Chemical degradation vs. creep loading vs. hydration
processes in cement-based materials immerged in
seawater characterized with a multiscale model

Marinelle El-Khoury, Frédéric Grondin, Emmanuel Roziere, Rachid Cortas,
Fadi Hage Chehade

► To cite this version:

Marinelle El-Khoury, Frédéric Grondin, Emmanuel Roziere, Rachid Cortas, Fadi Hage Chehade. Chemical degradation vs. creep loading vs. hydration processes in cement-based materials immerged in seawater characterized with a multiscale model. Mechanics of Time-Dependent Materials, 2025, 29 (3), pp.63. 10.1007/s11043-025-09792-x. hal-05505829

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# Chemical degradation vs. creep loading vs. hydration processes in cement-based materials immersed in seawater characterized with a multiscale model

Marinelle El-Khoury<fnref n="1" />,<fnref n="2" />,<fnref n="3" /> · Frédéric Grondin<fnref n="1" /> · Emmanuel Roziere<fnref n="1" /> · Rachid Cortas<fnref n="4" /> · Fadi Hage Chehade<fnref n="3" />

Received: 14 October 2024 / Accepted: 30 May 2025 / Published online: 14 July 2025
© The Author(s) 2025

## Abstract

Upon contact with seawater, concrete undergoes degradation caused by the diffusion of aggressive ions into its porous network and their reaction with cement hydration products. In addition, time-dependent deformations occur resulting from long-term operational use and mechanical loading. The analysis of these coupled chemomechanical phenomena is complex and requires the development of innovative approaches. A micromechanical model has been developed to analyze these phenomena at the microscopic scale. A multiscale approach has been performed for the evaluation of their effects in mortars. Creep loading has opposite effects compared to chemical degradation due to seawater ingress and the evolution of cement hydration at early ages. After 3 days of loading, the model can reproduce the experimental measurements as the chemical reactions occur slowly, but differences are larger during the first 3 days. The contradictory effects of the formed phases balance each other out, resulting in similar creep behavior in tap water and seawater. This indicates that to limit the failure risk of offshore concrete structures it is necessary to reduce the loading at the early stages during the first days of seawater attack.

**Keywords** Creep · Seawater · Chemomechanical · Cement · Offshore · Homogenization

## 1 Introduction

This study aims to provide a descriptive model for offshore concrete structures that can be used for the design optimization and the development of sustainable marine constructions. Offshore energy is a major concern of our time, aiming to prevent climate change, reduce greenhouse gas emissions (GHGs), and protect the marine environment while meeting the

<figure><img src="image_2.png" /></figure>

F. Grondin

<footnote n="1">¹ Nantes Université, Ecole Centrale Nantes, CNRS, GeM, UMR 6183, F-44000 Nantes, France</footnote>

<footnote n="2">² Laboratoire Matériaux et Durabilité des Constructions (LMDC), Université de Toulouse, INSA/UPS Génie Civil, 135 Avenue de Rangueil, Toulouse 31077 Cedex 04, France</footnote>

<footnote n="3">³ Centre de Modélisation, Ecole Doctorale des Sciences et Technologie, Université Libanaise, Hadath, Lebanon</footnote>

<footnote n="4">⁴ Centre de Recherches Scientifiques en Ingénierie (CRSI), Université Libanaise, Beyrouth, Lebanon</footnote>
increasing energy demands. This topic is firmly embedded in the Sustainable Development Goals (SDGs) of the United Nations (UN). The durability, safety, and sustainability of these infrastructures play a crucial role, particularly in the transportation and oil and gas industries. Failures in these sectors can lead to catastrophic consequences such as explosions, extensive gas leaks into seawater, contamination of seawater, harm to wildlife, and significant economic repercussions. In today's world, offshore constructions are rapidly expanding, particularly in the energy and transportation sectors, encompassing structures like wind farms, petroleum platforms, and bridges. Concrete is commonly employed in such constructions, especially due to its relatively low cost, making it imperative to thoroughly examine the performance of concrete in marine environments.

The marine environment is characterized by its salinity and the ions in the water. The marine environment derives its complexity from a diverse range of major ions, including chloride, sulfate, magnesium, and sodium. This coupling of ions notably distinguishes it from sulfate-related degradations at equivalent sulfate concentrations (Santhanam et al. 2006). Several investigations have centered on the impact of either chloride (Al-Amoudi et al. 1995; De Weerdt et al. 2019; Zhang et al. 2013) or external sulfate attack (Cefis and Comi 2014; Ragoug et al. 2019) on concrete performance and durability. However, limited attention has been paid to the intricate nature of the marine environment, characterized by the interaction of numerous ions and the cumulative chemical attacks arising from the incorporation of all seawater ions (De Weerdt et al. 2023, 2014; Santhanam et al. 2006), particularly by performing in situ exposures in natural seawater sites (De Weerdt et al. 2014; Touil et al. 2022) or laboratory representative tests (El-Khoury et al. 2023a, 2022; Islam et al. 2010).

This effect of the ions combination is of high importance as it can create opposite effects on the behavior of cementitious materials such as “protective” layers formations and damaged layers (De Weerdt et al. 2023; El-Khoury et al. 2022; Guillon 2004). Hence, when evaluating the resilience of concrete structures in oceanic environments, it is essential to take into account the presence of all ions in the seawater (El-Khoury et al. 2022).

