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FFT-based model for irradiated aggregate microstructures in concrete


The concrete biological shield of light water reactors is exposed to neutron and gamma irradiation throughout its lifetime, which results in the long-term degradation of the concrete’s mechanical properties. Under neutron irradiation, the concrete’s aggregates are subjected to radiation-induced volumetric expansion (RIVE), which strongly depends on the mineral content of the aggregate and exhibits the largest expansion in silicate-bearing minerals. In this work, the authors used the fast Fourier transform-based code Microstructure-Oriented Scientific Analysis of Irradiated Concrete (MOSAIC) in 2D to model the expansion of five different aggregates provided by the Japan Concrete Aging Management Program (JCAMP). Comparable rock specimens were irradiated at the JEEP-II test reactor. The model uses realistic aggregate microstructure reconstruction based on high-resolution characterization images. The model accounts for anisotropic RIVE, thermal expansion, and the associated initiation and propagation of damage. The RIVE models are calibrated based on expansion data in the literature. The authors assume that damage occurs exclusively at interfaces between the particles that compose an aggregate and that these interfaces also exhibit swelling. Using a micromechanical model, the evolution of Young’s modulus with RIVE is calculated for each aggregate and compared with Russian irradiation data. The modeled linear expansion agrees well with the experimentally measured expansion. The model also predicts that anisotropic RIVE and thermal expansion result in an earlier onset of damage with neutron fluence than in the isotropic case.

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This work is supported by the US Department of Energy’s (DOE’s) Office of Nuclear Energy Light Water Reactor Sustainability Program under contract number DE-AC05-00OR22725 and Effects of Irradiation on Bond Strength in Concrete Structures project (31310018S0021) of the US Nuclear Regulatory Commission. The authors also acknowledge support from the US Department of Energy’s Nuclear Energy University Program under contract number DE-NE0008886. The authors thank JCAMP and researchers Maruyama (Nagoya University), Takizawa (Mitsubishi Research Institute), and Kontani (Kajima Corporation) for sharing the unirradiated rock specimens that made this research possible. This cooperation is supported by the Civil Nuclear Energy Research and Development Working Group, a joint organization between DOE and the Japanese Agency for Natural Resources and Energy of the Ministry of Economy, Trade and Industry. This research is also supported by the Horizon 2020 European Commission Project ACES (“Improved assessment of NPP concrete structures toward ageing”). The authors thank Susan Ennaceur and Albert Migliori from Alamo Creek Engineering for providing preliminary estimates of the Young’s modulus for some of the aggregates by using Resonance Ultrasound Spectroscopy.

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Appendix A: Anisotropy in RIVE

In the book by Denisov et al. [7], the dimensional change is given with respect to the lattice parameters a, b, and c of a given mineral. The unit cell is represented by the lattice vectors \(\vec a\), \(\vec b\), and \(\vec c\), which are not necessarily orthogonal, depending on the crystal system of the mineral. Therefore, a coordinate transformation must be applied to the tensor of dimensional changes to obtain the dimensional changes in the “sample” Cartesian frame represented by the orthogonal unit vectors \(\vec x\), \(\vec y\), and \(\vec z\) in MOSAIC. The chosen orientation of the crystal lattice with respect to the Cartesian system follows the convention described in [57]. Following their approach, the structure rank-two tensor A needed for the transformation is a function of angles \(\alpha\) (between \(\vec b\) and \(\vec c\)), \(\beta\) (between \(\vec a\) and \(\vec c\)), and \(\gamma\) (between \(\vec a\) and \(\vec b\)) with each component corresponding to the cosine of the angle between two directions:

$$\begin{aligned} A = \begin{pmatrix} \frac{f}{sin\alpha } &{} 0 &{} 0\\ \frac{cos\gamma -cos\alpha cos\beta }{sin\alpha } &{} sin\alpha &{} 0\\ cos\beta &{} cos\alpha &{} 1 \end{pmatrix}, \end{aligned}$$


$$\begin{aligned}&f = (1 - cos\alpha ^2 - cos\beta ^2 - cos\gamma ^2 \nonumber \\&\quad + 2\,cos\alpha \,cos\beta \,cos\gamma )^{\frac{1}{2}}. \end{aligned}$$

