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Mesoscopic and nonlinear aspects of dynamic and fatigue failure (experimental and theoretical results)
J.-L. Lataillade and O.B. Naimark1
LAMEFIP ENSAM, Bordeaux University, Talence, 33405, France 1 Institute of Continuous Media Mechanics UrB RAS, Perm, 614013, Russia
Statistically based phenomenology of mesodefects allowed the interpretation of damage kinetics as the scaling transitions in defect ensembles and the explanation of limited steady-state crack velocity, transition to the branching regime, scaling aspects of dynamic and fatigue failure. Qualitative different nonlinearity for damage kinetic equation was established for the slip bands and microcrack controlled stages of fatigue failure. This allows the original interpretation of the 4th Paris law in the terms of scaling universality and the explanation of the crack closure effect as the consequence of the anomalous energy absorbtion at the crack tip zone during scaling transitions in mesodefects ensemble. Presented results reflect the long-term collaboration of research teams of the Laboratory of Physical Foundation of Strength of the Institute of Continuous Media Mechanics of RAS (Russia) and LAMEFIP ENSAM of the Bordeaux University (France).
1. Some aspects of dynamic and fatigue failure
The recent experimental study of dynamic crack propagation revealed the limiting steady state crack velocity, a dynamical instability to microbranching [1, 2], the formation ofnon-smooth fracture surface [3], and the sudden variation of fracture energy (dissipative losses) with a crack velocity [4]. This renewed interest was the motivation to study the interaction of mesodefects at the crack tip area (process zone) with a moving crack. The still open problem in the crack evolution is the condition of crack arrest that is related to the question whether a crack velocity smoothly approaches to zero as the loads is decreased from large values to the Griffith point [5]. There is also problem at the low end of crack velocity. How a crack that is initially at rest might achieve its steady-state.
The increasing interest is observed in the use of mesoscopic approach in the condition of cyclic loading of the high-strength brittle materials (ceramics, intermetalics) and traditional materials (metals, alloys) after the structure reforming due to the compacting of metal (alloy) powder or granules. Such materials have the improved specific strength at high temperatures and simultaneously reveal the pronounced lack of damage tolerance due to the distinct features of fatigue crack propagation related to the nonlinear behavior of defects. Fatigue damage is one of the major life limiting factors for most structural components subjected to variable loading during service [6, 7]. The natural tendency in the
fatigue failure is the development of the so-called “damage tolerant” approach based on the prediction of crack propagation rate to inspectable flaw size. There is difference in the remaining life estimation for low cycle fatigue (LCF) and high cycle fatigue (HCF). LCF cracks are typically of an inspectable size early enough in total life so that there is a considerable fraction of life remaining during which an inspection can be made. HCF, on other hand, requires a relatively large fraction of life for initiation to an inspectable size, or the creation of damage, which can be detected. This results in a very small fraction of life remaining for propagation.
There are two main problems of fatigue failure — long and short fatigue crack growth. Physics-based model description of fatigue crack nucleation and growth are needed in order to obtain a reliable life prediction model. The following most striking features of fatigue phenomena have been recognized:
1) a ductile metal can break in a fatigue test like a brittle material without any appreciable external deformation;
2) fine cracks, which would not influence noticeable the static strength, substantially impair its fatigue limit.
Several types of fatigue damage have been identified in [8]: persistent slip bands (PSB); roughness profile of extrusion; microcracks formed at the interfaces between PSB and matrix, in the valleys of surface roughness of PSB surface profile; fatigue damage at grain boundaries. Most
© J.-L. Lataillade and O.B. Naimark, 2004
of the damage causing defects range from 1 ^m to 1 mm which is below the in-service non-destructive evaluation (NDE limit ~ 1 mm) inspection limit. It is generally observed that a component in service seems to spend about 80 % of its life-time in the region of short crack growth. Hence studies on nucleation and growth kinetics of these cracks become a necessary part of assessing the total life.
2. Statistical model
Structural parameters associated with typical mesode-fects were introduced using the following parameters
Sik = svt vk, Sik = 1l2s(vi lk + lVk), (1)
for microcracks and microshears. Here v is unit vector normal to the base of a microcrack or slip plane of a microscopic shear; l is a unit vector in the direction of shear; s is the microcrack volume or the shear intensity for microshear. The average of the “microscopic” tensor sik gives the macroscopic tensor of defect density
Pik = n(sik), (2)
which coincides with the deformation induced by defects. The dimension analysis of damage accumulation and the hypothesis of statistical self-similarity were used for the description of the statistics of nucleation and growth of defects (microshears, microcracks) responsible for the damage-fai-lure transition. Self-similarity of fatigue damage accumulation means [9] that only mean size of defects increases in time, but the distribution function of defects is invariant in the terms of some dimensionless variables. The established fact of self-similarity allowed one to propose the quantitative relationships between characteristic variables in some fatigue laws and interpretation of power-type constitutive equations [10]. Self-similar features take the place when the dominant mechanism subjects the evolution of complex system. As it was shown in [11] the self-similar mechanism of fatigue crack can be linked with the self-similar morphology of fatigue bands with the number of cycle.
