Modeling the mechanical behavior and damage development of concrete materials under neutron irradiation

Abstract

Evaluation of concrete structures in nuclear power plants (NPPs) under long-term irradiation exposure is important to ensure a safe and reliable operation of NPPs during extended service life from 40 to 80 years. In this study, a comprehensive multiscale framework of theoretical models was developed to predict the deformation and degradation of mechanical properties of concrete materials subject to long-term neutron irradiation. The generalized self-consistent model and the Mori–Tanaka model were used to characterize the mechanical properties of concrete with multiple phases and multiple scale internal structures. The overall expansion and degradation of mechanical properties of concrete resulted from neutron irradiation as well as elevated temperature were estimated using a composite damage mechanics approach. The neutron radiation-induced degradation, volumetric expansion of aggregates, thermal strains, and shrinkage of cement paste were considered in the comprehensive model. The model can be used as a predictive tool for the effect of long-term neutron irradiation on concrete used in NPPs.

Graphical abstract

Keywords

concrete / composite mechanics / damage / neutron radiation / mechanical properties

1 Introduction

Certain concrete structures in nuclear power plants (NPPs), such as biological shielding structures, are under a high nuclear radiation environment, and thus the radiation levels in the structures are expected to be high with license renewal to extend NPP operation time [1]. Research data [2] indicated that nuclear radiations can cause degradation of the physical and mechanical properties of concrete which could impair the structural and shielding capacities of safety-related concrete structures in a NPP. Significant effort has been made by research communities to address durability issues of nuclear irradiated concrete. The research results will be beneficial to the assessment process to further extend the operation period of existing reactors conducted by NPP regulatory agencies.

Many experimental data and analysis about the effect of nuclear irradiation on concrete were published decades ago, and many details of these tests are not well documented [3]. The related experimental research has been very limited recent years. Maruyama et al. [4] conducted a comprehensive and detailed experimental study regarding the cementitious material under neutron and gamma irradiation which provided the most comprehensive experimental data set of nuclear irradiated concrete to date. Information and analysis of the experimental data provided very valuable insight into the impact of nuclear irradiation on concrete materials. It was found that the radiation induced volumetric expansion (RIVE) of aggregates is the primary cause of the degradation of nuclear irradiated concrete [2,5,6]. Supplementary to experimental studies, modeling work is still needed to characterize, simulate and predict the deterioration mechanisms of aged concrete structures in NPPs and evaluate their performance in the future.

A coupled hygro-thermo-mechanical model was first introduced to investigate the degradation of nuclear irradiated concrete structures under nuclear [7–9]. In this model, the continuum scalar damage model was adopted to evaluate the coupled thermo-chemo-mechanical and radiation damage. Finite element method was applied to obtain the results numerically. A similar approach was also used for the coupled damage-creep analysis of a concrete biological shielding wall [10]. The radiation damage in these models was introduced based upon experimental data of the overall behavior of irradiated concrete, and the actual irradiation induced damage mechanisms that occurred within the concrete was not considered. In another word, what happened in the concrete during the nuclear irradiation was not described, which is important because concrete is a multiphase material and each phase behaviors differently under the same level of irradiation. To this end, micromechanical models [11,12] were developed based on analytical approach for analysis of radiation effects in the concrete, and the RIVE of aggregates were taken into account to predict the expansion or damage of nuclear irradiated concrete. Two/ three-dimensional mesoscale finite element models [13–16] were proposed to handle this problem as well, in which aggregate and cement paste were described by two different mechanical constitutive models considering nuclear radiation effects. Another approach called the rigid body-spring network model [17] was applied to evaluate the degradation of concrete properties due to the RIVE of aggregates. There are also prediction models proposed for coupled radio-thermo analysis in concrete bioshield wall [18,19], and concrete strength prediction models considering heat, moisture, radiation transport, and hydration of cement [20].

Concrete is a heterogeneous composite with multiple constituents spanning multiple scales, and different types of damage can be generated in different constituents under nuclear irradiation which should be considered systematically. In our preceding study [11], the damage development in concrete material is not fully considered. In the work, the internal structures of concrete at multiple scales are simplified and characterized based on generalized self-consistent (GSC) model and the Mori–Tanaka model. A prediction framework is built through the development of a multiscale and multiphase analytical model for the mechanical behavior and degradation of concrete exposed to radiation for long-term operation. The framework can integrate radiation-induced degradation mechanisms, and be applied to various concrete materials with different mix designs used in different NPPs.

2 Multiscale model for the internal structure of concrete

In this study, concrete is characterized into a multiphase composite with four different scale levels. The four scale levels are concrete, mortar, cement paste, and clinker as shown in Fig.1. GSC model and the Mori–Tanaka model are used to describe the internal structures of concrete and to obtain the effective properties of the composite at each scale. The obtained values at the lower scale level will be used as the input for the composite material at the upper scale level. The model can take into account the contributions of all constituent phases in the concrete material. The detailed descriptions of the models at all scale levels are provided in the following sections.

2.1 Mesostructures and models of concrete and mortar

At the mesoscale level, concrete can be treated as a matrix-inclusion composite material: mortar matrix with coarse aggregates acting as inclusions. Similarly, for the mortar, the fine aggregates can be considered as the inclusions where the cement paste is the matrix. With this approach, these two scale levels can be modeled using the GSC model to estimate the effective properties of the mortar, and subsequently, the effective properties of the concrete.

2.1.1 Generalized self-consistent model

The GSC model was first developed for the effective elastic moduli and thermal conductivity of composite materials [21]. The model has been used by the authors to study some durability issues of concrete, including shrinkage [22], thermal conduction [23], moisture transport [24], thermal damage [25], and radiation effects [11,18,19]. The model is briefly introduced here and more details can be found in the previous studies.

As shown in Fig.2, a heterogeneous and multiphase composite material can be partitioned into multiple elements which are further simplified to be spherical, and each element is a concentrically layered sphere. The radius ratios for the phases in each element are constant. The basic three-phase model shown in Fig.2(c) is applied to describe the whole composite material. The three phases are the inclusion, the matrix, and the effective medium. The inclusion and the matrix are considered as two constituents, and the effective medium surrounding the basic element represents the composite material.

The effective strain, bulk modulus, and shear of concrete can be calculated based on the GSC model using the following equations.

(1){\epsilon }_{\mathrm{e}\mathrm{f}\mathrm{f}}=\frac{K_1{\epsilon }_1{f}_1\left(3K_2+{4G}_2\right)+K_2{\epsilon }_2(1-f_1)(3K_1+{4G}_2)}{K_2\left(3K_1+{4G}_2\right)-4f_1{G}_2(K_2-K_1)},

(2)K_{\mathrm{e}\mathrm{f}\mathrm{f}}=K_2+\frac{f_1(K_1-K_2)}{1+\left(1-f_1\right){\displaystyle \frac{K_1-K_2}{K_2+{\displaystyle \frac{4}{3}}{G}_2}}},

(3){C_1\left(\frac{G_{\mathrm{e}\mathrm{f}\mathrm{f}}}{G_2}\right)}^2+2C_2\left(\frac{G_{\mathrm{e}\mathrm{f}\mathrm{f}}}{G_2}\right)+C_3=0,

where K_i is bulk modulus of phase i (MPa); G_i is shear modulus of phase i (MPa); ε_i is normal strain of phase i (dimensionless); f_i is volume fraction of phase i (dimensionless); C₁, C₂, and C₃ are equation coefficients (dimensionless), their expressions can be found in Appendix A in Electronic Supplementary Material.

The subscript i with a value of 1, 2, or ‘eff’ represents the inclusion, matrix, or effective medium, respectively. Determination of f_i can be found in Subsection 2.3. When the deformations of the inclusion and matrix shown in Fig.2(c) do not match each other, there will be a nonzero internal pressure P at the interfaces due to the volumetric mismatch

(4)P=\frac{12K_1{K}_2{G}_2\left(1-f_1\right)}{K_2\left(3K_1+{4G}_2\right)-4f_1{G}_2\left(K_2-K_1\right)}\left({\epsilon }_1-{\epsilon }_2\right).

