Fracture Mechanics: Understanding Theoretical Fracture Strength and Cracks, Summaries of Engineering

Download Fracture Mechanics: Understanding Theoretical Fracture Strength and Cracks and more Summaries Engineering in PDF only on Docsity! FRACTURE  Brittle Fracture: criteria for fracture.  Ductile fracture.  Ductile to Brittle transition. Fracture Mechanics T.L. Anderson CRC Press, Boca Raton, USA (1995). Fracture Mechanics C.T. Sun & Z.-H. Jin Academic Press, Oxford (2012). MATERIALS SCIENCE & ENGINEERING Anandh Subramaniam & Kantesh Balani Materials Science and Engineering (MSE) Indian Institute of Technology, Kanpur- 208016 Email: anandh@iitk.ac.in, URL: home.iitk.ac.in/~anandh AN INTRODUCTORY E-BOOK Part of http://home.iitk.ac.in/~anandh/E-book.htm A Learner’s GuideA Learner’s Guide Theoretical fracture strength and cracks  Let us consider a perfect crystalline material loaded in tension. Failure by fracture can occur if bonds are broken and fresh surfaces are created.  If two atomic planes are separated the force required initially increases to a maximum (Fmax) and then decreases. The maximum stress corresponding to Fmax is the theoretical strength t .  This stress is given by: A pp li ed F or ce ( F ) → r →a0 Cohesive force 0 a E tTFS     E → Young’s modulus of the crystal   → Surface energy  a0 → Equilibrium distance between atomic centres Fma x 0  This implies the theoretical fracture strength is in the range of E/10 to E/6*.  The strength of real materials is of the order of E/100 to E/1000 (i.e. much lower in magnitude). Tiny cracks are responsible for this (other weak regions in the crystal could also be responsible for this).  *For Al: E=70.5 GPa, a0=2.86 Å, (111)= 0.704 N/m.  t = 13.16 GPa  Cracks play the same role in fracture (of weakening) as dislocations play for plastic deformation. By Energy consideration   2 E TFS  By atomistic approach For many metals  ~ 0.01Ea0 Breaking of Liberty Ships Cold waters Welding instead of riveting High sulphur in steel Residual stress Continuity of the structure Microcracks  The subject of Fracture mechanics has its origins in the failure of WWII Liberty ships. In one of the cases the ship virtually broke into two with a loud sound, when it was in the harbour i.e. not in ‘fighting mode’.  This was caused by lack of fracture toughness at the weld joint, resulting in the propagation of ‘brittle’ crack. The full list of factors contributing to this failure is in the figure below.  The steel of the ship hull underwent a phenomenon known as ‘ductile to brittle transition’ due to the low temperature of the sea water (about which we will learn more in this chapter). 2a A crack in a material What is a crack?Funda Check  As we have seen crack is an amplifier of ‘far-field’ mean stress. The sharper the crack-tip, the higher will be the stresses at the crack-tip. It is a region where atoms are ‘debonded’ and an internal surface exists (this internal surface may be connected to the external surface).  Cracks can be sharp in brittle materials, while in ductile materials plastic deformation at the crack-tip blunts the crack (leading to a lowered stress at the crack tip and further alteration of nature of the stress distribution).  Even void or a through hole in the material can be considered a crack. Though often a crack is considered to be a discontinuity in the material with a ‘sharp’ feature (i.e. the stress amplification factor is large).  A second phase (usually hard brittle phase) in a lens/needle like geometry can lead to stress amplification and hence be considered a crack. Further, (in some cases) debonding at the interface between the second phase and matrix can lead to the formation of an interface crack.  As the crack propagates fresh (internal) surface area is created. The fracture surface energy required for this comes from the strain energy stored in the material (which could further come from externally applied loads). In ductile materials energy is also expended for plastic deformation at the crack tip.  A crack reduces the stiffness of the structure (though this may often be ignored). Hard second phase in the materialThough often in figures the crack is shown to have a large lateral extent, it is usually assumed that the crack does not lead to an appreciable decrease in the load bearing area [i.e. crack is a local stress amplifier, rather than a ‘global’ weakener by decreasing the load bearing area]. ~ 2a a Characterization of Cracks Cracks can be characterized looking into the following aspects.  