3.2 Opening Loading Mode 145 and the primary slip system in Figure 3.10(a)correspond to a moment just before the activation of the secondary slip.The interaction of primary and secondary dislocations leads to a formation of sessile dislocations,i.e.,either to the Lommer-Cottrell barriers (in fcc metals)or to the [001]dislocations (in bcc metals).This means a start of the slip irreversibility,since the pri- mary dislocations become locked in between the secondary (newly created) barrier and the primary(microstructural)barrier.Consequently,they cannot return to the Frank-Read source during the reverse half-cycle.Moreover,the secondary slip always reduces the back stress of the pile-up (regardless to a creation of the secondary barrier).This corresponds to an increase in the compressive yield stress.These micromechanisms are schematically depicted in Figure 3.10(b)(ATB is the increment of the compressive yield stress,GB is the grain boundary,SD is the sessile dislocation and SS is the secondary slip system).Both processes substantially reduce the reverse plasticity,and leave a residual tensile plastic strain Yrb,i.e.,initiate the ratcheting [278,290. The first experimental observation of a direct connection between the sec- ondary slip activity and the ratcheting process was reported by Lorenzo and Laird 291.Due to superimposed internal and external stresses,the creation of barriers is related to a peak of the applied true stress.If the peak is ten- sile (compressive),the ratcheting causes the elongation (contraction)of the specimen. =1102030405060 ratcheting N=1=10=20=3040 5060 onset of ratcheting Figure 3.9 The scheme of the hysteresis behaviour of cyclic softening materials before and after the onset of ratcheting.The ratcheting starts after reaching the critical range of plastic strain Aepe In order to quantify the residual plastic deformation associated with the loop disclosure,let us consider the reverse slip of the pile-up in more detail. The Burgers vector density B(r)within the pile-up can be expressed as B(a)=T工 TCO Va2 -12 where co =u/2(1-v)],v is the Poisson's ratio,x is the coordinate along the pile-up and a is the length of the pile-up.In the first half-cycle the slip
3.2 Opening Loading Mode 145 and the primary slip system in Figure 3.10(a) correspond to a moment just before the activation of the secondary slip. The interaction of primary and secondary dislocations leads to a formation of sessile dislocations, i.e., either to the Lommer–Cottrell barriers (in fcc metals) or to the [001] dislocations (in bcc metals). This means a start of the slip irreversibility, since the primary dislocations become locked in between the secondary (newly created) barrier and the primary (microstructural) barrier. Consequently, they cannot return to the Frank–Read source during the reverse half-cycle. Moreover, the secondary slip always reduces the back stress of the pile-up (regardless to a creation of the secondary barrier). This corresponds to an increase in the compressive yield stress. These micromechanisms are schematically depicted in Figure 3.10(b) (ΔτB is the increment of the compressive yield stress, GB is the grain boundary, SD is the sessile dislocation and SS is the secondary slip system). Both processes substantially reduce the reverse plasticity, and leave a residual tensile plastic strain γrb, i.e., initiate the ratcheting [278, 290]. The first experimental observation of a direct connection between the secondary slip activity and the ratcheting process was reported by Lorenzo and Laird [291]. Due to superimposed internal and external stresses, the creation of barriers is related to a peak of the applied true stress. If the peak is tensile (compressive), the ratcheting causes the elongation (contraction) of the specimen. N= 1 10 20 30 pc ratcheting N= 1=10=20=30 40 40 50 60 onset of ratcheting 50 60 Figure 3.9 The scheme of the hysteresis behaviour of cyclic softening materials before and after the onset of ratcheting. The ratcheting starts after reaching the critical range of plastic strain Δεpc In order to quantify the residual plastic deformation associated with the loop disclosure, let us consider the reverse slip of the pile-up in more detail. The Burgers vector density B(x) within the pile-up can be expressed as B(x) = τ πc0 x √a2 − x2 , where c0 = μ/[2π(1 − ν)], ν is the Poisson’s ratio, x is the coordinate along the pile-up and a is the length of the pile-up. In the first half-cycle the slip