The three-phase model can be future applied to composite material with multiple phases. Fig.3 illustrates the algorithm of the generalization from three-phase to four-phase. The derivation details for Eqs. (1)–(4) are shown in Appendix A in Electronic Supplementary Material.

2.1.2 Mesoscale models for concrete and mortar

It should be emphasized that the GSC model is good for characterizing composite with strong phase association features. The inclusions should have low angularities and can be randomly distributed particles, and the matrix must be the phase surrounding the inclusions. Therefore, two scale levels are considered for concrete in mesoscale level using the GSC model as shown in Fig.1. The first one at the mesoscale level is for concrete, in which the coarse aggregates are inclusions and the mortar is the matrix. The second one is for the mortar, in which the fine aggregates are inclusions and the cement paste is the matrix.

2.2 Microstructure of cement paste

The microstructure of cement paste at the micrometer level involves various hydration products which are quite different from the mesostructure of concrete at the millimeter level described in the previous section. The Mori–Tanaka model instead of GSC model is used for cement paste.

2.2.1 Mori–Tanaka model

The Mori–Tanaka model [26,27] is capable of modeling a porous media containing microstructural constituents with various morphologies, making it suitable for modeling the effective properties of cement paste. As shown in Fig.4, a heterogeneous composite material is generalized to an equivalent homogeneous medium (EHM) with different inclusions within the material. The inclusions can be simplified and categorized into three groups: elliptical inclusions with various orientations (P₁), spherical inclusions (P₂), and elliptical inclusions with a known identical orientation (P₃). All of them are embedded in the matrix phase (M). The effective stiffness tensor can be calculated based on the stiffness tensors of the constituents and the volume fractions of the phases, and another tensor will be needed in the formulation to consider the shapes of the phases [28,29]. The effective stiffness tensor of the composite material can be expressed as

(5)C_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}=\sum _{p=P_1,P_2,P_3,M}{f}_p{C}_p:A_p,

(6)\begin{array}{rl}{A}_p=& {\left[\mathrm{I}+S_p:\left(C_{\mathrm{M}}^{-1}:C_p-\mathrm{I}\right)\right]}^{-1}:\\ & {\left\{\sum _{p=P_1,P_2,P_3}{f}_p{\left[\mathrm{I}+S_p:\left(C_{\mathrm{M}}^{-1}:C_p-\mathrm{I}\right)\right]}^{-1}\right\}}^{-1},\end{array}

where f_p is volume fraction of phase p (unitless); C_p is fourth order stiffness tensor of phase p; A_p is fourth order strain concentration tensor of phase p; I is symmetric fourth order unity tensor; S_p is fourth order Eshelby tensor of phase p.

The Eshelby tensor S_p quantifies the response of an ellipsoidal inclusion to a homogeneous static strain. The shape factor of the inclusion is described by the ratio between the axis length of revolution and the length of an orthogonal axis. For the sake of simplicity, the average Eshelby tensor is used. The stiffness tensor of the composite material shown in Fig.4 is given as

(7)\begin{array}{rl}{\mathit{C}}_{\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}}=& \{{\int }_{\theta =0}^{\pi }{\int }_{\phi =0}^{2\pi }{f}_{P_1}{\mathit{C}}_{{\mathrm{P}}_1}:[\mathbf{I}+{\mathit{S}}_{{\mathrm{P}}_1}\left(\theta ,\phi \right):\\ & {\left({\mathit{C}}_{\mathrm{M}}^{-1}:{\mathit{C}}_{{\mathrm{P}}_1}-\mathbf{I}\right)]}^{-1}\frac{\mathrm{s}\mathrm{i}\mathrm{n}\theta }{4\pi }\mathrm{d}\theta \mathrm{d}\phi +\sum _{p=P_2,P_3}{f}_p{\mathit{C}}_p:\\ & {\left[\mathbf{I}+{\mathit{S}}_p:\left({\mathit{C}}_{\mathrm{M}}^{-1}:{\mathit{C}}_p-\mathbf{I}\right)\right]}^{-1}+f_{\mathrm{M}}{\mathit{C}}_{\mathrm{M}}\}:\\ & \{{\int }_{\theta =0}^{\pi }{\int }_{\phi =0}^{2\pi }{f}_{{\mathrm{P}}_1}[\mathbf{I}+{\mathit{S}}_{{\mathrm{P}}_1}\left(\theta ,\phi \right):\\ & {\left({\mathit{C}}_{\mathrm{M}}^{-1}:{\mathit{C}}_{{\mathrm{P}}_1}-\mathbf{I})\right]}^{-1}\frac{\mathrm{s}\mathrm{i}\mathrm{n}\theta }{4\pi }\mathrm{d}\theta \mathrm{d}\phi +\\ & \sum _{p=P_2,P_3}{f}_p{\left[\mathbf{I}+{\mathit{S}}_p:\left({\mathit{C}}_{\mathrm{M}}^{-1}:{\mathit{C}}_p-\mathbf{I}\right)\right]}^{-1}+f_{\mathrm{M}}\mathbf{I}{\}}^{-1},\end{array}

where \theta ,\phi are Euler angles which describe the orientation of a rigid body with respect to a fixed coordinate system.

Comparing Fig.4 and Fig.3, it is clear that the Mori–Tanaka model is good for those materials with multiple randomly distributed phases, while the GSC model is suitable for the internal structures with phase associations. The Mori–Tanaka model will be applied to cement paste in the next section, where the specific phases in cement paste will be assigned to the inclusions shown in Fig.4.

2.2.2 Micro-scale level model for cement paste

The hardened cement paste can be treated as a multiphase composite. The phases considered are clinker, and calcium silicate hydrate (C-S-H) gel. C-S-H gel is categorized into three different forms: low density (LD) C-S-H, high density (HD) C-S-H and ultrahigh density (UHD) C-S-H [30]. The LD C-S-H is considered as the matrix. The UHD C-S-H forms surrounding the clinker. The HD C-S-H locates between them. A similar approach shown in Fig.3 is used: the clinker and UHD C-S-H are treated as the sub-inclusion which combines with HD C-S-H are considered as the equivalent inclusion. Fig.5 also shows the theoretical model of hardened cement paste including each component’s morphology. The clinker can be modeled as the average of an oblate spheroid and a prolate spheroid with an aspect ratio of 0.81 [31]. The orientations of clinker, HD, and UHD C-S-H are considered to be parallel with same aspect ratio. The ettringite shows an intrinsic needle-like morphology and is considered as a prolate spheroid with an aspect ratio of 0.69. Calcium hydroxide (CH) is modeled as a prolate spheroid with an aspect ratio of 0.7. The voids are assumed to be spherical.