Taking into account the large number of mesoscopic defects and the influence of thermal and structural fluctuations, that are involved in the damage accumulation process, the formulation of statistical problem concerning the distribution function of defects was proposed in [12] in terms of the solution of the Fokker-Plank equation in the phase space of characteristic mesodefect variables. As it was shown in
[12] the distribution function of defects can be represented in the form of stationary solution of the Fokker-Plank equation W = Z_1 exp(- E/Q), where Z is the normalization constant. The key question in such formulation is the definition of the energy of defect in the condition of the interaction of defect with the external stress and structural stress induced by the surrounding defects and the dispersion properties of the system given by the value of Q. According to the presentation of the microcracks (microshears) as the
dislocation substructure the energy can be written in the form
E = E0 - Hiksik +asik > (3)
where the quadratic term represents the own energy of defects (1) and the term Hiksik describes the interaction of the defects with the external stress <3ik and with the ensemble of the defects in the effective field induced by defects:
Hik = ®ik + XPik = °ik + Xn(sik), (4)
where a, X are the material constants.
The average procedure gives the self-consistency equation for the determination of the defect density tensor
Pik = n j SikW(S v>l)dsik• (5)
For the dimensionless variables pik = 1/nja/Qpik, % = Va/Qsik, 6ik = aik/ijQv, self-consistency equation has the form
that includes the single dimensionless material parameter 8 = a/Xn. The dimension analysis allowed us to estimate that
a ~ G/V0, X ~ G, n ~ R”3. (7)
Here G is the elastic modulus, V0 ~ r03 is the mean volume of the defect nuclei, R is the distance between defects. Finally we obtain for 8 the value 8 ~ (R/r0)3 that is in the correspondence with the hypothesis of statistical self-similarity of the defect distribution on the different structural levels.
The solution of the self-consistency equation (6) was found for the case of the uniaxial tension and simple shear
[13] (Fig. 1). The existence of characteristic nonlinear behavior of the defect ensemble in the corresponding ranges of 8 (8 >8* = 1.3, 8c < 8 < 8*, 8 >8*= 1) was established, where 8c and 8* are the bifurcation points. It was shown [14] that the above ranges of 8 are characteristic for the quasibrittle (8 < 8c = 1), ductile (8c < 8 < 8*) and nanocrystalline (8 > 8* = 1.3) responses of materials. It is evidence from this solution that the behavior of the defect ensemble in different ranges of 8 is qualitative different. The replace of stable material response for fine grain materials to the metastable one for the ductile materials with the intermediate grain size occurs for the value of 8 = 8* = 1.3, when the interaction between the orientation modes of the defects has more pronounced character. It means that the metastability has the nature of the orientation ordering in the defect ensemble.
3. Phenomenology of a solid with defects
The statistical description allowed us to propose the model of a solid with defects based on the appropriate free
0 g1 gc a
Fig. 1. Characteristic responses of materials on defect growth
energy form (Fig. 2). The simple phenomenological form of the part of the free energy caused by defects (for the uniaxial case p = pzz, a = azz, 8 = 8zz) is given by sixth order expansion, which is similar to the well-known Ginzburg-Landau expansion in the phase transition theory [15]
F = 1/2 A(1 - 8/8*)p2 -1/4 Bp4 -
-1/6 C (1 -8/8 c) p6 - Dap + x(V tp )2. (8)
The gradient term describes the non-local interaction in the defect ensemble; A, B, C, D are positive phenomenological material parameters, x is the non-locality coefficient. The damage kinetics is determined by the evolution inequality
8F18t = (dF/ dp) p + (dF/ 38)8 < 0 , (9)
that leads to the kinetic equation for the defect density p and scaling parameter 8
p = -rp (A(1 ~8/8*)p - BpZ +
+ C(1 -8/8c)p5 - Da)-d/dxl(xdp/dxl)), (10)
8 = -r8(- A(28*)p2 -C(68c)p6), (11)
where rp, r8 are the kinetic coefficients.
p -
e ^ ^ t>2 , \ V
! >
Ay
X
1 1 1— 1 \ 1 I a.
1 1 1 1 1 1 1 1 j !
Pi CM Q. o Q. Gc U P
Fig. 2. Free energy dependence on stress and defect density for 8 < 8c ~ 1
These equations describe the defect accumulation kinetics and the influence of this kinetics on the change of scaling characteristics of nonlinear system “a solid with defects”.