146 3 Fatigue Fracture SS F-R GB F-R SB GB 上出上上山 B OB 入 △B (a) (b) Figure 3.10 Incomplete closure of hysteresis loops produced by the secondary slip mechanism:(a)the moment just before a secondary slip activation(point B'),and (b)disclosure of the loop caused by the secondary barrier of pile-up dislocations emitted from the F-R source is restricted only by the microstructural barrier.Consequently,the related shear strain a B()zd=_ (3.8) a2-x2 0 0 When the maximum applied stress Tmax is reached,the secondary slip cre- ates the sessile dislocation.One can assume that the probability P(x)of its location at the point r inside the pile-up is proportional to the density B(r), i.e.,P(r)=DB(x).The probability of finding the dislocation barrier any- where in between the F-R source and the primary obstacle(grain boundary) must be equal to 1,and,therefore,the following relation holds: 1=Pe→1- ()B(). 2 (3.9) After the creation of the secondary barrier,only dislocations located be- tween the F-R source and that barrier can move back to the F-R source.These dislocations are positioned in the range (0,y),where y is the coordinate of the secondary obstacle occurring with the probability P(y).This means that, according to Equations 3.9 and 3.8,the total reverse shear displacement can be expressed as
146 3 Fatigue Fracture B D D’ C’ A B cc D C A B (a) (b) B’ F-R GB F-R GB SS SB Figure 3.10 Incomplete closure of hysteresis loops produced by the secondary slip mechanism: (a) the moment just before a secondary slip activation (point B’), and (b) disclosure of the loop caused by the secondary barrier of pile-up dislocations emitted from the F-R source is restricted only by the microstructural barrier. Consequently, the related shear strain γr ∼ a 0 B(x)xdx = τ πc0 a 0 x2 √a2 − x2 dx. (3.8) When the maximum applied stress τmax is reached, the secondary slip creates the sessile dislocation. One can assume that the probability P(x) of its location at the point x inside the pile-up is proportional to the density B(x), i.e., P(x) = DB(x). The probability of finding the dislocation barrier anywhere in between the F-R source and the primary obstacle (grain boundary) must be equal to 1, and, therefore, the following relation holds: 1 = a 0 P(x)dx ⇒ 1 = τD πc0 a 0 x √a2 − x2 dx ⇒ P(x) = 1 a B(x). (3.9) After the creation of the secondary barrier, only dislocations located between the F-R source and that barrier can move back to the F-R source. These dislocations are positioned in the range (0, y), where y is the coordinate of the secondary obstacle occurring with the probability P(y). This means that, according to Equations 3.9 and 3.8, the total reverse shear displacement can be expressed as
3.2 Opening Loading Mode 147 rb P(y)B(x)xdxdy=coa Va2-y Va2-T2 dxdy.(3.10) After integrating Equations 3.8 and 3.10 one obtains rb/r=0.41. This result shows that the secondary barrier prohibits at least a half of dis- locations from returning back to the F-R source in the unloading half-cycle, which leads to a disclosure of the hysteresis loop.Obviously,this demands a sufficient density of primary dislocations to be generated during the cyclic softening in order to allow the creation of secondary barriers in the pref erentially oriented large grains.From a macroscopic point of view it corre- sponds to a critical amount of a global cyclic plastic strain,i.e.,the critical plastic strain range Aspe associated with the onset of ratcheting during the ramp-loading(generally during the global cyclic softening stage).This is a micromechanical interpretation of initial delays in the cyclic creep process observed for many metallic materials and various cyclic ratios [286,288.The existence of ratcheting in case of symmetrical loading can be elucidated in a similar way.Due to a transverse contraction of the specimen (component), the peaks of the true stress come up in the tensile half-cycles.Because the creation of the first barriers is related to these peaks,the cyclic creep follows the tensile direction and elongates the specimen. 61 N=12345 shake-down ratcheting till fracture ratcheting 50..00 4321 (a) stable loop (b) Figure 3.11 The scheme of the hysteresis behaviour when the critical plastic strain range is exceeded in all loading cycles:(a)the loading starts in the tensile direction, and(b)the loading starts in the compressive direction(the shake-down behaviour) When the work load is reached just in the first loading cycle and,si- multaneously,the critical value Aspe is exceeded,the kinetics of ratcheting strongly depends on the starting loading direction and the plastic strain range in the following cycles 292].Such multislip plastic behaviour can already be described by continuum plasticity theories involving kinematic harden- ing [293,294].The starting tensile(compressive)half-cycle produces the elon- gation (contraction)after a completion of the first cycle-see Figure 3.11.The