In addition, offshore structures are exposed to mechanical loadings for long periods. The chemomechanical evaluation of cementitious materials seeks to explore the interplay between chemical deterioration caused by environmental factors (such as air, groundwater, seawater, tap water, etc.) and the mechanical characteristics of cement-based materials (including strength, stiffness, shrinkage, creep, etc.). In recent years, this field of study has gathered significant attention due to the growing need for construction materials that are both sustainable and durable (El-Khoury et al. 2021; El Bitouri et al. 2016; Hou et al. 2021; Kuhl et al. 2004a; Yao et al. 2020). Creep refers to the time-dependent behavior of cement-based materials under sustained loadings. Creep of cement-based materials arises from the viscous behavior of Calcium Silicate Hydrates (C-S-H) (Rhardane 2018; Tsitova et al. 2023; Youssef Namnoum et al. 2019). Indeed, the observed creep phenomenon is linked to microscopic and nanoscopic phenomena. For instance, these include microdiffusion of water and the reorganization of C-S-H clusters (Acker and Ulm 2001; Ulm and Acker 1998; Ulm et al. 1999). Creep deformations can reach values two to six times higher than the initial elastic deformation measured after the application of the load (Bazant 2001). Therefore, creep is of significant influence on the long-term performance and safety of structures, including offshore structures when creep is associated with chemical reactions. Creep is likely to generate some damage (Bazant 2001; Betten 2005) and creep-damage coupling has been previously reported (Omar et al. 2009; Pijaudier-Cabot et al. 2004). The model suggested by Wang et al. (Wang et al. 2021) coupled damage, chemistry, and transport; they studied chloride diffusion by considering the effects of sulfate attack and temperature on chloride diffusivity in concrete, but this model did not include calcium leaching. Aiming for relevance in structural applications, a poromechanical creep model was proposed by Sellier et al. (Sellier et al.
2016). It considered nonlinearity, multiaxiality, hydration, temperature, and drying effects. The chemomechanical coupling between mechanical loading of cementitious materials and calcium leaching has been previously examined by (Kuhl et al. 2004b, 2004a; Nguyen et al. 2007b, 2007a). Experiments have shown a loss of stiffness and strength during leaching (Nguyen et al. 2007a). Also, concrete becomes more ductile after leaching (Bernard et al. 2003; Lacarrière et al. 2006; Nguyen et al. 2007a; Torrenti et al. 2008). Therefore, a strong coupling exists between calcium leaching and the mechanical behavior.

The literature review shows that the combined effects of the chemical attack of seawa-
ter and creep deformations have not been previously studied in detail. Indeed, the creep
phenomenon was associated with the chemical attack of seawater only at the cement-paste
level (El-Khoury et al. 2023a). Moreover, this calculation of creep considers all components
present in the microstructure of the cement paste at the age of loading (El-Khoury et al.
2023b). However, hydration could occur after loading and could influence the creep evo-
lution. Indeed, creep at early ages is highly influenced by the strength evolution due the
cement hydration (Farah et al. 2013). In addition, if damage appears at the loading time at
early ages the creep displacement is higher than for a material loaded at the end of hydration.
This means that initial cracks have a higher effect on the strength than the hydration process.
Nevertheless, in the case of concrete in contact with seawater at early ages, few studies have
been proposed. The authors (El-Khoury et al. 2024) have developed an experimental proto-
col to assess the creep displacement under these conditions but the competition between all
phenomena could not be characterized during the process. Then, a first model was devel-
oped to couple creep and chemical degradation due to seawater but without considering the
hydration evolution during the test (El-Khoury et al. 2023a, 2021).

Based on these last studies, the novelty of this study lies in the combination of chemical (seawater attack and cement hydration) and mechanical phenomena (specifically primary creep behavior), while taking into account the interplay among the diverse ions inherent in seawater (El-Khoury et al. 2022). This work has considered all the ions present in seawater and their reactions with the cement paste. Furthermore, at the mechanical scale, the novelty relies in considering the phases’ evolution during the loading period. Finally, by using this model we are able to simulate multiple formulations of cement pastes and mortars without the need of performing exhaustive experimental programs.

First, the chemomechanical model is described, including the main assumptions and equations. Then, the experimental program for the validation of simulations is given in detail in terms of materials, exposure conditions, and testing procedures. Finally, simulation results are analyzed and discussed.

## 2 Background

2.1 Chemical reactions in cement paste in contact with seawater

Considering that the main chemical transformations in concrete occur in the cement matrix, the study focused on the mortar and cement paste. The cement paste is a heterogeneous material composed of four main constituents of clinker ($C_3S$, $C_2S$, $C_3A$, $C_4AF$), gypsum ($C_3S H_2$), and other nonreacting materials under a very small volume fraction (less than 2%). The five main phase react with water ($H$) to create hydration products as portlandite ($CH$), calcium silicate hydrates ($C_x-S-H_y$), ettringite ($C_6A \bar{S} H_{32}$), sulfoaluminates ($C_3(A, F)S H_6$), ferrite ($FH_3$), and monosulfates ($3C_4A \bar{S} H_{12}$) (see equations in the Appendix).
Chemical reactions with seawater ions are numerous and complex. Here are the main evolutions of the phases in the cement paste. The action of chloride can lead to the formation of Kuzel (Eq. (1)) and Friedel (Eq. (2)) salts:

$$C_3A + Ca(OH)_2 + 2Cl^- + 10H_2O \rightarrow C_3A.CaCl_2 \cdot 10H_2O + 2OH^- \quad (1)$$

$$C_3A + 0.5Ca(OH)_2 + 0.5CaSO_4 + 2Cl^- + 10H_2O \rightarrow C_3A \cdot (CaCl_2)_{0.5} \cdot (CaSO_4)_{0.5} \quad (2)$$

