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Int. J. Mech. Eng. Res. & Tech 2016
M A Nasser and S T Hasan, 2016
ISSN 2454 – 535X www.ijmert.com
Vol. 2, No. 1, February 2016
© 2016 IJMERT. All Rights Reserved
Research Paper
ANALYTICAL SOLUTION OF ISOTHERMAL
FATIGUE IN SOLID CYLINDER
M A Nasser1,2* and S T Hasan1
*Corresponding Author: M A Nasser,  maaltememy69@yahoo.com
empirical isothermal fatigue modeling produced for solid cylindrical subjected to constant
temperatures superimposed with sinusoidal mechanical load applied to the cylinder at different
levels, based on the experimental mechanical properties and boundary conditions. Linear
equations are developed to describe the severity of the temperature gradient, thermal stresses,
and stress and strain intensity factors through the solid cylinder wall with time. Results shown
the effects of the temperature explained by the von-Mises thermal stresses increases with
increase in temperature, the high stress at 400 °C, due to hardness increased of the material
indicated by high modulus of elasticity. The mechanical stress is more effectively than thermal
loading and the stress intensity factor decrease with temperature, except at 400 °C, due to the
increases of both thermal and mechanical stresses give raise the highest values, and the strain
intensity factor calculation is bonded between the 500 °C and 400 °C temperature, influenced
by the modulus of elasticity of the material at that temperature.
Keywords: Isothermal fatigue, Solid cylinder, von-Mises thermal stress, Analytical solution
INTRODUCTION
cycle needs to be identified, so that fatigue
life may be predicted, usually with reference
to the maximum temperature in the loading
cycle on the assumption that this represents
the most damaging condition likely to be
experienced.
Nowadays many industries deal with
components which are subjected to higher
loads at elevated temperatures than before,
due to the increasing requirements regarding
weight and performance. The simplest
process to check the behaviour of the material
at high temperature is the Isothermal Fatigue
(IF),by operating a cyclic fatigue at constant
and uniform temperature to define stress-strain
Several investigators studied theoretically
the infinite long solid cylinder model in different
loading conditions.Ishikawa [1], studied the
theoretical transient thermal stresses when the
1
Faculty of Arts, Computing, Engineering and Sciences, Sheffield Hallam University, Sheffield, United Kingdom.
2
Board of Technical Education, Technical College, Baghdad, Iraq.
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Int. J. Mech. Eng. Res. & Tech 2016
M A Nasser and S T Hasan, 2016
surface is subjected to a sudden temperature
drop after a sudden temperature rise. Lopez
Molina and Trujillo [2], estimated the thermal
stresses when a constant heat flux is applied
to its surface employing hyperbolic heat
conduction model. Jun Zhao et al. [3],
presented an analysis of transient thermomechanical study for a solid cylinder material
under rapid exposure to the connective
medium of different temperatures, obtaining
the analytical solution formula by using the
separation of variables method and Moncef
Aouadi [4], studied the thermoplastic-diffusion
interactions when applying a thermal shock on
the solid cylinder surface employing the
Laplace transform and numerical Laplace
inversion in the solution.
generated experimentally and the physical
properties are details in Tables 1 and 2 [5].
Transient Temperature
Formulation
To analyse the solid cylinder subjected to
constant thermal loading at the outer surface,
the temperature distribution is estimated to
evaluate the thermal stresses induced due to
the thermal loading. To find a fair theoretical
scenario similar to the experimental work, the
time relation is the good crosslink between
the temperature distribution and the applied
cyclic mechanical load, by using the transient
thermal loading introduced across the solid
cylinder wall thickness in which results vary
with position through the wall thickness and
time.