3.2 Opening Loading Mode 147 γrb ∼ a 0 y 0 P(y)B(x)xdxdy = τ πc0a a 0 y a2 − y2 y 0 x2 √ a2 − x2 dxdy. (3.10) After integrating Equations 3.8 and 3.10 one obtains γrb/γr = 0.41. This result shows that the secondary barrier prohibits at least a half of dislocations from returning back to the F-R source in the unloading half-cycle, which leads to a disclosure of the hysteresis loop. Obviously, this demands a sufficient density of primary dislocations to be generated during the cyclic softening in order to allow the creation of secondary barriers in the preferentially oriented large grains. From a macroscopic point of view it corresponds to a critical amount of a global cyclic plastic strain, i.e., the critical plastic strain range Δεpc associated with the onset of ratcheting during the ramp-loading (generally during the global cyclic softening stage). This is a micromechanical interpretation of initial delays in the cyclic creep process observed for many metallic materials and various cyclic ratios [286,288]. The existence of ratcheting in case of symmetrical loading can be elucidated in a similar way. Due to a transverse contraction of the specimen (component), the peaks of the true stress come up in the tensile half-cycles. Because the creation of the first barriers is related to these peaks, the cyclic creep follows the tensile direction and elongates the specimen. ratcheting till fracture N= 1 pc 234 5 (a) (b) stable loop 50 4 2 1 shake-down 3 ratcheting Figure 3.11 The scheme of the hysteresis behaviour when the critical plastic strain range is exceeded in all loading cycles: (a) the loading starts in the tensile direction, and (b) the loading starts in the compressive direction (the shake-down behaviour) When the work load is reached just in the first loading cycle and, simultaneously, the critical value Δεpc is exceeded, the kinetics of ratcheting strongly depends on the starting loading direction and the plastic strain range in the following cycles [292]. Such multislip plastic behaviour can already be described by continuum plasticity theories involving kinematic hardening [293,294]. The starting tensile (compressive) half-cycle produces the elongation (contraction) after a completion of the first cycle – see Figure 3.11. The
148 3 Fatigue Fracture ratcheting process proceeds only when the plastic strain range also remains sufficiently high in the next cycles(Figure 3.11(a)).This case corresponds to exceeding the so-called shakedown limit,usually expressed in terms of a criti- cal applied stress value.When the ratcheting is oriented into the compressive direction,however,it eventually stops after a certain number of cycles due to a decreasing true stress range (an increasing diameter of the specimen)-see Figure 3.11(b).Such behaviour is known as a plastic shakedown.When the material hardens and the plastic strain range is relatively small,a rather slow ratcheting,reversed to that of the first half-cycle,is usually observed(Fig- ure 3.12).As a rule,however,the loop stabilizes and the ratcheting shakes down after a certain number of cycles.This backward ratcheting is caused by high residual stresses which are created in the first half-cycle.During a certain period of a cyclic plastic deformation,these stresses gradually become relaxed by multislip. stable loop shake-down↑o reverse ratchet verse ratcheting shake-down 2 stable loop a (b) Figure 3.12 The scheme of the shake-down behaviour when the critical plastic strain range is exceeded only in the first loading cycle:(a)starting in the tensile direction, and(b)starting in the compressive direction In order to assess the ratcheting rate in the initial softening stage,the difference between the plastic strains produced in the loading and unloading parts of the cycle is to be determined.Let us consider that during each loading cycle the reverse slip is restricted to one half due to the secondary slip in Ao grains.This means that the reversed plasticity will be reduced proportionally to Ao/2.By using Equation 3.5,and respecting the reduction of the reversed plasticity,it leads to the following result: 2a △ce=p-pR≈± -） (3.11) =士1 业 The positive sign holds for R>-1 (elongation)and the negative sign for R<-1 (contraction).According to Equation 3.11 the ratcheting rate increases with both the increasing cycle asymmetry and the plastic strain