4. Collective modes of defects. Scaling transitions
The phase analysis shows that Eq. (10) in the region 8 > 8* (S1, Fig. 3) is of the elliptic type with periodic solutions and possesses p anisotropy determined mainly by the applied stress. As 8^8* the periodic solution transforms into a solitary-wave solution (S2) that is the image of persistent slip bands (PSBs) — the stacking dislocation structure with the order dislocation arrangement. In this case the solution has the form
p = 1 pa[1 -tahn(ZLW)], Lw = 4/pa (2x/A)1/2,
/ 2 (12) V = XA(pa - pm)/2C2,
where the wave amplitude pa, velocity V and the width of the wave front Lw are determined by the parameters of orientation transition, (pa - pm) is the jump in p in the course of an orientation transition.
A transition through the bifurcation point 8c is accompanied by the appearance of spatio-temporal structures (S3) of a qualitatively new type characterized by explosive (“blow-up”) accumulation of defects as t ^ tc in the spect-
Fig. 3. Collective modes in mesodefect ensemble
Fig. 4. Schematic representation of the experiment
rum of spatial scales. The “blow-up” self-similar solution is the precursor of the crack nucleation. This solution arises due to the pass of the critical point pc and reads
p = g(t )f £ = x/LC>
- (13)
g(t) = G(1 -Xc)
5. Experimental study of scaling transitions under dynamic fracture
5.1. Experimental setup
Direct experimental study of crack dynamics in the preloaded PMMA plane specimen was carried out with the usage of a high speed digital camera Remix REM 100-8 (time lag between pictures 10 ^s) coupled with photoelasticity method, Fig. 4 [16].
The pictures of stress distribution at the crack tip is shown in Fig. 5 for slow (V < VC) and fast (V > VC) cracks. The experiment revealed that the pass of the critical velocity VC is accompanied by the appearance of a stress wave pattern produced by the daughter crack growth in the process zone. Independent estimation of critical velocity from the direct measurement of crack tip coordinates and from pronounced stress wave Doppler pattern gives a correspondence with the Fineberg data (VC - 0.4VR) [17].
5.2. Characteristic crack velocity
The dependence of crack velocity on the initial stress is represented in Fig. 6. The characteristic velocity VC -= 330 m/s allowed the estimation the peak time tc to mea-
where Tc is the so-called “peak time” (p ^ ^ at t ^ Tc), sure the size of the mirror zone LC - 0.3 mm: tc = LC /VC
LC is the scale of localization, G > 0, m > 0 are the para-
meters of non-linearity, which characterize the free energy release rate for 8 < 8c.
The generation of the collective modes in defect ensemble localized on spatial scales allows the description of the appearance of dislocation substructures (PSBs, crack hotspots) in some universal way. The scales of localization of these modes (Lw in the case of the orientation transition and LC in the case of the crack hotspot) play the role of the current scaling parameters in the definition of the current value of 8- (Lw/LS)3 for the PSBs transition type and 8 = (LC/LS)3 for the crack hotspots transition. Here LS is the characteristic spacing between mentioned mesodefect substructures.
-1 -10- s. This result leads also to the explanation of linear dependence of the branch length on the crack velocity [26]. Actually, since the failure time for V > VC is approximately constant (tf - tc - 1 jws), there is a unique way to increase the crack velocity to extend the size of the process zone. The crack velocity V is linked with the size of the process zone LPZ by the ratio V = LPZ/1c.
Since the branch length is limited by the size of the process zone, we obtain the linear dependence of branch length on the crack velocity. This fact explains the sharp dependence (quadratic law) of the energy dissipation on the crack velocity established in [4]. In our experiments the dependence of the density of the localized damage zone on the stress was observed (Fig. 7).
V < VC V > V C
Fig. 5. Different modes of crack dynamics
Fig. 6. Stress dependence of crack velocity
5.3. Failure surface analysis
Fracture surface analysis revealed the correspondence of the sharp change of the crack dynamics for the velocities VC and VB, and the fractographic pattern. The fracto-graphic image of the fracture surface was studied in the velocity range V ~ 300-800 m/s when different regimes of the crack propagation were observed.
The first regime V ~ 220-300 m/s is characterized by the mirror surface pattern (Fig. 8). The increase of the crack velocity in this range leads to characteristic pattern on the mirror surface in the form of the so-called conic markings [18, 19]. The conic markings are the traces of the junctions of the main crack and the damage localization zones that nucleate in the process zone. These data reflect the influence of the damage kinetics on the characteristic size of above mentioned failure structures on the surface.
The second regime appears in the velocity range 300600 m/s and the surface pattern includes the numerous mirror zones.