According to Fig.5 and Eq. (7), the stiffness tensor of the sub-inclusion is

(8)\begin{array}{rl}{\mathit{C}}_{\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{c}}=& \left\{{\mathit{f}}_{\mathrm{c}\mathrm{l}\mathrm{i}\mathrm{n}}^{\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{c}}{\mathit{C}}_{\mathrm{c}\mathrm{l}\mathrm{i}\mathrm{n}}:{\left[\mathbf{I}+{\mathit{S}}_{\mathrm{c}\mathrm{l}\mathrm{i}\mathrm{n}}:\left({\mathit{C}}_{\mathrm{U}\mathrm{H}\mathrm{D}\mathrm{C}\mathrm{S}\mathrm{H}}^{-1}:{\mathit{C}}_{\mathrm{c}\mathrm{l}\mathrm{i}\mathrm{n}}-\mathbf{I}\right)\right]}^{-1}+{\mathit{f}}_{\mathrm{U}\mathrm{H}\mathrm{D}\mathrm{C}\mathrm{S}\mathrm{H}}^{\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{c}}{\mathit{C}}_{\mathrm{U}\mathrm{H}\mathrm{D}\mathrm{C}\mathrm{S}\mathrm{H}}\right\}\\ & :{\left\{{\mathit{f}}_{\mathrm{c}\mathrm{l}\mathrm{i}\mathrm{n}}^{\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{c}}{\left[\mathbf{I}+{\mathit{S}}_{\mathrm{c}\mathrm{l}\mathrm{i}\mathrm{n}}:\left({\mathit{C}}_{\mathrm{U}\mathrm{H}\mathrm{D}\mathrm{C}\mathrm{S}\mathrm{H}}^{-1}:{\mathit{C}}_{\mathrm{c}\mathrm{l}\mathrm{i}\mathrm{n}}-\mathbf{I}\right)\right]}^{-1}+{\mathit{f}}_{\mathrm{U}\mathrm{H}\mathrm{D}\mathrm{C}\mathrm{S}\mathrm{H}}^{\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{c}}\mathbf{I}\right\}}^{-1}.\end{array}

The stiffness tensor of the equivalent inclusion is

(9)\begin{array}{rl}{\mathit{C}}_{\mathrm{E}\mathrm{I}}=& \left\{f_{\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{c}}^{\mathrm{E}\mathrm{I}}{\mathit{C}}_{\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{c}}:{\left[\mathbf{I}+{\mathit{S}}_{\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{c}}:\left({\mathit{C}}_{\mathrm{H}\mathrm{D}\mathrm{C}\mathrm{S}\mathrm{H}}^{-1}:{\mathit{C}}_{\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{c}}-\mathbf{I}\right)\right]}^{-1}+f_{\mathrm{H}\mathrm{D}\mathrm{C}\mathrm{S}\mathrm{H}}^{\mathrm{E}\mathrm{I}}{\mathit{C}}_{\mathrm{H}\mathrm{D}\mathrm{C}\mathrm{S}\mathrm{H}}\right\}\\ & :{\left\{f_{\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{c}}^{\mathrm{E}\mathrm{I}}{\left[\mathbf{I}+{\mathit{S}}_{\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{c}}:\left({\mathit{C}}_{\mathrm{H}\mathrm{D}\mathrm{C}\mathrm{S}\mathrm{H}}^{-1}:{\mathit{C}}_{\mathrm{s}\mathrm{u}\mathrm{b}\mathrm{i}\mathrm{n}\mathrm{c}}-\mathbf{I}\right)\right]}^{-1}+f_{\mathrm{H}\mathrm{D}\mathrm{C}\mathrm{S}\mathrm{H}}^{\mathrm{E}\mathrm{I}}\mathbf{I}\right\}}^{-1}.\end{array}

The stiffness tensor of the hardened cement paste is

(10)\begin{array}{rl}{\mathit{C}}_{\mathrm{c}\mathrm{p}}=& \{f_{\mathrm{L}\mathrm{D}}{\mathit{C}}_{\mathrm{L}\mathrm{D}\mathrm{C}\mathrm{S}\mathrm{H}}+f_{\mathrm{c}\mathrm{a}\mathrm{p}}{\mathit{C}}_{\mathrm{c}\mathrm{a}\mathrm{p}}:{\left[\mathbf{I}+{\mathit{S}}_{\mathrm{c}\mathrm{a}\mathrm{p}}^{\mathrm{M}}:\left({\mathit{C}}_{\mathrm{L}\mathrm{D}\mathrm{C}\mathrm{S}\mathrm{H}}^{-1}:{\mathit{C}}_{\mathrm{c}\mathrm{a}\mathrm{p}}-\mathbf{I}\right)\right]}^{-1}\\ & +f_{\mathrm{E}\mathrm{I}}{\mathit{C}}_{\mathrm{E}\mathrm{I}}:{\int }_{\theta =0}^{\pi }{\int }_{\phi =0}^{2\pi }{\left[\mathbf{I}+{\mathit{S}}_{\mathrm{E}\mathrm{I}}^{\mathrm{M}}\left(\theta ,\phi \right):\left({\mathit{C}}_{\mathrm{L}\mathrm{D}\mathrm{C}\mathrm{S}\mathrm{H}}^{-1}:{\mathit{C}}_{\mathrm{E}\mathrm{I}}-\mathbf{I}\right)\right]}^{-1}\frac{\mathrm{s}\mathrm{i}\mathrm{n}\theta }{4\pi }\mathrm{d}\theta \mathrm{d}\phi \\ & +f_{\mathrm{A}\mathrm{F}\mathrm{t}}{\mathit{C}}_{\mathrm{A}\mathrm{F}\mathrm{t}}:{\int }_{\theta =0}^{\pi }{\int }_{\phi =0}^{2\pi }{\left[\mathbf{I}+{\mathit{S}}_{\mathrm{A}\mathrm{F}\mathrm{t}}^{\mathrm{M}}\left(\theta ,\phi \right):\left({\mathit{C}}_{\mathrm{L}\mathrm{C}\mathrm{S}\mathrm{H}}^{-1}:{\mathit{C}}_{\mathrm{A}\mathrm{F}\mathrm{t}}-\mathbf{I}\right)\right]}^{-1}\frac{\mathrm{s}\mathrm{i}\mathrm{n}\theta }{4\pi }\mathrm{d}\theta \mathrm{d}\phi \\ & +f_{\mathrm{C}\mathrm{H}}{\mathit{C}}_{\mathrm{C}\mathrm{H}}:{\int }_{\theta =0}^{\pi }{\int }_{\phi =0}^{2\pi }{\left[\mathbf{I}+{\mathit{S}}_{\mathrm{C}\mathrm{H}}^{\mathrm{M}}\left(\theta ,\phi \right):\left({\mathit{C}}_{\mathrm{L}\mathrm{D}\mathrm{C}\mathrm{S}\mathrm{H}}^{-1}:{\mathit{C}}_{\mathrm{C}\mathrm{H}}-\mathbf{I}\right)\right]}^{-1}\frac{\mathrm{s}\mathrm{i}\mathrm{n}\theta }{4\pi }\mathrm{d}\theta \mathrm{d}\phi \}\\ & :\{f_{\mathrm{L}\mathrm{D}}\mathbf{I}+f_{\mathrm{c}\mathrm{a}\mathrm{p}}{\left[\mathbf{I}+{\mathit{S}}_{\mathrm{c}\mathrm{a}\mathrm{p}}^{\mathrm{M}}:\left({\mathit{C}}_{\mathrm{L}\mathrm{D}\mathrm{C}\mathrm{S}\mathrm{H}}^{-1}:{\mathit{C}}_{\mathrm{c}\mathrm{a}\mathrm{p}}-\mathbf{I}\right)\right]}^{-1}+{\int }_{\theta =0}^{\pi }{\int }_{\phi =0}^{2\pi }{f_{\mathrm{E}\mathrm{I}}\left[\mathbf{I}+{\mathit{S}}_{\mathrm{E}\mathrm{I}}^{\mathrm{M}}\left(\theta ,\phi \right):\left({\mathit{C}}_{\mathrm{L}\mathrm{D}\mathrm{C}\mathrm{S}\mathrm{H}}^{-1}:{\mathit{C}}_{\mathrm{E}\mathrm{I}}-\mathbf{I}\right)\right]}^{-1}\frac{\mathrm{s}\mathrm{i}\mathrm{n}\theta }{4\pi }\mathrm{d}\theta \mathrm{d}\phi \\ & +{\int }_{\theta =0}^{\pi }{\int }_{\phi =0}^{2\pi }{f_{\mathrm{A}\mathrm{F}\mathrm{t}}\left[\mathbf{I}+{\mathit{S}}_{\mathrm{A}\mathrm{F}\mathrm{t}}^{\mathrm{M}}\left(\theta ,\phi \right):\left(C_{\mathrm{L}\mathrm{D}\mathrm{C}\mathrm{S}\mathrm{H}}^{-1}:{\mathit{C}}_{\mathrm{A}\mathrm{F}\mathrm{t}}-\mathbf{I}\right)\right]}^{-1}\frac{\mathrm{s}\mathrm{i}\mathrm{n}\theta }{4\pi }\mathrm{d}\theta \mathrm{d}\phi \\ & +{\int }_{\theta =0}^{\pi }{\int }_{\phi =0}^{2\pi }{f_{\mathrm{C}\mathrm{H}}\left[\mathbf{I}+{\mathit{S}}_{\mathrm{C}\mathrm{H}}^{\mathrm{M}}\left(\theta ,\phi \right):\left({\mathit{C}}_{\mathrm{L}\mathrm{D}\mathrm{C}\mathrm{S}\mathrm{H}}^{-1}:{\mathit{C}}_{\mathrm{C}\mathrm{H}}-\mathbf{I}\right)\right]}^{-1}\frac{\mathrm{s}\mathrm{i}\mathrm{n}\theta }{4\pi }\mathrm{d}\theta \mathrm{d}\phi {\}}^{-1}.\end{array}

