Mechanics.Of.Materials.Saouma Episode 11 pdf

Draft 14.1 Maximum Circumferential Tensile Stress. 5 12. If the pair of SIF is inside the fracture loci, then that crack cannot propagate without sufficient increase in stress intensity factors. If outside, then the crack is locally unstable and will continue to propagate in either of the following ways: (a) With an increase in the SIF (and the energy release rate G), thus resulting in a global instability, failure of the structure (crack reaching a free surface) will occur. (b) With a decrease in the SIF (and the energy release rate G), due to a stress redistribution, the SIF pair will return to within the locus. Victor Saouma Mechanics of Materials II Draft 6 MIXED MODE CRACK PROPAGATION Victor Saouma Mechanics of Materials II Draft Chapter 15 FATIGUE CRACK PROPAGATION 1 When a subcritical crack (a crack whose stress intensity factor is below the critical value) is subjected to either repeated or fatigue load, or is subjected to a corrosive environment, crack propagation will occur. 2 As in many structures one has to assume the presence of minute flaws (as large as the smallest one which can be detected). The application of repeated loading will cause crack growth. The loading is usually caused by vibrations. 3 Thus an important question that arises is “how long would it be before this subcritical crack grows to reach a critical size that would trigger failure?” To predict the minimum fatigue life of metallic structures, and to establish safe inspection intervals, an understanding of the rate of fatigue crack propagation is required. Historically, fatigue life prediction was based on S − N curves, Fig. 15.1 (or Goodman’s Diagram) Figure 15.1: S-N Curve and Endurance Limit using a Strength of Material Approach which did NOT assume the presence of a crack. 15.1 Experimental Observation 4 If we start with a plate that has no crack and subject it to a series of repeated loading, Fig. 15.2 between σ min and σ max , we would observe three distinct stages, Fig. 15.3 1. Stage 1 : Micro coalescence of voids and formation of microcracks. This stage is difficult to capture and is most appropriately investigated by metallurgists or material scientists, and compared to stage II and III it is by far the longest. 2. Stage II : Now a micro crack of finite size was formed, its SIF’well belowK Ic ,(K<<K Ic ), and crack growth occurs after each cycle of loading. Draft 2 FATIGUE CRACK PROPAGATION Figure 15.2: Repeated Load on a Plate Figure 15.3: Stages of Fatigue Crack Growth 3. Stage III : Crack has reached a size a such that a = a c , thus rapid unstable crack growth occurs. 5 Thus we shall primarily be concerned by stage II. 15.2 Fatigue Laws Under Constant Amplitude Loading 6 On the basis of the above it is evident that we shall be concerned with stage II only. Furthermore, fatigue crack growth can take place under: 1. Constant amplitude loading (good for testing) 2. Variable amplitude loading (in practice) 7 Empirical mathematical relationships which require the knowledge of the stress intensity factors (SIF), have been established to describe the crack growth rate. Models of increasing complexity have been proposed. 8 All of these relationships indicate that the number of cycles N required to extend a crack by a given length is proportional to the effective stress intensity factor range ∆K raised to a power n (typically varying between 2 and 9). 15.2.1 Paris Model 9 The first fracture mechanics-based model for fatigue crack growth was presented by Paris (Paris and Erdogan 1963) in the early ’60s. It is important to recognize that it is an empirical law based on experimental observations. Most other empirical fatigue laws can be considered as direct extensions, or refinements of this one, given by da dN = C (∆K) n (15.1) which is a straight line on a log-log plot of da dN vs ∆K,and ∆K = K max − K min =(σ max − σ min )f(g) √ πa (15.2) Victor Saouma Mechanics of Materials II Draft 15.2 Fatigue Laws Under Constant Amplitude Loading 3 a is the crack ength; N the number of load cycles; C the intercept of line along da dN and is of the order of 10 −6 and has units of length/cycle; and n is the slope of the line and ranges from 2 to 10. 