5.4. Scaling properties of failure
The scaling properties of failure has spurred great interest in the context of the general problem of disordered media when the self-affinity of the failure surface was established in the terms of the universality of the so-called roughness exponent. The self-affinity of the fracture surface was established first by Mandelbrot [20] as the existence of the power law of the distance r measured within the horizontal plane for the points at which the heights h(r) are measured. This defines the surface-roughness index Z as h(r) ^ rq. Experimental data of the past decades on the measurement of the roughness exponent revealed the self-affinity of the fracture surface and it was established that the pattern of the fracture surface can be considered as scale invariant objects with the roughness index Z ~ 0.8± 0.05 [21, 22]. This fact allowed the determination of the length scales r > r0 where the roughness exponent is the invariant. The
Fig. 7. Stress dependence of localized damage zone density
range of the scales r > r0 for many materials show the universal scaling properties (the roughness exponent Z ~ ^ 0.8). For the scales r < r0 the roughness exponent can change. The scale invariant properties for r > r0 means the transition from the roughness statistics caused by the initial structural heterogeneity (size of blocks, grains) to the statistics given by the collective properties of defects under transition from damage to fracture.
In our experiments the roughness profile was determined for the PMMA fracture surface (Fig. 9) using the laser scanner system.
Fig. 8. Failure surface for slow (V < VC) (a) and fast (VB > V > VC) (b) cracks
1.5
-1.0 r i___i_i__i_I__i_i__i_i_I__i_i__i_i__I_i__i_i__i_I__i_i__i_i_
0 20 40 60 80 100
Length, mm
Fig. 9. Profile of fracture surface roughness for PMMA
The analysis of the roughness data in the term of the roughness exponent showed the dependence of the scaling properties on the regime of the crack propagation. However, the group of specimens was found with the exponent Z ~ 0.8. This fact allowed us to assume the existence of the regime of crack propagation with universal scaling index close to Z ~ 0.8. The existence of different scaling indexes for other regimes of crack propagation reflects the variety of the behavior of investigated nonlinear system. As it was shown, the crack dynamics in quasi-brittle materials is subject to two attractors. The first attractor is given by the intermediate asymptotic solution of the stress distribution at the crack tip. The self-similar solution (9) describes the blow-up damage kinetics on the set of spatial scales and determines the properties of the second attractor. This attractor controls the system behavior for V > VB when there is a range of angles with a > ac. The universality of the roughness index can be considered also as the property of this attractor. In the transient regime VB > V > VC the influence of two attractors can appear. This reason can be considered as a mechanism of the roughness index dispersion on the scale r > r0. The scaling properties of failure were studied also under the recording stress dynamics using the polarization scheme coupled with the laser system. The stress temporal
history was measured in the marked point deviated from the main crack path on the fixed (4 mm) distance. This allowed the investigation of the correlation property of system using the stress phase portrait a ~ a for slow and fast cracks (Figs. 10, 11). These portraits display the periodic stress dynamics (Fig. 11) that in the correspondence with the local ellipticity of Eq. (8) for a <a c (V <V C) and the stochastic dynamics (Fig. 11) for V > VC corresponding to the second type of the attractor. In the transient regime V ~ VC the coexistence of two attractors can appear that can lead to the intermittency effect as the possible reason for the scaling index dispersion.
The recording of the temporal stress history in the marked point for V > VC revealed the appearance of finite amplitude stress fluctuations, which reflect the qualitative new structural changes in the process zone for the fast crack (Fig. 11). The scaling properties as the above attractor properties were studied in the term of the correlation integral calculated from the stress phase pattern using the formula [23]
1 m ! \
C(r) = Hm—- ^H(r-L -Xj|)= rv,
m m i, j=1 1 1
where xi, Xj are the coordinates of the points in the a ~ a space; H (...) is the Heaviside function. The existence of the scales r > r0 with the stable correlation index was established for the regimes V < VC and VB > V > VC. The values of the correlation indexes in these regimes show the existence of two scaling regimes with the deterministic (V = = 200 m/s, v ^ 0.8) and stochastic (V = 426.613 m/s, v ^ 0.4) dynamics. The extension of the portions with a constant indexes determines the scale of the process zone LPZ. The length of the process zone increases with the growth of the crack velocity in the range VB > V > VC with the maintain of the scaling property of the dynamic system. Numerical simulation of the damage kinetics in the process zone allowed us to conclude that this scaling is the consequence of the subjection of failure kinetics to the blow-up self-similar solution which determines the collective behavior of defect ensemble in the process zone [16, 24, 25].
