
i.e. gasses fail explosively). In the United States, as
with many other countries, it is the law that vessels
over a certain size and pressure (15 PSI) be built to
Code, in the United States that Code is the ASME
Boiler and Pressure Vessel Code (BPVC), these
vessels also require an Authorized Inspector to sign
off on every new vessel constructed and each vessel
has a nameplate with pertinent information about the
vessel such as maximum allowable working pressure,
maximum temperature, minimum design metal
temperature, what company manufactured it, the date,
its registration number (through the National Board),
and ASME's official stamp for pressure vessels (U-
stamp), making the vessel traceable and officially
an ASME Code vessel.
Figure: Industrial pressure vessel
Main Features of Pressure Vessel
The main features of pressure vessel are given below Shape of pressure vessel
Pressure vessels can theoretically be almost any
shape, but shapes made of sections of spheres,
cylinders, and cones are usually employed. A
common design is a cylinder with end caps
called heads. Head shapes are frequently either
hemispherical or dished more complicated shapes
have historically been much harder to analyze for
safe operation and are usually far more difficult to
construct.
Figure: Types of pressure vessels
Theoretically, a spherical pressure vessel has
approximately twice the strength of a cylindrical
pressure vessel with the same wall thickness, and is
the ideal shape to hold internal pressure. However, a
spherical shape is difficult to manufacture, and
therefore more expensive, so most pressure vessels
are cylindrical with 2:1 semi-elliptical heads or end
caps on each end. Smaller pressure vessels are
assembled from a pipe and two covers. For
cylindrical vessels with a diameter up to 600 mm
(NPS of 24 in), it is possible to use seamless pipe for
the shell, thus avoiding many inspection and testing
issues, mainly the nondestructive examination of
radiography for the long seam if required. A
disadvantage of these vessels is that greater diameters
are more expensive, so that for example the most
economic shape of a 1,000 liters (35 cu ft),
250 bars (3,600 psi) pressure vessel might be a
diameter of 91.44 centimeters (36 in) and a length of
1.7018 meters (67 in) including the 2:1 semi-
elliptical domed end caps,[1-2].
Materials for the pressure vessel
Many pressure vessels are made of steel. To
manufacture a cylindrical or spherical pressure
vessel, rolled and possibly forged parts would have to
INTERNATIONAL JOURNAL OF PROFESSIONAL ENGINEERING STUDIES Volume 9 /Issue 3 / NOV 2017
IJPRES
be welded together. Some mechanical properties of
steel achieved by rolling or forging, could be
adversely affected by welding, unless special
precautions are taken. In addition to adequate
mechanical strength, current standards dictate the use
of steel with a high impact resistance, especially for
vessels used in low temperatures. In applications
where carbon steel would suffer corrosion, special
corrosion resistant material should also be used.
Some pressure vessels are made of composite
materials, such as filament wound composite
using carbon fiber held in place with a polymer.
Due to the very high tensile strength of carbon fiber
these vessels can be very light, but are much more
difficult to manufacture. The composite material may
be wound around a metal liner, forming a composite
overwrapped pressure vessel. Other very common
materials include polymers such as PET in
carbonated beverage containers and copper in
plumbing.
Safety Features
The safety feature of pressure vessel are given below
Leak before burst Leak before burst describes a pressure vessel
designed such that a crack in the vessel will grow
through the wall, allowing the contained fluid to
escape and reducing the pressure, prior to growing so
large as to cause fracture at the operating pressure.
Safety valves
Safety valve is a pressure relief valve which
automatically releases a substance from a boiler,
pressure vessel or other system when the pressure
exceeds preset limits.
Pressure Vessel Closures
Pressure vessel closures are pressure retaining
structures designed to provide quick access to
pipelines, pressure vessels, pig traps, filters and
filtration systems. Typically pressure vessel closures
allow maintenance personnel.
Composite Materials
Introduction
Composite materials have been widely used to
improve the performance of various types of
structures. Compared to conventional materials, the
main advantages of composites are their superior
stiffness to mass ratio as well as high strength to
weight ratio. Because of these advantages,
composites have been increasingly incorporated in
structural components in various industrial fields.
Some examples are helicopter rotor blades, aircraft
wings in aerospace engineering, and bridge structures
in civil engineering applications. Some of the basic
concepts of composite materials are discussed in the
following section to better acquaint ourselves with
the behavior of composites.