Its connection with the external free surface: (i) completely internal, (ii) internal cracks with connections to the outer surfaces, (iii) Surface cracks. Cracks with some contact with external surfaces are exposed to outer media and hence may be prone to oxidation and corrosion (cracking). We will learn about stress corrosion cracking later.  Crack length (the deleterious effect of a crack further depends on the type of crack (i, ii or iii as above).  Crack tip radius (the sharper the crack, the more deleterious it is). Crack tip radius is dependent of the type of loading and the ductility of the material.  Crack orientation with respect to geometry and loading. Behaviour described Terms Used Crystallographic mode Shear Cleavage Appearance of Fracture surface Fibrous Granular / bright Strain to fracture Ductile Brittle Path Transgranular Intergranular  Fracture can be classified based on: (i) Crystallographic mode, (ii) Appearance of Fracture surface, (iii) Strain to fracture, (iv) Crack Path, etc. (As in the table below).  Presence of chemical species at the crack tip can lead to reduced fracture stress and enhanced crack propagation. Classification of Fracture (based on various features) Brittle Shear Rupture Ductile fracture Little or no deformation Shear fracture of ductile single crystals Completely ductile fracture of polycrystals Ductile fracture of usual polycrystals Observed in single crystals and polycrystals Not observed in polycrystals Very ductile metals like gold and lead neck down to a point and fail Cup and cone fracture Have been observed in BCC and HCP metals but not in FCC metals Here technically there is no fracture (there is not enough material left to support the load) Cracks may nucleate at second phase particles (void formation at the matrix-particle interface) S lip P la ne Cle av ag e p la n e Types of failure in an uniaxial tension test ‘Early Days’ of the Study of Fracture   C.E. Inglis (seminal paper in 1913)[1]  A.A. Griffith (seminal paper in 1920)[2]  Stress based criterion for crack growth (local) → C.E. Inglis.  Energy based criterion for crack growth (global) → A.A. Griffith (Work done on glass very brittle material). [1] C.E. Inglis, Stresses in a plate due to the presence of cracks and sharp corners, Trans. Inst. Naval Architechts 55 (1913) 219-230. [2] A.A. Griffith, The phenomena of rupture and flow in solids, Philos. Trans. R. Soc. Lond. A221 (1920) 163-198. → Fat paper! E cohesive    ca E f 04     For a crack to propagate the crack-tip stresses have to do work to break the bonds at the crack-tip. This implies that the ‘cohesive energy’ has to be overcome.  If there is no plastic deformation or any other mechanism of dissipation of energy, the work done (energy) appears as the surface energy (of the crack faces).  The fracture stress (f) (which is the ‘far field’ applied stress) can be computed using this approach.  f → fracture stress (applied “far-field”)   → crack tip radius  c → length of the crack  a0 → Interatomic spacing Griffith’s criterion for brittle crack propagation  We have noted that the crack length does not appear ‘independently’ (of the crack tip radius) in Inglis’s formula. Intuitively we can feel that longer crack must be more deleterious.  Another point noteworthy in Inglis’s approach is the implicit assumption that sufficient energy is available in the elastic body to do work to propagate the crack.  (‘What if there is insufficient energy?’)  (‘What if there is no crack in the body?’). Also, intuitively we can understand that the energy (which is the elastic energy stored in the body) should be available in the proximity of the crack tip (i.e. energy available far away from the crack tip is of no use!).  Keeping some of these factors in view, Griffith proposed conditions for crack propagation: (i) bonds at the crack tip must be stressed to the point of failure (as in Inglis’s criterion), (ii) the amount of strain energy released (by the ‘slight’ unloading of the body due to crack extension) must be greater than or equal to the surface energy of the crack faces created.  