Fracture Toughness Prediction Of Brittle & Ductile Materials

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 fatigue-crack 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 crack-tip 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 fatigue-crack 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 time-dependent, 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 time-dependent aspect of strength results from the growth of defects under stress and environment, resulting in time-dependent 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].

1. 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.

2. 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.

   1. Fatigue-crack 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.

3. MATERIALS TO BE USED TO OUR WORK & MATERIAL PROPERTIES

   1. 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

   2. 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

   3. Mechanical properties of Aluminium: Material: 2024-T3 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

4. Stress Intensity factor:

   The local stresses near a crack depend on the product of the nominal stress ( ) and the square root of the half-flaw length. The relationship is called the stress intensity factor (K).

   Units: – MNm-3/2 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 Mode-I 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

5. Fracture Analysis

   1. 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 Mode-I (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 Mode-I (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 Mode-I (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.

      1. 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.

      2. Residual bending Strength:

         1. 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

         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)

6. LIFE PREDICTION

   1. Fatigue design and life prediction:

      The marked sensitivity of fatigue-crack growth rates to the applied stress intensity in intermetallics and ceramics, both at elevated and especially ambient temperatures, presents unique challenges to damage-tolerant design and life- prediction methods for structural components fabricated from these materials. For safety-critical applications involving most metallic structures, such procedures generally rely on the integration of data relating crack-growth 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.

      1. 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.

      2. 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	1.87E-05	1.457E-15	3.56E-04
         1.414	1.85E-05	1.448E-15	3.53E-04
         1.416	1.83E-05	1.440E-15	3.51E-04
         1.418	1.81E-05	1.432E-15	3.48E-04
         1.420	1.79E-05	1.424E-15	3.46E-04
         1.422	1.77E-05	1.407E-15	3.44E-04
         1.424	1.75E-05	1.399E-15	3.42E-04
         1.426	1.73E-05	1.391E-15	3.39E-04
         1.428	1.71E-05	1.384E-15	3.37E-04
         1.430	1.69E-05	1.376E-15	3.34E-04

      3. 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

         4.00E-04

         LIFE PREDICTION

         3.00E-04

         2.00E-04

         1.00E-04

         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

7. 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 .

8. 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: 2278-0181

Vol. 1 Issue 5, July – 2012