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# Couplings between the Mechanical and Physicochemical Behavior of Concrete: An Example of Basic Creep

Jean Michel Torrenti

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Jean Michel Torrenti. Couplings between the Mechanical and Physicochemical Behavior of Concrete: An Example of Basic Creep. Civil Engineering Sciences, 2025, 1, 7 p. 10.34133/cesci.0006. hal-05246025

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REVIEW ARTICLE

# Couplings between the Mechanical and Physicochemical Behavior of Concrete: An Example of Basic Creep

Jean Michel Torrenti¹,²

¹Université Gustave Eiffel, Marne-la-Vallée 77454, France. ²Ecole Supérieure des Ingénieurs de la Construction de Paris, Arcueil 94110, France.

Address correspondence to: jean-michel.torrenti@univ-eiffel.fr

This paper illustrates the need to consider the coupling between physical and chemical phenomena and the mechanical behavior of concrete for durability considerations. The basic creep of concrete is an example of a phenomenon that can affect the service life of concrete structures. It is coupled with physicochemical phenomena such as cement hydration, temperature, mechanical damage, and leaching. It is very important to consider these couplings to limit early-age cracking and correctly assess the long-term behavior of concrete structures such as dams, bridges, or nuclear power plants.

**Citation:** Torrenti JM. Couplings between the Mechanical and Physicochemical Behavior of Concrete: An Example of Basic Creep. Civil Eng. Sci. 2025;1:Article 0006. https://doi.org/10.34133/cesci.0006

Submitted 15 January 2025
Revised 14 February 2025
Accepted 13 April 2025
Published 2 September 2025

Copyright © 2025 Jean Michel Torrenti. Exclusive licensee Tsinghua University. No claim to original U.S. Government Works. Distributed under a Creative Commons Attribution License (CC BY 4.0).

## Introduction

The main drawback of concrete is its use of cement, which requires nonrenewable energy, and aggregates, which are natural resources. Its main quality is durability, which results in a very long life cycle and therefore limits the impact of its use. However, concrete sometimes encounters durability issues. Deteriorations due to the corrosion of the reinforcement, freeze-thaw cycles, internal swelling reactions, sulfate attack, and the consequences of creep and shrinkage deformations are examples of durability problems affecting concrete structures. In western Europe, expenses related to these problems were estimated at 5 billion euros per year in 1998 [1]. Owing to these high costs and the need for sustainability to limit environmental impacts, many projects are being conducted on the subject of durability.

The causes of these types of deterioration can be grouped into 3 categories, namely, physical, chemical, and mechanical, with strong couplings between these categories. These couplings are highly important for correctly predicting the durability of concrete structures.

Large deflections of concrete bridges are examples of durability problems that can affect concrete structures, sometimes leading to failure, as in the case of the Palau bridge [2-4]. These deflections are due to delayed deformation: creep and shrinkage of the concrete and relaxation of the prestressing steel. This paper focuses on basic creep, meaning that drying and its consequences for delayed strains are not considered. It shows a few examples of the couplings between basic creep and physical or chemical phenomena.

## Early Age

Cracking at an early age is an important problem that occurs when shrinkage (autogenous and thermal strains) is restrained

by boundary conditions, which has been studied for a long time and for which recent recommendations are available [5]. The constitutive modeling of thermomechanical couplings was proposed by Ulm and Coussy [6], Gawin et al. [7,8], and Cervera et al. [9,10]. This modeling has been applied to several structures, including bridges [11], dams [12-14], containment vessels of nuclear power plants [15], and, in general, massive structures [12,16-18]. Recently, the phase-field method has been used to model thermochemo-mechanical couplings [19,20].