The second condition can be written as: dc dU dc dU s    Us → strain energy  U → surface energy (Energy per unit area: [J/m2])  dc → (‘infinitesimal’) increase in the length of the crack (‘c’ is the crack length) We look at the formulae for Us and U next. Essentially this is like energy balance (with the ‘=‘ sign) → the surface energy for the extended crack faces comes from the elastically stored energy (in the fixed displacement case)  The strain energy released on the introduction of a very narrow elliptical double ended crack of length ‘2c’ in a infinite plate of unit width (depth), under an uniform stress a is given by the formula as below.   E UUU a crackwithcrackwithout 22 s c U energy elastic in Reduction    This is because the body with the crack has a lower elastic energy stored in it as compared to the body without the crack (additionally, the body with the crack is less stiffer). Also, the assumption is that the introduction of a crack does not alter the far-field stresses (or the load bearing area significantly).  Notes:  The units of Us is [J/m] (Joules per meter depth of the crack→ as this is a through crack).  Though Us has a symbol of energy, it is actually a difference between two energies (i.e. two states of a body→ one with a crack and one without).  Half crack length ‘c’ appears in the formula.  E is assumed constant in the process (the apparent modulus will decrease slightly).  a is the ‘far field’ stress (this may result from displacements rather than from applied forces see note later). Should be written with a ve sign if U = (Ufinal  Uinitial) For now we assume that these stresses arise out of ‘applied’ displacements  Now we have the formulae for Us & U to write down the Griffith’s condition: dc dU dc dU s   f a E   2 c 2   LHS increases linearly with c, while RHS is constant.  The ‘equal to’ (=) represents the bare minimum requirement (i.e. the critical condition) → the minimum crack size, which will propagate with a ‘balance’ in energy (i.e. between elastic energy released due to crack extension and the penalty in terms of the fracture surface energy).  The critical crack size (c*): (Note that ‘c’ is half the crack length internal)  A crack below this critical size will not propagate under a constant stress a.  Weather a crack of size greater than or equal to c* will propagate will depend on the Inglis condition being satisfied at the crack-tip.  This stress a now becomes the fracture stress (f)→ cracks of length c* will grow (unstably) if the stress exceeds f (= a) 2 * 2 a fE c    *f c E 2     At constant c (= c*) when  exceeds f then specimen failsGriffith )1(c E 2 2*     f Plane strain conditions  E a 2 s c 2 c U     cf  2 c U                   E a f 22c c 4 U crack a of onintroducti theonenergy in Change   c →  U → 0 *        cdc Ud *c 0c 0 0 An alternate way of understanding the Griffith’s criterion (energy based) cγ U f4 E a 22 s c U    This change in energy (U) should be negative with an increase in crack length (or at worst equal to zero). I.e. (dU/dc) ≤ 0.  At c* the slope of U vs c curve is zero [(dU/dc)c* = 0]. This is a point of unstable equilibrium.  With increasing stress the value of c* decreases (as expected→ more elastic strain energy stored in the material). Stable cracks Unstable cracks For ready reference Negative slope Positive slope c → U → * 1c * 2c Increasin g str ess Griffith versus Inglis criteria ca E f 04    Inglis *f c E 2     Griffith result samethe give criterion Inglis and sGriffith' 8a If 0    0 3a Griffith's and Inglis criterion give the same result the 'Dieter' cross-over criterion If    2 f * E 2 c       a E c f          2 0 * 4  For very sharp cracks, the available elastic energy near the crack-tip, will determine if the crack will grow.  On the other hand if available energy is sufficient, then the ‘sharpness’ of the crack-tip will determine if the crack will grow. A sharp crack is limited by availability of energy, while a blunt crack is limited by stress concentration.  Historically (in the ‘old times’ ~1910-20) fracture was studied using the Inglis and Griffith criteria.  