The tensor transformation follows:

$$\begin{aligned} T'_{ij} = a_{ik}a_{jl}T_{kl}, \end{aligned}$$

where \(T'_{ij}\) are the components of the transformed dimensional change tensor \(T'\) (in the Cartesian system), \(a_{ik}\) and \(a_{jl}\) are the components of tensor A, and \(T_{kl}\) are the components of the dimensional change tensor T in the crystal lattice system. Lattice angles \(\alpha\), \(\beta\), and \(\gamma\), as well as RIVE redistribution coefficients, were implemented in the IMAC database for each mineral whenever available in the literature. When dimensional change data were not provided, isotropic RIVE was assumed.

The crystal orientation with respect to the reference Cartesian frame is accounted for by performing a rotation of \(T'\) using the provided Euler angles to obtain the final tensor \(T^c\). The Euler angles are randomly generated to ensure a random orientation for each particle in the 3D space. When simulations are performed in 2D, as in this work, users can specify the direction of the third axis to perform a projection of the 3D property tensor on the 2D simulation plane. Finally, each component of the eigenstrain tensor was set to RIVE multiplied by the corresponding component of the rotated tensor:

$$\begin{aligned} \epsilon _{ij}^{*} = T^c_{ij}\,\epsilon _{RIVE}^{*}, \end{aligned}$$

where \(\epsilon _{RIVE}^{*}\) is the RIVE of the specific mineral.

Appendix B: Micromechanical estimate of the stiffness of an aggregate

This appendix briefly reviews the general framework of homogenization of random media in linear elasticity, then derives a pattern-based generalized self-consistent scheme to estimate the stiffness of a given aggregate. The morphological model is inspired by the main features of the aggregate microstructure generated in Sect. 2. Finally, the case of a nonnatural initial state is considered.

More detailed examples of homogenization and morphological representative patterns are provided by [58,59,60].

1.1 Mean-field homogenization: introduction

This section considers a representative elementary volume (REV) \(\Omega\) of a composite material. The local behavior is linear elastic:

$$\begin{aligned} \varvec{\sigma }(\underline{x})={\mathbb {C}}(\underline{x}) :\varvec{\varepsilon }(\underline{x}), \end{aligned}$$

where \({\mathbb {C}}(\underline{x})\) is the heterogeneous stiffness tensor and \(\underline{x}\) is the position vector. An average, macroscopic, strain \(\varvec{E}\) is prescribed to this domain through uniform strain boundary conditions, \(\underline{\xi }(\underline{x})\), such that:

$$\begin{aligned} \underline{\xi }(\underline{x})=\varvec{E} \cdot \underline{x},\ \underline{x} \in \partial \Omega . \end{aligned}$$

Together with the strain–displacement relation and the stress field equilibrium equation, these equations define an elasticity problem. Because the equations are linear, the strain field solution depends linearly on the macroscopic strain:

$$\begin{aligned} \varvec{\varepsilon }(\underline{x})={\mathbb {A}}(\underline{x}) :\varvec{E}, \end{aligned}$$

introducing the strain localization tensor \({\mathbb {A}}(\underline{x})\). The macroscopic stress \(\varvec{\Sigma }\) is then obtained as the spatial average of the microscopic stress:

$$\begin{aligned} \varvec{\Sigma } = \left\langle \varvec{\sigma }(\underline{x}) \right\rangle _{} = \frac{1}{|\Omega |} \int _{\Omega }^{} \varvec{\sigma }(\underline{x}) \, \mathrm {d} \Omega . \end{aligned}$$

Combining Eqs. (16), (13), and (15) shows that the effective behavior is linear elastic. The effective stiffness is identified as:

$$\begin{aligned} {\mathbb {C}}^{eff} = \left\langle {\mathbb {C}}(\underline{x}) :{\mathbb {A}}(\underline{x}) \right\rangle _{}. \end{aligned}$$

When the composite material REV can be split into phases (i.e., domains where the stiffness tensor is uniform), the effective stiffness is given as:

$$\begin{aligned} {\mathbb {C}}^{eff} = \sum _i f_i {\mathbb {C}}_i :\left\langle {\mathbb {A}}(\underline{x}) \right\rangle _{i}, \end{aligned}$$

i iterating over phases and \(f_i\) is the fraction of phase i. Thus, the complete determination of the strain localization tensor field is not required—only averages over each phase are. These averages can be estimated through homogenization schemes built from a statistical description of the composite material morphology.