2.2.3 The theoretical model for clinker

To consider the effects of the cement type, clinker can be considered as an individual sub-phase of cement paste with four phases. The phases considered in the composite model are belite (C₂S), alite (C₃S), tricalcium aluminate (C₃A), and ferrite (C₄AF) as shown in Fig.6 and Tab.1. The model for the clinker shown in Fig.6 is based on its microstructure [32]. The C₃S is modeled as the matrix, while C₂S, C₃A, and C₄AF are considered as individual spherical inclusions.

According to Fig.6 and Eq. (7), the stiffness tensor of clinker is

(11)\begin{array}{rl}& {\mathit{C}}_{\mathrm{c}\mathrm{l}\mathrm{i}\mathrm{n}}\\ =& \left\{\sum _{p={\mathrm{C}}_2\mathrm{S},{\mathrm{C}}_4\mathrm{A}\mathrm{F},{\mathrm{C}}_3\mathrm{A}}{f}_p^{\mathrm{c}\mathrm{l}\mathrm{i}\mathrm{n}}{\mathit{C}}_p:{\left[\mathbf{I}+{\mathit{S}}_p:\left({\mathit{C}}_{{\mathrm{C}}_3\mathrm{S}}^{-1}:{\mathit{C}}_p-\mathbf{I}\right)\right]}^{-1}+f_{{\mathrm{C}}_3\mathrm{S}}^{\mathrm{c}\mathrm{l}\mathrm{i}\mathrm{n}}{\mathit{C}}_{{\mathrm{C}}_3\mathrm{S}}\right\}\\ & :{\left\{\sum _{p={\mathrm{C}}_2\mathrm{S},{\mathrm{C}}_4\mathrm{A}\mathrm{F},{\mathrm{C}}_3\mathrm{A}}{f}_p^{\mathrm{c}\mathrm{l}\mathrm{i}\mathrm{n}}{\left[\mathbf{I}+{\mathit{S}}_p:\left({\mathit{C}}_{{\mathrm{C}}_3\mathrm{S}}^{-1}:{\mathit{C}}_p-\mathbf{I}\right)\right]}^{-1}+f_{{\mathrm{C}}_3\mathrm{S}}^{\mathrm{c}\mathrm{l}\mathrm{i}\mathrm{n}}\mathbf{I}\right\}}^{-1}.\end{array}

2.3 Volume fractions of the phases

At every scale level, the volume fractions for each constituent in it are required for the composite models. The total volume of the cement paste (V_cp) is:

(12)V_{\mathrm{c}\mathrm{p}}=V_{\mathrm{c}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{k}\mathrm{e}\mathrm{r}}+V_{\mathrm{C}\mathrm{H}}+V_{\mathrm{A}\mathrm{F}\mathrm{m}}+V_{\mathrm{A}\mathrm{F}\mathrm{t}}+V_{\mathrm{C}\mathrm{S}\mathrm{H}}+V_{\mathrm{c}\mathrm{a}\mathrm{p}},

where subscripts cp = cement paste; clinker = unreacted clinker; CH = CH crystals; AFm = hydrated calcium aluminates phases; AFt = ettringite; CSH = C-S-H matrix; and cap = capillary pores. The volume fractions of each term are listed in Appendix B in Electronic Supplementary Material.

At the mortar and concrete levels, the volume fractions of the constituents are determined based on the mix design of concrete. The volume fractions for sand (f_s) and cement paste (f_cp) are calculated as

(13)f_{\mathrm{s}}=\frac{V_{\mathrm{s}}}{V_{\mathrm{s}}+V_{\mathrm{c}}+V_{\mathrm{w}}}=\frac{W_{\mathrm{s}}/{\rho }_{\mathrm{s}}}{W_{\mathrm{s}}/{\rho }_{\mathrm{s}}+W_{\mathrm{c}}/{\rho }_{\mathrm{c}}+W_{\mathrm{w}}/{\rho }_{\mathrm{w}}},f_{\mathrm{c}\mathrm{p}}=1-f_{\mathrm{s}},

where V_s, V_c, and V_w are volume of sand, cement, and water in concrete, respectively (cm³); W_s, W_c, and W_w are weight per unit volume of concrete for sand (fine aggregates), cement, and water, respectively (g/cm³); {\rho }_{\mathrm{s}} is the density of sand (g/cm³).

The volume fractions for gravel (f_g) and mortar (f_m) are calculated as

(14)\begin{array}{rl}& f_{\mathrm{g}}=\frac{V_{\mathrm{g}}}{V_{\mathrm{s}}+V_{\mathrm{c}}+V_{\mathrm{w}}+V_{\mathrm{g}}}=\frac{W_{\mathrm{g}}/{\rho }_{\mathrm{g}}}{W_{\mathrm{s}}/{\rho }_{\mathrm{s}}+W_{\mathrm{c}}/{\rho }_{\mathrm{c}}+W_{\mathrm{w}}/{\rho }_{\mathrm{w}}+W_{\mathrm{g}}/{\rho }_{\mathrm{g}}},\\ & f_{\mathrm{m}}=1-f_{\mathrm{g}},\end{array}

where V_g is volume of gravel (coarse aggregates) in concrete (cm³); W_g is weight per unit volume of concrete for gravel (g/cm³); {\rho }_{\mathrm{g}} is gravel density (g/cm³).

3 Degradation of concrete

In general, long-term neutron irradiation leads to notable volume expansion of concrete as well as reduction of its mechanical properties, such as strength and stiffness when neutron fluence is higher than 1 × 10¹⁹ n/cm². Gamma ray always presents with neutron radiation, but its effects on concrete, including aggregates and cement paste, are quite small [35,36].

3.1 Deformation of the constituent phases

3.1.1 Radiation induced volumetric expansion

Aggregates used in concrete usually have high SiO₂/quartz content. Refs. [2,3] shows that quartz will significantly expand in volume exposed to fast neutron fluence in excess of 1 × 10¹⁹ n/cm². Pronounced volume expansion can also be observed for many kinds of aggregates, especially for siliceous aggregates, when fast neutron fluence is greater than 1 × 10¹⁹ n/cm². The RIVE of several ceramic phases (quartz is one of them) and minerals is caused by the crystalline-to-amorphous transition [37,38]. The process is called amorphization or metamictization. Different aggregates expand at different rates under the same level of neutron fluence. Thus, when test data are available for the specific type of aggregate used in a concrete under consideration, the test data can be analyzed by curve fitting, and an equation for the aggregate expansion can be obtained and used in the analysis. By calculating the accumulation of amorphous fraction of materials induced by irradiation [39], a general equation for the expansive strain of aggregate {\epsilon }_{\mathrm{a}} due to neutron irradiation was developed

(15){\epsilon }_{\mathrm{a}}={\epsilon }_{\mathrm{u}}(1-\frac{1}{\sqrt{\mathcal{A}+\left(1-\mathcal{A}\right)\mathrm{e}\mathrm{x}\mathrm{p}\left(2\mathcal{B}\right(1-\mathcal{A}\left)N\right)}}),

where {\epsilon }_{\mathrm{u}} is saturation value of dimensional change of aggregate (unitless); \mathcal{A} is temperature-dependent crystallization efficiency parameter (unitless), varying in the range between 0 and 1; \mathcal{B} is normalization factor for neutron fluence (cm²/n); N is neutron fluence (n/cm²).