Fig. 10. Phase portrait for slow cracks
Fig. 11. Phase portrait for fast cracks
Fig. 12. TEM image of fatigued crystal of Cu [8]: saturation; matrix vein structure (a); ladder structure (b); labyrinth structure (c); cell structure (d); Ypl = 10-3(b); 5 -10-3 (c); 1.45 -10-2 (d)
6. Energy absorbing and scaling transitions in fatigue failure
6.1. Structural aspects of fatigue
Crack initiation, as well as the whole fatigue process, is controlled by the cyclic plastic deformation and fatigue crack coordinates are indicated by the positions of the higher cyclic plastic deformation than average. In a wide range of deformation conditions the cyclic plastic deformation are localized within the stacks of highly active primary slip planes forming persistent slip bands (PSBs), while the surmounting material accommodates an appropriate two orders of magnitude smaller plastic strain amplitude. The PSBs are imbedded into a second phase commonly known as “matrix”, which consists of irregularly arranged dislocation reach regions — “veins” (Fig. 12). The width of veins is about 1.5 jwm in copper fatigued at room temperature [26]. The dislocation density within the veins is of the order of 1015 m-2 which corresponds to a mean dislocation spacing of 30 nm. The veins are separated by channels, which are relatively dislocation-free and are of a size comparable to that of the veins. At the early stage of fatigue the veins contribute to rapid hardening by partly impeding dislocation motion on the primary slip system. Increasing the number of cycles leads to an increase in both the dislocation
density within the veins and the number of veins per unit volume.
6.2. Saturation plateau
The early stage of HCF is followed by a “saturation plateau”, where structural changes take place within the matrix to accommodate high values of plastic strains because the dislocation veins in the matrix can not accommodate strain in excess of approximately 10-4. PSBs structure is generated due to the initial blocking of glide dislocations and the formation of parallel wall (ladder) structures which occupy about 10 %, by volume, of the PSBs. The PSBs are composed of a large number of slip planes which form a flat lamellar structure. In strain-controlled experiments, the co-existence of matrix and PSB goes along with plateau in the cyclic stress-strain curve. If the applied strain amplitude is raised, this is accommodated by an increase in the volume fraction occupied by the PSBs [8].
6.3. Transition to failure
According to TEM observations of cyclically deformed Cu the labyrinth and cell dislocation structures are formed after saturation plateau [27] and can be considered as the dislocation arrangement precursor of fatigue crack nuclea-tion and early crack growth occur in the PSBs.
Fig. 13. Scaling transitions in PSBs controlled fatigue stage
The final stage of fatigue damage corresponds to an increase in the peak resolved shear stress.
The description of damage kinetics in terms of Eqs. (10), (11) reveals the specific system behavior in the range of scaling parameter 8C < 8 < 8* and 8 < 8C that can be qualified as the condition of the self-criticality [29]. It means that intensive order parameter — defect density tensor pik influences on the correlation properties of the nonlinear system (in term of the 8 kinetics) and provides the condition of continuous ordering of dislocation substructures on different spatial scales. The existence of two ranges of 8 characterizes qualitative difference of relaxation mechanisms, which provide different dissipation ways: the orientation ordering (PSBs dislocation structures) and dissipative structures as the crack hotspot nucleation.
The scenario of defect evolution in the range of orientation-scaling transition 8C < 8 < 8* leads to the qualitative change of relaxation properties and, as the consequence, the energy absorbing. The anomaly of the relaxation properties follows from (10) as the consequence of orientation transition in defect ensemble leading to the PSBs generation. To present the right part of Eq. (10) as the expansion of dF/dp (the non-locality term is dropped for the simplicity) in the vicinity of point pS (Fig. 13)
dF_
dp
rdF^
dp
1
+ —
I p=ps
y G=GC
'a2 f dp2
(p - ps) + ••• (14)
I p=ps
yG=G S
Eq. (10) reads in this case:
3 2 F dp2
(p — ps)
(15)
I p=ps
yG=G S
^eff
'N-1
V
V
d 2 F dp 2
I p=ps
yG=G S
we obtain that Tpff ^ ^ for 8^8*. In reality the finite increase of effective relaxation time occurs due to the appearance of the second minimum for the free energy related to more oriented state of defects. Two minimums are separated by the metastability barrier AF and the relaxation time can be estimated as
AF
kT
(17)
where T0 ~ 1012 is the Debay time; kT is the Boltzmann factor. The kinetics of 8 given by Eq. (11) provides the continuous ordering dislocation substructures with the long-range correlation properties.
The high level of the structural relaxation time in respect of the loading time Tl =8—1 coupled with the self-similar features of structure rearrangement in the scaling transition regime explain the anomaly of the energy absorbing under cycle load starting from some characteristic level of strain. This stain corresponds to the saturation stress aS providing the start of the scaling transition (the path ADHF, Fig. 13) and the saturation effect as the anomaly of energy absorbing in the condition of orientation-scaling transition in dislocation substructures.