At an early age, weak coupling is assumed: The temperature and hydration degree fields are first calculated, and then the mechanical consequences are estimated. The evolution of the temperature and degree of hydration fields are obtained through the simultaneous solution of the energy balance equation, which includes the release of heat due to the hydration reaction (Eq. 1) and the Arrhenius relation that accounts for thermoactivation (Eq. 2):

$$Cth \frac{dT}{dt} + k \nabla^2 T = L \frac{d\xi}{dt} \quad (1)$$

where L is the latent heat of hydration (J/m³), k is the thermal conductivity [W/(m·K)], ξ is the degree of hydration, T is the temperature in kelvin, and Cth is the volumetric heat capacity [J/(m³·K)].

$$\frac{d\xi}{dt} = \tilde{A}(\xi)\exp\left(-\frac{Ea}{RT}\right) \quad (2)$$

where $\tilde{A}(\xi)$ is the normalized affinity, $Ea$ is the activation energy (J/mol), and R is the constant of perfect gas [8.314 J/(mol·K)]. The identification of the normalized affinity $A(\xi)$ implies, for a given temperature history $T(t)$, the evolution of the degree of hydration $\xi(t)$. On the macroscopic scale, the hydration kinetic can be followed only indirectly through the following induced effects: heat generation or strength evolution.
The coupling with the mechanical problem can be obtained, for instance, through Eq. 3:

$$ \dot{\sigma} = C(\xi)\dot{\epsilon}_{el} = C(\xi)(\dot{\epsilon} - \dot{\epsilon}_{au} - \dot{\epsilon}_{th} - \dot{\epsilon}_{bc}) \quad (3) $$

where $\sigma$ is the stress tensor, $\epsilon$ is the strain tensor, $C$ is the stiffness tensor depending on $E(\xi)$, the elastic modulus and Poisson's ratio are assumed to be constant, $\epsilon_{el}$ is the elastic strain, $\epsilon_{au}$ is the autogenous shrinkage, $\epsilon_{th}$ is the thermal shrinkage, and $\epsilon_{bc}$ is the basic creep. Damage and/or plastic strains could also be introduced in Eq. 3.

Considering creep at an early age in Eq. 3 is a major improvement in the accuracy of early-age stress and risk of cracking estimation. It is also important in the case of the estimation of the crack width in reinforced structures via codes such as [21], which inspired the next version of the European Eurocode 2 [22,23].

In massive structures, drying can generally be neglected at early ages. In this case, only basic creep is considered. Basic creep corresponds to the creep of concrete when there is no drying.

From the basic creep tests performed by Laplante on ordinary and high-performance concrete (Fig. 1A), the importance of considering creep when estimating deformation is clear (Fig. 1B) [24].

Creep at an early age has been studied by many authors on experimental [25-27] and modeling [28-31] fronts. There are

<figure><img src="image_3.png" /><figcaption>Fig. 1. (A) Stress history of ordinary concrete. The compressive stresses are negative. (B) Comparison between experimental and simulated evolutions of strains (without and with creep) for the ordinary concrete.</figcaption></figure>

still discussions about the fact that the creep compliance in tension is not the same as that in compression [30,32-34]. If a difference exists, one possible explanation is a coupling between microcracking induced by creep in compression and the self-desiccation of concrete [35].

Ulm et al. [36] reported that the rate of basic creep compliance has a linear evolution with time on a logarithmic scale. There are now several experimental pieces of evidence that the basic creep compliance $J_{bc}$ can be expressed, regardless of age $t$, via logarithmic evolution (Eq. 4) [37-40]:

$$ J_{bc}(t_0, t-t_0) = \frac{1}{C} \ln \left( 1 + \frac{t-t_0}{\tau(t_0)} \right) \quad (4) $$

where $C$ is homogeneous to the stiffness and independent of the age of loading $t_0$ and $\tau$ is a characteristic time that depends on $t_0$.