Recently, a micromechanical model was developed to estimate the expansion of irradiated aggregates with different mineral compositions [40]. The study found that the RIVE of aggregate mainly results from the RIVE of forming minerals and the generation of voids during the process.

3.1.2 Drying shrinkage of cement paste

Current research results available show that there is no obvious correlation between neutron radiation and shrinkage of cement paste [2,41]. Compared to the effect of neutron irradiation on aggregate, its effect on the cement paste is little. The shrinkage of cement paste is primarily due to dehydration under radiation heating and radiolysis caused by gamma ray. Among the constituents in cement paste, unreacted cement, CH, and other crystals have negligible volumetric changes, and the C-S-H shrinkage is responsible for almost all of the drying shrinkage of cement paste [22].

The effect of drying shrinkage can be considered in two different ways. In the case that the experimental study is available, an empirical model can be developed which can be applied to the analysis. If no shrinkage test data for the concrete are available, shrinkage models can be used. For instance, the current model was initially developed for concrete shrinkage and can be used to model and calculate the drying shrinkage of concrete [22]. The control parameter for the shrinkage of concrete is the shrinkage of LD C-S-H, which can be considered as a function of pore relative humidity.

3.1.3 Thermal strains

Under radiation heating, the thermal expansion of concrete (ε_T) is

(16){\epsilon }_{\mathrm{T}i}={\alpha }_i^{\mathrm{T}}\mathrm{\Delta }T,

where {\alpha }_i^{\mathrm{T}} is coefficient of linear thermal expansion (CTE) for the ith constituent phase in concrete (strain/°C); \mathrm{\Delta }T is temperature increment (°C).

For the calculation of volumetric changes due to temperature variation, the equation for linear thermal strain, Eq. (16), can be used; that is, the volumetric strain (\delta) is the trace of the strain tensor

(17)\delta =\frac{\mathrm{\Delta }V}{V}={\epsilon }_{xx}+{\epsilon }_{yy}+{\epsilon }_{\mathrm{z}\mathrm{z}}=3{\epsilon }_{\mathrm{e}\mathrm{f}\mathrm{f}}.

The effect of neutron radiation on the CTE can be evaluated in conjunction with two possible mechanisms: irradiation itself and neutron irradiation heating. The results of some experimental studies [2] show that the differences between the CTE of neutron irradiated concrete and that of temperature-exposed concrete are very small, which implies that the direct impact of neutron irradiation is not significant. Therefore, the present model assumes that neutron radiation will not affect {\alpha }_i^{\mathrm{T}}. The CTEs for constituent phases in cement paste under the NPP normal operating temperature (65 °C) are constant, and they are shown in Tab.2. CTEs of aggregates are temperature dependent and their values can be found in Refs. [42,43]. These two mechanisms within concrete materials and its constituent phases are well discussed in our previous study as well [11], and the change of CTE due to the neutron radiation under NPP operating conditions is not investigated further in this study.

3.2 Degradation of mechanical properties

3.2.1 Aggregate

Neutron irradiation can reduce the elastic modulus of aggregates significantly, as shown in Fig.7. The reduction of concrete properties may be a result of the reduction of the elastic modulus of the aggregate and damage in the cement paste. A study found that the Young’s modulus of the aggregate decreases exponentially with RIVE under neutron radiation and the degradation depends on the silicate content of the aggregate [40]. Computational methods can be used for modeling the behavior of aggregates if the degradation mechanisms are fully understood [50].

3.2.2 Hardened cement paste

The collected test data has shown that there is no clear correlation between neutron irradiation and the mechanical properties of hardened cement paste compared to aggregates [51]. Based on these test data collected in Fig.8, it is assumed that the elastic modulus of hardened cement paste is unchanged by neutron irradiation [52].

Elevated temperature can lead to the reduction of elastic modulus of concrete. The degradation mechanisms include phase transformations of the constituents and micro-cracking at the aggregate-cement paste interface caused by the incompatibility between their thermal deformations. More details about the thermal degradation of concrete can be found in our previous study [25]. Under the normal environment temperature of biological shielding (65 °C), there is no phase transformation in the cement paste. Therefore, the mechanical properties of the cement paste constituents are considered to be constant during the normal operating condition. Their values are shown in Tab.2.

Degradation induced by the volumetric mismatch, including RIVE, thermal strains, and shrinkage of cement paste in the vicinity of the aggregate, will be modeled in the following section.

3.2.3 Damage development in cement paste

As analyzed in above sections, the RIVE of aggregate could result in damage to the surrounding cement paste. This is the primary damage mechanism of concrete under neutron irradiation. To evaluate the overall degradation of concrete, the damage in the hardened cement paste needs to be quantified first. As shown in Eq. (4), the incompatibility between the deformation of the aggregates and the cement paste will generate an internal pressure P at the aggregate-cement paste interface. Since the aggregate phase expands and the cement paste phase shrinks, {\epsilon }_1-{\epsilon }_2 is a positive value for nuclear irradiated concrete. As a result, P will be a positive pressure applied to the aggregate-cement paste interface. As shown in Fig.9 (a), the pressure leads to a tensile stress in the cement paste, and may result in cracking of the cement paste surrounding the aggregate. In the case of the spherical model for aggregate as used in the GSC model, the radial and tangential stresses in the cement paste are

(18)\mathrm{R}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}:{\sigma }_{\mathrm{r}}=-P\frac{r_{\mathrm{s}}^3}{r^3},

(19)\mathrm{T}\mathrm{a}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{i}\mathrm{a}\mathrm{l}\mathrm{s}\mathrm{t}\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{s}:{\sigma }_{\theta }={\sigma }_{\varphi }=P\frac{r_{\mathrm{s}}^3}{{2r}^3},

where r_s is aggregate radius (cm).

These two stresses decrease gradually with the cube of r.

To consider the degradation of the cement paste induced by the volumetric mismatch at the interface, the Drucker–Prager plastic failure criterion is adopted as the criterion for the cement paste damage. This is a simplified characterization of the damage process in the cement paste surrounding aggregate.

(20)\alpha I_1+\sqrt{J_2}=k,

where I₁ is the first invariant of the stress tensor (MPa); J₂ is the second invariant of the deviatoric stress tensor (MPa²); \alpha and k are constants in terms of the tensile strength (f_ct) and compressive strength (f_{\mathrm{m}\mathrm{L}}^{\prime }) of cement paste: \alpha =\left({\displaystyle \frac{1}{\sqrt{3}}}\right)\left(f_{\mathrm{c}\mathrm{c}}^{\prime }{-f}_{\mathrm{c}\mathrm{t}}\right)/\left(f_{\mathrm{c}\mathrm{t}}+f_{\mathrm{c}\mathrm{c}}^{\prime }\right), k=\left({\displaystyle \frac{2}{\sqrt{3}}}\right)\left(f_{\mathrm{c}\mathrm{c}}^{\prime }{f}_{\mathrm{c}\mathrm{t}}\right)/\left(f_{\mathrm{c}\mathrm{t}}+f_{\mathrm{c}\mathrm{c}}^{\prime }\right).

When the cement paste in the vicinity of aggregates starts to undergo plastic yielding (r = r_s), by substituting Eqs. (18) and (19) into Eq. (20), the critical interface pressure (P_c) for the plastic yielding initiation of cement paste is obtained:

(21)P_{\mathrm{c}}=\frac{{4f}_{\mathrm{c}\mathrm{c}}^{\prime }{f}_{\mathrm{c}\mathrm{t}}}{3\left(f_{\mathrm{c}\mathrm{t}}+f_{\mathrm{c}\mathrm{c}}^{\prime }\right)}.