Gypsum ($CaSO_4$) and ettringite ($C_3A \cdot 3CaSO_4 \cdot 32H_2O$) can be formed due to sulfate ions presence (De Weerdt and Justnes 2015; Santhanam et al. 2003) via the following reactions:

$$MgSO_4 + Ca(OH)_2 \rightarrow CaSO_4 + Mg(OH)_2 \quad (3)$$

$$CaSO_4 + C_3A + 32H_2O \rightarrow C_3A \cdot 3CaSO_4 \cdot 32H_2O \quad (4)$$

Calcium carbonate ($CaCO_3$) precipitates on the surface of concrete as aragonite, vaterite, and calcite (El-Khoury et al. 2022) by the reaction of $CH$ with $CO_2$ dissolved in seawater as follows:

$$Ca(OH)_2 + CO_2 + H_2O \rightarrow CaCO_3 + 2H_2O \quad (5)$$

Seawater also contains $Mg^{2+}$ ions, leading to the formation of a brucite layer on the surface of the structure (De Weerdt and Justnes 2015; El-Khoury et al. 2022):

$$Mg^{2+} + Ca(OH)_2 \rightarrow Mg(OH)_2 + Ca^{2+} \quad (6)$$

Thus, the formation of brucite and $CaCO_3$ layers can act as protective layers that partially clog the material porosity close to the surface and limit the diffusion of ions into the cement paste.

Thaumasite is often assumed to form at low temperatures (De Weerdt et al. 2014), but experiments performed in room temperature (25 °C) (Santhanam et al. 2003) and studies in warm climate regions (Diamond and Lee 1999) have shown that low temperature is not necessary for thaumasite formation. The formation of thaumasite directly involves a C-S-H gel, with a sulfate source and a carbonate source (Santhanam et al. 2003):

$$3CaO \cdot 2SiO_2 \cdot 3H_2O + 2(CaSO_4 \cdot 2H_2O) + CaCO_3 + CO_2 + 23H_2O \rightarrow 2Ca_3[Si(OH)_6](CO_3)(SO_4) \cdot 12H_2O \quad (7)$$

## 2.2 Experimental characterization of cement paste exposed to seawater

The description of cement paste in the model was based on existing knowledge on possible chemical reactions and experiments and a microstructural characterization of cement-paste specimens exposed to seawater for several months. In this section, the seawater-ion reactions with the phases of cement paste are summarized. In this section, microstructural characterization is performed on seawater-immersed cement samples to explain the input data of the numerical model.

Cylindrical cement paste samples were prepared using an Ordinary Portland Cement (OPC) of CEM I 52.5 N, with a diameter of 25 mm, using a water-to-cement (w/c) ratio of 0.5. The selection of this ratio was aimed at achieving a paste that was both workable and stable, similar to cement mortars presented in Sect. 3.2. OPC had a specific gravity of 3.16,
<figure><img src="image_3.png" /><figcaption>**Fig. 1** XRD Patterns and (h, k, l) planes (Miller indices) observed at the specimens surface and center at the age of 6 months (Ref: reference sample, SW: seawater exposed sample, Ett: Ettringite, CH: Calcium hydroxide, Ar: Aragonite, C: Calcite, V: Vaterite, Gyp: Gypsum, Bru: Brucite)</figcaption></figure>

a fineness modulus of 400 m²/kg and it was formed by 60% of C₃S, 16% of C₂S, 8% of C₃A, 11% of C₄AF, 4% of Gypsum, and 2% of other constituents. Cement-paste samples were divided into three parts (surface, center, and intermediate zone) to distinguish the formation of phases on the surface and the center of the sample. The chemical composition of the surface (0–2 mm) and the central zone was evaluated using X-ray diffraction (XRD) analysis. The analysis was conducted using an Aeris XRD operating with a CuKα source (λ₁ = 1.5405980 Å and λ₂ = 1.5444260 Å) and a 0.38-mm slit. The scan was performed within a range of 7° to 70°. XRD analysis examined cement paste crushed to powders and sieved at 80 µm. This characterization helped in identifying crystalline phases. The XRD analysis results performed on the surface and the center of paste samples after 6-months exposure are extracted from (El-Khoury et al. 2022) and presented in Fig. 1. The experimental method and more details are presented in (El-Khoury et al. 2022).

The graphs provide confirmation of the presence of brucite formation on the surface of samples exposed to seawater (Eq. (6)). However, no brucite peak was observed in the reference samples or in the center of the samples exposed to both seawater and the reference solution.

The peaks corresponding to aragonite and vaterite were exclusively detected on the surface of samples exposed to seawater. Aragonite and vaterite have been linked to the carbonation process of C-S-H. (Slegers and Rouxhet 1976). The seawater would actually favor the formation of aragonite among possible calcium carbonate polymorphs (De Weerdt and Justnes 2015). Calcite was observed at the surface of both the control samples and the one exposed to seawater. However, no calcite was detected in the center of the samples exposed to seawater or tap water.

These results show the presence of a brucite-rich layer and CaCO₃-rich layer near the surface (Eq. (5)) and Eq. (6)). Indeed, the carbonation actually takes place in the outermost part of cement-based materials exposed to seawater and De Weerdt et al. observed calcium carbonate in the 0–1 mm section of concrete specimens exposed to seawater for 16 years (De Weerdt et al. 2016).