At an early age, Eq. 4 is still valid [29,41,42], even if, at a very early age, additional phenomena may influence creep, and a deviation from the logarithmic trend has been reported in previous studies [29,42-45]. Equation 4 can be modeled via a rheological model. On the basis of the microprestress theory [46], basic creep at an early age could be modeled using only a dashpot, provided that its viscosity evolves linearly with time (Eq. 5). With this assumption, the evolution of the strain in the case of a constant load corresponds exactly to the logarithmic evolution (Eq. 6):

$$ \eta = C(t + \tau - t_0) \quad (5) $$

$$ \sigma = \eta \dot{\epsilon} = C(t + \tau - t_0) \dot{\epsilon} \Rightarrow \quad d\epsilon = \sigma \frac{1}{C} \frac{dt}{(\tau + t - t_0)} \Rightarrow \epsilon = \frac{\sigma}{C} \ln \left( 1 + \frac{t - t_0}{\tau(t_0)} \right) \quad (6) $$

Figure 2 gives an example of the application of Eq. 4 to basic creep at an early age. In tests performed by Briffaut et al. [26], concrete samples of ordinary concrete were loaded

<figure><img src="image_4.png" /><figcaption>Fig. 2. Comparison of experiments and Eq. 4 for the Briffaut tests [26]. The value of parameter $C$ is equal to 82 GPa and is the same regardless of the loading age $t_0$. The parameter $\tau(t_0)$ evolves with $t_0$ and is equal to 0.0014, 0.0021, 0.0064, and 0.010 d at 24, 32, 64, and 120 h, respectively.</figcaption></figure>
at 24, 32, 64, and 120 h. The evolution of the basic creep strains follows Eq. 4.

## Basic Creep and Damage

When concrete is subjected to high continuous loading, rupture can occur [47]. This phenomenon could be explained by damage corresponding to microcracking: During a creep test, microcracks are created, and the quantity of microcracks evolves rapidly when the stress level is greater than 80% of the compressive strength [48]. Interactions between crack propagation and creep have been observed by several authors [49,50].

In a classical damage model, such as the Mazars model [51], the mechanical damage variable $D_c$ is associated with the mechanical degradation process of concrete induced by the development of microcracks. It is defined as the ratio between the microcrack area and the whole material area and represents the degradation of the current Young's modulus. It evolves with an equivalent strain $\tilde{\epsilon}$ that is a function of the positive part of the strains. Mazzotti and Savoia [52] proposed adding some of the creep strains to this equivalent strain (Eq. 7). The evolution law of the damage $D_c$ is given in Eq. 8, where $\epsilon_D$ is the damage tensile strain threshold and A and B are constant material parameters.

$$ \tilde{\epsilon} = \sqrt{\langle \epsilon + \beta \epsilon_c \rangle_+ : \langle \epsilon + \beta \epsilon_c \rangle_+} \quad (7) $$

$$ D_c = 1 - \frac{\epsilon_D(1-A)}{\tilde{\epsilon}} - \frac{A}{\exp[B(\epsilon_D - \tilde{\epsilon})]} \quad \text{if } \tilde{\epsilon} > \epsilon_D \quad (8) $$

The relationships among the apparent stress $\sigma$, effective stress $\tilde{\sigma}$, and damage $D_c$ are given in Eq. 9. The coupling between creep and damage occurs because creep strains are considered to be driven by effective stresses. This means that only the uncracked material creeps [53], and damage is also affected by basic creep strain (Eq. 7).

$$ \sigma = (1 - D_c) \tilde{\sigma} \quad (9) $$

The model is compared with Rossi tests where concrete samples are loaded at 34%, 55%, and 78% of the compressive strength $f_{cm}$ [50]. The model is fitted with the experimental results at 34% and 55% using a common set of parameters and predicts the behavior at 78% (Fig. 3).

<figure><img src="image_5.png" /><figcaption>Fig. 3. Comparison of the model with Rossi creep tests [50]. Parameters of the model: $t_0 = 64$ d, $E(t_0) = 30$ GPa, $f_{cm} = 43$ MPa, $\tau(t_0) = 3.9$ d, $\epsilon_D = 122$ µm/m, $C = 231.9$ GPa, $\beta = 0.08$, $A = 0.85$, and $B = 1,000$.</figcaption></figure>