The birth of fracture mechanics (~1950+) led to the concepts of stress intensity factor (K) and energy release rate (G). Due to Irwin and others. Fracture Mechanics  G is defined as the total potential energy () decrease during unit crack extension (dc): Concept of Energy Release Rate (G) dc d G   The potential energy is a difficult quantity to visualize. In the absence of external tractions (i.e. only displacement boundary conditions are imposed), the potential energy is equal to the strain energy stored:  = Us.* * It is better to understand the basics of fracture with fixed boundary conditions (without any surface tractions). dc dU G s With displacement boundary conditions only  Crack growth occurs if G exceeds (or at least equal to) a critical value GC: CGG  For perfectly brittle solids: GC = 2f (i.e. this is equivalent to Griffith’s criterion). Understanding the stress field equation                          2 3 2 1 22    SinSinCos r K I xx r f K Ixx    2 )( →  cYK I  0 ‘Shape factor’ related to ‘Geometry’ Indicates mode I ‘loading’ Half the crack length  “KI (the Stress Intensity Factor) quantifies the magnitude of the effect of stress singularity at the crack tip”[1].  Quadrupling the crack length is equivalent to doubling the stress ‘applied’. Hence, K captures the combined effect of crack length and loading. The remaining part in equation(1) is purely the location of a point in (r, ) coordinates (where the stress has to be computed).  Note that there is no crack tip radius () in the equation! The assumptions used in the derivation of equations (1-3) are:   = 0,  infinite body,  biaxial loading.  ‘Y’ is considered in the next page. [1] Anthony C. Fischer-Cripps, “Introduction to Contact Mechanics”, Springer, 2007. (1) The Shape factor (Y)  It is obvious that the geometry of the crack and its relation to the body will play an important role on its effect on fracture.  The factor Y depends on the geometry of the specimen with the crack.  Y=1 for the body considered in Fig.1 (double ended crack in a infinite body).  Y=1.12 for a surface crack. The value of Y is larger (by 12%) for a surface crack as additional strain energy is released (in the region marked in orange colour in the figure below), due to the presence of the free surface.  Y=2/ for a embedded penny shaped crack.  Y=0.713 for a surface half-penny crack. Summary of Fracture Criteria Criterion named after & [important quantities] Comments Fracture occurs if Relevant formulae Inglis Involves crack tip radius Griffith Involves crack length Irwin [K] Concept of stress intensity factor. KI > KIC (in mode I) - [G] Energy release rate based. Same as K based criterion for elastic bodies. Material KIC [MPam]** Cast Iron 33 Low carbon steel 77 Stainless steel 220 Al alloy 2024-T3 33 Al alloy 7075-T6 28 Ti-6Al-4V 55 Inconel 600 (Ni based alloy) 110 * We have already noted that fracture toughness is a microstructure sensitive property and hence to get ‘true’ value the microstructure has to be specified. ** Note the strange units for fracture toughness! [1] Fracture Mechanics, C.T. Sun & Z.-H. Jin, Academic Press, Oxford (2012). Fracture Toughness* (KIC) for some typical materials [1] Is KIC really a material property like y? Funda Check  Ideally, we would like KIC (in mode I loading, KIIC & KIIIC will be the corresponding material properties under other modes of loading$) to be a material property, independent of the geometry of the specimen*. In reality, KIC depends on the specimen geometry and loading conditions.  The value KIC is especially sensitive to the thickness of the specimen. A thick specimen represents a state that is closer to plane strain condition, which tends to suppress plastic deformation and hence promotes crack growth (i.e. the experimentally determined value of KIC will be lower for a body in plane strain condition). On the other hand, if the specimen is thin (small value ‘t’ in the figure), plastic deformation can take place and hence the measured KIC will be higher (in this case if the extent of plastic deformation is large then KI will no longer be a parameter which characterizes the crack tip accurately). $ Without reference to mode we can call it KC. * E.g. Young’s modulus is a material property independent of the geometry of the specimen, while stiffness is the equivalent ‘specimen geometry dependent’ property..  To use KIC as a design parameter, we have to use its ‘conservative value’. Hence, a minimum thickness is prescribed in the standard sample for the determination of fracture toughness.  