Friedel's salt was identified in attacked samples at the peak (2θ = 11.193°) confirming the reaction of chloride ions with C₃A and Ca(OH)₂ (Eq. (1)). Friedel's salt was observed both on the surface and in the center of the exposed samples, providing confirmation of the diffusion of chloride ions throughout the sample. This observation shows that the action of chloride should be modeled near the center of the SW sample.
The gypsum peak expected at $(2\theta = 20.731^{\circ})$ could not be observed. An ettringite peak was detected in both series of samples, with no evidence that its formation was favored in seawater. Sulfate-attack tests on concrete highlight crystallization pressures due to ettringite formation leading to expansion, while longitudinal expansion remained very low (El-Khoury et al. 2022). Thus, sulfate attack actually had less influence than chloride ingress in seawater, and this confirms the findings in (Zhang et al. 2013). The extent of the sulfate-enriched layer's thickness has been documented as notably lower than the depth impacted by chloride diffusion, as noted in the study by De Weerdt et al. (De Weerdt et al. 2014). The important concentration of chloride in seawater triggers the creation of Friedel's salt from AFm, subsequently reducing its availability for the formation of ettringite. Therefore, based on the findings of De Weerdt et al. (De Weerdt et al. 2014), the effect of sulfate attack should be simulated in the intermediate zone.

The formation of thaumasite involves a C-S-H gel, with a sulfate source and a carbonate source. This formation will be considered in the simulation based on the findings of (De Weerdt et al. 2014; Diamond and Lee 1999; Santhanam et al. 2003).

These findings are in agreement with results of previous studies (De Weerdt et al. 2014) and were considered in the model hypothesis for simulating a cement paste attacked by seawater.

# 3 Multiscale modeling formulation

## 3.1 Model algorithm at the cement paste's scale

The simulation of the chemomechanical coupling at the cement-paste scale was performed by associating two codes: the hydration code CemPP (Hilloulin et al. 2016) for the chemical reactions and the finite-element code Cast3M (Verpaux et al. 1998) for the simulation of the creep behavior. The code CemPP v 0.2 is based on the CEMHYD3D v3 code developed at NIST (Bentz 1995).

First, a representative elementary volume (REV) of the cement paste is defined with two conditions:

1. It has to be sufficiently small to distinguish all hydrate and anhydrate phases.

2. It has to be sufficiently large to represent a real cement-paste specimen.

A cement-paste volume of $100 \mu m^3$ was used and divided into 10 000 voxels of unit size $1 \mu m^3$ to accurately consider the smallest phases as well as the capillary porosity. The cement properties (mineralogical composition and kinetic parameters) are the necessary input parameters for the code CemPP.

According to the chemical reactions' identification from the literature and XRD findings, since the chemical reaction time is short compared to the diffusion time of the ions, the concrete attacked by seawater can be schematized in a multilayer form as presented in Fig. 2. Each layer represents the effect of a chemical reaction. Based on an experimental study done by De Weerdt et al. (De Weerdt et al. 2014) on ordinary concrete immersed for 10 years in a tidal area of the Trondheim Fjord and the experimental campaign results presented in Sect. 2.2, it can be concluded that the attack forms several zones. Each layer's depth can increase with the exposure time or decrease if the product is not stable. These attack fronts cause a formation of carbonate- and magnesium-rich surface layers, other layers where sulfate attack takes place, and a layer near the center caused by the action of chloride ions. The determination of the layer's depth evolution with exposure time needs an extensive assessment. To simplify the model, a strong assumption was considered here: the thickness
<figure><img src="image_4.png" /><figcaption>**Fig. 2** The multilayer microstructure modeled representing the SW microstructure</figcaption></figure>

of the chemically modified layers was considered constant and equal to 15 µm throughout the immersion time. However, phase modifications occurred over time (using “Box replace pass”), leading to the evolution of products formation and chemical reactions (El-Khoury et al. 2023a, 2021). Indeed, with the hydration evolution, the hydration products volume fraction evolves with time. Using “Box replace pass”, the percentage of reactant is replaced by the seawater reaction product phase. This is how at each iteration time $t_i$ replacements are performed and volume fractions of the phases evolve. Consequently, the volume fractions of products and reactants also evolved with immersion time. The identification of the layers and their relative thicknesses was based on the work of De Weerdt et al. (De Weerdt et al. 2014), who studied ordinary concrete immersed for 10 years in a tidal area of the Trondheim Fjord.

Layer I simulates a brucite-rich layer and layer II models a CaCO$_3$-rich layer. Layer III simulates the consumption of the main hydration product C-S-H and the formation of thaumasite. Layers IV and V focused mainly on sulfate action forming a gypsum-rich layer and an ettringite-rich layer, respectively. Layer VI represents the chloride action. Finally, layer VII represents the intact hydrated layer that is not reached by ions from seawater.

From age 0 to the beginning of immersion in seawater, only the hydration process is modeled. Then, from the immersion time in seawater (age = 3 days), the coupling between seawater attack and hydration was modeled (using “Box replace pass” in each layer). This box allows simulating the phases changes at a full replacement rate (El-Khoury et al. 2023a). Indeed, reactions due to seawater attack occur quickly, that makes it impossible to follow experimentally any reaction-front progress and also impossible to define any kinetics of reactions. Hence, a simplified mechanism has been used based on total replacements of reactive phases by phases produced by the chemical reaction due to seawater attack. The beginning (chosen at the age of 3 days (El-Khoury et al. 2022)) and the end of the immersion time must be defined as input parameters for the chemical model in order to trigger the chemical attack caused by the seawater. As output data of this “Hydration-Seawater Attack” model, multiple 3D volumes of chemically modified cement paste ($img_i$ at $t_i$) were generated. Then, at each time $t_i$, mechanical calculations were performed, using the finite-element code Cat3M. Considering the assumption of plane stresses, by assuming that main stresses effects occur in the section, 2D sections were extracted from the 3D volume generated by CemPP, on which mechanical calculations were performed. The mechanical evaluation consisted in determining the strength, the Young’s modulus, the Poisson’s ratio, and the creep displacement.