Fibers
Fibers are the principal constituent in a fiber-
reinforced composite material. They occupy the
largest volume fraction in a composite laminate and
share the major portion of the load acting on a
composite structure. Proper selection of the type,
amount and orientation of fibers is very important,
because it influences the following characteristics of
a composite laminate.
Matrix
In a composite material the fibers are surrounded by a
thin layer of matrix material that holds the fibers
permanently in the desired orientation and distributes
an applied load among all the fibers. The matrix also
plays a strong role in determining the environmental
stability of the composite article as well as
mechanical factors such as toughness and shear
strength.
INTERNATIONAL JOURNAL OF PROFESSIONAL ENGINEERING STUDIES Volume 9 /Issue 3 / NOV 2017
IJPRES
Classification of composite materials
Composite Material: A material composed of two or
more constituents is called composite material.
Composites consist of two or more materials or
material phases that are combined to produce a
material that has superior properties to those of its
individual constituents. The constituents are
combined at a macroscopic level and or not soluble in
each other. The main difference between composite
and an alloy are constituent materials which are
insoluble in each other and the individual constituents
retain those properties in the case of composites,
whereas in alloys, constituent materials are soluble in
each other and forms a new material which has
different properties from their constituents.
Composite materials in general are categorized based
on the kind of reinforcements or the surrounding
matrix. There are four commonly accepted types of
composite materials based on reinforcements
Advantages of composite materials
The advantages of composites over the conventional
materials are high strength to weight ratio, high
stiffness to weight ratio, high impact resistance,
better fatigue resistance, ], Good thermal
conductivity, Low Coefficient of thermal expansion.
As a result, composite structures may exhibit a better
dimensional stability over a wide temperature range,
high damping capacity, [6].
Limitations of composite materials
The limitations of composites are
a) Mechanical characterization of a composite
structure is more complex than that of a metallic
structure.
b) The design of fiber reinforced structure is difficult
compared to a metallic structure, mainly due to the
difference in properties.
c) The fabrication cost of composites is high, rework
and repairing are difficult.
Applications of composite materials
The common applications of composites are
extending day by day. Nowadays they are used in
medical applications too. There is a research going on
the use of composite material in steam industry and
for storing liquids and gases on high pressure, Some
other fields of applications are [7]
Automotive : Drive shafts, clutch plates, fiber
Glass/Epoxy leaf springs for heavy trucks and
trailers, rocker arm covers, suspension arms and
bearings for steering system, bumpers, body panels
and doors.
Problem Definition
The metallic pressure vessels are having good
strength but due to their high weight to strength ratio
and corrosive properties they are least preferred in
aerospace as well as oil and gas industries. These
industries are in need of pressure vessels which will
have low weight to strength ratio without affecting
the strength in this paper a pressure vessel with wall
thickness of 98mm and diameter of 1879mm is used
with different light weight FRP material.
Project Background
In the below point the background of the project is
stated
(a) Brief study of pressure vessel types and working
is discussed in this project.
(b) Modeling of pressure vessel is done in solid
works 2016 design software with wall thickness of
98mm & diameter of 1879mm.
(c) Pressure vessel is assigned five different materials
such as one general material stainless steel and four
fibers reinforced composite material such as E-glass
epoxy, S2 glass, Kevlar-49 and Saffil.
INTERNATIONAL JOURNAL OF PROFESSIONAL ENGINEERING STUDIES Volume 9 /Issue 3 / NOV 2017
IJPRES
(d) Working Pressure 2MPa is applied on the inner
section wall of pressure vessel.
(e) Stress, strain, deformation values as a result due
to pressure is noted and concluded which material
can sustain max pressure against stress strain and
deformation.
Introduction to SOLIDWORKS
Solid Works is mechanical design automation
software that takes advantage of the familiar
Microsoft Windows graphical user interface. It is an
easy-to-learn tool which makes it possible for
mechanical designers to quickly sketch ideas,
experiment with features and dimensions, and
produce models and detailed drawings. A Solid
Works model consists of parts, assemblies, and
drawings.
Design and Scaling of Pressure Vessel
Material Used For This Project: 1. Stainless Steel
2. E-Glass Epoxy (Glass Fiber)
3. S2-Glass (Glass Fiber)
4. Kevlar-49 (Carbon Reinforced Polymer
Fiber)
5. Saffil(5%Sio2-Al2o3 Fiber)
Scaling
No matter what shape it takes, the minimum mass of
a pressure vessel scales with the pressure and volume
it contains and is inversely proportional to
the strength to weight ratio of the construction
material (minimum mass decreases as strength
increases).