7. The Paris law of crack kinetics in HCF
The microscopic mode of fatigue crack growth is strongly affected by the slip characteristics of material, characteristic microstructural dimensions, applied stress level and the extent of near tip plasticity. In ductile solids, cyclic crack growth is observed as a process of intense localized deformation in slip bands near the crack tip which leads to the creating of new crack surfaces by shear decohesion. A number of mechanisms has been proposed to clarify the linkage of above stages with crack growth kinetics.
The important feature of cyclic loading conditions when the onset of crack growth from pre-existing defects can occur at stress intensity values that are well below the quasistatic fracture toughness. This observation was used as a physical basis for the Paris model [31 ] when small scale yielding assumption allowed the formulation of the crack kinetics as
da
dN
CAKm
(18)
and to introduce the definition of the relaxation time
in the term of the stress intensity factor range defined as
AK = Kmax — Kmin , where Kmax and Kmin respectively
are the maximum and minimum stress intensity factors, C and m are empirical constants which a functions of material
properties and microstructure. This formula predicted the Paris exponent of m = 4 in agreement with experiments for most metals [28].
Since the crack growth kinetics is linked with the temporal ability of material to the energy absorbing at the crack tip area the understanding of the saturation nature can be the key factor for the explanation of the 4th power universality. It was shown that the saturation nature can be considered as a consequence of the anomaly of energy absorbing in the course of structural transition in dislocation system with the creation of PSBs and long-range interaction of dislocation substructures due to the internal stresses, which provide the “self-critical” regime of structure evolution at some constant aS value of external stress. The saturation plateau is very pronounced feature of the structure controlled regime with the low sensitivity to the applied stress starting from some critical value aa. Since the damage kinetics along the path DHF leads finally to the nucleation of crack hotspots (according to the solution (6)) after the second bifurcation point at 8 = 8C, the stress controlled regime corresponds to the set of states D, ..., H, ..., F, ... . The kinetics of this path is given by the 4th order difference in the power for the branches similar 0b and Cd and qualitative dependence of damage rate reads p ~ Aa4!. This result supports the phenomenological law proposed by Paris for the HCF crack growth kinetics.
It is interested to note that this channel (scaling transitions due to the generation of the oriented dislocation substructure similar to PSBs) is very powerful in the sense of the energy absorbing. For instance, similar to the Paris law, the 4th power law 8 = AaA of the linkage of plastic strain rate 8p on the stress amplitude aA, was established at the steady-state plastic wave front for the wide class of shocked materials [30].
8. Retardation of fatigue crack growth. Crack closure effect
The retardation effects in the crack response (crack closure) upon the applied stress can be also considered as the consequence of the anomaly of elastic energy absorbing (related to the factor AK) in the area consisting the crack and the crack tip zone undergoing the orientation-scaling transition with characteristic time Tpff that differs from the loading time Tl ~ 8_1: Tpff > Tl. To consider the influence of structural transitions in dislocation ensemble on the temporal evolution of the stress (stress intensity factor) at the crack tip (process) zone in respect to the loading time in “zero-tension” strain-controlled periodic test. The deformation properties of material with defects is given by the equation for total strain
and the kinetic equations (10), (11) for the defect density p and structural scaling parameter 8. Effective compliance of the material in the process zone can be estimated as
38
G—ff = —= G"1 +^
dp
da
da
(20)
8 = G 1a + p
(19)
and the second term in (20) reveals the pronounced temporal sensitivity depending on the level of stress (strain) related to the different branches of curve in Fig. 13 for the 8C < 8 < 8* range of scaling parameter 8. For the loading path 0b the defect accumulation provides small deviation of the compliance parameter from G —1 due to the low dislocation density that can be estimated by the first term of the expansion (8) and very fast relaxation to the equilibrium value on the 0b branch for the stress a < aa. The range of stress aa < a < ad (for corresponding total strain) is sensitive to the qualitative new kinetics (12) that is controlled by the p-non-linearity: the generation of collective modes (PSBs) and scaling transitions with a “coarsening” of dislocation substructures.
The anomalous increase of the effective compliance parameter G,—1 and essential growth of the effective relaxation time Tpff provide the sharp decrease of the stress (stress intensity factor) in the process zone and the anomalous absorbing of the elastic energy from the material surrounding crack into the crack tip area. These two physical reasons can be considered as possible explanation of the retardation effects in the crack advance and crack closure in the presence of far-field tensile load.
