Nag2222

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS Int. J. Numer. Anal. Meth. Geomech. (2013) Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/nag.2222 Macroscale and mesoscale analysis of concrete as a multiphase material for biological shields against nuclear radiation V. A. Salomoni1,*,†, C. E. Majorana1, B. Pomaro1, G. Xotta1 and F. Gramegna2 1 Department of Civil, Environmental and Architectural Engineering, University of Padua, Via F. Marzolo, 9 35131 Padua, Italy 2 INFN, National Institute of Nuclear Physics, National Laboratories of Legnaro (PD), Viale dell’Università, 2 35020 Legnaro, Padua, Italy SUMMARY The overall thermo-hygro-mechanical behavior of concrete is to be investigated, because its bearing capacity is required together with its shielding properties, specifically when concrete structures are exposed to highenergy neutron fluxes, which represent the next generation facilities designed for the production of high energy radioactive ion beams in physics research. Irradiation in the form of either fast and thermal neutrons, primary gamma rays or gamma rays produced as a result of neutron capture, are learnt to affect concrete as well as neutron fluences of the order of 1019 n/cm2 and gamma radiation doses of 1010 rad seem to become critical for concrete strength. The collection of data on concrete samples, variously exposed to neutron radiation, has allowed for defining a law for radiation damage within the FEM research code NEWCON3D, assessing the 3D coupled thermo-hygro-mechanical behavior of concrete, modeled as a multiphase porous medium, both at the macroscale and the mesoscale level. The required damage law is thought to be a function of the neutron flux impinging the concrete shielding wall, and a good estimate of this quantity has been provided by means of a Monte Carlo code developed by CERN and the National Institute of Nuclear Physics of Milan, Italy; this code handles radiation transport calculations and represents at this day one of the most reliable procedures for dealing with the interaction of radiation and matter. The suggested procedure for the radiation damage evaluation has allowed for discussing on differences between mesolevel and macrolevel approaches. Stochastic contour maps of the expected radiation field, properly interfaced with the numerical FE code, have allowed for obtaining a more precise evaluation of the radiation damage front as well as its evolution in time. Copyright © 2013 John Wiley & Sons, Ltd. Received 23 May 2012; Revised 24 June 2013; Accepted 29 July 2013 KEY WORDS: shielding; radioactive ion beams; damage; multiscale approach 1. INTRODUCTION A recently developed extension of the F. E. research code NEWCON3D, performing fully coupled hygro-thermo-mechanical 3D analyses for cementitious materials, has been adopted to study the effects of nuclear radiation on a concrete shielding for the specific neutron source of a study case, in conjunction with a Monte Carlo code developed by CERN and the National Institute of Nuclear Physics (INFN) of Milan, Fluka [1], used to describe the radiation field (neutron fluence and deposited energy) that the mechanical field is dependent on. In fact, radiation in the form of either fast and thermal neutrons, primary gamma rays or gamma rays produced as a result of neutron capture, can affect concrete. Nuclear radiation may influence structural and mechanical properties of materials significantly; Hilsdorf et al. [2] collecting published *Correspondence to: V. A. Salomoni, Department of Civil, Environmental and Architectural Engineering, University of Padua, Via F. Marzolo, 9 – 35131 Padua, Italy. † E-mail: [email protected] Copyright © 2013 John Wiley & Sons, Ltd. V. A. SALOMONI ET AL. experimental data on the effect of nuclear radiation on the properties of concrete, stand out that up to neutron fluences of the order of 1019 n/cm2 the effects of the irradiation are relatively small, while higher fluences may have detrimental effects on concrete strength and modulus of elasticity. On the other hand, thermal expansion coefficient, thermal conductivity, and shielding properties are proved to be little affected by radiation. Changes in the properties of concrete appear to depend primarily on the behavior of concrete aggregates that can undergo a volume change when exposed to radiation. Radiation damage in concrete aggregates is caused by changes in the lattice structure of the minerals in the aggregates [3, 4]. Fast neutrons are mainly responsible for the considerable growth, caused by atomic displacements, that has been measured in certain aggregates (e.g., flint). Quartz aggregates, made of crystals with covalent bonding, seem to be more affected by radiation than calcareous aggregates that contain a weaker ionic bonding. Neutron fluences of