Elastic moduli reduction in SiC-SiC tubular specimen after high heat flux neutron irradiation measured by resonant ultrasound spectroscopy⋆
Introduction
Silicon carbide (SiC) – based ceramic composite cladding is an accident-tolerant fuel cladding concept for light water reactors (LWRs) [1] being widely studied because of its unique benefits under both normal and accident conditions, including relatively low neutron absorption and high-temperature strength [[2], [3], [4], [5]]. Development of SiC-based cladding for fission reactors will require extensive evaluation and assessment of its mechanical properties at different stages of exposure to the reactor environment. Although extensive post-irradiation examination (PIE) studies have been conducted in the past to understand the effects of neutron irradiation on SiC-SiC specimens, almost all the studies involved flat specimens [6]. From the point of view of SiC cladding development, PIE studies on tubular specimens irradiated under conditions relevant to reactor environments are essential. For example, it is important to understand how neutron irradiation affects SiC mechanical properties under a high heat flux caused by the presence of both hot fuel and coolant water. It has been shown that in a SiC fiber–reinforced–SiC matrix composite (SiC-SiC) cladding, the differential swelling across the thickness of the tube generates tensile stresses at the inner region of the cladding and compressive stresses at the outer region of the cladding [[7], [8], [9]]. The study presented herein is aimed toward understanding the irradiation behavior of SiC-SiC tubular specimen.
One of the difficulties with PIE studies is the limited number of available test specimens because of the limited volume of reactor irradiation for material testing. Therefore, a solution is to perform material evaluations of the same specimens before and after irradiation, which allows researchers to establish property correlations that constitute the key knowledge base for the material involved. Thus, considering the limited number of specimens available and the probabilistic nature of failure in ceramics, it is not ideal to perform destructive studies, which are often needed to understand the mechanical properties. There are several ultrasonic techniques that are used to determine mechanical properties, such as the pulse-echo [10], through-transmission and impulse excitation techniques [11]. Pulse-echo and through-transmission techniques evaluate the elastic properties of a material by determining the speed of sound which is directly connected to the elastic properties and density of the material. Impulse excitation technique evaluates the elastic properties of a material by measuring its resonance frequencies, which are functions of the elastic properties, mass and geometry of the material. This technique is based on the same principle as resonant ultrasound spectroscopy (RUS). Since RUS can employ a large number of resonance frequencies to determine the elastic constants, it tends to be more accurate than the other techniques. In this study, we explored nondestructive RUS [12] for evaluating the elastic properties of a SiC-SiC composite tubular specimen before and after irradiation.
The origin of RUS can be traced to the 1960s, when Fraser and LeCraw [13] of Bell Telephone Laboratories (now Nokia Bell Labs) successfully determined the Lamé constant and Poisson's ratio of several materials using a shear mode transducer. This technique, then called the resonant sphere technique, was used for determining the elastic properties of geological samples [14,15]. In the 1990s, Migliori and other researchers applied it to determine the elastic properties of several materials and detect defects [12,[16], [17], [18], [19], [20]]. With the rapid advancement of computing technology, the capability of RUS also increased. The technique has been applied to determine the elastic constants of anisotropic materials [[21], [22], [23], [24]].
The resonance frequencies of a body are the same as the stationary points of a Lagrangian function [12]. Therefore, resonance frequencies of a specimen with known dimensions and elastic constants can be calculated by the Lagrangian minimization technique. The Lagrangian of a free body is given aswhere
v: spatial velocity of infinitesimal element,
ρ: mass density,
εij: strain tensor component,
Cijkl: elastic stiffness tensor component,
dV: infinitesimal volume,
V: total volume
In Eq. (1), the first term inside the integral represents the kinetic energy and the second term represents the potential energy of the infinitesimal section of the body. The velocity term can be written as v2 = ωu2 = ωuiui, where ui is the displacement vector component. The strain tensor component εij can be expressed aswhere xj is the jth component of the position vector. With these substitutions for velocity term and strain tensor, the Lagrangian is expressed as
The displacements uj, which minimize the Lagrangian, give the natural modes of vibration of the body. There are analytical solutions for a very few geometries, such as a solid cylinder, rectangular parallelepiped and sphere. Therefore, in most cases, the approximate solution must be obtained by numerical methods. Rayleigh-Ritz methods are commonly employed for this purpose. The Rayleigh-Ritz method involves expanding the displacement component ui in a set of basis functions selected according to the geometry of the body and simplifying. The resulting equation is of the form of a typical eigenvalue problem:where I is the identity matrix, a is the eigenvector, and Γ is the potential energy term expressed using the basis functions and elastic constants of the material. The equation is solved for ω which is the resonance frequency, and the eigenvector a which corresponds to the mode of vibration.