9. Dissipative effects in HCF
9.1. Heat production mechanisms during fatigue process
Since the fatigue phenomenon is generally caused by the cyclic plastic strain, the plastic strain energy plays an important role in damage process. Fatigue cracks generally initiate from surface defects or discontinuities and are predominantly influenced by the surface stress system. Slip under partly or totally reversed rapidly applied repeated stress cycles is sharply concentrated due to the collective effects in the condition of orientation-scaling transition. The local temperature increases on individual slip planes or within a cluster.
The significance of an energy approach is in its ability to unify microscopic and macroscopic test data and subsequently to facilitate the detection of manifestation of damage. The idea to relating fatigue to intrinsic dissipation seems to be relevant. Infrared thermography readily detects the occurrence of both initiation and propagation of failure. The work done on the system by plastic deformation is identified as the major contribution to heat effect. However, a general acceptance exists that not all the mechanical work produced by the plastic deformation can be converted into
O
100
CD
...r; .'*♦..................................•
.... " \ ‘^Vrii ll:i is-C ,
. |r .• , ......... ......... ‘
0)
CL
| -100
* * u .♦¥!*!* *
* :*a* st** **!
y.’vIV’’”: > :•: * . ** *,*•’*' a :*‘ -~r
...........‘V- •* ” ; i “‘A . “ “
-0.4 0 0.4
Temperature, °C
-1
Log (R/Rmax)
Fig. 14. Determination of the correlation dimension for local temperature signal: a — phase space; b — correlation integral. The stress amplitude is 480 MPa
the thermal energy as well as configuration entropy stored in the material microscopic structure.
The internal dissipation term constitutes a significant part of the non-linear coupled thermomechanical analysis and the infrared thermography technique can quantify this intrinsic dissipation. The intrinsic dissipation can be considered as accurate indicator of structure evolution and damage manifestation. This technique allows the accurate illustration of crack initiation, the onset of its unstable propagation through the material and flaw coalescence, when the increasing microcracking is generated by cyclic loading. Particularly in the case of fatigue testing, the infrared thermography technique evidences the initiation of a crack and its propagation through the material. Signal processing techniques have been efficiently used to extract more quantitative information [33].
Taking into account to analyze the kinetics of damage-failure transitions for HCF as the property of collective behavior of defects with the usage of the infrared technique for the study of the structural scaling the estimation of the energy balance equation can be given using the results of developed phenomenology. The definition of driving forces related to the defect density tensor pik and the structural
scaling parameter 8 allows us to determine the dissipation function D in the form
1 dT
dxk
+ aikdik —
dF .
98;t
dF
fyi,
-p ik
-f* * 0,
(21)
where aik is the Cauchy stress tensor; dik is the Eulerian strain rate tensor; 8ik is the strain tensor. For the simplification the effects related to the thermo-elasticity are dropped. The energy balance equation for the considered situation reads
Pcp
3T
dt
XAT +
a ikdik—
9F . dF * dF .
(22)
where cp is the heat capacity; p is the density; X is the heat conductivity coefficient.
10. Long-correlation properties of materials subjected to fatigue load
The scenario of structure evolution at the crack tip area for different stages of HCF allowed the definition of the Paris regime as the steady-state system behavior and the universality of the power index in the Paris law can be linked with the self-similar character of structure rearrangement. This behavior can be described on the limited set of structural variables, for instance, using the ratio of characteristic scales in the PSBs pattern: the width of PSBs and the spacing between ones. The existence of this type of scaling reflects the stable dissipative ability of system in the period of time that can be classified as the life-time. The corresponding correlation property of system for this regime can be used for the general estimation of the existence of different stages of material response on the fatigue load in any conditions. The change of correlation properties due to the coarsening of the dislocation substructure can lead to the sharp decrease of correlation dimension that can be considered as the precursor of critical crack advance. These changes in the dispersion and long correlation properties is analyzed in [34, 35] for the condition of HCF in 35CD4 steel using the infrared data of the temperature dynamics and the scaling characteristics of structure morphology at the crack tip area.
Taking into account the remark concerning the qualitative changes in the dissipative ability when the collective modes of defects can subject the system behavior, we consider briefly the change of the dispersion properties of such systems. The continuous systems, described, for instance, by the nonlinear equation (22), reveal the so-called spatial-temporal chaos as the important feature of the dispersion property of system in the condition, as in our case, of structural (scaling) transitions in dislocation substructure. The
+
dispersion and scaling properties of such systems can be analyzed, as in the Section 5, in the phase space of experimentally observed variables.