the order of 1019 n/cm2 and gamma radiation doses of 1010 rad seem to become critical for concrete strength. The mechanism is explained by the reaction of OH ions of the alkaline solution in the micropores of concrete with SiO2 in aggregates; first, the Si–O bonding breaks, next follows the expansion of the aggregates by hydration of SiO2. The consumption of OH ions leads to the dissolution of Ca2+ ions into the solution. The Ca2+ ions then react with hydrated SiO2 gels to generate calcium silicate. Rigid calcium silicate shells are therefore formed on the surfaces of the aggregates by successive reactions with OH and Ca2+ ions, and the alkaline solution is possible to penetrate into the aggregates through the calcium silicate shells and to dissolve SiO2. Because the rigid shells prevent the deformation of the aggregates, the expansion pressure generated by the penetration of the solution is accumulated in the aggregates, thus leading to cracks in the cement paste and expansion of concrete. The previous observations support the importance of modeling concrete at the mesoscale level, distinguishing in the multiphase material system the role of aggregates, cement paste, and interfacial transition zone (ITZ) [5] for a more realistic representation of the building material, as well as for understanding the contribution of aggregates and ITZ in the characterization and definition of radiation shielding properties. The physical problem additionally requires to take into account the collateral effect represented by the development of heat within the shielding, as a consequence of absorbed radiation, as long as the power density is not negligible, that is, for values of energy flux density above 1010 MeV/(cm2 s) [6]. The study case is represented by a next generation nuclear facility (currently under design) for the INFN at the National Laboratories of Legnaro in Padua, Italy: the Selective Production of Exotic Species (SPES) project; the research structure is expected to produce neutron-rich unstable nuclei, called ‘exotic beams’, by fission reactions of a primary radioactive proton beam on an uranium– carbonium target, in a dedicated underground bunker. First, as an extension of previous works and results [7, 8], complex scenarios of irradiation cycles are accounted for at a macroscale level; then, a portion of the concrete wall directly impinged by the radioactive beam has been examined at the mesoscale level, discussing the role of each phase of concrete as a composite material on its final shielding characteristics. 2. THE RADIATION FIELD To account for radiation, two approaches are possible: the deterministic transport theory and the probabilistic-stochastic theory through the Monte Carlo approach, which is the one followed here. As regards to the former, the transport equation (so-called ‘Boltzmann equation’) is understood to describe the attenuation of radiation in matter; the Boltzmann equation is an integro-differential equation that is obtained by considering the balance (production minus losses) of inflow and outflow particles through the surface of an elementary closed volume, in steady state conditions for the radiation field, and for this reason it is considered at the basis of the ‘transport theory’; it describes in the most general form the physical phenomenon of radiation attenuation in a shielding, distinguishing among the amount of absorbed, scattered, and secondarily produced radiation, that are the different possible forms of removal of the incident particles [9]. Copyright © 2013 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2013) DOI: 10.1002/nag MAMENUCRADIO Hence, neutron transport calculations can be treated numerically by solving the Boltzmann equation; the unknown is the neutron flux, a spatially, directionally, and energetically dependent quantity, so the numerical resolution implies discretization in space, angles of propagation, and energy of radiation. Since the early 90s, the Monte Carlo technique was improved to face the same problem in a stochastic way and nowadays many physical phenomena have been implemented in Fluka or similar codes with stochastic nature. A high knowledge in shielding properties of media is anyway required, in terms of a so-called ‘cross-section’ exhibited by the atoms of the medium towards the different removal mechanisms. Particularly, the concept of the Monte Carlo technique in solving the particle transport problem is to use a large number of particle tracks or histories, random in nature, to estimate an average particle behavior. Provided that we have n variables x, y, z,…, distributed according to (normalized) functions f’(x, y, z,…), g’(x, y, z,…), h’(x, y, z,…), where f′ðxÞ ¼ f ðxÞ (1) xmax ∫ f ðxÞ dx xmin then the mean or average of a function of those variables A(x, y, z,…), over an n-dimensional domain D, is A ¼ ∫ ∫ ∫ …∫ Aðx; y; z; …Þ f ′ðx; y; z; …Þg′ðx; y; z; …Þh′ðx; y; z; …Þ dx dy dz…: (2) xyz To overcome direct integration, it is possible to sample N values of A, by sampling N sets of variables xi, yi, zi,… with probability f ′
