The chemical–physical deterioration process referred to as sulfate attack can affect the cementitious materials, typically inside massive structures, over decades of their service life. Due to the difficulties in coupling chemistry and materials mechanics, exacerbated by the lack of experimental data, the mathematical modeling of the sulfate attack represents a challenging topic. In this paper, a one-dimensional (1D) diffusion–reaction model, based on a peridynamic approach, is proposed to simulate this deterioration process exclusively from the chemical point of view, emphasizing the interactions between two aluminate phases and the sulfate ions. Parametric analyses are carried out to assess the effect on this process of the sulfate concentration at the boundary, of the aluminate concentration within the sample, of the reaction rate constants depending on the temperature and on the activation energy. Peridynamic simulations, carried out in a MATLAB® platform, have been critically compared to the available theoretical solution and to the finite element predictions provided by COMSOL Multiphysics®. The numerical results obtained in the present 1D scenario confirm the capability of the proposed model to describe the sulfate attack as a diffusion–reaction problem with multiple species, which is of crucial interest to improve the engineering design of new concrete infrastructures and to plan the maintenance interventions for the existing ones.
PouyaJNejiMDe WindtL, et al. Investigating chemical and cracking processes in cement paste exposed to a low external sulfate attack with emphasis on the contribution of gypsum. Constr Build Mater2024; 413: 134845.
2.
YinGJShanZQMiaoL, et al. Finite element analysis on the diffusion-reaction-damage behavior in concrete subjected to sodium sulfate attack. Eng Fail Anal2022; 137: 106278.
3.
ComiCFedeleRPeregoU. A chemo-thermo-damage model for the analysis of concrete dams affected by alkali-silica reaction. Mech Mater2009; 41(3): 210–230.
4.
ComiCPeregoU. Anisotropic damage model for concrete affected by alkali-aggregate reaction. Int J Damage Mech2011; 20(4): 598–617.
5.
IkumiTSeguraI. Numerical assessment of external sulfate attack in concrete structures. A review. Cem Concr Res2019; 121: 91–105.
6.
RanBOmikrine-MetalssiOFen-ChongT, et al. Impact of leaching and chlorides on sulfate attack for cement paste. Constr Build Mater2023; 376: 130881.
7.
QiaoDMatsushitaTMaenakaT, et al. Long-term performance assessment of concrete exposed to acid attack and external sulfate attack. J Adv Concr Technol2021; 19(7): 796–810.
8.
Le BescopPSoletC. External sulphate attack by ground water: experimental study on CEM I cement pastes. Revue européenne de génie civil2006; 10(9): 1127–1145.
9.
MousavinezhadSToledoWKNewtsonCM, et al. Rapid assessment of sulfate resistance in mortar and concrete. Materials2024; 17(19): 4678.
10.
EsselamiRWilsonWTagnit-HamouA. An accelerated physical sulfate attack test using an induction period and heat drying: first applications to concrete with different binders including ground glass pozzolan and limestone filler. Constr Build Mater2022; 345: 128046.
11.
TixierRMobasherB. Modeling of damage in cement-based materials subjected to external sulfate attack. I: formulation. J Mater Civ Eng2003; 15(4): 305–313.
12.
TixierRMobasherB. Modeling of damage in cement-based materials subjected to external sulfate attack. II: comparison with experiments. J Mater Civ Eng2003; 15(4): 314–322.
13.
SamsonEMarchandJ. Modeling the transport of ions in unsaturated cement-based materials. Comput Struct2007; 85(23–24): 1740–1756.
14.
ColomboMComiC. Hydro-thermo-mechanical analysis of an existing gravity dam undergoing alkali–silica reaction. Infrastructures2019; 4(3): 55.
15.
ScrofaniABarchiesiEChiaiaB, et al. Fluid diffusion related aging effect in a concrete dam modeled as a Timoshenko beam. Math Mech Complex Syst2023; 11(2): 313–334.
16.
BarchiesiEHamilaN. Maximum mechano-damage power release-based phase-field modeling of mass diffusion in damaging deformable solids. Z Angew Math Phys2022; 73: 35.
17.