10 Equation 15.1 can be rewritten as : ∆N = ∆a C [∆K(a)] n (15.3) or N =  dN =  a f a i da C [∆K(a)] n (15.4) 11 Thus it is apparent that a small error in the SIF calculations would be magnified greatly as n ranges from 2 to 6. Because of the sensitivity of N upon ∆K, it is essential to properly determine the numerical values of the stress intensity factors. 12 However, in most practical cases, the crack shape, boundary conditions, and load are in such a combination that an analytical solution for the SIF does not exist and large approximation errors have to be accepted. Unfortunately, analytical expressions for K are available for only few simple cases. Thus the stress analyst has to use handbook formulas for them (Tada et al. 1973). A remedy to this problem is the usage of numerical methods, of which the finite element method has achieved greatest success. 15.2.2 Foreman’s Model 13 When compared with experimental data, it is evident that Paris law does not account for: 1. Increase in crack growth rate as K max approaches K Ic 2. Slow increase in crack growth at K min ≈ K th thus it was modified by Foreman (Foreman, Kearney and Engle 1967), Fig. 15.4 Figure 15.4: Forman’s Fatigue Model da dN = C(∆K) n (1 −R)K c − ∆K (15.5) Victor Saouma Mechanics of Materials II Draft 4 FATIGUE CRACK PROPAGATION 15.2.2.1 Modified Walker’s Model 14 Walker’s (Walker 1970) model is yet another variation of Paris Law which accounts for the stress ratio R = K min K max = σ min σ max da dN = C  ∆K (1 −R) (1−m)  n (15.6) 15.2.3 Table Look-Up 15 Whereas most methods attempt to obtain numerical coefficients for empirical models which best approximate experimental data, the table look-up method extracts directly from the experimental data base the appropriate coefficients. In a “round-robin” contest on fatigue life predictions, this model was found to be most satisfactory (Miller and Gallagher 1981). 16 This method is based on the availability of the information in the following table: da dN ∆K R =-1 R=.1 R=.3 R=.4 17 For a given da dN and R,∆K is directly read (or rather interpolated) for available data. 15.2.4 Effective Stress Intensity Factor Range 18 All the empirical fatigue laws are written in terms of ∆K I ; however, in general a crack will be subjected to a mixed-mode loading resulting in both ∆K I and ∆K II . Thus to properly use a fatigue law, an effective stress intensity factor is sought. 19 One approach, consists in determining an effective stress intensity factor ∆K eff in terms of ∆K I and ∆K II , and the angle of crack growth θ 0 . In principle each of the above discussed mixed-mode theories could yield a separate expression for the effective stress intensity factor. 20 For the case of maximum circumferential stress theory, an effective stress intensity factor is given by (Broek 1986): ∆K Ieff =∆K I cos 3 θ 0 2 − 3∆K II cos θ 0 2 sin θ 0 (15.7) 15.2.5 Examples 15.2.5.1 Example 1 An aircraft flight produces 10 gusts per flight (between take-off and landing). It has two flights per day. Each gust has a σ max = 200 MPa and σ min = 50 MPa. The aircraft is made up of aluminum which has R =15 kJ m 2 ,E=70GPa, C =5× 10 −11 m cycle ,andn = 3. The smallest detectable flaw is 4mm. How long would it be before the crack will propagate to its critical length? Assuming K = σ √ πa and K c = √ ER, then a c = K 2 c σ 2 max π = ER σ 2 max π or a c = (70 ×10 9 )(15 ×10 3 ) (200 ×10 6 ) 2 π =0.0084m =8.4mm (15.8) Victor Saouma Mechanics of Materials II Draft 15.3 Variable Amplitude Loading 5 ⇒ N =  a f a i da C[∆K(a)] n =  a f a i da C (σ max − σ min ) n    (∆σ) n ((πa) 1 2 ) n =  8.4×10 −3 4×10 −3 da (5 ×10 −11 )    C (200 −50) 3    (∆σ) 3 (πa) 1.5    ((πa) .5 ) 3 = 1064  .0084 .004 a −1.5 da = −2128a −.5 | .0084 .004 = 2128[− 1 √ .0084 + 1 √ .004 ] =10, 428 cycles (15.9) thus the time t will be: t = (10,428) cycles × 1 10 flight cycle × 1 2 day flight × 1 30 month day ≈ 17.38 month ≈ 1.5 years. If a longer lifetime is desired, then we can: 1. Employ a different material with higher K Ic , so as to increase the critical crack length a c at instability. 2. Reduce the maximum value of the stress σ max . 3. Reduce the stress range ∆σ. 4. Improve the inspection so as to reduce tha ssumed initial crack length a min . 15.2.5.2 Example 2 21 Repeat the previous problem except that more sophisticated (and expensive) NDT equipment is available with a resolution of .1mmthusa i = .1mm t = 2128[− 1 √ .0084 + 1 √ .0001 ] = 184, 583cycles t = 1738 10,428 (189, 583) = 316 months ≈ 26 years! 