g′
h′ … and divide the sum by N N ∑ Aðxi ; yi ; zi ; …Þ SN ¼ i¼1 (3) N The central limit theorem guarantees that, for large values of N, the normalized sum of N independent and identically distributed random variables tends to a Gaussian distribution with mean A and variance σ2A/N N ∑ Aðxi ; yi ; zi …Þ lim SN ¼ lim N→∞ i¼1 N→∞ N ¼ A: (4) The Monte Carlo technique results in the following: a) an integration method that allows to solve multidimensional integrals, describing statistically an average function A, by sampling many values of A according to the probability distributions of the random variables that A depends on; b) and a computer simulation of a physical process, in that the function A is the particle density and the samples are the single experiences of each primary particle from the time it leaves its source until it is absorbed or passes out of the domain, that is, particle ‘histories’, which are generated by simulating the random nature of the particle interaction with the medium. The transport of a single particle, that is, the particle tracks, in fact, can be described as a sequence of collisions occurring at discrete points in space and subsequent displacement of the particle from one collision point to the next; the choice of an interaction type at each collision point, the choice of a new energy and new direction, if scattering occurs, or the possible production of secondary Copyright © 2013 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2013) DOI: 10.1002/nag V. A. SALOMONI ET AL. particles, are all stochastic variables of the process, with their probability distribution, which is randomly selected at each step of the particle path. The main assumptions for a Monte Carlo particle transport analysis are as follows [1]: a) the absorber media are static, homogeneous, isotropic and amorphous; b) the particle transport is a Markovian process, that is, the fate of a particle depends only upon its actual properties, not on previous events; c) and particles do not interact with each other but only with individual atoms, nuclei or molecules. The accuracy and reliability of a Monte Carlo estimator depends both on the models on which the probability density functions of the statistical variables are based on the number N of samples or histories. Fluka belongs to these Monte Carlo-based numerical tools; the code is suitable for a wide range of applications: cosmic ray physics, neutrino physics, accelerator design, and particle physics: calorimetry, tracking and detector simulation, shielding design, dosimetry and radioprotection, and hadrontherapy are a few. In general, for a problem to be fully determined, the following steps with Fluka are mandatory: definition of the geometry of the problem; assignment of a unique material to all regions of the domain; request of the desired scorings; and setting of the parameters, accuracy, and conditions on how holding calculations. The first point, in particular, is quite onerous: the geometry of the problem must be created by means of elemental volumes and Boolean operations between them through the use of the Fluka combinatorial geometry package. The visualization and debugging step-by-step are necessary in order to check the created geometry, because particles’ tracks and destiny depend on the crossed regions, which are volumes with assigned material, and need to be uniquely defined. The code provides itself the chemical composition of the most common elements and compounds (air, water, concrete, biological tissues, …) in stored libraries, but new materials can be defined. Typical Fluka outputs are as follows: particle flux, fluence, current, track length; particle/energy spectrum; energy deposition; radioactive decay of residual nuclei, referred to user-defined irradiation and cooling time profiles; and absorbed dose. In this way, via the use of such a code, neutron fluences and energy deposition have been obtained for each specific problem and geometry (as reported in the following), accounting for an environment under 0,25 mSv/h ambient dose equivalent [10], which is the limit dose prescribed by the National Standards on radioprotection. Hence, the hygro-thermal and the radiation fields are partly decoupled, being the results by Fluka used for updating the thermal state of concrete: power density values are in fact collected for evaluating the real thermal transients and temperature rising because of absorbed radiation possibly affecting the hydrogen content in concrete, which is meant to be the main responsible for neutron attenuation in the medium. Full coupling with the radiation field is then restored from the mechanical point of view, as explained in the next section. 