2022 Volume 62 Issue 10 Pages 2126-2131
A multiscale model based on micromechanics of short crack growth has been proposed and proved to be able to predict the fatigue strength of steels via experiments with thin specimens in our previous work. The present study aims to validate the applicability of our model in predicting fatigue strength in long crack problems. A camera with high resolution was set up and successfully captured the crack growth process during the fatigue test. Very good agreements for both fatigue life and crack growth process were demonstrated by comparing the experimental results and the predictions from the proposed model, indicating that our model can be served as a generalised foundation to accurately predict the fatigue behaviours of ferrite-pearlite steel for both the short and long crack growth problems.
In recent decades, researchers have strived to understand fatigue mechanisms better and develop fatigue life prediction methods by considering various length scales of crack growth mechanics.1,2,3,4,5,6) There are different definitions for the length scales of fatigue crack due to its variable growth characteristics among multiple spatial length scales. For instance, Miller7,8) described the crack length scales as three regimes: (i) the microstructurally short crack (SC), (ii) the physically short crack (PSC), and (iii) the mechanically long crack (LC). SC was defined as a crack size less than 0.5 mm in length, while LC was the crack with lengths more than 0.5–1 mm.7,8) McDowell9) defined microstructural SC to be in the order of 1 to 4 grain diameters. Suresh and Ritchie10) distinguished SC in three cases: (a) when its length was small compared to microstructural dimension, (b) when its length was small compared to the scale of local plasticity, usually ~0.1–1 mm in low-strength steels and 0.01 mm in high-strength steels, or (c) when it was simply physically small (e.g. ~0.5–1 mm). Generally, there has not been a unified view to distinguish the exact regimes between short and long cracks.
On the other hand, many efforts have been focused on developing the fatigue crack growth model of SC and LC despite the absence of a unified definition of crack length scales. Chapetti and Jaureguizahar11,12,13) proposed an integrated fracture mechanics approach to predict the fatigue crack growth in the SC regime using SC and LC thresholds. Wang et al.14) modified Chapetti’s model by considering the elastoplastic behaviour of the SCs. Several variables were added to Chapetti’s model to calculate fatigue crack growth rates (CGRs) at various stress ratios.14) Santus and Taylor15) provided a better description of the physically SC growth model and showed that the CGRs estimated in the LC regime were lower than those of the SC. Bang et al.16) improved the two-parameter fatigue crack growth model and the modelling framework by Noroozi et al.17,18) to account for both the SC and LC growth behaviours. However, there has not been a unified model to predict fatigue behaviours in both SC and LC problems, which does not require a number of adjustable fitting parameters.
Recently, we proposed a multiscale modelling strategy for predicting the fatigue strength of steel, considering the effect of grain boundaries (GBs) against fatigue crack growth.19) According to our experimental observations, the total fatigue life was estimated only based on the crack growth.20,21) The multiscale model included (i) Submodel-1: macroscale finite element analysis (FEA), (ii) Submodel-2: microstructure, and (iii) Submodel-3: crack growth. This model only required (a) loading conditions, (b) monotonic tensile properties, and (c) microstructural information of steels as input data. Noteworthy, no additional adjustable parameters were needed.
Although our model19) showed accurate prediction results of fatigue strength of thin specimens (1.9 mm) in the SC regime based on micromechanics of short crack growth, the applicability of the proposed model has not been validated in the LC regime. Therefore, this work intends to focus on validating the prediction accuracy of our model in the LC regime to further clarify its applicability in both SC and LC problems.
In this paper, we introduce the modelling framework of our model in Section 2, which includes: (i) Submodel 1: macroscopic FEA model, (ii) Submodel 2: microstructure model, and (iii) Submodel 3: crack growth model. The model validation is strictly performed by comparing the predicted results with the SC and LC experimental ones using two thin and two thick specimens in Section 3. Finally, the conclusions are provided in Section 4.
Since the purpose of the present study is to validate the applicability of our model in the LC regime, the modelling process, which is the main topic in our previous work,19) will not be introduced in detail currently. Instead, only a framework of the model will be presented in this section.
According to our previous experimental observation,20,21) the proposed model estimated the overall fatigue life only based on the crack growth. This model simplified the complicated 3D phenomenon of the actual fatigue cracks growth as a “2D problem with two steps”. The first step was to consider that fatigue cracks always initiate from the surface plane of structural components. The second step was to estimate the fatigue life by assuming crack growth in the inside plane. In addition, the crack shapes were assumed to be semi-elliptical due to a lack of experimental evidence to show the exact crack shape in the microstructural SC regime.