The temperature dynamics given by Eq. (22) can be plotted in the phase space (T ~ T) as the phase trajectory. The phase space of continuous systems has infinite dimension. Each point of this functional space corresponds to the current distribution of variables, for instance temperature. The generation of collective variables, corresponding to the self-similar solution for temperature distribution, decreases the system dimension. In this case the evolution of continuous dissipative systems is accompanied by the “compression” of the phase space. It means that after characteristic time the phase trajectories are concentrated in some subset of the initial (infinite dimension) phase space. The subset of system states corresponding to the collective modes is linked with the definition of the attractor space (the attractor dimension). The procedure of the definition of attractor dimension based on the analysis of experimental sequences of system variable was proposed in [36]. The idea of methods is the following: the steady-state evolution of system in the term of observable variables of continuous system is recorded as the experimental sequence following in the equal period of observation t: T(t) = Yl9 T(t + t) = Y29..., T(t + (n - 1)t) = Yn,.... Some subset of the sequence with the size k is selected and all vectors belonging to the subset with the dimension k is analyzed w(n) = {Yn-k+!, Yn-k_2, ..., YnYJ . The pairs of vectors w(n) and w(n) for the numbers n and n are introduced for the definition of distance between these vectors
pk (n, n) = | w(n) - w(n ) and the calculation of correlation integral Ck (l)
N
Ck (l) = lim N -2 £©[l -P k (n, nO],
N ,
n, n
where l is some selected value; N is the total number of elements in the sequence set; ©(z) is the Heaviside function defined as ©( z) = 0 for z < 0 and ©( z) = 1 for z > 0. For small l the correlation integral has the limit Ck (l) ~ lak. The value ak can be estimated from the plot ln Ck (l) ~ ln l for the set of k, starting from k = 1.
The stochastic properties of the temperature evolution (one pixel represents approximately 100x100 ^m on the specimen surface ) in fatigue loading were analyzed using the Grassberger and Procaccia algorithm [36]. To investigate the disturbances caused by the local plastic flow and/or damage, several films were recorded for the average temperature in the area ~ 1 cm2 with a framing frequency of 400 Hz for different stress amplitude (from 400 to 610 MPa).
It was established that for stress amplitude 480 MPa the pronounced difference between mean and local signal is observed. The correlation dimension of the local signal is
Fig. 15. 3D profile of the surface (the higher, the lighter) of specimen (the field of view is 70 x 50 |um). Marks show the extrusions on the surface
about 2.10 ±0.02 that can be linked with the deterministic chaos in the damage evolution under fatigue loading (Fig. 14). It is important to note that the measurement of surface roughness recorded by the high resolution interfe-rometer-profiler New View-5000 [35] revealed the generation of PSB on the specimen surface (Fig. 15).
11. Concluding remarks
The interaction of the main crack with the ensemble of the defects is the subject of intensive experimental and theoretical studies that revealed some unresolved puzzles in the problem of failure. The long-standing problem is the limiting velocity of a crack. The linear elastic theory predicted that a crack should continuously accelerate up to the Rayleigh wave speed VR , however, the experiments on a number of brittle materials showed that the crack will seldom reach even the half of this value. A view of the dissipation process was suggested where the main role in the explanation of qualitative new mentioned effects was assigned to the collective modes of the defect ensemble and the interaction of these modes with a moving crack. These phenomena demonstrate the qualitative new features of a crack behavior caused by the interaction of the crack with the ensemble of defects in the so-called the process zone. Similar approach that was proposed by the authors for the study of crack dynamics and wave fronts of shocked metals, the nonlinear theory of damage-failure transitions for the conditions of high cyclic fatigue were developed and experimental research based on the high resolution infrared technique of crack nucleation and propagation was carried out. Statistically based phenomenology of mesodefects allowed the interpretation of damage kinetics as the scaling transitions in defect ensembles and to establish the self-similar laws responsible for the PSBs and microshear (microcrack) controlled fatigue failure. It was shown that the growth of dislocation density occurs as the orientation transition in dislocation substructures and is characterized by the change of initial scaling properties in defect ensemble due to the long-range interaction in newly created dislocation substructures (veins - PSBs - dislocation cells... transitions). Qualitative different nonlinearity for damage kinetic equation was estab-
k 2 (Yn -k+1 Yn'-k+1)
i=1
lished for the slip bands and microcrack controlled stages. This allows the original interpretation of the 4th power index in the Paris law in the terms of scaling universality and the explanation of the crack closure effect as the consequence of the anomalous energy absorbing at the crack tip (process) zone during the scaling transitions in dislocation substructures. The linkage of scaling transitions with different stages of fatigue damage was studied experimentally in fatigue bending test for steel using high resolution digital infrared camera CEDIP Jade III and interferometer-profiler New View.
Acknowledgments
The significant part in presented work was carried out with participation of Thierry Palin-Luc, Nicolas Saintier, Sergey Uvarov, and Oleg Plekhov. The research was supported by the projects of the French Ministry of Education, the Russian Foundation of Basic Research (project No. 0201-00736) and ISTC projects (No. 1181, 2146).
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