Although S∼N curve-based approaches are widely followed for fatigue evaluation of nuclear reactor components and other safety critical structural systems, there is a chance of large uncertainty in estimated fatigue lives. This uncertainty may be reduced by using a more mechanistic approach such as physics based three-dimensional (3D) finite element (FE) methods. In a recent paper (Barua et al., 2018, ASME J. Pressure Vessel Technol., 140(1), p. 011403), a fully mechanistic fatigue modeling approach which is based on time-dependent stress–strain evolution of material over the entire fatigue life was presented. Based on this approach, in this work, FE-based cyclic stress analysis was performed on 316 nuclear grade reactor stainless steel (SS) fatigue specimens, subjected to constant, variable, and random amplitude loading, for their entire fatigue lives. The simulated results are found to be in good agreement with experimental observation. An elastic-plastic analysis of a pressurized water reactor (PWR) surge line (SL) pipe under idealistic fatigue loading condition was performed and compared with experimental results.

## Introduction

The current design codes and standards [15] for assessing fatigue life of nuclear reactor components are usually based on stress/strain versus life (S ∼ N) curves. In stress versus life approach stress within the elastic limit are applied and stress amplitude versus life is determined. In strain versus life approach (e.g., Coffin–Manson rule) significant plastic strain is applied and the number of cycles to failure is determined as a function of plastic strain. In recent years, several improvements to fatigue design (S ∼ N) curves have been recommended by various researchers. Some of the improvements are considering the effects of environments [6,7], surface finish [8], mean stress [9], and hold time [10]. To assess fatigue damage under variable/random amplitude loading using the fatigue design (S ∼ N) curves, currently, the linear damage accumulation rule [11] along with rainflow cycle counting method [12] is used. Although the S ∼ N curve-based fatigue evaluation approaches allow engineers to quickly assess the components' design lives, these curves are empirical curves and are estimated based on final life of a fatigue-test specimen without considering cumulative and time-dependent damage evolution in a material. Basically, they are not based on firm mechanistic understanding of how the stress–strain behavior of material evolves over time and its impact on overall fatigue lives. Moreover, as the S ∼ N curves are generated from uniaxial fatigue test data and may not truly represent the multiaxial stress–strain state at the component level. Furthermore, the linear damage accumulation rule does not take into account the nonlinear and time-dependent material hardening/softening of material. Additionally, as many different codes and standards [13] use different approaches (e.g., based on stress or strain range methods), it is sometimes difficult to decide which method to use. By adopting more mechanistic-based approaches for fatigue evaluation, the issues associated with present fatigue life evaluation methods can be greatly reduced. Moreover, with the current availability of advanced computation tools, such as the finite element (FE) method, along with high-performance computing, it may be possible to model a component or an overall system more mechanistically.

## Finite Element Implementation of Evolutionary Cyclic Plasticity Model

The proposed evolutionary cyclic plasticity model is developed based on Chaboche [2123] or Armstrong–Frederick [24] type analytical expressions for nonlinear kinematic hardening and can be expressed as
$dα=23C1vdεpl−γ1vαdP$
(1)
where $C1v$ is a proportional constant that gives a linear relation between the increment in the back stress,$dα$, and the increment in the plastic strain, $dεpl$, while $γ1v$ describes the rate at which the back stress decreases with the increase in accumulated effective plastic strain, $dP$. At any instant, the von Mises yield function $fv$, corresponding to the evolutionary cyclic plasticity model, can be expressed as
$fvσ,α=32S−α:S−α−σvy=0$
(2)

where $σ$ is the stress tensor at that instant, $α$ is the corresponding back stress tensor, $S$ is the deviatoric stress tensor, and $σvy$ is the yield stress. Note that the subscript v is used to demonstrate that the material parameters are not constant as they are in the case of conventional cyclic plasticity model, rather they are variable and can be function of time or fatigue cycle/block or any other physical state (e.g., APSE). The complete details of the model development and time-dependent material parameter estimation technique are discussed elsewhere [15,20].

The cyclic plasticity model was incorporated into the developed FE code for Chaboche-type models in the abaqus/Standard environment [25]. A user subroutine was developed to enable the use of time- or APSE-dependent material properties in the implementation of the evolutionary cyclic plasticity model into abaqus. The time- or APSE-dependent material properties such as elastic modulus, yield stress, and kinematic hardening for 316 SS can be found elsewhere [20].

## Experimental

Uniaxial fatigue tests under constant, variable, and random amplitude loading were conducted on 316 SS base metal using small hourglass specimens. All the tests were performed in air at 300 °C using a hydraulic-controlled mechanical testing system test frame. The details of the specimen geometry and test setup can be found in a previous Argonne report [15]. A 0.5% strain amplitude was applied during constant amplitude test. During the variable amplitude test, a repetitive block consists of 12 cycles with different strain amplitudes was applied (Fig. 1(a)), while a repetitive block of random strain inputs was applied to the specimen during the random-amplitude fatigue test (Fig. 1(b)). The strain inputs during random amplitude test were selected based on a matlab-based random number generator. During all the strain-controlled fatigue experiments, 316 SS material exhibited significant stress hardening followed by stress softening.