The model involves three submodels: (i) Submodel 1: macroscopic FEA model, (ii) Submodel 2: microstructure model, and (iii) Submodel 3: crack growth model, as shown in Fig. 1.19) Specifically, Submodel 1 aims to evaluate the strain distribution of the specimen by taking into account the test conditions (specimen geometry, boundary conditions) and materials’ monotonic tensile properties (yield strengths, ultimate tensile strengths, and reduction in area). Submodel 2 focuses on microstructural modelling in the surface plane and inside plane. Submodel 3 introduces the theory to predict the microstructurally SC growth behaviour interacting with GBs, which was first presented by Tanaka et al.22) based on Dugdale’s model23) and the continuous distribution of moving dislocations theory proposed by Mura.24)
The framework of the multiscale model for predicting the fatigue strengths of steels [13]. (Online version in color.)
In the multiscale model, crack growth rate da/dN can be formulated as:19)
| (1) |
The SC and LC experiments using two thin and two thick specimens machined by three kinds of ferrite-pearlite steels (denoted as A, B and C) were performed to validate the proposed model in this section. Firstly, the mechanical properties and microstructural information of test steels are described in Section 3.1. Secondary, the details of specimen geometries, experimental procedures and results are shown in Section 3.2. Finally, the predicted results of the model with the experimental ones are compared in Section 3.3.
3.1. Test SteelsTable 1 summarises the chemical compositions of A, B, and C. Figure 2 shows (a) the optical micrographs of test steels, (b) the inverse pole figure (IPF) map, (c) the distributions of equivalent grain diameters, (d) the distributions of pearlite band thicknesses, (e) the distributions of ferrite grain angles, and (f) the distributions of grain misorientations of each steel. In addition, Table 2 shows monotonic tensile properties, cyclic yield strength, average grain size, and estimated friction strengths of the test steels. The detailed method of retrieving this information can be found in our previous work.19)
| Steel | C | Si | Mn | P | S | Al | N |
|---|---|---|---|---|---|---|---|
| A | 0.18 | 0.15 | 1.00 | <0.002 | 0.0005 | 0.019 | 0.0008 |
| B | 0.14 | 0.36 | 1.54 | 0.014 | 0.0020 | – | – |
| C | 0.09 | 0.15 | 1.00 | <0.002 | 0.0005 | 0.019 | 0.0008 |
Test steels: (a) optical micrographs, (b) IPF maps from EBSD, (c) distributions of the equivalent diameters of ferrite grains, (d) distributions of pearlite band thicknesses, (e) distributions of ferrite grain angle, and (f) distributions of grain misorientations. (Online version in color.)
| Steel | Monotonic yield strength [MPa] | Cyclic yield strength [MPa] | Tensile strength [MPa] | Reduction in area [–] | Average grain size [μm] | Friction strength to move dislocations [MPa] | |
|---|---|---|---|---|---|---|---|
| Ferrite | Pearlite | ||||||
| A | 216 | 285 | 430 | 0.72 | 56.6 | 62.3 | 86.8 |
| B | 368 | 345 | 538 | 0.78 | 15.4 | 71.1 | 99.2 |
| C | 215 | 255 | 375 | 0.73 | 46.1 | 59.0 | 82.17 |
The geometric configurations of two thin dull-notched specimens machined by steel A and B for three-point bending test (denoted as A-Dull 3PB and B-Dull 3PB) and two thick sharp-notched specimens machined by steel B and C for tension/compression test (denoted as B-Sharp T/C and C-Sharp T/C) are shown in Fig. 3. Due to the existence of notches, the effective specimen thicknesses against crack growth of thin and thick specimens were considered as 1.9 mm and 4.5 mm, respectively.
The configuration of specimens and test conditions (R: stress ratio, rN: notch root radius, Kt : stress concentration factor): (a) thin dull-notched specimens (A-Dull 3PB and B-Dull 3PB), and (b) thick sharp-notched specimens (B-Sharp T/C and C-Sharp T/C). (Online version in color.)
Load control conditions (sinusoidal waveform) were applied to the thin specimens with a frequency of 20 Hz and a stress ratio of R = 0.1, and to the thick specimens with a frequency of 10 Hz and a stress ratio of R = −1 at room temperature. The setup of the three-point bending (3PB) tests for thin specimens and tension/compression (T/C) tests for thick specimens are shown in Figs. 4(a) and 4(b), respectively.
Overview of the setup of fatigue tests: (a) three-point bending test (3PB) and (b) tension/compression test (T/C). (Online version in color.)