THEORETICAL ANALYSIS
Table 1: The Mechanical Properties
of AISI 420 Experimentally Generated
To present the details of the analysis of the
isothermal fatigue assuming the material is
uniform, homogeneous and isotropic with
physical and thermal properties are constant
and independent of temperature. Also, the
temperature distribution varies only with the
radial coordinate within the cylinder, and the
cylinder is initially at a uniform temperature. In
this study the selection of AISI 420 martensitic
stainless steel, where the strength data
Property
300 °C
400 °C
500 °C
600 °C
518
495
484
445
Tensile strength, VTS
- (MPa)
621
595
581
494
Modulus of
Elasticity, E-(GPa)
121.6
193.36
78.06
87.08
poisons ratio, υ
0.24
0.24
0.24
0.24
0.2% yield strength,
ys - (MPa)
Table 2: The Physical Properties of AISI 420
Property
300 °C
400 °C
500 °C
600 °C
Density, ρ-(Kg /m 3)
78 00
780 0
7800
7800
Linear thermal expansion
coefficient, α-(1/°C)
10.8 x 10-6
11.7 x 10-6
11.7 x 10-6
12.2 x 10-6
Thermal conduction coefficient k(W/m.°C)
24.9
24.9
24.9
24.9
Specific heat, c-(J.Kg.°C)
460
460
460
460
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Int. J. Mech. Eng. Res. & Tech 2016
M A Nasser and S T Hasan, 2016
The transient temperature distributions for
a solid cylinder geometric model are given by
using the solution of the general conduction
heat equation:
...(3)
where (T0, Ta) are the initial and outer surface
temperatures, a is the outer radius. Also, n
are the positive roots of the equation:
...(1)
...(4)
where; D is the coefficient of diffusion (i.e.,
In this work four roots ( ) are used to give an
acceptable stability for the temperature
measurements.
are being the thermal conduction
coefficient of the cylinder material, the mass
density, and the specific heat.
Thermal and Combined Stresses
Formulation
Carslaw and Jaeger, has modelled a solid
cylinder using the application of Laplace
transformation in a cylindrical region and the
Bessel's function of the first and second kinds
employing the boundary conditions of the case
studied [6].
An elastic relation for resultant thermal stresses
have been derived by Timoshinko and Goodier
[7], induced the long circular solid cylinder due
to non-symmetrical temperature distribution
under zero axial strain (z = 0).
In this case study the outer surface of the
solid cylinder is subjected to constant
temperature and the initial temperature
assumed to be at a selected temperature.
The general thermal stress formulas for a
solid cylinder in plain strain is given by:
The initial and general boundary conditions
are:
...(5)
....(2)
...(6)
mech
Ta
r
...(7)
Ta
r
From the substitutions of the temperature
distribution in the general formulas in this case
studied the thermal stresses estimation is
given by:
a
The distribution of temperature T(r,t) through
the solid cylinder wall in this case is:
...(8)
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Int. J. Mech. Eng. Res. & Tech 2016
M A Nasser and S T Hasan, 2016
The combined stress is given by adding
thermo-elastic-plastic stress to the applied
mechanical loading:
...(9)
...(15)
STRESS AND STRAIN
INTENSITY FACTORS
FORMULATION
...(10)
The stress intensity factor has been derived
by Tada et al. [8], induced in a solid cylinder
externally circumferential cracked under
uniaxial tensile stress.
where;
...(11)
...(16)
The accumulative influence of the thermal
stress component is generally high strain
process especially at high temperatures and
additionally increased when combined with
mechanical loading takes place, to understand
such behaviour. Thermo-Elastic-Plastic
analysis introduced by using the von-Mises
Plasticity criterion given by:
...(17)
The strain intensity factor can be obtained
by the division of the stress intensity factor by
the modulus of Elasticity.
...(18)
where; c is the crack length and a is the outer
radius of the solid cylinder.
...(12)
von-Mises stress is given by;
RESULTS AND DISCUSSION
The temperature distribution through the solid
cylinder wall is shown in Figure 1, where the
temperature is increased through the cylinder
wall, reaching to the steady-state and
equalized to the outer surface supplied
temperature, depending on the solid cylinder
dimension (outer radius) and the mechanical
and thermal properties of the material (where
at zero radius and time the results goes to
infinity). The thermal radial and hoop stresses
distribution in the outer surface of the solid
cylinder at different time intervals are illustrated
in Figures 2 and 3 shows the maximum
...(13)
The mechanical load is formulated by
employing a sinusoidal mechanical stresses
based on the applied stress range oriented
with axis of the cylinder making the ratio of the
minimum applied stress to the maximum,
equal to (
and frequency (f = 5
Hz) by using the formulas:
...(14)
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Int. J. Mech. Eng. Res. & Tech 2016
M A Nasser and S T Hasan, 2016
compression stress at starting time and rapidly
decreased with time reaching to zero and
increases with increasing temperature, except
the highest compression stress at 400 °C, due
to the increased modulus of elasticity results
from the increased hardeness of the material.