15.2.5.3 Example 3 Rolfe and Barsoum p.261-263. 15.3 Variable Amplitude Loading 15.3.1 No Load Interaction 22 Most Engineering structures are subjected to variable amplitude repeated loading, however, most experimental data is based on constant amplitude load test. Thus, the following questions arrise: 1. How do we put the two together? 2. Is there an interaction between high and low amplitude loading? 1. Root Mean Square Model (Barsoum) da dN = C(∆K rms ) n (15.10) ∆K rms =   k i=1 ∆K 2 i n (15.11) where ∆K rms is the square root of the mean of the squares of the individual stress intensity factors cycles in a spectrum. 2. Accurate “block by block” numerical integration of the fatigue law ∆a = C(∆K) n ∆N (15.12) solve for a instead of N. Victor Saouma Mechanics of Materials II Draft 6 FATIGUE CRACK PROPAGATION 15.3.2 Load Interaction 15.3.2.1 Observation 23 Under aircraft flight simulation involving random load spectrum: • High wind related gust load, N H • Without high wind related gust load, N L N H >N L , thus “Aircraft that logged some bad weather flight time could be expected to possess a longer service life than a plane having a better flight weather history.” 24 Is this correct? Why? Under which condition overload is damaging! 15.3.2.2 Retardation Models 25 Baseline fatigue data are derived under constant amplitude loading conditions, but many structural components are subjected to variable amplitude loading. If interaction effects of high and low loads did not exist in the sequence, it would be relatively easy to establish a crack growth curve by means of a cycle-by-cycle integration. However crack growth under variable amplitude cycling is largely complicated by interaction effects of high and low loads. 26 A high level load occurring in a sequence of low amplitude cycles significantly reduces the rate of crack growth during the cycles applied subsequent to the overload. This phenomena is called Retardation, Fig. 15.5. Figure 15.5: Retardation Effects on Fatigue Life 27 During loading, the material at the crack tip is plastically deformed and a tensile plastic zone is formed. Upon load release, the surrounding material is elastically unloaded and a part of the plastic zone experiences compressive stresses. 28 The larger the load, the larger the zone of compressive stresses. If the load is repeated in a constant amplitude sense, there is no observable direct effect of the residual stresses on the crack growth behavior; in essence, the process of growth is steady state. 29 Measurements have indicated, however, that the plastic deformations occurring at the crack tip remain as the crack propagates so that the crack surfaces open and close at non-zero (positive) levels. 30 When the load history contains a mix of constant amplitude loads and discretely applied higher level loads, the patterns of residual stress and plastic deformation are perturbed. As the crack propagates through this perturbed zone under the constant amplitude loading cycles, it grows slower (the crack is retarded) than it would have if the perturbation had not occurred. After the crack has propagated through the perturbed zone, the crack growth rate returns to its typical steady-state level, Fig. 15.6. 15.3.2.2.1 Wheeler’s Model Victor Saouma Mechanics of Materials II Draft 15.3 Variable Amplitude Loading 7 Figure 15.6: Cause of Retardation in Fatigue Crack Grwoth 31 Wheeler (Wheeler 1972) defined a crack-growth retardation factor C p : da dN retarded = C p  da dN  linear (15.13) C p =  r pi a oL + r poL − a i  m (15.14) in which r pi is the current plastic zone size in the i th cycle under consideration, a i is the current crack size, r poL is the plastic size generated by a previous higher load excursion, a oL is the crack size at which the higher load excursion occurred, and m is an empirical constant, Fig. 15.7. 