## Basic Creep and Temperature

Basic creep is also affected by temperature. Here, we consider temperatures below 100 °C. The first consequence of temperature is an effect on the kinetics of basic creep. As proposed by Benboudjema and Torrenti [15] and Sellier et al. [54], the coefficient C (Eq. 4) is affected by a thermoactivation corresponding to the activation energy Q of water viscosity (Eq. 10):

$$ C_T = C \exp\left[Q\left(\frac{1}{T} - \frac{1}{293}\right)\right] \quad (10) $$

The second consequence is possible damage when temperatures are higher than a threshold, as proposed by Vidal et al. [55]. The damage evolves via Eq. 11:

$$ D_T = 1 - \exp\left[-\left(\frac{T - T_{\text{ref}}}{\Delta T_k}\right)\right] \quad (11) $$

where $T_{ref}$ and $\Delta T_k$ are parameters of the evolution of damage. Owing to the damage induced by temperature $D_T$, Eq. 9 becomes

$$ \sigma = (1 - D_c)(1 - D_T)\tilde{\sigma} \quad (12) $$

The model is also compared (Fig. 4) to Okada's tests where the temperatures are between 20 and 80 °C [56]. The parameters of the basic creep model are determined via tests at 20 °C. The other parameters were fitted via Vidal's tests.

## Creep and Leaching

Leaching is a phenomenon that can affect concrete in the very long term. This corresponds to the dissolution of hydrates (portlandite first, then C-S-H) when the concrete is in contact with water, where the pH is lower than that of the interstitial solution in the concrete. The kinetic of this phenomenon is slow, but it has been observed in dams and is a concern for the storage of nuclear waste because its service life is longer than that of classical concrete structures.

Brook's creep tests, which are known for their very long duration (30 years), were performed in a water tank [57]. The samples were cylinders with a height of 76 mm and a radius of only 38 mm. Leaching is therefore certain to have occurred on a substantial part of the samples because the water in the tank was changed every 3 months [58].

<figure><img src="image_6.png" /><figcaption>Fig. 4. Comparison of the model with Okada's basic creep tests [56]. Values of the parameters of the model: $t_0 = 90$ d, $E(t_0) = 35$ GPa, $f_{cm} = 39.4$ MPa, $\tau(t_0) = 7.4$ d, $\epsilon_D = 99$ µm/m, $C = 191$ GPa, $\beta = 0.08$, $A = 0.95$, $B = 1000$, $T_{ref} = 323$ K, $\Delta T_k = 383$ K, and $Q = 2,500$ K.</figcaption></figure>
Because leaching is driven by diffusivity in the leached zone, the degradation depth $x_d$ evolves with the square root of time (Eq. 12). Kamali et al. [59] have shown (Eq. 13) that the parameter $k$ in Eq. 12 evolves as a function of the water-to-cement (W/C) ratio of the concrete (Kamali's tested materials had a W/C between 0.25 and 0.6). The parameter $k_0$ in Eq. 13 is influenced by the size of the aggregates, the volume of cement paste, and the tortuosity effect of the aggregates [60]. Note also that coupling of leaching with temperature exists [61] but is not considered here. The parameter $k_0$ is assumed to be constant and equal to 0.1.

$$x_d = k\sqrt{t} \qquad (13)$$

$$k = k_0 \ln(8.6W/C) \qquad (14)$$

Thus, it is assumed that the leached part, which has lower mechanical performance [62,63], does not sustain stresses during the creep test. As a consequence, the stress applied to the core material increases (Eq. 15). The result is an apparent nonlinearity of the creep, and possibly, when the leached depth becomes greater than the radius of the samples, it is nonlinear due to coupling with damage. This phenomenon has been confirmed via a creep test where leaching was accelerated [64]. For the estimation of the creep strain, because the stress is not constant, the superposition principle should be applied (Eq. 15).

$$\sigma(t) = \sigma(t_0) \frac{\pi R^2}{\pi (R - k\sqrt{t})^2} \qquad (15)$$

$$\varepsilon(t, t_0) = \int_{t_0}^{t} J(\tau, t - \tau) d\sigma(\tau) \qquad (16)$$

Within this framework, Brooks tests were simulated. The first assumption is that the values of the basic creep parameters can be estimated on the basis of the initial parts of the curves, i.e., before 1,000 d, via Eq. 4. Beyond this age, leaching is considered by recalculating the stress at each time step and applying Eq. 15. Figure 5 shows that leaching, with a reasonable set of parameters, explains the nonlinearity of basic creep in the very long term.