148 3 Fatigue Fracture ratcheting process proceeds only when the plastic strain range also remains sufficiently high in the next cycles (Figure 3.11(a)). This case corresponds to exceeding the so-called shakedown limit, usually expressed in terms of a critical applied stress value. When the ratcheting is oriented into the compressive direction, however, it eventually stops after a certain number of cycles due to a decreasing true stress range (an increasing diameter of the specimen) – see Figure 3.11(b). Such behaviour is known as a plastic shakedown. When the material hardens and the plastic strain range is relatively small, a rather slow ratcheting, reversed to that of the first half-cycle, is usually observed (Figure 3.12). As a rule, however, the loop stabilizes and the ratcheting shakes down after a certain number of cycles. This backward ratcheting is caused by high residual stresses which are created in the first half-cycle. During a certain period of a cyclic plastic deformation, these stresses gradually become relaxed by multislip. (a) stable loop stable loop (b) shake-down reverse ratcheting reverse ratcheting shake-down 2 1 12 Figure 3.12 The scheme of the shake-down behaviour when the critical plastic strain range is exceeded only in the first loading cycle: (a) starting in the tensile direction, and (b) starting in the compressive direction In order to assess the ratcheting rate in the initial softening stage, the difference between the plastic strains produced in the loading and unloading parts of the cycle is to be determined. Let us consider that during each loading cycle the reverse slip is restricted to one half due to the secondary slip in Δα grains. This means that the reversed plasticity will be reduced proportionally to Δα/2. By using Equation 3.5, and respecting the reduction of the reversed plasticity, it leads to the following result: Δγcc = γp − γpR ≈ ± 2α μ (τmax − τ0) − 2α − Δα μ (τmax − τ0) = = ± Δα μ (τmax − τ0) = ± Δα μ 2 1 − Rτa − τ0 . (3.11) The positive sign holds for R > −1 (elongation) and the negative sign for R < −1 (contraction). According to Equation 3.11 the ratcheting rate increases with both the increasing cycle asymmetry and the plastic strain
3.2 Opening Loading Mode 149 range.Indeed,one can assume Aa o Asp,where Asp is the push-pull plastic strain range.This is in general agreement with experimental observations as well as with continuum plasticity models [295].It should be emphasized, however,that the continuum mechanics is unable to interpret the two,already discussed,micromechanically induced phenomena:the critical plastic strain range related to the onset of ratcheting at the end of a delay and the tensile cyclic creep in the case of symmetrical loading. Equation 3.11 was quantitatively verified by a simulation of initial ratch- eting stages in ultra-high-strength steels [296].Because AakAsp and the push-pull stresses o and strains s are proportional to the shear quantities r and y,Equation 3.11 can be rewritten in terms of o and s as k (max-go). (3.12) △ep According to Equation 3.12,the dependence Asee/Asp us omax should be linear.Thus,the linear interpolation of experimental points is plotted in Figure 3.13 along with associated scatter ranges. 10 Co=1920 MPa u=2180 MPa 8 60 2 0 1900 1950 2000 205021002150 2200 Omax [MPa] Figure 3.13 The normalized ratcheting rate as a function of the maximum loading stress.The straight line corresponds to the theoretical model The plastic strain range Asp in the experiment was of the order of 10-5. The extrapolated value oo =1905MPam1/2 for the zero ratcheting rate agrees well with the experimental one (oo=1920 MPam1/2).The constant k is of the order of 104 and Aa10-1.This means that,in every loading
3.2 Opening Loading Mode 149 range. Indeed, one can assume Δα ∝ Δεp, where Δεp is the push-pull plastic strain range. This is in general agreement with experimental observations as well as with continuum plasticity models [295]. It should be emphasized, however, that the continuum mechanics is unable to interpret the two, already discussed, micromechanically induced phenomena: the critical plastic strain range related to the onset of ratcheting at the end of a delay and the tensile cyclic creep in the case of symmetrical loading. Equation 3.11 was quantitatively verified by a simulation of initial ratcheting stages in ultra-high-strength steels [296]. Because Δα ≈ kΔεp and the push-pull stresses σ and strains ε are proportional to the shear quantities τ and γ, Equation 3.11 can be rewritten in terms of σ and ε as Δεcc Δεp ≈ k E (σmax − σ0). (3.12) According to Equation 3.12, the dependence Δεcc/Δεp vs σmax should be linear. Thus, the linear interpolation of experimental points is plotted in Figure 3.13 along with associated scatter ranges. 1900 2 4 0 6 8 10 1950 2050 2000 2100 2200 2150 max [MPa] cc / ap u 0 0 = 1920 MPa u = 2180 MPa Figure 3.13 The normalized ratcheting rate as a function of the maximum loading stress. The straight line corresponds to the theoretical model The plastic strain range Δεp in the experiment was of the order of 10−5. The extrapolated value σ0 = 1905MPa m1/2 for the zero ratcheting rate agrees well with the experimental one (σ0 = 1920MPa m1/2). The constant k is of the order of 104 and Δα ≈ 10−1. This means that, in every loading