Crack growth behaviour (i.e., the relationship between crack lengths and number of cycles) of the thick specimens (B-Sharp T/C and C-Sharp T/C) was observed during fatigue tests. An overview of the observation system is shown in Fig. 5. In order to clearly distinguish crack paths and accurately record crack lengths, the specimens were polished and etched with 2% nital, enabling crack tips as well as grain boundaries to be visible. A schematic diagram of the observation flow is shown in Fig. 5(a). The test was stopped at a pre-determined number of cycles and held at the maximum nominal stress for 20 seconds, during which time the crack initiated from the notch root was observed directly by a KEYENCE VW9000 microscope with a high-speed camera. The setup of the observation system can be seen in Fig. 5(b). The crack monitoring screen and the high-resolution observational result are shown in Figs. 5(c) and 5(d), respectively.
Observation system of the crack growth behaviour: (a) schematic diagram of the observation flow, (b) setup of the observation system, (c) crack monitoring screen, and (d) high-resolution observational result. (Online version in color.)
Several levels of nominal stress amplitudes, including those lower than the fatigue limits, were applied to the thin specimens. Two kinds of nominal stress amplitudes were applied to the thick specimens. The S–N curves based on the experimental results are shown in Fig. 6. It can be found that the S–N curves of the different specimens exhibit significant variances.
Experimental S–N curves. (Online version in color.)
In our previous model,19) the crack growth parameters C, n, and ΔCTSDth, which were utilized to calculate the crack growth rate da/dN, were determined by the experimental results as C = 0.045, n = 1.8, and ΔCTSDth = 6.3×10−5. Using the identified unified constants C, n, and ΔCTSDth, the predicted S–N curves for all the specimens (A-Dull 3PB, B-Dull 3PB, B-Sharp T/C and C-Sharp T/C) are shown in Fig. 7 by comparing them with the experimental ones.
Predicted results of S–N curves compared with experimental ones. (Online version in color.)
Interestingly, the fatigue lives and limits predicted by our model showed quite good agreement with the experimental ones, from thin specimens to thick specimens, even though they have various microstructures and loading conditions (i.e. notch radius, 3PB and T/C). Note that we only employed the unified crack growth parameters C, n, and ΔCTSDth without any adjustable parameters in our model.
In addition, the predicted relationships between crack depths and the number of cycles compared with experimental ones measured by the observation system (see Fig. 5) are shown in Fig. 8. The result indicates that our model can successfully predict the fatigue crack growth processes. In particular, it shows that nearly 90% of the total fatigue life was consumed within 1 mm in thick specimens (the effective thickness against crack growth is 4.5 mm).
Comparison between the experimental and predicted results for the crack growth process: (a) thick specimen B-Sharp T/C, σa = 140 MPa and (b) thick specimen C-Sharp T/C, σa = 108 MPa. (Online version in color.)
Although we did not intend to distinguish the regimes of SC and LC in our model, the accurate predicted results in the present study demonstrate that our model is applicable to both microstructurally SC and LC in ferrite-pearlite steel. Noteworthy, it is a unified model and does not require any adjustable parameters.
By comparing fatigue lives, fatigue limits and crack growth process from experimental results and those from predictions, the robustness and accuracy of our proposed multiscale model19) in predicting the fatigue strength for both SC and LC problems in ferrite-pearlite steel were demonstrated with success in this work.
Generally, our model includes (i) Submodel 1: macroscopic FEA model, (ii) Submodel 2: microstructure model, and (iii) Submodel 3: crack growth model. The required input data of this model are (a) loading conditions, (b) monotonic tensile properties, and (c) microstructural information of steels.
Note that we did not intentionally distinguish the regimes of SC and LC, nor did we use any adjustable parameters. Basically, we applied a unified theory, known as crack/grain-boundary interaction theory proposed by Tanaka,22) which has been only applied to simulate small crack growth26,27) but validated to be effective as one factor of our multiscale model for predicting the LC problem in this study. In other words, our model is applicable to both microstructurally SC and LC in ferrite-pearlite steel based on a multiscale model synthesis approach. Therefore, our model can be a generalised foundation to effectively predict the fatigue strength of ferrite-pearlite steel in both the microstructurally SC and LC regimes.
Funding: this work was supported by the Iron and Steel Institute of Japan (ISIJ) Research Promotion Grant. Part of this work was supported by Nanotechnology Platform of the Ministry of Education, Culture, Sports, Science and Technology of Japan (Grant number JPMXP09A21UT0302). The authors would like to express their gratitude to the project academic support specialist Shigeru Ohtsuka and technical staff Masahiro Fukukawa for their help with the EBSD analyses.
