Universality behind Basquin’s law of fatigue
One of the most important scaling laws of time dependent fracture is Basquin’s law of fatigue, namely, that the lifetime of the system increases as a power law with decreasing external load amplitude, , where the exponent has a strong material dependence. We show that in spite of the broad scatter of the Basquin exponent , the fatigue fracture of heterogeneous materials exhibits intriguing universal features. Based on stochastic fracture models we propose a generic scaling form for the macroscopic deformation and show that at the fatigue limit the system undergoes a continuous phase transition when changing the external load. On the microlevel, the fatigue fracture proceeds in bursts characterized by universal power law distributions. We demonstrate that in a range of systems, including deformation of asphalt, a realistic model of deformation, and a fiber bundle model, the system dependent details are contained in Basquin’s exponent for time to failure, and once this is taken into account, remaining features of failure are universal.
pacs:46.50.+a, 62.20.Mk, 61.82.Pv
Disordered media subject to sub-critical external loads present a time dependent macroscopic response and typically fail after a finite time zapperi_alava_statmodfrac . Such time dependent fracture evidently plays a crucial role in a large variety of physical, biological, and geological systems, such as the rupture of adhesion clusters of cells in biomaterials under external stimuli schwarz_adhesionbio_prl_2004 , the sub-critical crack growth due to thermal activation of crack nucleation santucci_prl_subcrit_2004 ; sornette_thermal_prl_2005 , creep nechad_sornette_prl_2005 and fatigue fracture of materials sornette_fusefatigue_prl1992 ; farkas_crystal_fatigue_prl_2005 , and the emergence of earthquake sequences marone_nature_healing_1998 . One of the most important scaling laws of time dependent fracture is the empirical Basquin law of fatigue which states that the lifetime of samples increases as a power law when the external load amplitude decreases, basquin_1910 . The measured values of the Basquin exponent typically vary over a broad range indicating a strong dependence on material properties basquin_1910 ; krajcinovic_damagebook_1996 ; krajcinovic_1 .
In this Letter we study the fatigue fracture of heterogeneous materials focusing on the underlying microscopic mechanism of the fatigue process and its relation to the macroscopic time evolution. We develop two generic models of time dependent fracture, namely, a fiber bundle model and a discrete element approach, which both capture the most important ingredients of the fatigue failure of disordered materials. Analytic solutions and computer simulations reveal that the models recover the Basquin law of fatigue, whose exponent is determined by the damage process. We show that, as a consequence of healing, a finite fatigue limit emerges at which the system undergoes a continuous phase transition from a regime where macroscopic failure occurs at a finite time to another one exhibiting only partial failure in the system having an infinite lifetime. Based on analytic solutions, we propose a generic scaling form for the macroscopic deformation. On the microlevel the fatigue of the material is accompanied by an avalanche activity where bursts of local breakings are triggered by damage sequences. We demonstrate analytically that the microscopic bursting activity underlying fatigue fracture is characterized by universal power law distributions which implies that the non-universality of the Basquin exponent at the macro-level is solely due to the specific degradation process of the material.
First we consider a mean field model of fatigue fracture, namely, a fiber bundle model (FBM) where fibers fail either due to immediate breaking or to ageing kun_asphalt_jstat_2007 . For the load redistribution after failure events, equal load sharing is assumed so that all the fibers carry the same load sornette_prl_78_2140 . During the evolution of the system, a fiber breaks instantaneously at time when the load on it exceeds the local tensile strength (). All intact fibers accumulate damage due to the load that they have experienced and break when exceeds the local damage threshold (). The accumulated damage up to time is obtained by integrating over the entire loading history of the specimen , where is a scale parameter, while the exponent controls the rate of damage accumulation krajcinovic_damagebook_1996 ; krajcinovic_1 . To capture damage recovery in the model due to healing of microcracks krajcinovic_damagebook_1996 or thermally activated rebinding of failed contacts schwarz_adhesionbio_prl_2004 ; marone_nature_healing_1998 , we introduce a memory term in the above damage law of exponential form whose characteristic time scale defines the memory range of the system marone_nature_healing_1998 ; schwarz_adhesionbio_prl_2004 ; sornette_thermal_prl_2005 . Hence, during the time evolution of the bundle, the damage accumulated over the time interval heals. Assuming independence of the two breaking thresholds and , the macroscopic evolution of the system under a constant external load can be cast into the form
where and denote the cumulative distributions of and , respectively. We solved Eq. (1) analytically obtaining the load on the intact fibers at a constant external load , with the initial condition , where denotes the solution of the constitutive equation sornette_prl_78_2140 . Here denotes the ultimate strength of the material. The most important input parameters of the model calculations are , and , which govern the damage accumulation.