32 Thus there is retardation as long as the current plastic zone is contained within the previously gen- erated one. Figure 15.7: Yield Zone Due to Overload 15.3.2.2.2 Generalized Willenborg’s Model 33 In the generalized Willenborg model (Willenborg, Engle and Wood 1971), the stress intensity factor K I is replaced by an effective one: K eff I = K I − K R (15.15) in which K R is equal to: K R = φK w R (15.16) φ = 1 − K max,th K max,i s oL − 1 (15.17) Victor Saouma Mechanics of Materials II Draft 8 FATIGUE CRACK PROPAGATION K R = K w R = K oL max  1 − a i − a oL r poL − K max,i (15.18) and a i is the current crack size, a oL is the crack size at the occurrence of the overload, r poL is the yield zone produced by the overload, K oL max is the maximum stress intensity of the overload, and K max,i is the maximum stress intensity for the current cycle. 34 This equation shows that retardation will occur until the crack has generated a plastic zone size that reaches the boundary of the overload yield zone. At that time, a i − a oL = r poL and the reduction becomes zero. 35 Equation 15.15 indicates that the complete stress-intensity factor cycle, and therefore its maximum and minimum levels (K max,i and K min,i ), are reduced by the same amount (K R ). Thus, the retardation effect is sensed by the change in the effective stress ratio calculated from: R eff = K eff min,i K eff max,i = K min,i − K R K max,i − K R (15.19) because the range in stress intensity factor is unchanged by the uniform reduction. 36 Thus, for the i th load cycle, the crack growth increment ∆a i is: ∆a i = da dN = f(∆K, R eff ) (15.20) 37 In this model there are two empirical constants: K max,th , which is the threshold stress intensity factor level associated with zero fatigue crack growth rate, and S oL , which is the overload (shut-off) ratio required to cause crack arrest for the given material. Victor Saouma Mechanics of Materials II [...]... plasticity 11 16.2.2 Causes of Plasticity The permanent displacement of atoms within a crystal resulting from an applied load is known as plastic deformation It occurs when a force of sufficient magnitude displaces atoms from one equilibrium position to another, Fig 16.6 The plane on which deformation occurs is the slip plane 12 τ τ τ τ τ τ 111 1111 1111 11 0000000000000 111 1111 1111 11 0000000000000 111 1111 1111 11... another, Fig 16.6 The plane on which deformation occurs is the slip plane 12 τ τ τ τ τ τ 111 1111 1111 11 0000000000000 111 1111 1111 11 0000000000000 111 1111 1111 11 0000000000000 111 1111 1111 11 0000000000000 111 1111 1111 11 0000000000000 111 1111 1111 11 0000000000000 τ τ Figure 16.6: Slip Plane in a Perfect Crystal Following a similar derivation as the one for the theoretical (normal) strength (Eq 12.19), it can be... 1 0 1 0 1 0 σ E2 1 E +E2 1 1 E 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00 E2 E1 ε 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0σ 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 111 000 E 111 111 000 000 111 111 000 000 111 111 000 000 111 000 111 000 111 000 111 000 E 1 0 1 0 1 0 ε 2 1 0 1 0 1 0 1 0 1 0 1 0 1 Figure 16.12: a) Rigid Plastic with Linear Strain Hardening; b) Linear Elastic, Perfectly Plastic; c) Linear... Figure 16 .11: Ideal Viscous (Newtonian), and Quasi-Viscous (Stokes) Models 1 0 1 0 σ 1 0 1 0 1 0 E 1 0 1 0 1 0 1 1 0 1 ε0 1 0 1 0 dσ=Εdε 1 0 1 0 1 0 1 11 0 00 E 1 11 0 00 1 11 0 00 1 0 1 0 1 0 1 0 1 0 σ E 1 E 1 0 1 0 1 0 1 0 1 0 1 0 1 0 σ=Ε ε E 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 ε0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 σ E2 1 E +E2 1 1 E 11 00 11 00 11 00 11 00 11 00 11 00 11 00 11 00... function of the displacement velocity, Newtonian dashpot, Fig 16 .11, where 22 dσ σ Victor Saouma = ηdε ˙ or = λε1/N ˙ (16.3-a) (16.3-b) Mechanics of Materials II Draft 16.3 Rheological Models σ 7 σ E σ ε 0 σ E ε 0 σ σ ε σ=Ε ε ε dσ=Εdε Figure 16.9: Linear (Hooke) and Nonlinear (Hencky) Springs εs σ σ σ ε 0 −ε s< ε < ε s σ σs σ ε 0 F 111 1111 0000000 M σ σs −σs < σ < σ s εs ε ε Figure 16.10: Strain Threshold . the SIF pair will return to within the locus. Victor Saouma Mechanics of Materials II Draft 6 MIXED MODE CRACK PROPAGATION Victor Saouma Mechanics of Materials II Draft Chapter 15 FATIGUE CRACK PROPAGATION 1. or refinements of this one, given by da dN = C (∆K) n (15.1) which is a straight line on a log-log plot of da dN vs ∆K,and ∆K = K max − K min =(σ max − σ min )f(g) √ πa (15.2) Victor Saouma Mechanics of Materials. the crack ength; N the number of load cycles; C the intercept of line along da dN and is of the order of 10 −6 and has units of length/cycle; and n is the slope of the line and ranges from 2