FedeleRMaierG. Flat-jack tests and inverse analysis for the identification of stress states and elastic properties in concrete dams. Meccanica2007; 42(4): 387–402.
18.
FedeleRMaierGMillerB. Identification of elastic stiffness and local stresses in concrete dams by in situ tests and neural networks. Struct Infrastruct Eng2005; 1(3): 165–180.
19.
YinGJZuoXBTangYJ, et al. Numerical simulation on time-dependent mechanical behavior of concrete under coupled axial loading and sulfate attack. Ocean Eng2017; 142: 115–124.
20.
CefisNComiC. Chemo-mechanical modelling of the external sulfate attack in concrete. Cem Concr Res2017; 93: 57–70.
21.
CefisNTedeschiCComiC. External sulfate attack in structural concrete made with Portland-limestone cement: an experimental study. Can J Civ Eng2021; 48(7): 731–739.
22.
QinSSZhangMZouDJ, et al. A failure thickness prediction model for concrete exposed to external sulfate attack. Constr Build Mater2024; 416: 135202.
23.
LucianiJMoraPVirmontJ. Nonlocal heat-transport due to steep temperature-gradients. Phys Rev Lett1983; 51: 1664–1667.
24.
MahanGClaroF. Nonlocal theory of thermal-conductivity. Phys Rev B1988; 38: 1963–1969.
25.
LebonGGrmelaM. Weakly nonlocal heat conduction in rigid solids. Phys Lett A1996; 214: 184–188.
26.
SillingSA. Reformulation of elasticity theory for discontinuities and long-range forces. J Mech Phys Solids2000; 48(1): 175–209.
27.
SillingSAAskariE. A meshfree method based on the peridynamic model of solid mechanics. Comput Struct2005; 83(17–18): 1526–1535.
28.
JaviliAMorasataROterkusE, et al. Peridynamics review. Math Mech Solids2019; 24(11): 3714–3739.
SteinmannPde VilliersAMMcBrideA, et al. Configurational peridynamics. Mech Mater2023; 185: 104751.
31.
dell’IsolaFCorteAGiorgioI. Higher-gradient continua: the legacy of Piola, Mindlin, Sedov and Toupin and some future research perspectives. Math Mech Solids2016; 22(4): 852–872.
32.
Dell’IsolaFAndreausUPlacidiL. At the origins and in the vanguard of peridynamics, non-local and higher-gradient continuum mechanics: an underestimated and still topical contribution of Gabrio Piola. Math Mech Solids2015; 20(8): 887–928.
33.
AlibertJJSeppecherPdell’IsolaF. Truss modular beams with deformation energy depending on higher displacement gradients. Math Mech Solids2003; 8(1): 51–73.
34.
CarcaterraAdell’IsolaFEspositoR, et al. Macroscopic description of microscopically strongly inhomogenous systems: a mathematical basis for the synthesis of higher gradients metamaterials. Arch Rational Mech Anal2015; 218(3): 1239–1262.
35.
PolizzottoCFuschiPPisanoA. Nonlocal strain gradient elastic beam models with two-step differential approach and decoupling of standard and extra boundary conditions, I. Math Mech Complex Syst2022; 10(3): 205–231.
36.
EremeyevVANaumenkoK. Wave dispersion relations in peridynamics: influence of kernels and similarities to nonlocal elasticity theories. Int J Engn Sci2025; 211: 104256.
37.
FedeleRPlacidiLFabbrocinoF. A review of inverse problems for generalized elastic media: formulations, experiments, synthesis. Contin Mech Thermodyn2024; 36(6): 1413–1453.
38.
HuangDLuGWangC, et al. An extended peridynamic approach for deformation and fracture analysis. Eng Fract Mech2015; 141: 196–211.
39.
HanFLubineauGAzdoudY. Adaptive coupling between damage mechanics and peridynamics: a route for objective simulation of material degradation up to complete failure. J Mech Phys Solids2016; 94: 453–472.
40.
GerstleWSillingSReadD, et al. Peridynamic simulation of electromigration. Comput Mater Continua2008; 8(2): 75–92.
41.