3. THE MATHEMATICAL MODEL The basic features and equations of NEWCON3D (see e.g. [11–13]) are briefly recalled in the following. Concrete is treated as a multiphase system where the micropores of the skeleton are partially filled with liquid water, both in the form of bound or absorbed water and free or capillary water, and partially filled with a gas mixture composed of dry air (noncondensable constituent) and water vapor (condensable), supposed to behave like an ideal gas. When higher than standard temperatures are taken into account several phenomena are considered within the code, dealing with concrete as a porous medium: heat conduction, vapor diffusion, and liquid water flow in the voids. As regards the mechanical field, the model couples shrinkage, creep, damage, and plasticity effects within the constitutive law of the material. Copyright © 2013 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2013) DOI: 10.1002/nag MAMENUCRADIO The coupled system of differential equations for dealing with humidity transport and heat transport can be written in the form ∂h ∂t k ∂hs ∂t ∇T C∇h ρCq ∂T ∂t K ∂T ∂ε þ αmT ¼ 0 ∂t ∂t ∇T Λ∇T (5) ∂Qh ¼0 ∂t (6) where k is the cotangent of the isotherm slope, C is the (relative humidity) diffusivity diagonal
matrix,  ∂h  dhs the self-dessication, K the hygrothermic coefficient, m ¼ f 1 1 1 0 0 0 gT , α ¼ ∂ε  v T;w equals the change in humidity h because of the unit change of volumetric strain εv at constant moisture content w and temperature T, ρCq is the thermal capacity, Λ the thermal conductivity diagonal matrix, and Qh the outflow of heat per unit volume of solid. The last term in Eq.(5) represents the coupling term for connecting hygrothermal and mechanical responses [11]. The linear momentum balance equation for the whole multiphase medium, neglecting inertial forces, is divσ þ ρm g ¼ 0 (7) where σ is the total stress tensor, ρm the density of the multiphase medium (concrete plus water species), and g an acceleration related to gravity. As regards to the mechanical field, considering that concrete is treated as a viscoelastic damaged material following the Maxwell chain model [14, 15], the constitutive relationship can be written as [16] t n Dðeε ðt Þފ∫ D′ ∑ E μ ðt’e Þ e½yμ ðt’v Þ σ ðt Þ ¼ ½ 1 0 yμ ð t v Þ Š μ¼1  dεðt Þ dε0 ðt Þ  (8) in which the damage factor is simply considered as a stress multiplier, depending on the current strain only (precisely, on the strain dependent on stresses ε ðt Þ ¼ εðt Þ, with t being fixed). Particularly, in Eq. (8), D is the upgraded scalar nonlocal radio-chemo-thermo-mechanical damage (see later), ε0 refers to imposed deformations, not related to stresses (e.g., due to effects of chemical nature), eε is the equivalent strain eε ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi  2 ∑ hεi iþ hxiþ ¼ jxj þ x 2 (9) being εi the principal strains, D′ is given by 2 D′ 1 1 ν 6 6 6 6 6 ¼6 6 6 6 4 1 ν 0 0 0 ν 0 0 0 0 0 0 2 ð1 þ ν Þ 0 0 2ð1 þ νÞ 0 1 2ð1 þ νÞ 3 7 7 7 7 7 7 7 7 7 5 (10) (ν being the Poisson’s ratio), yμ(t) are called reduced times and may be expressed as  yμ ðt Þ ¼ t=τ μ qμ ðμ ¼ 1; 2; …; N Þ (11) with qμ positive exponents ≤1 and Eμ, representing the μ-th elastic modulus of the Maxwell unit; functions Eμ can be typically determined by least squares methods when analytical expressions for Copyright © 2013 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2013) DOI: 10.1002/nag V. A. SALOMONI ET AL. the compliance function is available [17]; t’ is the time of first application of load and subscripts e and v refer to equivalent hydration and to humidity and temperature effects on creep velocity, respectively; τμ indicates creep or relaxation times. For additional details, the reader is referred to [16]. Because the damaging mechanisms are different in uniaxial tension and compression experiments, the damage parameter Dm, 0 ≤ Dm ≤ 1 (the subscript stands for mechanical contributions only), is decomposed according to the Mazars model [18, 19] into two parts, dt for traction and dc for compression D m ¼ αt d t þ αc d c (12) where αt and αc are weighting coefficients defined in [19]. Chemo-mechanical damage has been introduced for the first time in [20]; thermo-chemical effects have been also taken into account in multiplicative way, as proposed by Gerard et al. [21] and Nechnech et al. [22]: a damage parameter Dtc, 0 ≤ Dtc ≤ 1, describes thermo-chemical material degradation at elevated temperatures (mainly because of micro-cracking and cement dehydration) resulting in reduction of the material strength properties, so that the total effect of the mechanical and thermo-chemical damages acting at the same time is multiplicative, that