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 Total Downloads : 1188
 Authors : Dr.B.Balakrishna, S.Hari Krishna
 Paper ID : IJERTV1IS5453
 Volume & Issue : Volume 01, Issue 05 (July 2012)
 Published (First Online): 03082012
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Fracture Toughness Prediction Of Brittle & Ductile Materials
Dr.B.Balakrishna1, S.Hari krishna2
1Associate professor, Department of mechanical engineering, University College of engineering, JNTU Kakinada, A.P, INDIA.
2PG student, Department of mechanical engineering, University College of engineering, JNTU Kakinada, A.P, INDIA.
Abstract
The mechanisms of fatiguecrack propagation are examined with particular emphasis on the similarities and differences between cyclic crack growth in ductile materials, such as metals, and corresponding behavior in brittle materials, such as ceramics. Which promote crack growth, and mechanisms of cracktip shielding behind the tip (e.g., crack closure), which impede it. Brittle & ductile materials fail in a time dependent manner in service and how to estimate the lifetimes that can be expected for such materials. In addition, we describe procedures to evaluate the confidence with which these lifetime predictions can be applied. The widely differing nature of these mechanisms in ductile and brittle materials and their specific dependence upon the alternating and maximum driving forces (e.g., K and Kmax) provide a useful distinction of the process of fatiguecrack propagation in different classes of materials; moreover, it provides a rationalization for the effect of such factors as load ratio and crack size.Major aspect of the failure is stress intensty factor calculted and next which the Residual stress is calcuated to the ductile and brittle,And finally calculates the life prediction for the ductile and brittle materials and Results to be compared ,Good agrement to the materials.
Keywords Brittle and Ductile materials, Stress intensity factor, Residual stregth, Fracture toughness, Fracture mechanics, Life prediction.
1. Introduction
Techniques to determine reliability of components fabricated from brittle materials (e.g., ceramics and glasses) have been extensively developed over the last 30 years.[17] Reliability is generally defined as the
probability that a component, or system, will perform its intended function for a specified period of time.[8] Accordingly, the two overarching principles influencing reliability are the statistical nature of component strength and its timedependent, environmentally en hanced degradation under stress. The statistical aspect of strength derives from the distribution of the most severe defects in the components (i.e., the strengthdetermining flaws).[9 15] The timedependent aspect of strength results from the growth of defects under stress and environment, resulting in timedependent component failure.[1619]. These concepts have lead to a lifetime prediction formalism that incorporates strength and crack growth as a function of stress. Predicted reliability, or lifetime, is only meaningful, however, when coupled with a confidence estimate. Therefore, the final step in the lifetime prediction process must be a statistical analysis of the experimental results.[2, 2022].

General Considerations
The most basic assumption made in this is that the material whose lifetime is of interest is truly brittle; that means there are no energy dissipation Mechanisms (e.g., plastic deformation, internal friction, phase transformations, creep) other than rupture occurring during mechanical failure. It has been well documented that brittle materials fail from flaws that locally amplify the magnitude of stresses to which the material is subjected.[9, 10,14,15,23] These flaws, e.g., scratches, pores, pits, inclusions, or cracks, result from processing, handling, and use conditions.
For a given applied or residual stress, the initial flaw distribution determines whether the material will survive application of the stress or will immediately fail. Similarly, the evolution of the flaw population with time determines how long the surviving material will remain intact.
1.1 Strength:
Because the flaw population under stress defines the initial strength of a brittle material, it is necessary to characterize this distribution or, equivalently, the Distribution of initial strengths.

FUNDAMENTALS OF FRACTURE:
Simple fracture is the separation of a body into two or more pieces in response to an imposed stress that is static (i.e., constant or slowly changing with time) and at temperatures that are low relative to the melting temperature of the material. The applied stress may be tensile, compressive, shear, or torsional; the present discussionwill be confined to fractures that result from uniaxial tensile loads. For engineering materials, two fracture modes are possible: ductile and brittle. Classification is based on the ability of a material to experience plastic deformation. Ductile materials typically exhibit substantial plastic deformation with high energy absorption before fracture. On the other hand, there is normally little or no plastic deformation with low energy absorption accompanying a brittle fracture. Ductile and
brittle are relative terms; whether a particular fracture is one mode or the other depends on the situation. Ductility may be quantified in terms of percent elongation and percent reduction in area . Furthermore, ductility is a function of temperature of the material, the strain rate, and the stress state. The disposition of normally ductile materials to fail in a brittle manner
Any fracture process involves two stepscrack formation and propagation in response to an imposed stress.