## Finite Element Modeling of Fatigue Experiments

One of the major tasks in mechanics-based fatigue modeling is to develop an FE modeling framework based on the evolutionary cyclic plasticity model. The FE modeling framework can then be used for extrapolating uniaxial fatigue test-based material behavior to a multiaxial domain for stress analysis and fatigue evaluation of realistic reactor components, which are ideally subjected to multiaxial loading. Compared to the conventional FE model, the evolutionary cyclic plasticity FE model would be able to predict the cyclic hardening and softening behavior of a component. It is assumed that, similar to the conventional tensile-test-based FE model, which is extensively used by industry for stress analysis of metallic components subjected to monotonic loading transients, the proposed evolutionary cyclic plasticity-based FE model framework would be able to simulate a component subjected to cyclic loading. Also, similar to the tensile-test-based FE model, it is assumed that the translation from a uniaxial to multiaxial fatigue-test-based model is isotropic and based on the assumption that the metallic components are homogeneous, with material behavior being similar in all directions. However, before the new FE model can be used for component-level stress analysis, the FE framework must be validated with experimental test cases.

A single 3D eight-node brick element representing the gauge section (0.5 in) was used for FE simulation of the fatigue experiments. A single element was used to reduce the computational time for simulating thousands of fatigue cycles. In the proposed FE model, the cross section of the 3D brick element (hexahedral eight-node linear brick element: C3D8) was considered equal to the nominal cross section of the specimen. The geometry information of the actual specimen and FE modeled equivalent specimen is shown in Fig. 2. Simulations representing strain-controlled fatigue tests with constant, variable, and random amplitude loadings were performed by applying corresponding deformation in the z-direction, as shown by the arrows in Fig. 2.

### Time-Based Modeling.

Two 3D-FE simulations representing the constant- and variable-amplitude experiments using time-dependent material properties were presented. The 3D-FE simulated axial stress along with the experimentally observed stress for the entire life of constant- and variable-amplitude fatigue specimens are shown in Figs. 3(a) and 4(a), respectively. Magnified versions of Figs. 3(a) and 4(a), demonstrating initial stress hardening and then softening followed by stabilized cycles, are shown in Figs. 3(b) and 4(b), respectively. Figures 3(b) and 4(b) demonstrate that model predicts not only the stress hardening but also the stress softening with significant accuracy. The model also predicts the stabilized cycles, as seen in Figs. 3(b) and 4(b), which represent a quasi-stable state during fatigue. Most importantly, as shown in Figs. 3(c) and 4(c), it accurately predicts the fast stress drop toward the end of the fatigue life of the specimen, which represents unstable or rapid crack propagation.

### Accumulated Plastic Strain Energy-Based Modeling.

A time-based approach was used for constant- or variable-amplitude loading test cases, but could not be used to model fatigue behavior under random loading. This is due to the complexity in tracking the time associated with load reversals. To solve this issue, a more versatile approach based on APSE is used. In this work, APSE-based 3D-FE modeling was performed for all the fatigue test cases such as constant-, variable-, and random-amplitude tests. Note that, for the APSE-based modeling, the material parameters estimated from the variable-amplitude test were used. The predicted axial stress profiles from the APSE-based 3D-FE simulation of constant- and variable-amplitude fatigue tests were compared with the experimentally observed stress profiles in Fig. 5. As seen from the figures, the simulated stress profile exhibits all the characteristic behavior (initial hardening followed by softening and stabilized cycles and rapid crack propagation toward the end of fatigue life) of 316 SS under constant- and variable-amplitude fatigue loads. However, the APSE-based prediction is not as close to the experimental data as the time-based prediction. For example, the predicted fatigue life from simulation was found to be 6151 cycles, while the experimentally observed fatigue life was 6914 cycles under constant amplitude loading. However, considering the robustness it provides in predicting material behavior under any arbitrary loading, including random-amplitude, the APSE-based prediction is considered to be a reasonable prediction.

Figure 6(a) shows the 3D-FE simulated axial stress profiles along with the experimentally observed stress history of random-amplitude test. The figure shows that the model can accurately predict all the characteristic behavior (initial hardening followed by softening and stabilized cycles and rapid crack propagation toward the end of fatigue life) of 316 SS under random-amplitude fatigue loads. A magnified version of Fig. 6(a) is shown in Fig. 6(b). The experimental and predicted stresses shown in this figure correspond to the strain input shown in Fig. 1(b) (magnified inset). The corresponding stress–strain hysteresis plot is shown in Fig. 6(c). Figure 6(c) depicts the intricacy in the stress–strain variation during random loading. Despite this variation, the FE model prediction of stress response is reasonably good.

### Summary of Finite Element Modeling Results in Context of Predicting Maximum Hardening Stress and Fatigue Life of Specimens.