is, the total damage D is defined by the following: D¼1 ð1 D m Þð 1 Dtc Þ: (13) The upgrade of the model has been developed by assuming that nuclear radiation can activate a damage process that combines with the mechanical and thermo-mechanical ones so that the above multiplicative relation is maintained and the total damage is redefined: D¼1 ð1 D m Þ ð1 Dtc Þ ð1 Dr Þ (14) in which Dr accounts for radiation damage, 0 ≤ Dr ≤ 1. In fact, as reported in literature [2], an observed decrease in concrete strength fcu is primarily due to neutron radiation, although there is evidence of some detrimental effect because of temperature increase. The experimental data vary over a wide range for a given neutron fluence: for a neutron fluence of 5 × 1019 n/cm2 the strength ratios range from 0.72 to 1.05 and from 0.65 to 1.05 for fcu/fcuo and fcu/fcuT, respectively, being fcuo the compressive strength of companion specimens neither irradiated nor temperature exposed and fcuT the one of not irradiated but heated samples. By differentiating between fast or slow neutrons action, Gray [23] found that for fast neutron fluences between 7 × 1018 and 3 × 1019 n/cm2 the modulus of irradiated concrete was between 10% and 20% less than that of not irradiated unheated concrete. Alexander [24] reported similar reductions in Young modulus’ values for slow neutron fluencies of about 2 × 1019 n/cm2. Hence, an enveloping curve of the collected experimental data has been specifically defined [7, 8] (Figure 1) so that up to a neutron fluence Φ of 1018 n/cm2 the radiation damage is zero, then it increases with the neutron fluence and finally stabilizes at the maximum value of Ec/Eco = 0,5 being Ec and Eco the modules of elasticity of concrete after neutron radiation and of untreated concrete, respectively. It was not necessary to extrapolate data in the numerical application. Accordingly, the lowest enveloping curve has been taken as reference (in favor of safety) to define the radiation damage parameter (1-Dr) as the reported ratio between the two elastic modules, in agreement with the effective stress theory. 4. NUMERICAL ANALYSES 4.1. Long-term cyclic analyses at macroscale level Previously performed analyses [7, 8] have evidenced that the attenuation process for the incident radiation on concrete is a phenomenon that evolves in time: neutron fluence [n/cm2] is a flux density Copyright © 2013 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2013) DOI: 10.1002/nag MAMENUCRADIO Ec/Eco= 483,59• -0,15 Figure 1. Modulus of elasticity of concrete after neutron radiation Ec related to modulus of elasticity of untreated concrete Eco (data from [25]). [n/(cm2 s)] integrated over the time duration of radiation; instead, power density [GeV/(cm3 s)] varies with the irradiation profile of the facility and in general is due to a constant component, the so-called prompt radiation, instantaneously provided when the facility works (zero otherwise), and a delayed component represented by the decay of radioactive particles, which is the time-variable amount. Consequently, the problem is numerically treated as a time dependent process where radiation damage and temperatures at the boundary need to be calculated for several time steps. The characteristics of the primary proton beam have come from the most serious exercise scenario designed by INFN for the SPES facility: a beam of 70 MeV energy, and 300 μA current, so that the proton flux p is the following: p¼ 300
10 6 ¼ 1; 87 1015 p=s: 1; 602
10 19 (15) In the facility under design at Legnaro Laboratories, the initial proton driver is a cyclotron with variable energy (15–70 MeV). The simulations [7, 8] have referred to the response of a concrete structure during about 50 years and by assuming a neutron flux density of 1012 n/(cm2 s), which is nearly twice the one expected for SPES and quite less than the radiation dose reached by nuclear cores [even of the order of 1019 n/(cm2 s)]. The radiation flux has been supposed to be directly transmitted, unaltered, to the inner surface of the shielding, being negligible the distance between the target and the concrete wall. The simulations have been then performed to take into account the collateral effect represented by the development of heat within the shielding, consequent to absorbed radiation; correspondingly, the coupled hygrothermal response of concrete has been evaluated. A not negligible temperature increase has been noticed, which is in agreement with the requirements of ANSI/ANS-6.4-1985, according to which radiation heat and subsequent thermal effects are to be taken into account for energy flux densities higher than the above threshold. The analyses have so justified also the working period of the facility assumed in the study, that is, 7 months, nearly 5000 h per year, longer