## Creep of Concrete Structures

The logarithmic evolution of basic creep explains the nonasymptotic evolution of deflections in some concrete structures,

<figure><img src="image_7.png" /><figcaption>Fig. 5. Comparison of Brooks' experimental results and creep modeling considering the leaching of concrete. $k_0 = 0.1$.</figcaption></figure>

such as bridges [4] and containment vessels of nuclear power plants [65]. To predict the delayed strains of structures, recent codes such as the B4 model [66] or the fib model code 2010 (MC2010) [67] split the delayed strains of concrete into 4 components: autogenous and drying shrinkage and basic and drying creep. In MC2010 (and in the recent MC2020), the basic creep is expressed as in Eq. 4.

Owing to this decomposition of the delayed strains and taking into account the temperature history and its influence on delayed deformations (as seen for basic creep), it is possible to correctly predict the long-term behavior of the Vercors experiment, which is a one-third mock-up of a containment vessel [68,69].

The relations of MC2010 are mean relations based on a limited number of material parameters, such as the compressive strength and the type of cement that is used. The nature and volume of aggregates, for example, are not considered. The variability around the mean relationships is expected to be at least ±30%. This is why it is very important to adjust the parameters of the delayed strains on the experimental results. Figure 6 presents a comparison of the basic creep of the concrete of the Vercors experiment with the direct application of the equation proposed in MC2010 and the same equation but with the parameters fitted to the experimental results. The importance of the fitting procedure is clear. It is expected that with this procedure, the variability of the delayed strains will be reduced to ±10%, and the prediction of the long-term behavior of prestressed concrete structures will be improved.

Note that the value of the parameter C in Eq. 4 (or its equivalent in MC2010) could also be evaluated in relation to the 28-d compressive strength [70].

The importance of correctly predicting delayed strains for large bridges or nuclear containments has been demonstrated [2,68]. For structures that are sensitive to delayed strains such as large-span segmentally erected box girders or prestressed containments of nuclear power plants, the proposed fitting procedure allows a better assessment of the deformation and limits future repairs due to the loss of prestressing. The best way is to use this fitted behavior in the design phase, but it is also possible to verify that the used concrete does not present delayed strains larger than expected in the design phase. During the service life of a structure, if the behavior of the used concrete is not known, it is nevertheless possible to use the measured structural deformations to adjust the MC2010 relations and use this fitting to predict the future behavior of

<figure><img src="image_8.png" /><figcaption>Fig. 6. Comparison of the measured and MC2010 models (nonfitted and fitted) of the basic creep of the Vercors concrete.</figcaption></figure>
the considered structure and anticipate possible future repairs. An example of this procedure for the problem of extending the service life of nuclear power plant containments is presented in [71].

## Conclusion

Basic creep is a good example of a coupled phenomenon that can affect durability and may affect the service life of concrete structures. It intervenes at an early age, is coupled with temperature, with leaching, and is a topic that is not discussed here, especially with respect to relative humidity [72]. Considering these couplings is very important for limiting early-age cracking and correctly assessing the long-term behavior of concrete structures such as dams, bridges, or nuclear power plants.

This illustrates the need to consider the coupling between physical and chemical phenomena and mechanical behavior for durability considerations. The correct prediction of cracking due to these durability problems, such as corrosion, sulfate attack, and swelling reactions, and the influence of crack width on these phenomena will certainly be a challenge in the future.

## Acknowledgments

The many stimulating discussions and collaborations with numerous colleagues over the years are acknowledged; special thanks to A. Aili, F. Benboudjema, M. Briffaut, B. Nedjar, R. Le Roy, and M. Vandamme.

**Competing interests:** The author declares that he has no competing interests.

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