The RUS technique is based on measuring the resonance frequencies of a specimen. In addition to measuring the elastic moduli, it can use any deviation in the resonance frequency spectrum from its typical characteristics for a defect-free specimen to detect defects or changes in the elastic properties of the specimen. A major advantage of this technique is that it can also determine the elastic properties of anisotropic materials.
The RUS technique is not entirely experimental, as Eq. (4) is solved for elastic constants through numerical methods as an inverse problem: for a set of elastic constants, the resonance frequencies are calculated, then these elastic constants are adjusted iteratively until the difference between the calculated frequencies and measured frequencies is minimized. The numerical techniques for calculating the frequencies and algorithms for optimization of the elastic constants [[25], [26], [27], [28], [29]] were developed to improve the convergence of the solution process and evaluate anisotropic materials. Even though anisotropic materials could be evaluated using these numerical techniques, the types of specimen geometries that could be evaluated were very limited. Later on, the incorporation of the finite element method for calculating the frequencies significantly expanded the domain of specimen geometries that could be studied [[30], [31], [32], [33], [34], [35]] through RUS. Even though usage of the RUS technique has been increasing, there are very few studies [[36], [37], [38]] that evaluate the elastic properties of specimens with a tubular geometry using RUS. And so far, no composite material specimens with tubular geometries have been studied using RUS. Moreover, the use of RUS for nuclear material studies has been very limited [[39], [40], [41], [42], [43]]. In the present study, RUS was applied to evaluate the elastic properties of orthotropic SiC-SiC tubular specimen before and after neutron irradiation. The purpose of this study was to assess RUS as a material characterization technique to evaluate complex materials intended for nuclear applications.
Section snippets
Materials and methods
Irradiation experiments were conducted for the purpose of understanding radiation effects on SiC-SiC tubular specimens irradiated at representative LWR temperatures and heat fluxes. The elastic properties of one of these specimens was evaluated using RUS before and after irradiation. The experiment involved irradiating miniature SiC-SiC tubular specimens for one cycle in the High Flux Isotope Reactor at Oak Ridge National Laboratory. The details of the experiments are provided in Petrie et al. [
Results
The RUS technique was validated for evaluating the elastic properties of SiC-SiC tubular specimen. The details of the validation procedure are presented in section 3.1. The results of evaluating the effects of irradiation on the elastic properties of the SiC-SiC tubular specimen are presented in section 3.2. Note that the SiC-SiC specimens used for RUS validation purposes and the SiC-SiC specimen used for the irradiation study are not identical: the specimens used for validation purposes had
Discussion
The results presented in Table 2 indicate a significant reduction in the elastic moduli of the irradiated SiC-SiC tubular specimen. There are three mechanisms that potentially lead to reductions in the elastic moduli of a composite specimen: (1) a decrease in the bulk elastic moduli of the constituents (mainly SiC fibers and matrix), (2) debonding at the fiber-matrix interphases and (3) microcracking within the material due to irradiation-induced stress. The first mechanism of reduction in the
Conclusion and future work
The work presented herein assessed the viability of RUS for evaluating irradiation effects on the elastic properties of SiC-SiC composite tubular specimen. The validation tests gave reasonable results for the elastic properties of the SiC-SiC specimens. For the PIE study, the elastic properties of the SiC-SiC specimen were successfully evaluated before and after irradiation. These first results on the effects of irradiation on different elastic constants of the SiC-SiC composite tubular
Acknowledgments
This research was supported by the Advanced Fuels Campaign of the Nuclear Technology R&D program within the United States Department of Energy (DOE) Office of Nuclear Energy. This work was prepared under contract DE-AC05-00OR22725 with Oak Ridge National Laboratory managed by UT Battelle LLC. A portion of this research used resources at the High Flux Isotope Reactor, which is funded by the DOE Office of Basic Energy Sciences. This work was also supported by the US Department of Energy (DOE)
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Notice: This manuscript has been authored by UT-Battelle, LLC, under contract DE-AC05-00OR22725 with the US Department of Energy (DOE). The US government retains and the publisher, by accepting the article for publication, acknowledges that the US government retains a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this manuscript, or allow others to do so, for US government purposes. DOE will provide public access to these results of federally sponsored research in accordance with the DOE Public Access Plan (http://energy.gov/downloads/doe-public-access-plan).
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Currently at University of Tennessee, Knoxville, TN, USA.