Fatiguecrack propagation in ductile metallic materials
Subcritical crack growth can occur at stress intensity K levels generally far less than the fracture toughness Kc in any metallic alloy when cyclic loading is applied. In simplified concept, it is the accumulation of damage from the cyclic plasticdeformation in the plastic zone at the crack tip that accounts for the intrinsic mechanism of fatigue crack growth at K levels below Kc. The process of fatigue failure itself consists of several distinct processes involving initial cyclic damage (cyclic hardening or softening), formation of an initial fatal flaw (crack initiation), macroscopic propagation of this flaw (crack growth), and final catastrophic failure.


MATERIALS TO BE USED TO OUR WORK & MATERIAL PROPERTIES

Mechanical properties of Al2O3 ceramics (99.5%)
Units
Density
Kg/m3
3.90
Poisons Ratio
–
0.22
UTS
MPa
262
Youngs
modulus
GPa
370
Flexural
strength
Mpa
379

Mechanical properties of 1045 steel:
Units
Density
Kg/m3
7.872 103
Poisons Ratio
–
0.29
UTS
MPa
621
Yield strength
MPa
382
Youngs
modulus
GPa
210
Elongation
%
16
Reduction in area
%
35

Mechanical properties of Aluminium: Material: 2024T3 Al alloy:
Units
/td>
Density
Kg/m3
Poisons Ratio
–
0.3
Yield strength
Mpa
355
Youngs modulus
GPa
71
Tensile stress
Mpa
80
Fracture toughness
Mpa
70.6


Stress Intensity factor:
The local stresses near a crack depend on the product of the nominal stress ( ) and the square root of the halfflaw length. The relationship is called the stress intensity factor (K).
Units: – MNm3/2 or MPam1/2
Geometry and loading conditions influence this environment through the parameter K, which may be determined by suitable analysis. This single parameter
K is related to both the stress level and crack size. The determination of stress intensity factor is a specialist task necessitating the use of a number of analytical and numerical techniques. The important point to note is that it is always possible to determine KI to a sufficient accuracy for any given geometry or set of loading conditions.
Thus, the form of fracture of ceramic materials is fundamentally brittle, with Mode I, Mode II,Mode
III. The ModeI stress intensity factor, KIc is the most often used engineering design parameter in fracture mechanics. Typically for most materials if a crack can be seen it is very close to the critical stress state predicted by the "Stress Intensity Factor". Various modes of failures are shown in fig
Modes of crack surface displacement

Fracture Analysis

Description of the Model:
The rectangular bar having dimensions of length 30 mm, height 5.75mm and width 2.88 mm. The crack was poisoned at the middle of the rectangular bar and it is an edge type. The length of the crack is 1.412 mm and the width is 3.23mm.
Figure 1: Shows the rectangular bar
5.2. Stress intensity factor figs
Figure 2: Nodal solution of a SEVNB specimen with a crack Length of a=1.412(CERAMICS)
figure 3. Nodal solution of a SEVNB speimen with a Crack length of a=1.412mm(STEEL)
Figur 4.Nodal solution of a SEVNB specimen with a crack length of a= 1.412 mm (ALUMINIUM)
Stress strain characteristics for brittle mate rials are different in two ways. (1) They fail by rupturing (separation of atomic planes) at the ultimate stress (Su) without any noticeable yielding (slip phenomena) before the rupture. It can be presumed that Syp value of brittle material is greater than Su (2) Brittle materials are generally stronger in compression than in tension and consequently, for brittle materials Su in compression (Suc) is greater than Su in tension (Sut). As a result of this Suc and Sut are the limiting stresses in mechanical de sign with brittle materials
5.3 STRESS INTENSITY FACTOR FORMULAS:
For ceramics:
There are numerous expressions which make it possible to calculate the Mode I critical stress intensity factor in bending tests, starting from the test load and the geometry of the notched beam. Guinea et al. [24] proposed the use of thefollowing equation.
For ModeI (Opening Mode) :
5.3.1 FOR CERAMICS
for different crack lengths, w = the height of the beam, B = beamdepth Where
( =
5.3.3 FOR ALUMINIUM:
For ModeI (Opening Mode)
KI is given for rectangular bar as follows:
K P f
I B W
3 S
KI is given for rectangular bar as follows:
f W 1.99 1 2.15 3.93 2.7 2
3
2 1 2 1 2
..1
where: P =critical applied load =705 N, S = length of the rectangular bar, Y()=geometry parameter for different crack length ,
, w = the height of the beam, B = beam depth.
Where
Y () = 2
F() = 0.83 0.31 +0.14 …………………2a
and
H() = 0.42+0.82 0.31.2b
5.3.2 FOR STEEL :
For ModeI (Opening Mode)
KI is given for rectangular bar as follows:
K1 =
where: P =critical applied load =705,
S = length of the rectangular bar
=geometry parameter, l= for different crack lengths, w = the height of the beam, B = beam depth,
where: P =critical applied load = 705 N, S = length of the rectangular bar, =geometry parameter for different crack length, w = the height of the beam, B = beam depth.