Scaling of stress in walls of vessel
Pressure vessels are held together against the gas
pressure due to tensile forces within the walls of the
container. The normal (tensile) stress in the walls of
the container is proportional to the pressure and
radius of the vessel and inversely proportional to the
thickness of the walls. Therefore, pressure vessels are
designed to have a thickness proportional to the
radius of tank and the pressure of the tank and
inversely proportional to the maximum allowed
normal stress of the particular material used in the
walls of the container.Because (for a given pressure)
the thickness of the walls scales with the radius of the
tank, the mass of a tank (which scales as the length
times radius times thickness of the wall for a
cylindrical tank) scales with the volume of the gas
held (which scales as length times radius squared).
The exact formula varies with the tank shape but
depends on the density, ρ, and maximum allowable
stress σ of the material in addition to the pressure P
and volume V of the vessel. (See below for the exact
equations for the stress in the walls.)
Spherical vessel
For a sphere, the minimum mass of a pressure vessel
Where, M= mass
P= pressure difference between ambient
(gauge pressure)
V= volume
ρ= density of the pressure vessel material
σ= maximum working stress that material
can tolerate
Other shapes besides a sphere have constants larger
than 3/2 (infinite cylinders take 2), although some
tanks, such as non-spherical wound composite tanks
can approach this.
Cylindrical vessel with hemispherical ends
This is sometimes called a "bullet" for its shape,
although in geometric terms it is a capsule.
For a cylinder with hemispherical ends,
Where, R is the radius
of pressure vessel, W is the middle cylinder width
only, and the overall width is W + 2R
INTERNATIONAL JOURNAL OF PROFESSIONAL ENGINEERING STUDIES Volume 9 /Issue 3 / NOV 2017
IJPRES
Cylindrical vessel with semi-elliptical ends
In a vessel with an aspect ratio of middle cylinder
width to radius of 2:1,
The other factors are constant for a given vessel
shape and material. So we can see that there is no
theoretical "efficiency of scale", in terms of the ratio
of pressure vessel mass to pressurization energy, or
of pressure vessel mass to stored gas mass. For
storing gases, "tank age efficiency" is independent of
pressure, at least for the same temperature.
So, for example, a typical design for a minimum
mass tank to hold helium (as a pressurant gas) on a
rocket would use a spherical chamber for a minimum
shape constant, carbon fiber for best possible M/pv,
and very cold helium for best possible
Stress in thin-walled pressure vessels
Stress in a shallow-walled pressure vessel in the
shape of a sphere is
Where σϴ is hoop stress, or stress in the
circumferential direction, σlong is stress in the
longitudinal Direction, p is internal gauge pressure; r
is the inner radius of the sphere, and the thickness of
the sphere wall. A vessel can be considered "shallow-
walled" if the diameter is at least 10 times
(sometimes cited as 20 times) greater than the wall
depth.
Stress in a shallow-walled pressure vessel in the
shape of a cylinder is almost all pressure vessel
design standards contain variations of these two
formulas with additional empirical terms to account
for wall thickness tolerances, quality control
of welds and in-service corrosion allowances.
For example, the ASME Boiler and Pressure Vessel
Code (BPVC) (UG-27) formulas are
Spherical shells:
Cylindrical shells:
Where E is the joint efficient, and all others variables
as stated above.
Modeling of Pressure Vessel
Modeling of pressure vessel is done on solid works
2016 and analyses for the stress and strain values
through ansys software, the modeling of pressure
vessel is done step wise which are given below.
Figure: Making boss extrude
Figure: Making shell for 98 mm wall thickness of
pressure vessel
Figure: Making boss extrude on edge
INTERNATIONAL JOURNAL OF PROFESSIONAL ENGINEERING STUDIES Volume 9 /Issue 3 / NOV 2017
IJPRES
Figure: Making fillet on edge
Figure: Making holes cut extrude
Figure: Making boss- extrude
Figure: Making fillet on edges
Figure: Making boss extrude for boilers foot
construction
Figure: Making cut extrude to remove material
Figure: Making boss extrude on edges
Figure: Making cut-extrude hole
Figure: Making boss extrude on other edge
Figure: Making cut extrude
INTERNATIONAL JOURNAL OF PROFESSIONAL ENGINEERING STUDIES Volume 9 /Issue 3 / NOV 2017
IJPRES
3D model presser vessel
SIMULATION
Simulation is a design analysis system. Simulation
provides simulation solutions for linear and nonlinear
static, frequency, buckling, thermal, fatigue, pressure
vessel, drop test, linear and nonlinear dynamic, and
optimization analyses.