durations resulting unacceptable for the material. We have now extended the analysis for a more complex scenario of irradiation cycles (70 MeV energy and 300 μA): 7 months of work, then 5 months of rest, for 5 years. Boundary conditions are assigned in terms of the following: (i) thermal fluxes at the inner faces of the target room; and (ii) heat sources for the internal elements (both from Fluka results in terms of deposited energy), Figure 2. The values are averaged on the different exposed faces, in particular the front face has been differentiated into three zones: 3b, 3b1, and 3b2; similarly, source elements have been distinguished between two zones: s1 and s2. The adopted material characteristics are listed in Table I. Copyright © 2013 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2013) DOI: 10.1002/nag V. A. SALOMONI ET AL. 2b 5b 3b s1 6b 3b1 3b2 s2 4b Figure 2. Model for the cyclic transient heat analysis and subdivision in zones. Table I. Material data of concrete for macroscale analyses. Elastic modulus Poisson’s ratio Permeability/g (isotropic) Thermal expansion coefficient Specific heat Heat conductivity (isotropic) Hygrothermic coefficient k Coefficient χ for h = 0 Damage parameters according to Mazars’ model Triggering of damage K0 At Bt Ac Bc Maxwell chain model Number of Maxwell units Creep parameters [26] (time in days) E0 φ1 m α n 20000 MPa 0.2 40 mm2/days 10 6°C 1 880 J/(kg°C) 0.13 × 10 2 J/(mm°C s) 0.005°C 1 0.004 0.1 × 10 1 2000 1.4 1545 3 8 70836 MPa 4.5 0.296 0.076 0.181 The assigned profiles in time for fluxes and sources are given by the average value of deposited energy for the corresponding zone (Figure 3; the referred face 7 is opposite to face 3b, not visible because approximately coinciding with the section plane of the concrete bunker of Figure 2). Copyright © 2013 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2013) DOI: 10.1002/nag Heat flux [J/(mm 2 s)] MAMENUCRADIO 5.E-06 5.E-06 4.E-06 4.E-06 3.E-06 3.E-06 2.E-06 2.E-06 1.E-06 5.E-07 0.E+00 face 7 face 3b face 3b1 face 3b2 face 4b face 5b face 6b face 2b 0 10 20 30 40 50 60 70 80 Heat source [J/(mm 3 s)] Time [months] 1.4E-07 1.2E-07 1.0E-07 source s1 8.0E-08 source s2 6.0E-08 4.0E-08 2.0E-08 0.0E+00 0 10 20 30 40 50 60 70 80 Time [months] Figure 3. Assigned heat fluxes and sources. Consider that for the total deposited energy a superposition of effects holds: it is the sum of energy due to prompt radiation and energy due to decay radiation, as recalled before, the latter depending on the implemented irradiation profile in Fluka. In Figure 4, the final contour maps of temperature are shown, as well as three different representative locations; the increase in temperature corresponding to each irradiation cycle is depicted in Figure 5, whereas Figure 6 shows the temperature distribution along the concrete walls at the end of the last cycle. The system is found to stabilize at 50°C after 5 cycles. Such a result is due to the fact that the system stores heat more quickly than it cools being heat fluxes and thermal sources never zero: in fact a heat component is always present due to radioactive decay from the so-called ‘delayed radiation’, many orders of magnitude lower to ‘prompt radiation’ (present only when the exotic beam is active) but anyway not negligible in terms of consequent heat development. Accordingly, the humidity distribution within a reference concrete volume 1 × 1 × 1.5 m3 has been reported in Figure 7, uniformly decreasing from 60% (assumed initial condition) to 45% after Figure 4. Five years simulation – final temperatures and sketch of three main locations for point graphs. Copyright © 2013 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2013) DOI: 10.1002/nag V. A. SALOMONI ET AL. Figure 5. Five years simulation – point graph evolution of temperature (continuous/dotted curves refer to correspondent locations – straight lines – of Figure 4). Figure 6. Temperature versus depth after 5^ cycle (continuous/dotted curves refer to correspondent locations – straight lines – of Figure 4). 5 years. Humidity gradients are evidently negligible along the concrete thickness, in agreement with the considered low thermal gradients. By also considering damage evolution (see the next section), the previously discussed phenomenon of accumulated heat is of great importance for evaluating a safety working scenario for the bunker and durability features for the building material: after 5 years, concrete is close to critical temperatures (about 100°C for Ordinary Portland Concretes (OPCs)), so that additional repeated cycles would represent a potential hazard for the structure. 