Test Procedure

The generalized simulation procedure for the fatigue behavior of specimens by using MD simulations is explained in detail with the aid of open-source LAMMPS software [30]. Create the specimen with the required dimensions with the aid of lattice, region, create_box, and create_atoms. After the completion of the specimen creation, select the suitable potential files for determining the interatomic interactions between the atoms of the specimen. Then, perform the energy-minimization process to obtain a stable structured specimen by using the conjugate gradient method. Consequently, the specimen equilibrates at the required temperature under Constant Number, Volume and Temperature (NVT) or Constant Number, Pressure and Temperature (NPT) ensemble. The Nose-Hoover thermostat controls the temperature throughout the process [31,32]. The entire simulation process is carried out at periodic boundary' conditions. Then, perform the uniaxial tensile test at a suitable strain rate and temperature with a timestep of 2 femtosecond in a particular direction. Fatigue test has been performed owing to the tensile test results. Fatigue test executes in two ways: (1) based on stress ratio (<7_mjn/rr_max) and (2) based on strain ratio (e_min/e_max). If the maximum stress or strain is more than the yield value, then the process is called the ratcheting process, or else, if the maximum stress or strain is less than the yield value, then the process is called the cyclic loading process. For the ratcheting process (stress-based cyclic loading), the maximum stress is considered from the tensile loading, which is above the yield stress value, and minimum stress is obtained from the stress ratio. The loading is carried out from zero to maximum stress value in required equal steps, and then, the maximum value is diminished to zero in equal steps. In the same manner, the lower half cycle is carried out from zero to minimum stress value and then taken up to zero in equal steps, which indicates the completion of one loading cycle. For distinct stress ratios, the stress-controlled asymmetric loading is shown in Figure 9.11. In a similar way, a number of loading cycles have to be performed on the specimen to investigate the fatigue behavior.

For strain-controlled cyclic loading process, the maximum strain value is considered with respect to the maximum stress of the tensile loading. The minimum strain

FIGURE 9.11 Plot of stress-controlled cyclic loading (asymmetric) for various stress ratios. (From Meraj. M. et al„ J. Mater. Eng. Perform.. 26, 5197-5205. 2017. With permission.)

is obtained from the maximum strain and strain ratios. Strain-based cyclic loading can be implemented using two approaches: one is constant-strain-controlled and the other is increasing-strain-controlled cyclic loadings. In constant-strain-controlled cyclic loading, the minimum strain value is zero and the loading is applied continuously by maximum strain value only (refer to Figure 9.7b). However, in the case of increasing-strain-controlled cyclic loading, the minimum strain is half of the maximum strain, and then, the strain ratio is 0.5 (refer to Figure 9.7a). In these modes, a number of loading cycles have to be done on the specimen to investigate the fatigue behavior. Open Visualization Tool (OVITO) software is used for visualization and analysis of the atomic simulation results [33].

Case Study: Fatigue Behavior of Cu Film through Nanoimpact Under Cyclic Loading by MD Simulation

Molecular dynamics simulations were carried out through LAMMPS software. Single crystal of FCC Cu specimen was created with a lattice constant of 3.61 A° to perform the nanoimpact test by using MD simulations. The dimensions of the Cu specimen were 144 A⁰ x 72 A⁰ x 2 A°. Embedded-atom method potential was used for the interaction of Cu atoms [34]. Sphere-type diamond indenter was chosen for nanoimpact. The timestep considered for the simulation process was 2.97 fs. Figure 9.12 illustrates the initial specimen of matrix zone (72 A° X 36 A⁰ X 2 A⁰) with the help of OVITO software.

Figure 9.12 shows that the last three atom regions (below black line) are in fixed mode and the top atoms are in the deforming zone while cyclic loading. The specimen is simulated in Z-direction. Loading rate and indenter size effects on fatigue behavior of Cu films have been analyzed using MD simulations [35].

FIGURE 9.12 Initial specimen of matrix zone (72 A⁰ x 36 A° x 2 A⁰). (From Liu, J.N. et al., Surf. Coat. Technol., 364, 204-210, 2019. With permission.)

Structural Evolution

In this section, the structural evolution of materials during fatigue process has been explained in detail under cyclic loading. During cyclic loading, the fatigue failure process involves the crack initiation, followed by crack growth and sudden fracture.

Structural Evolution of Pre-existing Crack in Single-Crystal Iron

The fatigue process of single-crystal iron is shown in Figure 9.13 [11]. The structural evolution of fatigue crack propagation under the common neighbor analysis (CNA) for single-crystal iron pre-existing crack ([0 0 1] (0 1 0)) is shown in Figure 9.13, where the atoms in black and ash-gray colors indicate the BCC and FCC structures, respectively.