Once P reaches P_c, the cement paste surrounding the aggregate will undergo plastic yielding and can be considered as the damaged zone with a small increase ∆P in P. The damage development process is shown in Fig.9(a). The thickness of the damaged zone is ∆x in the radial direction. In this case, the new interface pressure is P = P_c + ∆P and the new yield surface of the cement paste is r = r_s + ∆x. Based on Eq. (20), the ratio between ∆x and r_s is

(22)\frac{\mathrm{\Delta }x}{r_{\mathrm{s}}}=\sqrt[3]{1+\frac{3\mathrm{\Delta }P\left(f_{\mathrm{c}\mathrm{t}}+f_{\mathrm{c}\mathrm{c}}^{\prime }\right)}{{4f}_{\mathrm{c}\mathrm{c}}^{\prime }{f}_{\mathrm{c}\mathrm{t}}}}-1.

Then, a recursive equation describing the radius of yield surface of the cement paste at time t_n can be obtained

(23)r_{n+1}=r_n+\mathrm{\Delta }{x}_n=r_n\sqrt[3]{1+\frac{3\mathrm{\Delta }{P}_n\left(f_{\mathrm{c}\mathrm{t}}+f_{\mathrm{c}\mathrm{c}}^{\prime }\right)}{{4f}_{\mathrm{c}\mathrm{c}}^{\prime }{f}_{\mathrm{c}\mathrm{t}}}}.

The initial value for r_n is r₀ = r_s when P just reaches P_c. If the pressure is less than P_c at r_{n+1}, there will be no new distressed phase, and the damage initiates once the pressure increases to a level larger than P_c. The calculation process is explained below.

As shown in Eq. (4), the interface pressure P is expressed in terms of the elastic properties and deformations of the composite phases in the material, which means that ∆x/r_s is size independent. Therefore, the GSC model can be applied to the distressed materials, providing a proper treatment of damaged cement paste. The approach is named as composite damage mechanics [53,54].

In this theory, the damaged material is modeled as a composite material comprised of two different phases. One is the fully damaged phase and the other is the fully intact phase. The “composite damage mechanics” is the application of composite theory to a damage problem, for example, to use the GSC model to estimate the damage development in concrete. During the damage development process such as due to nuclear irradiation, all phases can be assumed to be linearly elastic and isotropic.

In this case, as shown in Fig.9, the damaged zone in cement paste is replaced by a damaged phase with reduced stiffness. The damaged phase in cement paste has a lower stiffness (E_d) than that of the intact cement paste (E₀) when the damage occurs. Thus, 0 < E_d/E₀ ≤ 1. E_d/E₀ is an important parameter in this theory, and a constant value is usually used for it. This is the ratio of elastic modulus of the fully distressed material and the intact material. The ratio is an input parameter for the analysis. This parameter depends on the composition of the cement paste, which is determined by the cement type and concrete mix design parameters. It allows the user to control the acceptable level of damage at the end state of degradation of the concrete. The overall damage is quantified using the volume fraction of the fully distressed phase. At the initial stage, the volume fraction is zero when there is no damage in the material; while at the end stage, the volume fraction is 1 (100%) when the damage is fully developed to the design limit, for instance when the neutron fluence reaches 1 × 10²⁰ n/cm². In the composite damage mechanics, all of the phases can carry loads and have their own nonzero elastic modulus, which is different from conventional scalar damage mechanics where the damaged phase has a zero modulus [55].

After these treatments, the distressed concrete can be modeled as a composite material with multiple phases. As one can see in Fig.10, the inclusion (black) is the aggregate phase, the first layer (red) is the damaged cement paste, and the second layer (blue) is the intact cement paste. The volume fraction of the damaged phase is a variable parameter which quantifies the internal damage development and can be obtained based on Eq. (23).

During the calculation process, small time steps should be used, which means that the incremental deformation of aggregate should be small enough to ensure the slow variation of the P with respect to time. If the damage criteria are not met, go to the next time step with an increased aggregate expansion until the damage is initiated. After that, from the previous time t_n to the current time t_{n + 1}, the inclusion phase in the model should be the combination of the original inclusion and damaged cement paste formed in all of the previous time steps. This new inclusion will have new mechanical properties and strains that can be calculated using the previously developed equations. There will be a new interface with radius r_{n + 1}. In the next time step, the calculation of the interface pressure and damage criterion should be performed at the new interface. The calculation process of the model is shown in Fig.11.

4 Model validation

To validate the proposed model, the post-irradiation residual properties of one specific concrete called Con-A tested at the Kjeller JEEP-II reactor [4] under a fast neutron fluence from 7.09 × 10¹⁸ to 9.62 × 10¹⁹ n/cm² (E > 0.1 MeV) was analyzed. However, the specimens in the group with the highest neutron radiation level are excluded due to some issues experienced in an accident during the test.

Aggregate expansion profiles are shown in Fig.12. GA is for coarse aggregate and GB is for fine aggregate. Other symbols (GC, GD, GE, and GF) refer to other types of aggregates and will not be used here. The properties used in this case are listed in Tab.2 and Tab.3. The chemical compositions of the cement are shown in Tab.4. The water–cement ratio is 0.5 and the mix design is shown in Tab.5. The curing time is one year. The temperatures during the test are in the range of 62.0–71.9 °C; and the highest temperature experienced by the concrete (71.9 °C) is used in the analysis. The strengths of cement paste are f_{\mathrm{m}\mathrm{L}}^{\prime } = 65 MPa and f_ct = 3.5 MPa. It was assumed that the degradation of the Young’s modulus of the aggregate is the same as the linear regression of the serpentine data collected by Elleuch et al. [41], as shown in Fig.13. E_d/E₀= 0.20 is determined using the proposed model reversely if the concrete is fully damaged at the highest radiation level.

Predictions of the dimensional change and the variation of Young’s modulus by the present model agree well with the experimental data, as shown in Fig.14 and Fig.15. In the calculation, the time step increment should be small enough to achieve a slow damage development with time. For this particular problem, 10000 time steps were used. Since the total irradiation time for this accelerated test is 299.36 d, the time step increment is 0.03 d.

5 Parametric analyses of the model

After the proposed model was validated using available test data, a case study was developed to analyze the effects of the model input parameters on the damage development of the cement paste, the dimensional change, and the elastic modulus of nuclear irradiated concrete. The cement paste damage is quantified using the volume fraction of the fully distressed cement paste in the whole cement paste (V_{\mathrm{c}\mathrm{p}}^{\mathrm{d}\mathrm{i}\mathrm{s}}{/V}_{\mathrm{c}\mathrm{p}}).

Concrete samples assumed to be made of ordinary Portland cement and crushed gravel are exposed to a fast neutron fluence of up to 1 × 10²⁰ n/cm²(E > 0.1 MeV). Various material parameters were used, including three different water–cement ratios: 0.3, 0.4, and 0.55; three different aggregate volume fractions: 0.6, 0.7, and 0.8; and two different aggregate expansion profiles, as shown in Fig.16. Profiles 1 and Profile 2 are volume change of the aggregate as a nonlinear function of neutron intensity as described by Eq. (15). The two profiles are based on the test data of aggregate GA and GE in Fig.12. These two profiles are the upper bound and lower bound of the data shown in Fig.12 (aggregate GF is excluded since it shows no expansion at all). They are used here for the parametric analysis of the model. A linear regression of the serpentine data shown in Fig.13 was used as the degradation trend of the Young’s modulus for the aggregate under neutron radiation. All parameters are listed in Tab.6. The selection of the input parameters was to cover the potential variation in the parameter values.