This implies that KIC is the value determined from ‘plane strain tests’. I seem totally messed up with respect to the proliferation of fracture criteria! How do I understand all this?  Essentially there are two approaches: global (energy based) and local (stress based).  For linear elastic materials the energy and stress field approaches can be considered equivalent. Q & A E c c )( 4 U energy in Change 2 2 ps    * ps f c E )( 2      Orowan’s modification to the Griffith’s equation to include “plastic energy” 232 )1010(~ )21(~ J/m J/m p 2 s     * p f c E 2     Ductile – brittle transition  Certain materials which are ductile at a given temperature (say room temperature), become brittle at lower temperatures. The temperature at which this happens is terms as the Ductile Brittle Transition Temperature (DBTT).  As obvious, DBT can cause problems in components, which operate in ambient and low temperature conditions.  Typically the phenomena is reported in polycrystalline materials. Deformation should be continuous across grain boundary in polycrystals for them to be ductile. This implies that five independent slip systems should be operatiave (this is absent in HCP and ionic materials).  This phenomenon (ductile to brittle transition) is not observed in FCC metals (they remain ductile to low temperatures).  Common BCC metals become brittle at low temperatures (as noted before a decrease in temperature can be visualized as an increase in strain rate, in terms of the effect on the mechanical behaviour).  As we have noted before a ductile material is one which yields before fracture (i.e. its yield strength is lower in magnitude than its fracture strength).  f ,  y → y (BCC) T → f DBTT y (FCC) No DBTT Griffith versus Hall-Petch *f c E 2     d k iy  Griffith Hall-Petch * '1 c k c E 2 *f      f ,  y → y d-½ → DBT T1 T2 T1 T2f Grain size dependence of DBTT Finer sizeLarge size Finer grain size has higher DBTT  better T1T2 >  Cracks developed during grinding of ceramics extend upto one grain  use fine grained ceramics (grain size ~ 0.1 m)  Avoid brittle continuous phase along the grain boundaries → path for intergranular fracture (e.g. iron sulphide film along grain boundaries in steels → Mn added to steel to form spherical manganese sulphide) Conditions of fracture Torsion Fatigue Tension Creep Low temperature Brittle fracture Temper embrittlement Hydrogen embrittlement Why do we need a large ductility (say more than 10% tensile elongation) material, while ‘never’ actually in service component is going to see/need such large plastic deformation (without the component being classified as ‘failed’). Funda Check  Let us take a gear wheel for an example. The matching tolerances between gears are so small that this kind of plastic deformation is clearly not acceptable.  In the case of the case carburized gear wheel, the surface is made hard and the interior is kept ductile (and tough).  The reason we need such high values of ductility is so that the crack tip gets blunted and the crack tip stress values are reduced (thus avoiding crack propagation).  → c → a0 c* Safety regions applying Griffith’s criterion alone Unsafe Safe 2 f * E 2 c    Unsafe Safe  → c → a0 Safety regions applying Inglis’s criterion alone    a E c f          2 0 * 4  → c → a0 c* 3a0 Griffith safe Inglis unsafe  safe Griffith unsafe Inglis safe  safe Griffith safe Inglis unsafe  unsafe Griffith unsafe Inglis unsafe  unsafe Griffith safe Inglis safe  safe Alloy KIC (MN/m3/2) SCC environment KISCC (MN/m3/2) 13Cr steel 60 3% NaCl 12 18Cr-8Ni 200 42% MgCl2 10 Cu-30Zn 200 NH4OH, pH7 1 Al-3Mg-7Zn 25 Aqueous halides 5 Ti-6Al-1V 60 0.6M KCl 20 http://en.wikipedia.org/wiki/Stress_corrosion_cracking  Another related phenomenon, which can be classified under the broad ambit of SCC is hydrogen embrittlement.  Hydrogen may be introduced into the material during processing (welding, pickling, electroplating, etc.) or in service (from nuclear reactors, corrosive environments, etc.).  Ductile fracture → ► Crack tip blunting by plastic deformation at tip ► Energy spent in plastic deformation at the crack tip Ductile fracture  → y r →  → y r → Sharp crack Blunted crack Schematic r → distance from the crack tip