STRESS INTENSITY FACTOR:
TABLE 1: for stress intensity values for 3 materials .
Crack Lengtha (mm)
Theoretical CERAMICS
Theoretical STEEL
Theoretical ALUMINIUM
1.412
717.810
360.58
728.55
1.414
717.814
361.11
727.63
1.416
717.816
361.64
726.71
1.418
720.090
362.17
725.79
1.420
720.090
362.17
725.79
1.422
722.098
363.23
723.95
1.424
723.102
363.76
723.03
1.426
724.106
364.29
722.11
1.428
725.108
364.82
721.18
1.430
726.113
365.35
720.27
Stress intensity factor
800
600
400
SIF Vs Crack length
CERAMICS STEEL
fc
(1 0.025.
K
. .a
2 0.06.
4 ). sec
2
200
0
ALUMINIUM
Where K=300 MPam , =geometry factor.
5.5.3 FOR ALUMINIUM:
1.41 1.42 1.43 1.44
Crack length
Figure 5 : shows the stress intensity values for CERAMICS, STEEL, ALUMINIUM.

Residual bending Strength:

for CERAMICS
Case1: Plastic collapse condition. b
Where b= residual bending strength (Mpa), P = maximum load at specimen breakage (N),
W = width of the plate (mm),t = test specimen thickness, l= various crack lengths (mm)
Case1: Plastic collapse condition
(w a) .
fc w
Where fc= residual strength, w= width of the plate, a= crack length, y= stress 350 N/mm2 for steel ( assumed)
Case2: Fracture toughness condition
K
fc . .a
Where K=280 MPam, =geometry factor,
Case2: Fracture toughness condition
K
(1 0.025. 2
0.06.
4 ). sec
2
fc . .a
Table 2 : Fracture collapse condition (case I)
Crack Length a (mm)
Residual strength (N/mm2) CERAMICS
Residual strength (N/mm2) STEEEL
Residual strength (N/mm2) ALUMINIU
M
1.412
31.308
2.3696
2.3696
1.414
31.352
2.3685
2.3685
1.416
31.397
2.3674
2.3674
1.418
31.441
2.3664
2.3664
1.420
31.485
2.3653
2.3653
1.422
31.530
2.3642
2.3642
1.424
31.574
2.3631
2.3631
1.426
31.618
2.3620
2.3620
Where K=370MPam ,=geometry factor
(1 0.025. 2

FOR STEEL:

0.06.
4 ). sec
2
ase1: Plastic collapse condition
(w a) .
fc w
y
Where fc= residual strength, w= width of the plate, a= crack length, = stress 350 N/mm2 for steel ( assumed)
caseII: Fracture toughness condition
1.428
31.663
2.3609
2.3609
1.430
31.707
2.3598
2.3598
Residual strength Vs Crack length
Residual strength
40
30
20
10
CERAMICS STEEL
ALUMINIU M
200
RESIDUAL STRENGTH
150
100
50
0
Residual strength Vs crack length
CERAMICS STEEL
ALUMINU M
1.41 1.42 1.43 1.44
CRACK LENGTH in mm
0
1.41 1.42 1.43 1.44
Crack length in mm
Figure6 :shows the Residual strength Vs Crack length(case I)
Table 3: Fracture toughness condition(case II):
Figure7:shows the Residual strength Vs Crack length(case II)