An elastic damage law has been assigned to each constituent of the material. The damage model adopted for the cement paste’s constituents was proposed by Fichant et al. (Fichant
et al. 1999, 1997) corresponding to the following problem in the volume V:

$$ div \cdot \overline{\bar{\sigma}}(\bar{y}) = \bar{0} \quad \forall \bar{y} \in V \qquad (8) $$

$$ \overline{\bar{\sigma}}(\bar{y}) = (1 - D) \overline{\overline{\overline{C}}}(\bar{y}, \overline{\overline{\epsilon}}(\bar{y})) : \overline{\overline{\epsilon}}^{e}(\bar{y}) \quad \forall \bar{y} \in V \qquad (9) $$

$$ \overline{\overline{\epsilon}}(\bar{y}) = \frac{1}{2}(\nabla \overline{u}(\bar{y}) + \nabla^t \overline{u}(\bar{y})) \quad \forall \bar{y} \in V \qquad (10) $$

$$ (\overline{\overline{\sigma}}(\bar{y}) \cdot \overline{\eta}) \cdot \overline{\eta} = \Sigma_{imp} \quad \forall \bar{y} \in \Gamma_1 \qquad (11) $$

$$ \overline{u}(\bar{y}) = \overline{U}_{imp} \quad \forall \bar{y} \in \Gamma_2, \qquad (12) $$

where $\bar{y}$ represents the coordinate vector of a particle, $\overline{\bar{\sigma}}$ the stress tensor, $\overline{\overline{\epsilon}}^e$ the elastic strain tensor, $\overline{\overline{\epsilon}}$ the total strain tensor, $\overline{\overline{\overline{C}}}$ the stiffness tensor, $\overline{u}$ the displacement vector, $\overline{\eta}$ the normal unit vector outside the surface $\Gamma_1$, $\Sigma_{imp}$ the imposed stress, and $D$ the damage scalar value. It was assumed that the damage tensor is isotropic and represented by: $\overline{\overline{D}} = \overline{\overline{D}}_I (\overline{\overline{I}}$ is the identity tensor).

The damage-loading surface $f$ is given by following where only the tensile eigenvalues of the strain tensor are considered:

$$ f(\bar{n}) = \epsilon_{EQ} - \kappa_0 = \sqrt{\langle \epsilon_{11} \rangle_+^2 + \langle \epsilon_{22} \rangle_+^2 + \langle \epsilon_{33} \rangle_+^2 - \kappa_0}, \qquad (13) $$

where $\epsilon_{EQ}$ represents the equivalent strain and $\langle \cdot \rangle_+$ the Macaulay brackets.

The following exponential form gives the damage law evolution due to tensile microcracking in an isotropic model:

$$ \begin{cases} f = 0 \text{ and } \dot{\epsilon}_{EQ} > 0 \rightarrow \dot{D} = \frac{\kappa_0(1+B_T\epsilon_{EQ})}{\epsilon_{EQ}} \exp[-B_T(\epsilon_{EQ} - \kappa_0)] \\ f < 0 \text{ or } \dot{\epsilon}_{EQ} = 0 \rightarrow \dot{D} = 0, \end{cases} \qquad (14) $$

where $\kappa_0$ represents the strain-damage threshold and $B_T$ the strain-softening slope parameter. Using the following equations, $\kappa_0$ and $B_T$ can be determined from the intrinsic mechanical properties of the constituents of the cement paste:

$$ \kappa_0 = \frac{f_t}{E} \qquad B_T = \frac{f_t}{\frac{G_f}{h} - \frac{f_t \kappa_0}{2}}. \qquad (15) $$

The input parameters of the micromechanical damage model for each constituent of the cement paste are: Young's modulus $E$, Poisson's ratio $\nu$, tensile strength $f_t$, and fracture energy $G_f$. The data, presented in the Appendix, were derived from the literature and considered intrinsic to the model (Rhardane et al. 2020a). Therefore, the model has the originality of being used for any cement-paste composition without the need for calibration and this has been tested and verified in the work of (Rhardane 2018; Rhardane et al. 2020a, 2020b).

The algorithm in Fig. 3 shows a graphical summary of the chemomechanical model at each computational time $t_i$ for cements and mortars. Hence, at each computational time $t_i$ a heterogeneous microstructure is generated, then the mechanical evaluation is performed.

To simulate a creep test, the generalized Kelvin-Voigt model, using four chains, was used to define the viscoelastic behavior. In the isotropic case, the creep compliance can be defined by the following problem by applying a constant stress on the boundary of the volume:

$$ \overline{\overline{\epsilon}}(\bar{y}) = J \otimes \overline{\overline{\sigma}}(\bar{y}) \quad \forall \bar{y} \in V \qquad (16) $$
<figure><img src="image_5.png" /><figcaption>Fig. 3 Algorithm summarizing the multiscale chemomechanical model at each computational time tᵢ</figcaption></figure>

$$ (\bar{\bar{\sigma}}(\bar{y}) \cdot \bar{\eta}) \cdot \bar{\eta} = \text{constant} \quad \forall \bar{y} \in \Gamma_1, \qquad (17) $$

where J(t) represents the compliance factor defined by:

$$ J(t) = \sum_{i=1}^{n} \frac{1}{\kappa_i} \left( 1 - e^{-\frac{\kappa_i}{\eta_i} t} \right) \qquad (18) $$

in which $\tau_i = \eta_i / \kappa_i$ represents the characteristic times of the Kelvin–Voigt chains ($\tau_1 = 0, 1$ day, $\tau_2 = 1$ day, $\tau_3 = 10$ days, and $\tau_4 = 100$ days), and $\kappa_i$ the elementary stiffness of the phase i. A strong assumption was made at this scale: only C-S-H have a nonnegligible contribution to creep (Rhardane 2018). The other constituents of the cement paste were then considered elastic with a restraining role on the deformation of C-S-H. This assumption used the solidification theory established by (Bazant and Prasannan 1989) for mature cementitious materials. The elementary stiffness of the constituents are summarized in the Appendix (Rhardane 2018).