An alternative approach is to prescribe an average, macroscopic, stress \(\varvec{\Sigma }\) through uniform stress boundary conditions:

$$\begin{aligned} \varvec{\sigma }(\underline{x}) \cdot \underline{n}(\underline{x}) = \varvec{\Sigma } \cdot \underline{n}(\underline{x}),\ \underline{x} \in \partial \Omega , \end{aligned}$$

where \(\underline{n}(\underline{x})\) is the unit normal tensor at the boundary, pointing outward. The stress localization tensor \({\mathbb {B}}(\underline{x})\) is then defined as:

$$\begin{aligned} \varvec{\sigma }(\underline{x})={\mathbb {B}}(\underline{x}) :\varvec{\Sigma }, \end{aligned}$$

and the effective compliance tensor as:

$$\begin{aligned} {\mathbb {S}}^{eff} = \left\langle {\mathbb {S}}(\underline{x}) :{\mathbb {B}}(\underline{x}) \right\rangle _{}, \end{aligned}$$

where \({\mathbb {S}}(\underline{x})={{{\mathbb {C}}}^{-1}}(\underline{x})\) is the compliance tensor field.

1.2 Derivation of a generalized self-consistent scheme

This section considers the specific case of a 3D polycrystalline material with IPI (Fig. 5). The IPI (denoted by the subscript IPI) stiffness is isotropic. Each mineral phase (denoted by the subscript \(i=1...N\)) is characterized by a shape and an anisotropic stiffness tensor, and is present in the REV as many different particles differing only in orientation, because the orientation distribution, and thus the effective stiffness, is isotropic.

This morphological description suggests resorting to a generalized self-consistent scheme with morphologically representative patterns (MRPs) [60]. Thus, the IPI subdomains that separate two particles are split in two parts, each associated with its adjacent particle. The IPI is thus subdivided into N subdomains. The part that covers the particles that comprise mineral i is denoted by the subscript IPIi. The whole REV is now subdivided into MRPs, which are composite particles, composed of a mineral particle surrounded by an IPI layer.

The average strain localization tensor per phase in such an MRP is classically estimated by embedding the composite particle into an infinite medium whose stiffness is the sought effective stiffness \({\mathbb {C}}^{eff}\) of the REV. With uniform strain (\(\varvec{E}^0\)) boundary conditions at infinity, this defines an auxiliary problem (Fig. 12). Accounting for the very complex particle shapes and the anisotropic mineral stiffness in aggregates would require numerical computations to solve the auxiliary problem. To maintain a reasonable computing time, simplifications are introduced, which enable a semi-analytical resolution of the problem:

  • mineral particles are represented as spheres in the auxiliary problem, and

  • mineral stiffness is considered isotropic and estimated from the anisotropic stiffness tensor through a self-consistent scheme assuming spherical shapes and an isotropic orientation distribution).

Fig. 12
figure 12

Auxiliary problem related to mineral i and its associated IPI cover layer IPIi

The average strains in both the mineral particle and the IPI layer can be derived, from the layered spherical inclusion result ([61]):

$$\begin{aligned} \left\langle \varvec{\varepsilon } \right\rangle _{i} = {\mathbb {A}}_i^0 :\varvec{E}^0 \text {\quad and \quad } \left\langle \varvec{\varepsilon } \right\rangle _{IPI,i} = {\mathbb {A}}_{IPI,i}^0 :\varvec{E}^0, \end{aligned}$$

where \({\mathbb {A}}_i^0\) and \({\mathbb {A}}_{IPI,i}^0\) are strain localization tensors associated with the auxiliary problem whose expressions are too lengthy to be reported here; [61] provides more details on this derivation. The reference strain \(\varvec{E}^0\) prescribed on auxiliary problems can be related to the macroscopic strain \(\varvec{E}\) iterating over the N MRPs as:

$$\begin{aligned} \varvec{E}&=\left\langle \varvec{\varepsilon } \right\rangle _{} = \sum _{i=1}^{N} f_i \left\langle \varvec{\varepsilon } \right\rangle _{i} + f_{IPI,i} \left\langle \varvec{\varepsilon } \right\rangle _{IPI,i} \nonumber \\&\quad = \sum _{i=1}^{N} (f_i {\mathbb {A}}_i^0 + f_{IPI,i} {\mathbb {A}}_{IPI,i}^0) :\varvec{E}^0, \end{aligned}$$

where \(f_i\) and \(f_{IPI,i}\) denote the volume fractions of the respective phases in the REV. Substituting \(\varvec{E}^0\) in Eq. (22) yields estimates of the average strain localization tensor in every phase of the REV: minerals i and half IPI layers IPIi surrounding each mineral (\(i=1..N\)). Equation (18) yields a tensorial equation whose solution is the effective stiffness \({\mathbb {C}}^{eff}\); \({\mathbb {A}}_i^0\) and \({\mathbb {A}}_{IPI,i}^0\) both depend on \({\mathbb {C}}^{eff}\) through the resolution of the auxiliary problem. Two nonlinear equations on the effective bulk \(k^{eff}\) and shear \(g^{eff}\) moduli are then obtained, projecting the tensorial equation over the base tensors \({\mathbb {J}}\) and \({\mathbb {K}}\).

It remains to subdivide the IPI found in the REV into N ”phases”, each surrounding the particles that comprise one mineral, to evaluate the volume fractions \(f_{IPI,i}\). The radius of the sphere representing mineral i is denoted \(r_i\) (in the REV, all particles that comprise mineral i are assumed to be roughly the same size), and the thickness of the IPI layer is denoted \(t_{IPI,i}\) (half the thickness of the IPI separating mineral i from its neighbours). Straightforward volume calculations yield the ratio between the volume fractions in REV of the IPI part covering mineral i and of mineral i in the REV:

$$\begin{aligned} \frac{f_{IPI,i}}{f_i} = \left( 1+\frac{t_{IPI,i}}{r_i}\right) ^3 - 1. \end{aligned}$$

The IPI thickness is now assumed to be uniform across the whole REV, and all particles are assumed to be roughly the same size, regardless of the mineral type. This means that the thickness-to-radius ratio \(t_{IPI,i}/r_i\) is independent of the mineral type i. The volume fraction of the IPI that covers particles of mineral i is thus proportional to the volume fraction of mineral i:

$$\begin{aligned} f_{IPI,i} = \frac{f_{IPI}}{1-f_{IPI}} f_i, \end{aligned}$$

where \(f_{IPI}\) being is the total volume fraction of IPI in the REV.

1.3 Nonnatural initial state

1.3.1 General approach

The local behavior of REV \(\Omega\) is assumed to be linear elastic with a nonnatural initial state—that is, prestressed with a prestress field \(\varvec{\sigma }_p(\underline{x})\):

$$\begin{aligned} \varvec{\sigma }(\underline{x})={\mathbb {C}}(\underline{x}) :\varvec{\varepsilon }(\underline{x}) + \varvec{\sigma }_p(\underline{x}), \end{aligned}$$

or with an eigenstrain field \(\varvec{\varepsilon }_l(\underline{x})\):

$$\begin{aligned} \varvec{\sigma }(\underline{x})={\mathbb {C}}(\underline{x}) :\left( \varvec{\varepsilon }(\underline{x}) - \varvec{\varepsilon }_l(\underline{x}) \right) . \end{aligned}$$

The classical Levin’s theorem [62] shows that the macroscopic behavior has the same structure as the local one with the macroscopic prestress:

$$\begin{aligned} \varvec{\Sigma }_p = \left\langle \varvec{\sigma }_p(\underline{x}) :{\mathbb {A}}(\underline{x}) \right\rangle _{}, \end{aligned}$$

or the macroscopic eigenstrain:

$$\begin{aligned} \varvec{E}_l = \left\langle \varvec{\varepsilon }_l(\underline{x}) :{\mathbb {B}}(\underline{x}) \right\rangle _{}. \end{aligned}$$

The strain (respectively stress) localization tensor \({\mathbb {A}}(\underline{x})\) (respectively \({\mathbb {B}}(\underline{x})\)) that appear in these equations are the ones obtained in a natural initial state (Sect. B.1).