On the macrolevel the process of fatigue is characterized by the evolution of deformation of the specimen, which is related to as , where is the Young modulus of fibers. Neglecting immediate breaking and healing, Eq. (1) can be transformed into a differential equation for the number of broken fibers as , where . Using , for uniformly distributed threshold values the exact solution of the equation of motion Eq. (1) reads
where denotes the lifetime of the system. Equation (2) shows that damage accumulation leads to a finite time singularity where the deformation of the system has a power law divergence with an exponent determined by . It is important to emphasize that has a power law dependence on the external load in agreement with Basquin’s law of fatigue found experimentally in a broad class of materials basquin_1910 ; krajcinovic_damagebook_1996 ; krajcinovic_1 . The Basquin exponent of the model therefore coincides with that of the microscopic degradation law . Another interesting outcome of the derivation is that the macroscopic deformation of a specimen undergoing fatigue fracture obeys the generic scaling form , where the scaling function has the property , with and the scaling exponents are and . Figure 1 presents a verification of this scaling law on experimental data from asphalt specimens obtained at two different load values kun_asphalt_jstat_2007 . The good quality data collapse obtained by rescaling the two axis and the power law behavior of as a function of the time-to-failure demonstrates the validity of our scaling relation.
Healing dominates if for a fixed load the memory time is smaller than the lifetime obtained without healing . Then, a threshold load emerges below which the system relaxes, i.e., the deformation converges to a limit value with a characteristic relaxation time resulting in an infinite lifetime. Figure 2 presents the characteristic time scale of the system varying the external load over a broad range. The results from numerical simulations with the complete FBM (i.e., including immediate breaking and healing) are in excellent agreement with the measured lifetime of asphalt samples for , recovering also the Basquin exponent kun_asphalt_jstat_2007 . The regime below is of particular importance in geodynamics where memory effects take place during cyclic loading of rocks with a stress amplitude increasing from one cycle to the next memory_effect . It is important to note that approaching the fatigue limit from either side, the characteristic time scale diverges. Figure 3 shows that both the relaxation time and the lifetime follow a power law as a function of the difference from the fatigue limit with distinct exponents: and . We stress that the exponents neither depend on the disorder distributions ( and ) nor on the details of the damage law (, and ), i.e., they are universal implying a continuous phase transition at the fatigue limit between partial failure and macroscopic fracture (see Fig. 3).
Our calculations revealed that the Basquin law of lifetime emerges on the macrolevel as a consequence of the competition between the two microscopic failure mechanisms of fibers. Rewriting Eq. (1) in the form of the constitutive equation of simple FBMs as it can be seen that the slow damage process on the left hand side quasi-statically increases the load on the system: ageing fibers accumulate damage and break slowly one-by-one in the increasing order of their damage thresholds . After a number of damage breakings, the emerging load increment on the remaining intact fibers can trigger a burst of immediate breakings. Since load redistribution and immediate breaking occur on a much shorter time scale than damage accumulation, the entire fatigue process can be viewed on the microlevel as a sequence of bursts of immediate breakings triggered by a series of damage events happening during waiting times , i.e., the time intervals between the bursts. The microscopic failure process is characterized by the size distribution of bursts , damage sequences , and by the distribution of waiting times . At small loads most of the fibers break in long damage sequences, because the resulting load increments do not suffice to trigger bursts. Consequently, the burst size distribution has a rapid exponential decay. Increasing the total number of bursts increases linearly and a power law regime of burst sizes emerges with the well-known mean field exponent of FBM hansen_crossover_prl . When macroscopic failure is approached the failure process accelerates such that the size and duration of damage sequences decrease, while they trigger bursts of larger sizes , and finally macroscopic failure occurs as a catastrophic burst of immediate failures. Since in the limiting case of a large number of weak fibers breaks in the initial burst, we found that the distribution has a crossover to a smaller exponent , in agreement with Ref. hansen_crossover_prl . After the linear increase, the number of bursts has a maximum at and rapidly decreases to 1 as is approached. All these results are independent of , , and .
Since damage events increase the load on the remaining intact fibers until an immediate breaking is triggered, the size of damage sequences is independent of the damage characteristics and of the material, instead, it is determined by the load bearing strength distribution of fibers. Under broad conditions this mechanism leads to an universal power law form with an exponential cutoff , where . The damage law of the material controls the time scale of the process of fatigue fracture through the temporal sequence of single damage events. In damage sequences fibers break in the increasing order of their damage thresholds which determine the time intervals between consecutive fiber breakings. Analytic calculations showed that has an explicit dependence on as , however, the duration of sequences , i.e., the waiting times between bursts follow an universal power law distribution , where only the cutoff has -dependence (see Fig. 4).