The coupling between creep and the seawater attack could be considered as a weak form or as a strong form.

The first one considers that the creep calculation is performed only after the calculation of the chemical reactions, so the microstructure chemically modified is used as an input parameter but its evolution during the creep test is not considered.

The second one considers that the creep calculation is performed for an evolving microstructure due to the chemical reactions progress at the same time. For this one, the evolution of the volume fraction of each constituent due to hydration and chemical reactions with seawater has been considered.

Figure 4 shows the algorithm of the model taking into account the evolution of the phases. The creep load corresponds to 30% of the strength at the age of the load application (equal to 3 days in calculations) and was maintained during 100 days (creep test duration). To consider the chemical evolution of microstructure, 40 microstructures at different times were considered.

## 3.2 Model algorithm at the mortar's scale

After the calculation of the mechanical behavior of the cement paste a homogenization method was proposed in order to perform the transition to the upper scale (that of the mortar).

At the mortar scale, two continuous media form the material: the cement paste (considered homogeneous at the mortar scale) and the sand particles. Siliceous sand (0–2 mm) was used to avoid leaching and the reactions due to the calcium carbonate present in limestone
<figure><img src="image_6.png" /><figcaption>**Fig. 4** Algorithm showing the microstructures evolutions</figcaption></figure>

aggregates (Justnes 2003; Lee et al. 2008). The sand had a specific gravity of 2.61, a fineness modulus of 2.6 m²/kg, and a water absorption coefficient of 0.44%.

At the mortar scale, the cementitious matrix was considered homogeneous and its mechanical properties were defined by the average properties calculated from the lower scale on the heterogeneous cement paste. To consider the hydration evolution with time, the homogenization was applied on the results found at the cement-paste scale and calculated using the code including the hydration evolution.

A simulation was performed on a homogeneous cement paste with the same problem presented in Sect. 1. Therefore, only one material constituted the volume. Its properties (the Young's modulus, the Poisson's ratio, the tensile strength, the compressive strength, and the fracture energy) were calibrated to have the same stress–strain curve of the calculation made on the heterogeneous cement paste. This method allows us to obtain the properties of the mortar's matrix based on the calculation on the heterogeneous cement paste.

Regarding the elementary stiffness $\kappa_i$, the same method was used. The $\kappa_i$ adopted for the mortar's matrix are those that give the creep curve that fits well those of the heterogeneous cement matrix.

The transition to the mesoscopic scale is summarized in Fig. 5. The elastic properties of the sand (Young's modulus and Poisson's ratio) were obtained from an inverse analysis after a compressive test on a mortar beam: $E_s = 78$ GPa, $v_s = 0.2$ (Rhardane et al. 2020a). The sand particles represent 30% of the sample volume. The REV of the selected mortar was equal to 5 times the diameter of the largest inclusion and it was subdivided into 100 $\mu$m³ voxels, similarly to the REV adopted for the cement paste (El-Khoury et al. 2023a). For the sand used (0–2 mm), the size of the mortar microstructure adopted was 10 mm.

To simulate a creep test, the generalized Kelvin–Voigt model was used to define the viscoelastic behavior similarly to the calculation process done in Sect. 3.1, but using only two continuous media form the material: the cement paste (considered homogeneous) and the sand particles. The constant load applied was 30% of the mortar's strength at the loading time.
<figure><img src="image_7.png" /><figcaption>Fig. 5 Algorithm summarizing the transition to the mesoscopic scale</figcaption></figure>

# 4 Model validation and creep analysis

## 4.1 Materials, mixtures, and exposure solutions preparation

Mortar specimens were made with the CEM I 52.5 N OPC and a w/c ratio of 0.6 (El-Khoury et al. 2022). The mortar blend consisted of 500 kg/m³ of cement CEM I 52.5 N OPC, 307 kg/m³ of water, and 1305 kg/m³ of sand. Hollow cylindrical mortar specimens were cast with a height of 160 mm, an inner diameter of 20 mm, and an outer diameter of 70 mm. The presence of the hollow space in these cylindrical specimens facilitates the accelerated diffusion of ions and enhances the seawater attack by allowing exposure to the attacking solutions from both the inside and outside.

After preparing the mortar and paste mixtures, the specimens were kept in their molds for 24 h at a temperature of 20 °C and a relative humidity of 90%. Following this, the specimens were demolded after 24 h and a layer of waterproof epoxy resin was applied to both circular ends. At 48 h of age, a second layer of epoxy resin was added to ensure two-dimensional ion diffusion while preventing ion diffusion in the vertical direction. Once the specimens reached 3 days of age, they were immersed in exposure solutions designated as “SW” (seawater) for some samples, while others were placed in tap water denoted as “Ref” (reference).

Seawater reference-Composition Salinity was precisely measured at 35.16504 g/kg (Millero et al. 2008). In this campaign (El-Khoury et al. 2023a), a reformulation of the
<figure><img src="image_8.png" /><figcaption>**Fig. 6** Overview of the experimental approach employed to combine the effects of creep and seawater exposure</figcaption></figure>

seawater exposure was done in the laboratory: multiple salts were dissolved in deionized water; this solution was renewed every 14 days at first, then each month after stabilization of the mass curves of mortar specimens plotted in (El-Khoury et al. 2022). In this manner, the salinity and composition of the seawater remained constant throughout the testing period. The pH was not regulated since seawater is recognized as a tampon solution where the pH is naturally maintained (Lide 2004; Rozière 2012). In order to accelerate the process, a laboratory-reconstituted seawater with a salinity of 70 g/L, twice the normal reference level, was employed. Reference samples, immersed in tap water, were used to distinguish the effects of hydration from those of seawater attack. During the immersion time, all the samples were stored in their solutions (tap water and seawater) in a temperature-controlled room at 20 °C.