4.2. Mesoscale analyses A small portion of the wall in front of the source has been then modeled, both at the macroscale and at the mesoscale level, 7 cm long and with a cross-sectional area of 12 cm2. The adopted chemical compositions are indicated in Figure 8 and Figure 9; they are required by Fluka, together with the specific weight of the material, to define the different materials through Copyright © 2013 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2013) DOI: 10.1002/nag MAMENUCRADIO Figure 7. History of relative humidity in the first 1.5 m of the concrete wall (reference points with maximum and minimum temperature). Macro scale composition Element % by weight Hydrogen 0,64 Oxygen 45,36 Carbon - Sodium 1,76 Magnesium 3,66 Aluminium 5,88 Silicon 20,90 Phosphorus 0,09 Sulphur 0,09 Potassium 0,64 Calcium 12,66 Titanium 0,47 Iron 0,13 Nickel 7,64 Concrete density [g/cm3] 2,33 Figure 8. Sketch of the sample and chemical characterization for the macroscale analysis [27]. which particles travel. Particularly, for the macroscale problem, an ordinary concrete mixture has been considered (the same used for the previous macro analyses); see Figure 10. The same chemical composition of concrete for the macroscale approach has been adopted for modeling the cement paste (mesoscale), being the overall mixture mainly defined by the cement paste itself. The ITZ has been characterized as calcium idroxide according to literature [28, 30, 31]; a typical limestone aggregate composition has been assigned to the aggregate component. Reference material data have been taken from [5, 32]. Damage evolution on the directly impinged face (Figure 11) shows hence a typically stepped curve up to the maximum value of 12%, uniformly distributed along each sample section and rapidly decreasing within its length; damage values are hence limited and, for the considered neutron fluences, ascribed only to cumulated thermal effects. The main discrepancy through the ITZ, the aggregate-cement paste interface, has been envisaged in terms of energy deposition results (Figure 12): the order of magnitude for maximum values is the same as found in the macro analyses [1010 GeV/(cm3 s)], but strong discontinuities are here evident due to the presence of aggregates and ITZ. Copyright © 2013 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2013) DOI: 10.1002/nag V. A. SALOMONI ET AL. Figure 9. Sketch of the sample and chemical characterization for the mesoscale analysis [28, 29]. Macro scale parameters Parameters Meso scale parameters Parameters Elastic modulus [MPa] 20000 Cement Cementpaste paste Elastic modulus [MPa] ITZ 20000 10000 0,2 Poisson’s ratio 0,2 Poisson’s ratio 0,2 Hydraulic diffusivity [mm /d] 20 Hydraulic diffusivity [mm /d] 20 40 Thermal conductivity [W/(m·K)] 1,27 Thermal conductivity [W/(m·K)] 1,27 1,27 Thermal capacity [J/(kg·K)] 880 Thermal capacity [J/(kg·K)] 880 880 Boundary conditions: Aggregates: Initial Conditions: Translational restraints at the 5 faces not irradiated Elastic behaviour Temperature 20°C Elastic modulus: 70000 MPa Relative Humidity 60% Thermal load at the free face: Thermal Cycles: Hydraulic diffusivity: 0,01 mm2 /d Thermal conductivity: 2,55 [W/(m·K)] According to slide 20 (blue continous line): from 20° to 60°C in 4,5 years – decay for 10 years Figure 10. Macroscale and mesoscale parameters. Some crucial aspects hence arise, detectable thanks to a mesolevel approach only: once the component contains sufficient moderating material (hydrogen, e.g., for cement paste and ITZ, with their intrinsic water content) [6, 27], the attenuation of fast neutrons is dominated by a removal process with associated deposited energies and consequent temperature rising because of absorbed radiation that are lower than those generated within components with reduced hydrogen content Copyright © 2013 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2013) DOI: 10.1002/nag MAMENUCRADIO Damage 0.14 0.12 Damage [-] 0.1 0.08 0.06 0.04 0.02 0 0 500 1000 1500 2000 Time [days] Figure 11. Damage evolution, macroscale cyclic analysis (reference point above). Deposited energy [GeV /(cm3 s)] Distance [cm] Figure 12. Deposited energies for the mesoscale analysis. Copyright © 2013 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2013) DOI: 10.1002/nag V. A. SALOMONI ET AL. (i.e., aggregates); in fact, once the amount of hydrogen is insufficient to thermalize and consequently lead to the absorption by neutron capture of the neutron after its first collision, the intensity or flux density of the fast neutrons may be greater than that predicted by radiation transport equations: hence, it is proved again here that aggregates, with reduced moderating capabilities with respect to cement paste and ITZ, are subjected to higher temperature levels. As a consequence, thermal gradients will be encountered within concrete and specifically within ITZ, even independently on the different thermal conductivities for each component; additionally, being