BobaruFDuangpanyaM. A peridynamic formulation for transient heat conduction in bodies with evolving discontinuities. J Comput Phys2012; 231(7): 2764–2785.
De MeoDDiyarogluCZhuN, et al. Modelling of stress-corrosion cracking by using peridynamics. Int J Hydrogen Energy2016; 41(15): 6593–6609.
44.
LiuYJLiWJGuanJW, et al. Fully coupled peridynamic model for analyzing the chemo-diffusion-mechanical behavior of sulfate attack in concrete. Constr Build Mater2023; 409: 133874.
45.
LiuYJLiWJGuanJW, et al. Improved fully coupled peridynamic model for analyzing the compressive behaviour of concrete under sulfate attack. Ocean Eng2024; 310: 118790.
46.
JiangLZhangXHZhengBJ, et al. A reduced-order peridynamic differential operator for unsteady convection–diffusion problems. Eng Anal Bound Elem2024; 161: 1–10.
47.
BlezardR. Reflections on the history of the chemistry of cement. London: Society of Chemical Industry, 2000.
48.
CollepardiM. A state-of-the-art review on delayed ettringite attack on concrete. Cem Concr Compos2003; 25(4–5): 401–407.
49.
IdiartAELópezCMCarolI. Chemo-mechanical analysis of concrete cracking and degradation due to external sulfate attack: a meso-scale model. Cem Concr Compos2011; 33(3): 411–423.
50.
MatscheiTLothenbachBGlasserF. The AFm phase in Portland cement. Cem Concr Res2007; 37(2): 118–130.
51.
SteinfeldJIFranciscoJSHaseWL. Chemical kinetics and dynamics, vol. 26. Upper Saddle River, NJ: Prentice Hall, 1999.
52.
EnderleJD. Biochemical reactions and enzyme kinetics. In: EnderleJBronzinoJ (eds) Introduction to biomedical engineering. Oxford: Elsevier, 2012, pp. 447–508.
53.
GiorgioIdell’IsolaFAndreausU, et al. On mechanically driven biological stimulus for bone remodeling as a diffusive phenomenon. Biomech Model Mechanobiol2019; 18(6): 1639–1663.
54.
BarchiesiEHamilaN. A bone remodeling approach encoding the effect of damage and a diffusive bio-mechanical stimulus. Contin Mech Thermodyn2024; 36(4): 993–1012.
55.
RozièreELoukiliAEl HachemR, et al. Durability of concrete exposed to leaching and external sulphate attacks. Cem Concr Res2009; 39(12): 1188–1198.
56.
IkumiTCavalaroSHSeguraI, et al. Alternative methodology to consider damage and expansions in external sulfate attack modeling. Cem Concr Res2014; 63: 105–116.
57.
SunCChenJZhuJ, et al. A new diffusion model of sulfate ions in concrete. Constr Build Mater2013; 39: 39–45.
58.
BobaruFFosterJTGeubellePH, et al. Handbook of peridynamic modeling. Boca Raton, FL: CRC press, 2016.
59.
SelesonP. Improved one-point quadrature algorithms for two-dimensional peridynamic models based on analytical calculations. Comput Methods Appl Mech Eng2014; 282: 184–217.
60.
The MathWorksI. Matlab user’s guide (version r2023b). Natick, MA: The MathWorks, Inc., 2023.
61.
Al-AmoudiOSB. Attack on plain and blended cements exposed to aggressive sulfate environments. Cem Concr Compos2002; 24(3–4): 305–316.
62.
PtáčekPOpravilTŠoukalF. A brief introduction to the history of chemical kinetics. Intr Eff Mass Act Complex Discuss Wave Funct Instanton2018; 1: 1–25.
63.
CokerAK. Modeling of chemical kinetics and reactor design. Oxford: Gulf Professional Publishing, 2001.
64.
PunekarN. Enzymes: catalysis, kinetics and mechanisms. Singapore: Springer, 2018.
65.
DolmaireT. Workshop “Kinetic Limits and Probability”, La Sapienza, Roma, June 9–13, 2025: a description of the round table (to appear). Math Mech Complex Syst2025; 13: 459–470.