## 4.2 Chemomechanical creep test

Tests were conducted using a newly implemented device (El-Khoury et al. 2024). Mortar specimens were immersed in the liquid solution (tap water or seawater) at the age of 3 days and loaded at the age of 28 days (after 25 days of immersion). Plates were applied to the two extremities (lower and upper surface) of the specimen. Three LVDT sensors were placed at 120° on the upper plate to monitor the displacement. The load was monitored using a load sensor positioned on the upper plate. The main output data of this experiment are the applied load and the displacement evolutions.

Figure 6 presents a 3D perspective of the newly implemented device, and a summary of the experimental program adopted to combine creep and seawater attack.

The compressive strength was determined. To comply with the ASTM recommendation height = 2×Diameter, the samples were cut at the two ends and ground to obtain a flat loading surface and a total height of 140 mm. Three samples were tested and the average compressive strength was determined. During the compressive test, the load was applied at a rate of 0.5 MPa/s until failure. The creep load applied was equal to 30% of the strength of the samples.
<table><caption>**Table 1** Creep compliance of each layer in the cement paste REV when a load is applied at the age of 28 days for 100 days</caption><thead><tr><th>Layer</th><th>I</th><th>II</th><th>III</th><th>IV</th><th>V</th><th>VI</th><th>Hyd.</th><th>SW</th></tr></thead><tbody><tr><td>C-S-H (%)</td><td>30.44</td><td>30.51</td><td>0.00</td><td>29.47</td><td>30.60</td><td>30.63</td><td>30.64</td><td>26.62</td></tr><tr><td>J (μm/m/MPa) when loaded at the age of 28 days for 100 days</td><td>654.04</td><td>647.92</td><td>0.4951</td><td>574.42</td><td>700.17</td><td>652.77</td><td>714.29</td><td>484.06</td></tr></tbody></table>

<figure><img src="image_9.png" /><figcaption>**Fig. 7** Correlation between the creep compliance $J_{100}$ of each layer in the cement paste’s REV with the local C-S-H amount</figcaption></figure>

# 5 Results and discussion

## 5.1 Characterization of the creep behavior of evolving cement paste in seawater

The influence of the behavior of each attacked layer in cementitious materials on the overall creep cannot be evaluated experimentally, but its analysis is important to define the deformation potential associated with the cement’s chemical composition.

The model allowed calculating the phases’ amounts of the cement paste during its hydration and the chemical attack. The mechanical model calculated the creep displacement of each layer considering different chemical reactions. Table 1 and Fig. 7 show a correlation between the calculated creep compliance in the cement paste’s layers after 100 days of loading and the C-S-H amount present in the input REV at the loading age. These results highlight the competition at the mechanical level of each chemically modified layer and these findings converge with the competition of the phases formed discussed in (De Weerdt et al. 2023; El-Khoury et al. 2022, 2021; Guillon 2004). Indeed, even with the nearly similar C-S-H percentages present in the input microstructures, the creep was different for each layer. This finding shows the model’s capacity to estimate the mechanical behavior of the layers even though a strong hypothesis was used in this model: only C-S-H have a nonnegligible contribution to creep based on the fact that the viscoelastic behavior of other components is negligible in comparison with that of C-S-H (Rhardane 2018).

Figure 8 shows the results of the creep calculation considering the microstructure phase evolution due to hydration of cement and seawater chemical attack. This graph shows the creep evolutions of the control microstructure and the multilayer microstructure, as well as the variation of C-S-H percentage in 40 input microstructures. This result is compared with the case where the cement hydration does not evolve during the creep-loading period.
<figure><img src="image_10.png" /><figcaption>**Fig. 8** Comparison of the creep and CSH content variation of the reference layer and the layer attacked by seawater (multilayer) between the code without phase evolution and that with phase evolution</figcaption></figure>

This comparison highlights the relevance of considering the phases' variation, especially the effect of the formation of C-S-H throughout the evolution of hydration and seawater attack. First, the C-S-H content at the age of loading (3 days) was 9.9% (El-Khoury et al. 2023b). A maximum creep of 200 µm/m/MPa at 100 days was calculated when the hydration evolution was not considered and the paste was loaded at the age of 3 days. When the hydration evolution was considered, the creep compliance was higher. This is due to the increase of the amount of C-S-H during the loading period. It depends on the evolution of the hydration of cement, creating new C-S-H, and the dissolution of initial C-S-H according to some chemical reactions due to the seawater attack.

Figure 8 indicates that the reference microstructure records slightly higher creep compliance than the SW microstructure. This is due to the evolution of C-S-H amount difference between the two types of specimens. In the SW specimen, the C-S-H amount in layer III is equal to zero after chemical reactions. The amount of C-S-H recorded in the reference layer is at any time higher than the SW microstructure.

The evolution of the creep compliance following the loading at 3 days is correlated to the variation of C-S-H content during the 100 days of loading (Fig. 7). The model restores the difference between the behavior of the reference and the SW microstructures although it is based on the assumption that only the C-S-H amount influences the viscoelasticity and that layer thickness was fixed as an input parameter.

In the next section, these assumptions are validated by comparing the numerical results to experiments after the homogenization of the cement paste's properties.

## 5.2 Characterization of creep of mortar in seawater

Figure 9 shows a comparison between the calculated specific creep displacements for a loading at the age of 28 days and the values from experimental monitoring for reference and SW mortars. The results show a significant difference between simulations and experiments for SW specimens.