Finite Element Method
The software uses the finite element method (fem).
Fem is a numerical technique for analyzing
engineering designs. Fem is accepted as the standard
analysis method due to its generality and suitability
for computer implementation. Fem divides the model
into many small pieces of simple shapes called
elements effectively replacing a complex problem by
many simple problems that need to be solved
simultaneously.
Introductions to Ansys
ANSYS delivers innovative, dramatic
simulation technology advances in every major
Physics discipline, along with improvements in
computing speed and enhancements to enabling
technologies such as geometry handling, meshing and
post-processing. These advancements alone represent
a major step ahead on the path forward in Simulation
Driven Product Development.
Material applied and their properties:
Table: material properties
Mesh
Figure: meshing of pressure vessel
Boundary condition
Figure: Applying fixed support.
Load
Figure: Applying the load of 3 Mpa inside the
pressure vessel walls.
INTERNATIONAL JOURNAL OF PROFESSIONAL ENGINEERING STUDIES Volume 9 /Issue 3 / NOV 2017
IJPRES
RESULTS
Stainless Steel
Stress, deformation and strain on pressure vessel
made up of stainless steel
Stress
Figure: Stress on pressure vessel
Deformation
Figure: Deformation on pressure vessel
Strain
Figure: Strain on pressure vessel
E Glass Epoxy
Stress, deformation and strain on pressure vessel
made up of E Glass Epoxy.
Stress
Figure: Stress on pressure vessel
Deformation
Figure: Deformation on pressure vessel
Strain
Figure: Strain on pressure vessel
S2 Glass
Stress, deformation and strain on pressure vessel
made up of S2 Glass
Stress
Figure: Stress on pressure vessel
Deformation
Figure: Deformation on pressure vessel
INTERNATIONAL JOURNAL OF PROFESSIONAL ENGINEERING STUDIES Volume 9 /Issue 3 / NOV 2017
IJPRES
Strain
Figure: Strain on pressure vessel
Kevlar-49
Stress, deformation and strain on pressure vessel
made up of Kevlar-49
Stress
Figure: Stress on pressure vessel
Deformation
Figure: Deformation on pressure vessel
Strain
Figure: Strain on pressure vessel
Saffil
Stress, deformation and strain on pressure vessel
made up of saffil.
Stress
Figure: Stress on pressure vessel
Deformation
Figure: Deformation on pressure vessel
Strain
Figure: Strain on pressure vessel
Discussion
In this discussion the selection of best suitable
material against steel is analyzed by obtained results
of simulation. All the simulation results of materials
are tabulated below
Table: Simulation results of pressure
Selection of material for pressure vessel according
to stresses in the material
In this dissertation we are going to find the best
suitable material for pressure vessel which can
replace the stainless steel and shows the approximate
INTERNATIONAL JOURNAL OF PROFESSIONAL ENGINEERING STUDIES Volume 9 /Issue 3 / NOV 2017
IJPRES
stress value as compared to values of stainless steel
.to find out the best material suitable for pressure
vessel in replacement of steel. Simulations are carried
out for the analysis of pressure vessel with 98 mm
thickness of wall and the internal pressure applied
was 2 Mpa the simulation was carried out with taking
materials they are stainless steel,E-Glass epoxy, S2-
glass, Saffil and Kevlar - 49 the values of wall
thickness and internal pressure is kept constant for all
the materials and analyzed. In this simulation
analysis the stress of pressure vessel made up of
Kevlar-49 has shown less stress when compared to
all other material that is the maximum stress of
156.73MPa.Kevlar-49 has shown very less stress
when compared to stainless steel which shown
maximum stress of 157.71 MPa,and other materials
E-glass Epoxy 159.12MPa, S2 glass 159.12,
Saffil159.5MPa as shown in the figure 5.47. As per
the objective of this dissertation to find the best
alternative material to stainless steel has been
selected as per its maximum stress value comparing
with all material Kevlar-49 has been found suitable in
replacement of stainless steel. The variation in the
stress value of this two material is because Kevlar-49
is a light material but has high tensile strength to
weight ratio that is more than steel and it has a very
high amount of thermal protection.