These structures are formed due to the deformation of specimen by plastic slip. Fatigue crack growth does not occur in the specimen up to cycle 7, and from cycle 8 onward, fatigue growth is initiated by the blunting process at the crack tips. The specimen is deformed mainly by the blunting mechanism consequent of stress concentration at the crack tips. Shearing slip bands appear at the crack tips for cycle 9. Increase in loading from cycle 9 to cycle 10 has indicated that the slip bands regions are extended at the crack tips. From crack tips of cycle 10, fatigue crack propagation appears in a sharp form and originates at the edge dislocations in the specimen. In case of cycle 11, the movement of edge dislocations causes fatigue crack propagation, which transforms into the next stage (rapid crack growth). The deformation mechanics behind the rapid crack growth are void nucleation and crack cleavage. At the last cycle, that is cycle 12, the accumulation of voids in fatigue crack propagation leads to brittle rupture of the specimen in cycle 12 [11].

Grain Boundary Effect on the Crack Growth of BCC Iron

Grain boundaries have a significant aspect in crack propagation of fatigue during cyclic loading, to examine the grain boundary effect on fatigue crack growth. Two distinct grain boundary models have been considered, as shown in Figures 9.14 and 9.15. Figure 9.14 shows the growing procedure of <111> (112) crack propagation in a specimen of grain boundary X3 < 1 1 2> {1 1 1}. Cyclic loading is applied normal to the crack direction. Blunting of the crack tip starts from cycle 5, owing to stress

concentration. An increase in cyclic loading from cycle 5 to cycle 7 leads to stress hardening at the crack tip due to stress concentration along with a blunting deformation, and the brittle fracture exists at the crack. In cycle 8, the crack crosses the grain boundary with the existing deformation process, because the selected grain boundary does not show a significant change on fatigue crack propagation. Crack propagation is continuous in cycle 9 with high-density vacancies at the crack tip, which lead to void formation and crack growth. Full fracture of the specimen is observed in cycle 10. The grain boundary of X3 < 1 1 2> {1 1 1} model exhibits brittle nature by failing to the resistance of crack growth, owing to the stress concentration en route for stress hardening; lack of slip bands and voids nucleation lead to fracture of the specimen [11].

Figure 9.15 shows the crack propagation of fatigue for the grain boundary model of Z5 < 3 1 0> {0 0 1} during cyclic loading. The specimen is initially created with edge crack, and loading is applied normal to the crack direction. From sixth cycle onward, blunting action is initiated at the crack tip, owing to stress concentration. Stress

concentration is relaxed at the crack tip in cycle 7, owing to the arising of plastic shear deformation. Correspondingly, the ductile nature is exhibited by the formation of fee slip band in the plastic zone. In cycle 8, sharp crack propagation takes place at 45° to the initial crack of the specimen. Rise in shear force leads to the diffusion of the crack tip, with sharp crack growth in cycle 9. From cycle 10, the direction of ductile crack fracture is altered, and blunting is detected again at the crack tip. An increase in shear stress causes an expansion of the plastic deformation in terms of slip band. Furthermore, edge dislocations develop at high densities and limited hep slip bands are formed in the plastic region when the crack propagates into adjacent grain boundary. The rate of crack propagation decreases commencing cycles 11 and 12 by the grain boundary effect. A new slip system <1 1 1> {1 1 2} induced by shear stress with limited hep slip bands is formed. In cycle 13, the slip band of lower grain boundary traverse the interface totally, which directs to mutilate the interface. The rate of crack growth enhances based on blunting at the crack tip by stress concentration at another time. In cycle 14, fast crack growth is observed due to the void nucleation at the crack tip with respect to grain boundary and deformation caused by edge dislocations and vacancies. Finally, the slip bands induced in 15 grain boundary model reduce the stress concentration and boot out the stress hardening as well as diminish the rate of fatigue crack growth [11].

Crack Growth Rate

The crack broadening during cyclic loading is known as fatigue crack growth, and the crack grows in the material corresponding to the load application. The plot between the crack growth rate (da/dN) and stress intensity factor range (AK) is shown in Figure 9.16. In region I, the crack growth rate is slightly lesser than the lattice space, owing to the threshold region. In some cases, the crack propagation is not observed

below the threshold region [36]. The fatigue crack growth rate is noticed in region

II corresponding to stress intensity, and Paris proposed power law for finding the fatigue crack growth rate in region II during cyclic loading.

Paris power law:

where da/dN is the crack growth rate per cycle, a is the length of crack, N is the cycles number, ДК is the stress intensity factor (AK = К_тлх - K_min), and C and m are constants

where Y is the geometry factor and a_mjn/niax are the stress ranges. The relationship between crack growth rate and stress intensity factor has a linear variation, which is observed in region II of Figure 9.16. In region III, the crack growth rates are rapid because high stress intensity ranges are acting on the specimen; this leads to sudden failure of the specimen. The fatigue crack growth characteristics are as follows: region I exhibits a slow growth rate, region II displays a mid growth rate, and region

III shows a high growth rate. Microstructural and load ratio effects are higher in regions I and III than in region II. Generally, the plot obtained between da/dN vs. AK is of sigmoidal form.