With the objective to understand this difference, the curves were initialized after 3 days of loading. Figure 10 shows a good global agreement between the results provided by the original experimental protocol and those obtained by the multiscale model, even though assumptions were made.

The application of the 28-day loading shows similar long-term creep between the control and SW samples. This confirms the competition between the positive and negative effects of the chemomechanical coupling, which compensate each other at long term. This overall
<figure><img src="image_11.png" /><figcaption>**Fig. 9** Mortar creep at 28 days of age: experimental vs. numerical approaches</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>**Fig. 10** Long-term behavior of mortar samples loaded at 28 days of age: experimental vs. numerical approaches</figcaption></figure>

compensation was also observed in results provided by the original experimental protocol developed.

## 5.3 Discussion

The comparison between the numerical model and experimental measurements highlights the quick effect of seawater attack during the first days after immersion of the cementitious material in the solution. Considering that creep is induced by the viscoelastic behavior of C-S-H restrained by the other elastic phases of the cement paste, the creep is well simulated in comparison with experiments. This result validates the choice of the equations for the chemomechanical problem. The difference of creep during the first 3 days could be due to two effects:

- The damage due to chemical reactions could be very influential on the creep behavior. In the model, the chemical reactions give the new phases amounts in the cement paste but the intrinsic elastic properties were not modified. Damage can appear only if the strain threshold is reached under loading, but the new phases formed by chemical reactions do not decrease the strength sufficiently.

- The C-S-H dissolution was considered to be too fast in the model. Indeed, without experimental measurements of the kinetics of cement paste's phases dissolution, it was considered that at the first contact of the cement paste with the seawater all reactions were
instantaneous. A parametric study is underway to assess this kinetics of dissolution in order to improve the calculation.

# 6 Conclusion

This article primarily focuses on the coupling of creep, seawater attack, and hydration in cementitious materials. The analysis of the interactions between these phenomena is complex and needs the development of innovative measurement and modeling strategies. While experimental analysis of micromechanisms associated with degradation is feasible after unloading, monitoring its evolution during the creep-seawater test remains challenging. The influence of the behavior of each attacked layer in cementitious materials on the overall creep cannot be evaluated experimentally. Therefore, a multiscale chemomechanical model could be used to help in this analysis. The multiscale model employed in this study relied on the intrinsic mechanical properties of the phases and was used without any calibration. The chemical model developed allowed for the consideration of a high degree of attack and a significant thickness of chemically modified material, which would be difficult to achieve experimentally due to the slow nature of seawater attack and the challenges associated with characterizing chemically modified thicknesses in experiments.

The obtained results from the study indicate that short-term creep of structures in seawater is more pronounced compared to tap water. However, at long term, the adverse and positive effects of the chemomechanical coupling offset each other.

This research protocol plays a crucial role in the risk analysis of seawater attack combined with mechanical loading, aiming to facilitate the selection of appropriate materials for marine environments. This aligns with the global trend of supporting renewable energies. The results are promising and will contribute to the exploration of alternative solutions for offshore structures. This exploration focuses on the use of low-carbon materials, the assessment of their durability, and the optimization of their impacts on the environment, aquatic life, society, and the economy.

We should emphasize the limitations of the proposed descriptive model. The thickness of the modified layers is inherently tied to the variable “time,” as both delayed hydration and creep are time-dependent processes. In this work, the volume fractions of phases evolve over time within a constant Representative Elementary Volume. Furthermore, the application of sustained load could alter the microstructure, particularly affecting porosity, and the load would be applied to this microstructure, which has been modified both mechanically and chemically. In the proposed model, only chemical modifications of the microstructure were considered. Mechanical modifications were accounted for by adjusting the intrinsic phase properties, so when phases were modified, their mechanical properties changed accordingly. Therefore, a key area for future work would be extending the model to simulate the resulting microstructure of the creep process.

# Appendix

Equations (A1) to (A8) describe all the chemical reactions due to hydration:

$$C_3S + 5.3H \rightarrow C_{1.7}SH_4 + 1.3CH \quad (A1)$$

$$C_2S + 4.3H \rightarrow C_{1.7}SH_4 + 0.3CH \quad (A2)$$
$$C_3A + 6H \rightarrow C_3(A, F)SH_6 \quad (A3)$$

$$C_4AF + 10H \rightarrow C_3(A, F)SH_6 + FH_3 + CH \quad (A4)$$

$$C_3A + 3C\bar{S}H_2 + 26H \rightarrow C_6A\bar{S}H_{32} \quad (A5)$$

$$C_4AF + 3C\bar{S}H_2 + 30H \rightarrow C_6A\bar{S}H_{32} + FH_3 + CH \quad (A6)$$

$$2C_3A + C_6A\bar{S}H_{32} + 4H \rightarrow 3C_4A\bar{S}H_{12} \quad (A7)$$

$$2C_4AF + C_6A\bar{S}H_{32} + 12H \rightarrow 3C_4A\bar{S}H_{12} + 2FH_3 + 2CH \quad (A8)$$

**Acknowledgements** This work has received funding from the Carnot MERS institute (Marine Engineering Research for Sustainable, Safe and Smart Seas).

**Author contributions** M.E.-K. wrote the main manuscript text and realized the numerical and experimental tests. F.G. searched for the funding, developed the numerical algorithm and reviewed the manuscript. E.R. developed the experimental concept and reviewed the manuscript. R.C. reviewed the manuscript. F.H.C. reviewed the manuscript and proposed the research project.

**Funding information** Open access funding provided by Centrale Nantes.

**Data availability** No datasets were generated or analysed during the current study.

## Declarations

**Competing interests** The authors declare no competing interests.

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

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