For comparison with experimental results in this blog post, we refer to Ref. 1, in which the thermomechanical fatigue of a P91 (a common power plant steel) was investigated. Stress-strain curves were obtained, and material model parameters were taken from Ref. 2. It is worth noting that for the referenced work, a unified model (i.e., the viscoplastic strains are composed of both the plastic and creep components) is used. This will only affect the values of the creep part of the model, however.
At room temperature, the ultimate tensile strength of P91 steel is 600 MPa (Ref. 7), and the fatigue limit is approximately 420 MPa, taken from Ref. 6, along with the number of cycles to failure at varying stress levels. These can be used to determine a least-squares fit for the S-N curve as defined by the Chaboche fatigue model, Figure 1. Here, the load ratio R (R=\frac\sigma_min\sigma_max, ratio of minimum to maximum stress) is R = -1; i.e., \bar\sigma=0. For the parameter determination, M\bar(\sigma)=M_0 and \sigma_l(\bar\sigma)=\sigma_l0.
Ideally, more points in the intermediate region are needed to fully define the curve, but for a method demonstration, this will suffice. Since accounting for thermal effects is crucial, the S-N curve must be scaled. There are several ways to do this; the most proper would be to utilize curves at different temperatures, obtain material parameters for each temperature, and interpolate these. However, lacking such data, it is convenient to scale the ultimate tensile strength and fatigue limit according to reduction factors defined by Figure 3.24 in Ref. 1. The resulting S-N curves are shown in Figure 2. Note that these are all for R = -1.
The implementation of scaling these curves for temperature will be shown in the subsequent section. The computed cycles to failure for a selected number of stress levels, assuming an out-of-phase temperature cycling between 400ÂC and 500ÂC, defined by the setup in Ref. 2, is also included.
In Ref. 2, the specimen geometry is shown. For the simple computation of the cyclic stress-strain curves, it is sufficient to model only half of the narrow region as an axisymmetric model; see Figure 3. Note that the offset from the axial axis is due to the presence of a hole through the specimen.
The procedure to scale the ultimate tensile strength and fatigue limit with temperature is similar to that outlined above, although a reference value at room temperature is then scaled with a dimensionless reduction factor (because both the ultimate and fatigue strengths are to be scaled). The expression for the analytic function defining the Chaboche S-N curve (cf. Figure 1 and Figure 2) is shown in Figure 17, here defined at room temperature (293 K), as a function of the maximum stress.
Using COMSOL Multiphysics, these phenomena can be incorporated by use of the nonlinear material models described in this blog post. Especially important are the hardening parameters, which govern the cyclic stress-strain curve behavior over time. Using the capabilities to define custom functions for temperature and to then make material properties temperature-dependent, makes it possible to use the well-known Chaboche fatigue model to estimate the number of cycles to fatigue failure of a component subjected to simultaneous thermal and mechanical loading. This is easy to do using analytical expressions if the loading/temperature cycle is identical throughout the component life.
To calculate fatigue life, there are two approaches that are common in the industry. One is the Stress Life approach; it captures the material and geometry information in S-N curves. It is the more straightforward method, but it requires the availability of S-N curves. The required S-N curves are either specific to the component under test or at least to a standard probe of the used material (material S-N curves).
The second approach is the Strain Life analysis. Here, instead of S-N curves, the material and geometry information are given by parameters. This approach can be used when no S-N curves or data specific to the component under test are available. The material properties and geometry are described in a parametric way.
The paper presents a thermographic method of accelerated determination of the S-N curve. In the presented method, the S-N curve was developed based on energy-related parameter with the assumption of its dependency on the stress amplitude. The tests made on C45 steel and X5CrNi18-10 steel under reversed bending revealed that the S-N curve obtained by accelerated thermographic method fits inside the 95% confidence interval for the S-N curve obtained from the full test.
The S-N curve is the basic parameter defining fatigue strength properties of the material under cyclic load, particularly in the case of the so-called stress approach. The fatigue tests are long-lasting and expensive. For example, the minimum number of specimens required to obtain the S-N curve depends on the type of test program conducted and ranges from 12 to 24 for data on design allowables or reliability data (ASTM E 739 91 (reapproved 2004)). This requires testing of specimens at tens or hundreds of thousands of cycles, which significantly influences duration of tests. A test of an individual specimen depends on load frequency and can last even several days, thus making it expensive as well. Hence researchers struggle to develop new methods or to improve existing ones for rapid determining of S-N curve. Thanks to those methods it is possible to significantly cut cost of fatigue tests by reduction of the quantity of specimens and shortening test duration.
Effects of temperature changes induced by fatigue loading can be used for development of accelerated methods for determining the fatigue S-N curve. Fargione et al. [19] proposed for uniaxial loading that the integration of the area under the temperature rise curve over the entire number of cycles of a specimen (area under curve A, Figure 1, especially under curve A in Phase 2) remains constant and is independent of the stress amplitude, thus representing a characteristic of the material. Amiri and Khonsari [20, 21] used the rate of temperature rise in Phase 1 for prediction of fatigue failure.
However not all materials show temperature stabilization period during fatigue life in Phase 2. That was observed, for example, by Krewerth et al. [22], Plekhov et al. [23], and Wang et al. [24]. This fact significantly limited the use of previously developed thermographic methods of rapid determination of the fatigue curve.
Application of the proposed method was presented in [30], where it was used to determine the inclination of the S-N curve which is important from the point of view of the accuracy of the fatigue limit determination by the Locati method [31, 32]. This paper presents the improved version of the analysis of the accelerated test results, allowing outside determination of the inclination of the S-N curve determination of its position on the S-N plane.
The following software was installed on the computer: VirtualCAM allowing two-way communication between PC and the thermographic camera via USB interface, CIRRUS Front End which was the user interface used to control the CEDIP camera, and ALTAIR allowing downloading, storage, and advanced processing of thermographic images.
The T-N curve (Figure 9) obtained from gradually increasing loading test was cut into individual T-N curves for the different load levels (Figure 10). The reference level of the temperature was the initial temperature of the specimen.
Assuming that the relationship between energy-related parameter for full test and load at the level is described by formula (4)and putting (7) into (6),is obtained:where there are two unknown parameters and of the curve which can be determined numerically. Then it is possible to determine the values of at the level by using (7).
The S-N curves estimated for the individual specimens were presented in Figures 12 and 13. The individual estimated fatigue life data obtained from (10) for all three specimens has been used as data to estimate one S-N curve (Figures 14 and 15).
The presented new thermographic method enables rapid determination of the S-N curve. This method is based on the energy-related parameter with the assumption of its dependency on the stress amplitude.
Major findings and conclusions of the presented tests and analysis are summarized as follows:(1)The S-N curve obtained by accelerated thermographic method fits inside the 95% confidence interval for the S-N curve obtained from the full test for both tested steels.(2)The estimation of the parameters of the - curve and the S-N curve for each specimen for both materials is characterized by repeatability.(3)The accelerated thermographic test for one specimen lasted ca. 1.5 hours only. For comparison, the full test lasts several days.(4)The accelerated thermographic test can be made on one specimen. Three specimens for increasing the accuracy of estimates were assumed in this work. The full test requires a minimum of a dozen or so specimens.
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