Crack Length in Various Crack Orientations and Grain Boundaries

Crack growth rate of single-crystal iron has been indicated in crack length range during cyclic loading [11]. The orientations considered for crack growth rate are [0 0 1] (0 1 0), [0 0 -1](1 1 0), and [1 1 —2]( 1 1 1). The common point in all three orientations is that, initially, there is no crack growth up to a few numbers of cycles. The crack of [0 0 1](0 1 0) has a limited crack propagation, in which plastic deformation initiates with stress concentration, followed by blunting at crack tip, slip band formation, and ductile rupture. These processes restrict the crack growth up to a certain number of cycles; thereafter, an increase in cyclic loading causes crack growth propagation. The crack of [0 0 —1]( 1 1 0) presents rapid crack propagation with respect to cycle number because stress relaxation does not occur at the crack tip, owing to a lack of slip bands, and it exhibits brittle fracture. The crack of [1 1 -2](1 1 1) is also similar to the previous one, but in this case, crack growth initiates a little bit more than the crack of [0 0 —1](1 1 0). The crack of [0 0 1](0 1 0) has trivial crack propagation in contrast to the other two orientations. The crack length variation for grain boundary models of S3 < 112> [111] and S5 < 310> (001) is observed with the number of cycles [11]. £3 < 112> [111] exhibits big fatigue crack growth rate compared with 15 < 310> [001]. In the S3 < 112> [111] model, there is no grain boundary effect on the crack growth process of fatigue, but for the S5 < 310> {001} model, grain boundary delays the fatigue crack growth from cycle 10 onward.

Crack Growth Subjected to Stress Intensity Factor

The [0 0 1 ](0 1 0) crack exhibits steady crack growth rate in proportion to stress intensity range []. The cracks of [0 0 -1](1 1 0) and [1 1 —2]( 1 1 1) demonstrate rapid and unsteady crack growths based on stress intensity range. The crack growth rate values are calculated with the help of the Paris power law [37], and stress intensity range values are found with the help of the Griffith fracture toughness equation [38]. The rate of crack growth is low for two grain boundary models (13 <112> [111] and 15 < 31 ()> [001)) compared with the three cracks. The grain boundary model of Z5 < 31 ()> [001} exhibits stable crack growth rate in contrast to the grain boundary model of S3 < 112> {111}.

Effect of Temperature during Cyclic Loading

In general, fatigue at high temperatures takes place with the help of the diffusion process or dislocation movement, which leads to fast plastic deformation. The slow plastic deformation takes place for fatigue at low temperatures, owing to the resistance of dislocations mobility. At high temperatures, the specimens exhibit ductile nature, and this dwindles the crack growth rate. The fatigue crack propagations are subjected to initial crack orientations of the specimen. Figure 9.17 shows the atomic snapshots of the fatigue crack growth process for single-crystal Mg with an initial crack orientation of (1 -2 1 0) [ 1 0 -1 0] at different temperatures for cycle 15. Crack growth was clogged for low temperatures 10 and 100 К owing to the restriction of plastic deformation by slow dislocations movement. For room temperature, 300 K,

FIGURE 9.17 Atomic snapshots of fatigue crack growth process for single-crystal magnesium of crack orientation with (1 -2 1 0) [1 0 -1 0] at different temperatures for cycle 15 at (a) 10 K, (b) 100 K, (c) 300 K, and (d) 500 K. and crack growth by coagulation of voids in right side of the figure. (From Tang, T. et al.. Comp. Mater. Sci., 48,426M39, 2010. With permission.)

FIGURE 9.18 Crack growth rate of single-crystal magnesium specimen at different temperatures. (From Tang, T. et al., Comp. Mater. Sci., 48,426-439, 2010. With permission.)

crack propagation was stopped at cycle 3, whereas for 500 К temperature, the crack propagation took place up to cycle 13. At high temperature, the crack propagation mechanism was the nucleation, growth and amalgamation of voids ahead of crack tip and the void connection with main crack as showed in Figure 9.17 (i.e. at right side). The nucleation of void is owing to the stress concentration ahead the crack tip where the materials become soft resultant from high temperature [5].

The temperature effects of single-crystal magnesium for crack orientation of (1 -2 1 0) [0 0 0 1] with respect to crack length and number of cycles are shown in Figure 9.18. Increasing temperature causes fatigue crack growth process to decrease with respect to the number of cycles. With a rise in temperature, material exhibits ductile nature, and it restricts the fatigue crack growth process. The crack propagation length gradually diminishes from 40 nm at 10 К temperature to below 10 nm at 500 К temperature, as presented in Figure 9.18 [5].