The same in axial thermal stress distribution
shown in Figure 4, started with maximum
compression, then reduced with time to
constant and equal to outer surface stress
(
) rises to steady-state.
Figure 1: The Temperature Distribution
with Time at Different Temperature
Levels at Radius (1 mm) from
the Central Axis
The accumulative influence of thermal
stress component described by the
distributions of thermo-elastic-plastic
equivalents von-Mises stress through the solid
cylinder outer surface (outer radius) with time
are shown in Figure 5. The overall behavior
of the von-Mises thermal stress is starting
high value at the beginning (0.05 sec) due to
thermal shock, then rapidly decrease
reaching to steady-state at (1 sec). For that
the isothermal fatigue study will depend on
the steady-state section only. The effect of
increasing the temperature increases the
elastic-plastic thermal stress, except at 400
°C is the higher value due to the high modulus
of elasticity (E) magnitude. It is difficult to
predict the behaviour of the interactive effects
of thermo-mechanical loading through the
cylindrical coordinates in reality. The
distributions of the thermo-mechanical
combined stress for all testing temperatures
in the solid cylinder related to time are
presented in Figure 6. Thermo-mechanical
combined stress increased with increasing
the temperature due to the effect of the
thermal elastic-plastic stress on the cyclic
mechanical loading is rising up the value of
the stress effect. At 400 °C the stress value
Figure 2: The Thermal Radial Stresses
Distribution with Time at Different
Temperature Levels in the Outer Surface
of the Cylinder
Figure 3: The Thermal Hoop Stresses
Distribution with Time at Different
Temperature Levels in the Outer Surface
of the Cylinder
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Int. J. Mech. Eng. Res. & Tech 2016
M A Nasser and S T Hasan, 2016
is jumped regarding to the high thermal and
applied mechanical stresses. The cyclic
behaviour dominant on the combined stress
profile regarding to the constant thermal stress
value, especially over (1 sec), which is the
steady-state refers by isothermal fatigue case.
Figure 4: The Thermal Axial Stresses
Distribution with Time at Different
Temperature Levels in the Outer Surface
of the Cylinder
The distribution of the stress and strain
intensity factors as a result of the combined
stress range with the crack growth in the solid
cylinder at different temperature levels are
shown in Figures 7 and 8. The profiles illustrate
the incremental of the stress intensity factor
with crack propagation through the solid
Figure 5: The Thermal Von Mises Stresses
Distribution with Time at Different
Temperature Levels in the Outer Surface
of the Cylinder
Figure 7: The Stress Intensity Factor
Against the Crack Length at Different
Temperature Levels Started Form the
Outer Surface of the Cylinder
Figure 6: The Combined Stresses
Distribution with Time at Different
Temperature Levels in the Outer Surface
of the Cylinder
Figure 8: The Strain Intensity Factor
Against the Crack Length at Different
Temperature Levels Started Form the
Outer Surface of the Cylinder
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Int. J. Mech. Eng. Res. & Tech 2016
M A Nasser and S T Hasan, 2016
cylinder wall and decreases with increasing
temperature, except at 400 °C is maximized
due to high thermal and mechanical stresses.
Also the curves give an indication that the
theoretical estimation of the material behaviour
is bounded by the maximum stress occurred
at 400 °C and the minimum at 600 °C. While
the strain intensity factor is also increased with
crack propagation, but not always increases
by raising temperature. This is because of the
effect of the modulus of Elasticity of the material
on the profiles, which is clear significantly,
especially at 400 °C and the strain intensity
bounded between the maximum at 500 °C and
400 °C.The correlation between the theoretical
solution and the generated experimental stress
and strain intensity factors data [5], find out
convergence in less than 0.5 mm crack length
limit, which then diverge with the increase in
the crack length, more closely at the stress
intensity factor, but the behavior relation with
the increasing temperature is similarly, as
shown in Figures 9 and 10.