5.1 Water–cement ratio

For this portion of the parametric analysis, only the water–cement ratio was changed; all of the other input parameters were kept constant (aggregate fraction = 0.7, E_d/E₀ = 1/3, and the aggregate expansion follows Profile 1 in Fig.16). As one can see in Fig.17, a higher w/c results in less damage induced by neutron radiation in the cement paste. A higher w/c leads to a less densified concrete framework which can accommodate more volume expansion at the aggregate-cement paste interface. This result also explains why the expansion of concrete decreases with the increase of the w/c, as shown in Fig.18. As shown in Fig.19, the w/c has different effects on the elastic modulus of concrete at different ranges. When the neutron fluence is small, the damage in the cement paste is not severe, as shown in Fig.17, and a higher w/c ratio will reduce the damage in the cement paste. Thus, the reduction of elastic modulus of concrete due to nuclear irradiation will be smaller for the concrete specimen with a higher w/c. When the neutron fluence is high, the damage to the cement paste is already significant, and the differences among the three cases became small compared to the absolute value, as shown in Fig.17. On the other hand, a higher w/c ratio will reduce the stiffness of the cement paste, and thus, the reduction of the elastic modulus of concrete due to large neutron fluence will be larger for the concrete specimen with a higher w/c, as shown in Fig.19.

5.2 Aggregate fraction

In this case, only the aggregate fraction was changed; all of the other input parameters were kept constant (water–cement ratio = 0.4, E_d/E₀ = 1/3, and the aggregate expansion follows Profile 1 in Fig.16). As one can see in Fig.20, a higher aggregate fraction slightly reduces the percent of damaged cement paste. This result is because the interface pressure decreases with increasing aggregate fraction (Eq. (4)). With a higher volume fraction of aggregate, the confinement of surrounding cement paste is reduced, and thus, the interface pressure is lower. The elastic modulus of concrete under neutron radiation shows a smaller reduction with the increase in the aggregate fraction, as shown in Fig.21. This difference is because the modulus of elasticity of aggregate is higher than that of cement paste, with more aggregate, the relative modulus of concrete is higher. Of course, with an increasing neutron fluence level, the relative values of effective modulus decrease for all three cases. Since the expansion of concrete is mainly due to the expansion of aggregates, the overall expansion of concrete increases with the increase of aggregate fraction, as shown in Fig.22.

5.3 The effects of aggregate expansion

In this series of calculations, only the aggregate expansion was changed among the three specimens considered; all of the other input parameters were kept constant (w/c = 0.4, aggregate fraction = 0.7, and E_d/E₀ = 1/3). As shown in Fig.23 and Fig.24, when the aggregate expansion is slower and smaller (Profile 1), the damage development in the cement paste and the degradation of concrete is also slower and smaller. As one can see in Fig.25, the overall expansion of concrete mainly results from the expansion of aggregates.

6 Summary and conclusions

A framework of theoretical models was developed for the prediction of the mechanical damage and deformation of neutron irradiated concrete. The GSC model and the Mori–Tanaka model were used to characterize the mechanical properties of concrete with multiple phases and multiple scale internal structures. The models can take into account the degradation mechanisms at multiscale levels resulting from neutron irradiation and elevated temperature, and different spatial distributions of the multiphase constituents in concrete.

In this study, concrete is modeled as a multiscale and multiphase composite with four different scale levels: concrete, mortar, cement paste, and clinker in descending order of scale. The internal structures of concrete at the four scale levels were characterized by using two different types of models, the Mori–Tanaka model and the GSC model, based on the structural features at each level. In practice, the obtained properties at the lower scale level can be used as the input for the composite at the upper scale level. By this approach, the proposed composite model can take into account the contributions of all constituent phases in the concrete material. The damage of concrete due to nuclear irradiation was estimated using a composite damage mechanics approach based on a certain failure criterion. The Drucker–Prager plasticity is used as an example, and different constitutive models could be used to consider other mechanical behaviors in addition to damage, such as creep. The established model can consider the neutron radiation induced degradation and volumetric expansion of aggregates, thermal strains, and shrinkage of cement paste. It was validated using a set of experimental data of concrete specimens irradiated in a test reactor, and the model predicted elastic modulus and deformation of concrete agreed with the experimental values quite well.

Parametric analyses of the model input parameters, including the water–cement ratio, aggregate fraction, and aggregate expansion, were performed to analyze the effects of the parameters on the degradation and deformation of concrete under neutron irradiation. It is shown that aggregate expansion is important for the degradation of concrete under neutron irradiation as are the other parameters analyzed. For example, with an aggregate that exhibits large neutron irradiation induced expansion, the overall damage to the concrete is expected to be high, if we only consider the expansion of the aggregate. However, the overall damage might not be very high, if the stiffness of the surrounding cement paste is not high. The stiffness of the cement paste depends on the water–cement ratio, so the water–cement ratio is an important parameter to be considered.

The developed prediction framework can be applied to various concrete materials with different mix designs used in different NPPs. The parameters required for modeling can be obtained based on the experimental data, available models, or justified assumptions.

References

[1] EsselmanTBruckP. Expected Condition of Concrete at Age 80 Years of Reactor Operation. Technical Report, A13276-R-001. New York: Lucius Pitkin Incorporated, 2013

[2] Hilsdorf H , Kropp J , Koch H . The effects of nuclear radiation on the mechanical properties of concrete. ACI Special Publications, 1978, 55: 223–254

[3] Field K G , Remec I , Le Pape Y . Radiation effects in concrete for nuclear power plants—Part I: Quantification of radiation exposure and radiation effects. Nuclear Engineering and Design, 2015, 282: 126–143

[4] MaruyamaIKontaniOTakizawaMSawadaSIshikawaoSYasukouchiJSatoOEtohJIgariT. Development of soundness assessment procedure for concrete members affected by neutron and gamma-ray irradiation. Journal of Advanced Concrete Technology, 2017, 15(9): 440–523

[5] Pomaro B . A review on radiation damage in concrete for nuclear facilities: from experiments to modeling. Modelling and Simulation in Engineering, 2016, 2016: 4165746

[6] Rosseel T M , Maruyama I , Le Pape Y , Kontani O , Giorla A B , Remec I , Wall J J , Sircar M , Andrade C , Ordonez M . Review of the current state of knowledge on the effects of radiation on concrete. Journal of Advanced Concrete Technology, 2016, 14(7): 368–383

[7] Pomaro B , Salomoni V A , Gramegna F , Prete G , Majorana C E . Radiation damage evaluation on concrete within a facility for Selective Production of Exotic Species (SPES Project), Italy. Journal of Hazardous Materials, 2011, 194: 169–177

[8] Pomaro B , Salomoni V A , Gramegna F , Prete G , Majorana C E . Radiation damage evaluation on concrete shielding for nuclear physics experiments. Annals of Solid and Structural Mechanics, 2011, 2(2–4): 123–142

[9] Salomoni V A , Majorana C E , Pomaro B , Xotta G , Gramegna F . Macroscale and mesoscale analysis of concrete as a multiphase material for biological shields against nuclear radiation. International Journal for Numerical and Analytical Methods in Geomechanics, 2014, 38(5): 518–535

[10] Khmurovska Y , Štemberk P , Fekete T , Eurajoki T . Numerical analysis of VVER-440/213 concrete biological shield under normal operation. Nuclear Engineering and Design, 2019, 350: 58–66

[11] Jing Y , Xi Y . Theoretical modeling of the effects of neutron irradiation on properties of concrete. Journal of Engineering Mechanics, 2017, 143(12): 04017137

[12] Le Pape Y , Field K G , Remec I . Radiation effects in concrete for nuclear power plants, part II: Perspective from micromechanical modeling. Nuclear Engineering and Design, 2015, 282: 144–157

[13] Giorla A , Vaitová M , Le Pape Y , Štemberk P . Meso-scale modeling of irradiated concrete in test reactor. Nuclear Engineering and Design, 2015, 295: 59–73

[14] Giorla A B , Le Pape Y , Dunant C F . Computing creep-damage interactions in irradiated concrete. Journal of Nanomechanics & Micromechanics, 2017, 7(2): 04017001

[15] Pomaro B , Xotta G , Salomoni V A , Majorana C E . A thermo-hydro-mechanical numerical model for plain irradiated concrete in nuclear shielding. Materials and Structures, 2022, 55(1): 14