aggregates conductivity higher than cement paste and ITZ ones, initial gradients are expected to be amplified during the transient hygro-thermo-mechanical analyses. The results in terms of humidity (Figures 13 and 14) evidence the physical barrier exerted by the aggregate towards the flux of humidity; consequently, a slightly delay in drying appears when comparing humidity fluxes from the macroscale analysis, Figure 14: in fact, the effect due to the inclusion of aggregates is not only local but has even consequences on the global humidity distribution, as well as on its temporal variation. Contour maps of temperature (Figure 15) confirm the low thermal gradients within the sample’s length as in the macroscale analysis; anyway, locally the results are strongly different, as anticipated before (see Figures 16–18) so that temperature gradients arise within ITZ and higher values are Figure 13. Contour maps of relative humidity at the end of the 1^, 3^ and 5^ cycle (from top to bottom). Figure 14. Relative humidity evolution at a reference line section, cyclic mesoscale analysis. Copyright © 2013 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2013) DOI: 10.1002/nag MAMENUCRADIO Figure 15. Contour maps of temperature at the end of the 1^ and 5^ cycle. Figure 16. Temperature along a reference line section, end of 1^ cycle. Figure 17. Temperature along a reference line section, after 3^ cycle. encountered in the aggregates (lower ones in Figure 17 considering that we are in the cooling phase after the 3^ cycle, hence a temperature decrease is faster within aggregates). Correspondingly, damage is driven by aggregates and ITZ distribution and concentrates at ITZ itself (the weakest region of the composite material), Figure 19, so confirming that the interface is the first to be affected by damage triggering and evolution. Once again (and as expected), global damage values are close to the ones coming from the macroscale analysis (now about 6% higher), Figure 20, but the local situation allows for stating that preferential zones of damage concentration must be taken into account. So, even if for the considered scenario neutron fluences do not produce an additional damage contribution, the present results give already the starting situation if multiple effects are to be accounted for, as well as they evidence the necessity of modeling concrete at mesolevel if durability is to be assessed against nuclear shielding. In fact microcracks around aggregates create preferential paths to neutrons and possibly modify neutron transport in matter, so that the Monte Carlo simulations themselves should be relaunched for updating the scenario. Copyright © 2013 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2013) DOI: 10.1002/nag V. A. SALOMONI ET AL. Figure 18. Temperature along a reference line section, end of 5^ cycle. Damage Figure 19. Contour maps of damage at the end of the 1^, 3^, and 5^ cycle. Figure 20. Damage evolution at the reference point of Figure 10. 5. CONCLUSIONS The overall thermo-hygro-mechanical response of concrete structures subjected to high energy neutron fluxes has been investigated, starting from available data and previous numerical analyses, so to assess the durability performance of concrete shieldings under long-term scenarios and irradiation cycles (70 MeV energy and 300 μA): 7 months of work, then 5 months of rest, for 5 years. Copyright © 2013 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech. (2013) DOI: 10.1002/nag MAMENUCRADIO The interaction of radiation with matter has been evaluated in terms of deposited energy and neutron fluences via the Monte Carlo code Fluka and subsequently the former has been converted into heat sources, being known that a collateral effect is represented by the development of heat within the shielding for values of energy flux density above 1010 MeV/(cm2 s) [6]. On the other side, it is proved that neutron fluences higher than 1019 n/cm2 may have detrimental effects on concrete strength and modulus of elasticity [2], so that a correct assessment of concrete shielding performances has required the definition of an appropriate scalar damage parameter [7, 8] following the nonlocal damage approach by Mazars and Pijaudier-Cabot. A realistic prediction of concrete durability for nuclear radiation protection has been evidenced by a mesoscale modeling indicating that preferential weakness paths, possibly modifying neutrons transport within matter, are represented by the interfacial transition zone even when damage is due to thermal effects only. Hence, an appropriate concrete mixing as well as repeated simulations are needed to construct concrete structure with appropriate radiation shielding capabilities even against prolonged cyclic nuclear loads. 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