CONCLUSION
In this research the developed empirical
modeling solution based on adding the
experimental fatigue applied load to the
calculated thermo-elastic-plastic stresses
depends on the mechanical properties
generated under elevated tensile tests, gives
a good agreement with the experimental work
results. The conclusions collected from the
theoretical solutions are:
Figure 9: The Stress Intensity Factor
Correlation Between Theoretical
and Experimental Solutions
1 The effects of the temperature explained by
the von Mises thermal stress increase with
an increase in the temperature. The jump
of high stress at 400 °C is due to the high
modulus of elasticity (E). This effect
superimposed with the applied mechanical
cyclic stress, rises up the total combined
effects.
Figure 10: The Strain Intensity Factor
Correlation Between Theoretical
and Experimental Solutions
2 The effect of the mechanical stress on the
combined loading is present as the effective
load than the thermal loading.
3 The stress intensity factor calculation shows
the increase in temperature will decrease
the intensity factor due to reduction of the
strength of the material, which is indicated
by the reduction of mechanical cyclic
loading with increment of temperature.
Except at 400 °C, due to the increases of
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36
Int. J. Mech. Eng. Res. & Tech 2016
M A Nasser and S T Hasan, 2016
both thermal and mechanical stresses give
raise the highest values.
Resistance of Functionally Gradient Solid
Cylinders”, Journal of Materials Science
and Engineering, Vol. A 418, pp. 99-110.
4 The strain intensity factor calculation is
bonded between the upper bond at 500 °C
and the lowest at 400 °C temperature, due
to the significant effects of the modulus of
elasticity of the material at that temperature.
4. Lopez Molina J A and Trujillo M (2006),
“Thermal Stresses in an Infinitely Long
Solid Cylinder Using Green’s Function
and Hyperbolic Heat Equation”,
Proceeding of the 2 nd W SEAS
International Conference on Applied and
Theoretical Mechanics, November,
pp. 20-22, Venice, Italy.
5 The combination of stress and strain
intensity factor theoretical calculations with
the experimental output recorded data
shows the same behaviour with increasing
temperature, and there is a fair correlation
between the profiles at the beginning until
(0.5 mm) crack length after the divergence
increases, due to the combined effect of
cyclic softening with fast crack propagation
on the behaviour of the material.
5. Mohammed A N Ali (2013), “ThermoElastic-Plastic Analysis for Elastic
Component Under High Temperature
Fatigue Crack Growth Rate”, Ph.D.
Thesis of Sheffield Hallam University.
6. Moncef Aouadi (2006), “A Generalized
Thermoelastic Diffusion Problem for an
Infinitely Long Solid Cylinder”,
International Journal of Mathematics and
Mathematical Sciences, Article ID
25976, pp. 1-15.
REFERENCES
1. Carslaw H S and Jaeger J C (1959),
Conduction of Heat in Solids, 2nd Edition,
Oxford University Press, New York.
2. Hiromasa Ishikawa (1978), “A
Thermoelastoplastic Solution for a
Circular Solid Cylinder Subjected to
Heating and Cooling”, Journal of
Thermal Stresses, Vol. 1, pp. 211-222.
7. Tada H, Paris P and Irwin G (1985), “The
Stress Analysis of Cracks Handbook”, Del
Research Corporation.
8. Timoshenko S P and Goodier J N (1970),
Theory of Elasticity, International Student
Edition, McGraw-Hill Book Company.
3. Jun Zhao, Xing A, Yanzheng Li and
Yanghui Zhou (2006), “Thermal Shock
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37
Int. J. Mech. Eng. Res. & Tech 2016
M A Nasser and S T Hasan, 2016
APPENDIX
The Bessel functions are defined by the equations:
...(A.1)
...(A.2)
The details of stresses integrating are given below:
...(A.3)
So, at n = 0.
...(A.4)
and at the outer radius of the solid cylinder, r = b, n= 0;
...(A.5)
where;
...(A.6)
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