[16] Saklani N , Banwat G , Spencer B , Rajan S , Sant G , Neithalath N . Damage development in neutron-irradiated concrete in a test reactor: Hygro-thermal and mechanical simulations. Cement and Concrete Research, 2021, 142: 106349

[17] Sasano H , Maruyama I , Sawada S , Ohkubo T , Murakami K , Suzuki K . Meso-scale modelling of the mechanical properties of concrete affected by radiation-induced aggregate expansion. Journal of Advanced Concrete Technology, 2020, 18(10): 648–677

[18] Jing Y , Xi Y . Long-term neutron radiation levels in distressed concrete biological shielding walls. Journal of Hazardous Materials, 2019, 363: 376–384

[19] Jing Y , Xi Y . Modeling long-term distribution of fast and thermal neutron fluence in degraded concrete biological shielding walls. Construction & Building Materials, 2021, 292: 123379

[20] Maruyama I , Haba K , Sato O , Ishikawa S , Kontani O , Takizawa M . A numerical model for concrete strength change under neutron and gamma-ray irradiation. Journal of Advanced Concrete Technology, 2016, 14(4): 144–162

[21] ChristensenR M. Mechanics of Composite Materials. New York: Dover Publications, 2005

[22] Xi Y , Jennings H M . Shrinkage of cement paste and concrete modelled by a multiscale effective homogeneous theory. Materials and Structures, 1997, 30(6): 329–339

[23] Meshgin P , Xi Y . Multi-scale composite models for the effective thermal conductivity of PCM-concrete. Construction & Building Materials, 2013, 48: 371–378

[24] Eskandari-Ghadi M , Zhang W , Xi Y , Sture S . Modeling of moisture diffusivity of concrete at low temperatures. Journal of Engineering Mechanics, 2013, 139(7): 903–915

[25] Lee J , Xi Y , Willam K , Jung Y . A multiscale model for modulus of elasticity of concrete at high temperatures. Cement and Concrete Research, 2009, 39(9): 754–762

[26] Benveniste Y . A new approach to the application of Mori–Tanaka’s theory in composite materials. Mechanics of Materials, 1987, 6(2): 147–157

[27] Mori T , Tanaka K . Average stress in matrix and average elastic energy of materials with misfitting inclusions. Acta Metallurgica, 1973, 21(5): 571–574

[28] Chen J , Jin X , Jin N , Tian Y . A nano-model for micromechanics-based elasticity prediction of hardened cement paste. Magazine of Concrete Research, 2014, 66(22): 1145–1153

[29] Damien D , Wang Y , Xi Y . Prediction of elastic properties of cementitious materials based on multiphase and multiscale micromechanics theory. Journal of Engineering Mechanics, 2019, 145(10): 04019074

[30] Vandamme M , Ulm F J , Fonollosa P . Nanogranular packing of C–S–H at substochiometric conditions. Cement and Concrete Research, 2010, 40(1): 14–26

[31] Stora E , He Q C , Bary B . Influence of inclusion shapes on the effective linear elastic properties of hardened cement pastes. Cement and Concrete Research, 2006, 36(7): 1330–1344

[32] StutzmanP EBullardJ WFengP. Quantitative Imaging of Clinker and Cement Microstructure. Gaithersburg: National Institute of Standards and Technology, 2015

[33] Acker P . Swelling, shrinkage and creep: A mechanical approach to cement hydration. Materials and Structures, 2004, 37(4): 237–243

[34] Velez K , Maximilien S , Damidot D , Fantozzi G , Sorrentino F . Determination by nanoindentation of elastic modulus and hardness of pure constituents of Portland cement clinker. Cement and Concrete Research, 2001, 31(4): 555–561

[35] Hunnicutt W , Rodriguez E T , Mondal P , Le Pape Y . Examination of gamma-irradiated calcium silicate hydrates. Part II: Mechanical properties. Journal of Advanced Concrete Technology, 2020, 18(10): 558–570

[36] Rodriguez E T , Hunnicutt W A , Mondal P , Le Pape Y . Examination of gamma-irradiated calcium silicate hydrates. Part I: Chemical-structural properties. Journal of the American Ceramic Society, 2020, 103(1): 558–568

[37] Weber W J , Ewing R C , Wang L M . The radiation-induced crystalline-to-amorphous transition in zircon. Journal of Materials Research, 1994, 9(3): 688–698

[38] Weber W J , Ewing R C , Catlow C R A , De La Rubia T D , Hobbs L W , Kinoshita C , Matzke H , Motta A T , Nastasi M , Salje E K H . . Radiation effects in crystalline ceramics for the immobilization of high-level nuclear waste and plutonium. Journal of Materials Research, 1998, 13(6): 1434–1484

[39] Wang S X , Wang L M , Ewing R C . Irradiation-induced amorphization: Effects of temperature, ion mass, cascade size, and dose rate. Physical Review B: Condensed Matter, 2000, 63(2): 024105

[40] Le Pape Y , Sanahuja J , Alsaid M H F . Irradiation-induced damage in concrete-forming aggregates: Revisiting literature data through micromechanics. Materials and Structures, 2020, 53(3): 62

[41] Elleuch L F , Dubois F , Rappeneau J . Effects of neutron radiation on special concretes and their components. ACI Special Publications, 1972, 34: 1071–1108

[42] AlexanderMMindessS. Aggregates in Concrete. Boca Raton, FL: CRC Press, 2005

[43] BažantZ PKaplanM F. Concrete at High Temperatures: Material Properties and Mathematical Models. London: Addison-Wesley, 1996

[44] LeeJ. Experimental studies and theoretical modeling of concrete subjected to high temperatures. Dissertation for the Doctoral Degree. Boulder: University of Colorado at Boulder, 2006

[45] Vandamme M , Ulm F J . Nanogranular origin of concrete creep. Proceedings of the National Academy of Sciences of the United States of America, 2009, 106(26): 10552–10557

[46] Constantinides G , Ulm F J . The effect of two types of CSH on the elasticity of cement-based materials: Results from nanoindentation and micromechanical modeling. Cement and Concrete Research, 2004, 34(1): 67–80

[47] Monteiro P J M , Chang C T . The elastic moduli of calcium hydroxide. Cement and Concrete Research, 1995, 25(8): 1605–1609

[48] Speziale S , Jiang F , Mao Z , Monteiro P J M , Wenk H R , Duffy T S , Schilling F R . Single-crystal elastic constants of natural ettringite. Cement and Concrete Research, 2008, 38(7): 885–889

[49] Zohdi T I , Monteiro P J M , Lamour V . Extraction of elastic moduli from granular compacts. International Journal of Fracture, 2002, 115: 49–54

[50] Jenabidehkordi A . Computational methods for fracture in rock: A review and recent advances. Frontiers of Structural and Civil Engineering, 2019, 13(2): 273–287

[51] BiwerBMaDXiYJingY. Review of Radiation-Induced Concrete Degradation and Potential Implications for Structures Exposed to High Long-Term Radiation Levels in Nuclear Power Plants (NUREG/CR-7280, ANL/EVS-20/8). Washington, DC: U.S. Nuclear Regulatory Commission, 2020

[52] WilliamKXiYNausD. A Review of the Effects of Radiation on Microstructure and Properties of Concretes Used in Nuclear Power Plants (NUREG/CR-7171). Washington, D.C.: U.S. Nuclear Regulatory Commission, 2013

[53] Eskandari-Ghadi M , Xi Y , Sture S . Cross-property relations between mechanical and transport properties of composite materials. Journal of Engineering Mechanics, 2014, 140(7): 06014006

[54] Xi Y , Eskandari-Ghadi M , Suwito S , Sture S . Damage theory based on composite mechanics. Journal of Engineering Mechanics, 2006, 132(11): 1195–1204

[55] Ren H , Rabczuk T , Zhuang X . Variational damage model: A new paradigm for fractures. Frontiers of Structural and Civil Engineering, 2025, 19(1): 1–21

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