Results from the 3D-FE modeling are summarized here to provide an overall picture of the predicted results and to compare them with the experimental results. Table 1 compares the experimentally observed maximum hardening stress and fatigue life during all the fatigue tests with those predicted through time-based and APSE-based 3D-FE modeling. As seen from the table, the maximum hardening stress predicted through 3D-FE simulation is more than 90% accurate for all the fatigue test cases. To determine the fatigue life of the specimens, a failure criterion is required. As all the tests were performed uniaxially and under strain control, a drop in uniaxial stress amplitude in the direction of applied strain was used for determining a failure criterion for the fatigue tests. Conventionally, a 25% load-drop from the maximum load is used as a failure criterion of fatigue specimens. The experimental and predicted lives for all fatigue tests and the 3D-FE modeling cases are given in Table 1. As seen from the table, the accuracy in predicted life from time-based 3D-FE simulation is almost 100% for both constant- and variable-amplitude fatigue tests. Because the time-based approach cannot be used for predicting material behavior under random-amplitude loading, the APSE-based approach was used. The predicted lives from APSE-based modeling are found to be 89%, 93%, and 95% accurate for constant, variable, and random amplitude tests, respectively.

## Finite Element Modeling of a Pressurized Water Reactor Surge Line Pipe

The mechanics-based fatigue-modeling work is aimed at developing an FE modeling framework to estimate the life of nuclear reactor critical safety components. The overall picture of this modeling framework is shown in Fig. 7. As shown in the flowchart, mechanics-based fatigue modeling starts with uniaxial fatigue experiments followed by material model (based on an evolutionary cyclic plasticity model) development along with material model parameter estimations. The details of this work and related results are discussed in Refs. [15,20]. The next step is validation of the evolution cyclic plasticity model through analytical and 3D-FE modeling of the specimen. Results from the analytical modeling of 316 SS specimens under uniaxial fatigue loading are presented in Ref. [20]. The 3D-FE modeling results of the fatigue specimens are discussed in Sec. 4. In the final step, as seen in Fig. 7, the developed FE model framework and material model parameters are utilized to extrapolate uniaxial fatigue test-based material behavior to a multiaxial domain for structural (ST) analysis of nuclear reactor components subjected to multiaxial fatigue loading. This section presents some preliminary results from the final step of proposed fully mechanistic fatigue evaluation framework.

An elastic-plastic analysis of a PWR SL pipe was performed. The FE mesh of the PWR SL along with the boundary conditions and the direction of applied displacement is shown in Fig. 8(a). Cyclic displacements equivalent to $±$0.5% constant strain amplitude in z-direction were applied to nodes near one of the ends, as shown in Fig. 8(b). Application of 0.5% strain amplitude is considered reasonable. The profile of the applied cyclic displacement is shown in Fig. 8(c).

The APSE-based modeling approach was used for structural simulation of the PWR SL pipe. Figure 9 shows the contour plot of the von Mises stress at a typical instant. As shown in the figure, maximum stress concentration occurs in the elements that are directly subjected to the applied deformation. One of these elements, as shown in the magnified inset in Fig. 9, is selected as the element of interest for analyzing results from simulation. All the simulated stress–strain results presented are at the centroid of this element. Figure 10(a) shows the simulated strain in the direction (z-axis) of applied displacement as a function of fatigue cycles. As shown in the figure, the strain amplitude is 0.5%. Note that the simulation was performed only for 100 fatigue cycles. The simulated stresses in the principal coordinate system are shown in Figs. 10(b), 10(c), and 10(d) for maximum, mid, and minimum principal stress, respectively. The figures also compare the simulated principal stresses with the values calculated from the uniaxial fatigue experiment. Because of the multi-axiality, simulated principal stress amplitudes are very different from experimental amplitudes. However, the evolutionary cyclic plasticity model uses von Mises stress for checking the yield criteria during elastic-plastic analysis, von Mises stress should be used for comparing simulation results with experimental observations. The comparison between von Mises stress amplitudes of experimental observation and those of simulation is shown in Fig. 11. The plots indicate material hardening followed by softening in the simulated stress profile. This behavior is typical of 316 SS as observed during uniaxial fatigue tests. The value of maximum hardening von Mises stress from simulation is found to be 245.7 MPa, which is very close to the experimentally observed value (244.2 MPa). Note that, due to long simulation time (for example 29 h for 20 fatigue cycles using 1 cpu and 1 gpu) the PWR SL component was simulated for only 100 fatigue cycles. However, fully mechanistic determination of the fatigue life of PWR SL pipe requires elastic-plastic analysis of the component for thousands of fatigue cycles. Future work includes high performance computing implementation of the FE modeling framework so that large components/systems can be simulated for thousands of fatigue cycles for fully mechanistic fatigue evaluation.

## Acknowledgment

This research was supported through the U.S. Department of Energy's Light Water Reactor Sustainability program under the work package of environmental fatigue study, program manager Dr. Keith Leonard.

## Funding Data

• U.S. Department of Energy's Light Water Reactor Sustainability program.

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