Figure 5.47: Maximum stresses across pressure
vessel of different materials
Selection of material for pressure vessel according
to strain in the material
Here in this section we are discussing about the strain
values of materials which are being used for
manufacture of pressure vessel. in this dissertation
we are going to find which material has the ability to
replace stainless steel as a pressure vessel material in
this process we have given a fixed dimensions of
pressure vessel and the internal pressure for all the
materials and carried out the simulation of each
material from which we got all the values of
maximum strains in material that are Stainless Steel
0.00081782 mm/mm, E-Glass Epoxy
0.0020057mm/mm,S2-glass 0.0019618mm/mm,
Saffil 0.00059901mm/mm, Kevlar-49 0.0010187
mm/mm as shown in the figure 5.48 in this
simulation Kevlar-49 has shown strain of 0.0010187
mm/mm which is more than the stainless steel when
compare to strain value of it 0.00081782 mm/mm as
shown in the Table 5.2other materials which are
having less strain values then Kevlar-49 which makes
Kevlar-49 more suitable replacement for stainless
steel in pressure vessel. Stainless steel material has
more stiffness that is young’s modulus more than the
Kevlar -49 due to which the variation in the strain
values are shown in the analysis in spite of having a
same dimensions and application of load.
Figure: Maximum strains across pressure vessel of
different materials
Selection of material for pressure vessel according
to deformation in the material
The deformation analysis of pressure vessel using
five different materials is carried out using ANSYS
software by maintaining default dimensions of
pressure vessel and the load applied for all the
INTERNATIONAL JOURNAL OF PROFESSIONAL ENGINEERING STUDIES Volume 9 /Issue 3 / NOV 2017
IJPRES
materials. That are Stainless Steel, E-Glass Epoxy,
S2-glass, Saffil and Kevlar-49 from the result of this
analysis we are going to find the best material which
can replace stainless steel as a construction material
for pressure vessel. the values of analysis has
obtained by using ANSYS software the values are
Stainless Steel 0.95058mm,E-Glass
Epoxy2.2948mm,S2-glass 2.2446mm Saffil
0.66429mm Kevlar-49 1.1666mm as shown in the
Table 5.2. In this simulation Kevlar - 49 has shown
the deformation value of 1.1666mm under stress of
156.73MPa whereas stainless steel has shown the
deformation of 0.95058mm under the stress value of
157.71 MPa as shown in the figure 5.49While
comparing these deformation values of all the
material with respect to strains and applied load the
Kevlar-49 has selected as most suitable material in
replacement of stainless steel.
Figure: Total deformation across pressure vessel of
different materials
CONCLUSION
In this study modeling and analysis of pressure vessel
is performed. Further materials of these designs have
been changed and analyzed. ANSYS 14.5 software is
used to analyze the best suitable material which can
replace the stainless steel. The following tables has
been prepared, showing the comparison of
Effectiveness of all materials as shown in the table
5.2 the thickness of the pressure vessel walls were
taken as 98 mm and the pressure inside the pressure
vessel has been taken as 2 MPa.From the simulation
results it has been observed that Kevlar -49 fiber
material is showing least stress value compare to all
other materials.
(1) Kevlar -49 is also a very lightweight material
compared to other FRP & stainless steel materials.
(2) Kevlar-49 material has deformation
approximately same as steel at 2MPa pressure across
the pressure vessel. With the simulation of five
materials for pressure of 2MPa as a material for
pressure vessel of 98mm wall thickness and with the
tabulated values and graphs it has been observed that
for the construction of pressure vessel the
conventional stainless steel can be replaced with
Kevlar-49 composite material.
REFERENCES
(1) S.Xu & M.Yu, “Shakedown Analysis of Thick
Walled Cylinders Subjected to Internal Pressure with
the Infield Strength Criterion”, International Journal
of Pressure vessels& piping, Vol. *@, pp.706-712,
2005.
(2) D. Kozak, J. Sertic, “Optional Wall-Thickness Of
The Spherical Pressure Vessel With Respect To
Criterion About Minimal Mass And Equivalent
Stress, In: Annals Of The Faculty Of Engineering Of
Engineering Hunedoara, Tome Iv Vol. 4 (2), pp. 173-
178, 2006.
(3) B. Harris and A. R. Bunsell, "Impact Properties of
Glass Fiber/Carbon Fiber Hybrid Composites,"
Composites, vol. 6, pp. 197-201, 1975.
(4) T. Hayashi, "On the Improvement of Mechanical
Properties of Composites by Hybrid Composition," in
Proceedings Of The 8th International Reinforced
Plastics Conference, Paper No. 22, pp. 149-152,
1972.
(5) I.M Lavit & N.U. Trung, “Thermoelastoplastic
Deformations of A Thick Walled Cylinder With A
Radial Crack”, Journal of Applied Mechanics &
Technological Physics, Vol 49, pp. 491-499, 2008.