1.3.2 Voigt-Reuss-Hill estimate

The Voigt approximation assumes that the strain field is homogeneous in the whole REV:

$$\begin{aligned}&{\mathbb {A}}(\underline{x}) = {\mathbb {I}}\text {\,\quad } {\mathbb {C}}^{eff} = \left\langle {\mathbb {C}}(\underline{x}) \right\rangle _{} \text {\,\quad } \nonumber \\&\varvec{\Sigma }_p=\left\langle \varvec{\sigma }_p(\underline{x}) \right\rangle _{} = - \left\langle {\mathbb {C}}(\underline{x}) :\varvec{\varepsilon }_l(\underline{x}) \right\rangle _{}, \text {\quad and \quad } \nonumber \\&\varvec{E}_l=-{{{\mathbb {C}}^{eff}}^{-1}} :\varvec{\Sigma }_p. \end{aligned}$$

The Reuss approximation assumes that the stress field is homogeneous in the whole REV:

$$\begin{aligned} {\mathbb {B}}(\underline{x}) = {\mathbb {I}}\text {\,\quad } {\mathbb {S}}^{eff} = \left\langle {\mathbb {S}}(\underline{x}) \right\rangle _{} \text {\quad and \quad } \varvec{E}_l=\left\langle \varvec{\varepsilon }_l(\underline{x}) \right\rangle _{}. \end{aligned}$$

The Voigt-Reuss-Hill estimate of the effective stiffness or the effective eigenstrain is then the arithmetic average of the Voigt and Reuss estimates.

Appendix C: High-resolution chemical characterization

The aggregate samples were mounted in conductive epoxy and polished for chemical analyses. The samples were mapped using an mXRF Atlas unit (IXRF systems) with a Rh x-ray source operated at 50 kV, 600 \(\mu\)A, and 10 \(\mu\)m spot size. The maps were acquired by surveying 6 \(\times\) 6 mm areas with a dwell time of 300 ms, a resolution of 400 \(\times\) 400 pixels, a time constant of 1, and a pixel size of 15 \(\mu\)m. The resulting x-ray images and elemental maps are provided as Supplementary Information. The square marks in aggregates GB/E and GA/F were laser engraved to serve as a reference for aligning further measurements. The data were cropped to eliminate the engravings to create the maps displayed in Fig. 4.

The compositional data acquired with the mXRF were complemented with SEM-EDS maps of localized areas to obtain information on Na concentration because Na is both below the detection limit of the mXRF and a marker of some likely important minerals, such as albite (actually albitic alkali feldspar or plagioclase). Samples GB/E and GA/F were mapped by using an environmental scanning electron microscope (FEI Quanta FEG 450 ESEM) coupled with an energy dispersive x-ray spectrometer. The EDS maps were collected in low-vacuum mode with an acceleration voltage of 10.00 kV, a spot size of 5 nm, and a chamber pressure of 1 Torr. Images were taken with a working distance of 10 mm, dwell time of 50 \(\mu\)s, and a resolution of 1024 \(\times\) 800 pixels by using an EDAX light element EDS module. ZAF correction was applied to quantify the elemental maps. Samples GC/G, GD/H, and GE/J were analyzed using a Tescan MIRA3 GMH SEM equipped with an Ultim Max EDS Oxford Instruments EDS system. The maps were collected at an acceleration voltage of 15 kV and a beam intensity of 18 \(\mu\)A, covering an area of 1.328 \(\times\) 1.047 mm. The resolution was 512 \(\times\) 512 pixels, the frame count was 80, and the pixel dwell time was 200 \(\mu\)s. The maps were quantified in weight percent using the AZTEC software package. The acquired maps are provided as Supplementary Information.

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Cheniour, A., Li, Y., Sanahuja, J. et al. FFT-based model for irradiated aggregate microstructures in concrete. Mater Struct 55, 214 (2022).

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