The macroscopic lifetime of a finite system can be related to characteristic quantities of the microscopic failure process as , from which the average lifetime can be obtained in the form . In the load regime where the generic scaling laws of the distributions , , and prevail, this leads to the form in agreement with the Basquin law Eq. (2) of the system. The results demonstrate that the Basquin law of lifetime on the macro-scale is a fingerprint of the scale-free microscopic bursting activity, with the material dependence entering only through the damage law determining the waiting times between bursts. Experimentally, the microscopic fracture process underlying fatigue can be monitored by the acoustic emission technique and by direct optical observations zapperi_alava_statmodfrac ; santucci_prl_subcrit_2004 ; krajcinovic_damagebook_1996 ; krajcinovic_1 . Sub-critical cracking has recently been found to produce a power law distribution of step sizes of the advancing crack in agreement with our predictions on the size distribution of bursts santucci_prl_subcrit_2004 .
In order to study the effect of stress concentration and crack growth in fatigue fracture, we also developed a discrete element model (DEM) humberto in which we discretize a two-dimensional disc-shaped specimen in terms of randomly shaped convex polygons connected by elastic beams. The beams fail either due to immediate breaking or damage which are coupled in a single failure variable . Here describes the deformation state of the beam taking into account both stretching and bending , being the longitudinal deformation, and the bending angles at the two ends of the beam, and and denote the threshold values a beam can sustain under stretching and bending, respectively. As a consequence, the parameters , , and play the same role as their counterparts in our FBM. The time evolution of the system is followed by numerically solving the equations of motion of polygons. The breaking criterion is evaluated at each time step and beams which fulfil the condition are removed humberto . We study the fatigue fracture under diametric compression of discs with constant stress (Brazil test). Figure 2 shows that DEM provides also an excellent fit of the lifetime data of asphalt specimens kun_asphalt_jstat_2007 . DEM simulations revealed that in the presence of stress concentrations bursts are spatially correlated and they can be identified as sudden advancements of slowly growing cracks. DEM results on burst characteristics also show power law behavior as the mean field FBM, but with different exponents due to the two-dimensionality of the model. The localized stress concentration built up around cracks gives rise to higher values of the exponents of the size distribution of bursts , and of damage sequences , while for the waiting time distribution the DEM exponent falls very close to the mean field value (see Fig. 4). The results proved to be independent of the value of .
Although the exponent of Basquin’s law depends on the microscopic damage accumulation, we found an astonishing spectrum of universal features hidden behind this originally empirical law. On one hand we discovered in the experimentally relevant situation of finite damage memory a continuous phase transition between partial failure and macroscopic rupture. On the microscopic level of individual breaking events we showed that the separation of time scales of the two competing failure mechanisms leads to a bursting activity, where we disclosed several universal scaling laws in the distributions and determined their exponents as well in mean-field as in two dimensions. In summary our approach provides a direct connection between the microscopic mechanisms constituting the main ingredients of the model (i.e., immediate breaking, damage accumulation and healing of microcracks) and the macroscopic behavior of the fatigue process. The (macroscopic) exponent from Basquin’s law coincides with the (microscopic) exponent of the degradation law, namely . Following a slightly different pathway, our methodology is also capable to show explicitly the bridge between the (universal) mechanism related with the scale-free bursting activity at the micro-scale and the (non-universal) lifetime law of the material at the macro-scale.
This work opens up new experimental challenges. Our scaling relation of the macroscopic deformation should be verified on various types of materials, after which it could help to extract the relevant information from fatigue life measurements. For instance, it would be interesting to check our theoretical predictions with fatigue measurements performed at very low external loads, i.e., for . More precisely, in the infinite lifetime limit, , the experimental confirmation of the power-law variability with load of the relaxation time should certainly provide some considerable insight on the role of healing in the entire fatigue process. For similar reasons, it would be also interesting to verify the distinct lifetime behavior obtained from the model in the other limit of low external loads, . Finally, another interesting outcome from our study is the statistical behavior related with the bursting activity during the fatigue process of a given material. According to our analysis, both the size of damage sequences and magnitude of waiting times between bursts should obey universal power-law distributions that might reflect the intrinsic features of the typical restructuring events taking place at the microscopic level. As a possible monitoring technique, acoustic emission measurements could be conducted in conjunction with fatigue experiments to confirm our claim for universality behind Basquin’s law.
We thank the Brazilian agencies CNPq, CAPES, FUNCAP and FINEP, and the Max Planck prize for financial support. F. Kun was supported by OTKA T049209.
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