LIFE PREDICTION

Fatigue design and life prediction:
The marked sensitivity of fatiguecrack growth rates to the applied stress intensity in intermetallics and ceramics, both at elevated and especially ambient temperatures, presents unique challenges to damagetolerant design and life prediction methods for structural components fabricated from these materials. For safetycritical applications involving most metallic structures, such procedures generally rely on the integration of data relating crackgrowth rates (da=dN or da=dt ) to the applied stress intensity (1K or Kmax) in order to estimate the time or number of cycles Nf to grow the largest undetectable initial flaw ai to critical size ac, viz.

For steel:
Crack growth life has been predicted using Paris law. The formula for predicting the life:
Crack length
(mm)
Residual Strength
(N/mm2)
CERAMICS
Residual Strength
(N/mm2)
STEEL
Residual Strength
(N/mm2)
ALUMINIUM
1.412
175.903
0.9596
0.8956
1.414
175.779
0.9604
0.8975
1.416
175.654
0.9612
0.8996
1.418
175.530
0.9620
0.9016
1.420
175.405
0.9628
0.9036
1.422
175.281
0.9636
0.9056
1.424
175.156
0.9644
0.9076
1.426
175.032
0.9652
0.9096
1.428
174.907
0.9660
0.9116
1.430
174.783
0.9668
0.9136
dN da
c( K )m
where N= Crack life which is initialized to zero, c, m= Material constants (2 to 4),
K= Stress intensity factor.
K= , f a W
= Geometry
a = crack length , da = increment in crack length
=0.002, dN = increment in crack life.

FOR ALUMINIUM:
dN da
c( K )m
where
N= Crack life which is initialized to zero, c, m = Material constants (2 to 4),
K= Stress intensity factor.
P a a
= Geometry factor.
K f f
B W W , W
a = crack length , da = increment in crack length =0.002, dN = increment in crack life.
Crack Length (mm)
Life prediction
Ceramics
Life prediction
Steel
Life prediction Aluminium
1.412
1.87E05
1.457E15
3.56E04
1.414
1.85E05
1.448E15
3.53E04
1.416
1.83E05
1.440E15
3.51E04
1.418
1.81E05
1.432E15
3.48E04
1.420
1.79E05
1.424E15
3.46E04
1.422
1.77E05
1.407E15
3.44E04
1.424
1.75E05
1.399E15
3.42E04
1.426
1.73E05
1.391E15
3.39E04
1.428
1.71E05
1.384E15
3.37E04
1.430
1.69E05
1.376E15
3.34E04

FOR CERAMICS
CRACK LENGTH Vs LIFE PREDICTION
Where: C = Constant (2 & 4), ,
Stress intensity factor value, n= 3.6 & p= 1.9 dN = increment in crack life.
Table 4 :LIFE PREDICTION FOR 3 MATERIALS
4.00E04
LIFE PREDICTION
3.00E04
2.00E04
1.00E04
0.00E+00
1.41 1.42 1.43 1.44
CRACK LENGTH
CERAMI CS STEEL
ALUMIN IUM
Figur 8 : shows the Life prediction Vs Crack length



Summary and conclusions:
Although the mechanisms of cyclic fatigue in brittle materials are conceptually different from the well known mechanisms of metal fatigue, First we calculted & consentrated step to my work is stress itensity factor. Finally, the marked sensitivity of growth rates to the applied stress intensity in ceramics and intermetallics implies that projected lifetimes will be a very strong function of stress and crack size; this makes design and life prediction .

Acknowledgment:
It is with a feeling of great pleasure that I would like to express my most sincere heartfelt gratitude to Associate Prof, Dr.B.Balakrishna Dept. of Mechanical Engineering, JNT University, Kakinada .for suggesting the topic for my work and for his ready and noble guidance throughout the course of my Preparing the work. I thank you Sir for your help, inspiration and blessings. Last but not least I would like to express my gratitude to my parents and other family members, whose love and encouragement have supported me throughout my education.
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International Journal of Engineering Research & Technology (IJERT)
ISSN: 22780181
Vol. 1 Issue 5, July – 2012